N = 2 supergravity in four dimensions, or equivalently N = 1 supergravity in five dimensions, has an interesting set of BPS solutions that each correspond to a number of charged centers. This set contains black holes, black rings and their bound states, as well as many smooth solutions. Moduli spaces of such solutions carry a natural symplectic form which we determine, and which allows us to study their quantization. By counting the resulting wavefunctions we come to an independent derivation of some of the wallcrossing formulae. Knowledge of the explicit form of these wavefunctions allows us to find quantum resolutions to some apparent classical paradoxes such as solutions with barely bound centers and those with an infinitely deep throat. We show that quantum effects seem to cap off the throat at a finite depth and we give an estimate for the corresponding mass gap in the dual CFT. This is an interesting example of a system where quantum effects cannot be neglected at macroscopic scales even though the curvature is everywhere small. arXiv:0807.4556v1 [hep-th] 29 Jul 2008Note that the spacetime contribution (from the vertices) and the "internal" contributions (nodes) are easily and clearly separated in this computation. For more details, including a derivation (assuming the split attractor conjecture) the reader is referred to [23] and [15]. Simple Solution SpacesLet us describe some simple moduli spaces of solutions in order to have some feeling for the spaces we wish to quantize (in section 4). We begin with the simple case of the two centers solution and then discuss the three centers case. The Two Center CaseThe solution space for two centers, when it exists, is two dimensional. The constraint

We systematically construct the geometries dual to the 1+1 dimensional (0,4) conformal field theories that arise in the low-energy description of wrapped M5-branes in S 1 × CY 3 compactifications of M-theory. This includes a large number of multicentered black hole bound states asymptotic to AdS 3 × S 2 . In addition, we find many geometries that develop multiple, mutually decoupled AdS 3 × S 2 throats. We argue there is a useful one to one correspondence between the connected components of the space of solutions and particular limits of type IIA attractor flow trees. We point out that there is a thermodynamic instability of small supersymmetric BTZ black holes to localization on the S 2 , a supersymmetric and exactly solvable analog of the well known AdS-Schwarzschild localization instability, and identify this with the "Entropy Enigma" in four dimensions. We discuss the phase transition this suggests, and initiate the CFT interpretation of these results.

Gravitational entropy arises in string theory via coarse graining over an underlying space of microstates. In this review we would like to address the question of how the classical black hole geometry itself arises as an effective or approximate description of a pure state, in a closed string theory, which semiclassical observers are unable to distinguish from the "naive" geometry. In cases with enough supersymmetry it has been possible to explicitly construct these microstates in spacetime, and understand how coarse-graining of non-singular, horizon-free objects can lead to an effective description as an extremal black hole. We discuss how these results arise for examples in Type II string theory on AdS 5 ×S 5 and on AdS 3 ×S 3 ×T 4 that preserve 16 and 8 supercharges respectively. For such a picture of black holes as effective geometries to extend to cases with finite horizon area the scale of quantum effects in gravity would have to extend well beyond the vicinity of the singularities in the effective theory. By studying examples in M-theory on AdS 3 ×S 2 ×CY that preserve 4 supersymmetries we show how this can happen. This review is a revised and extended version of proceedings submitted for the Sower's Theoretical Physics Workshop 2007 "What is String Theory?" and of lecture notes for the CERN RTN Winter School on Strings, Supergravity and Gauge Theories [1].

We determine, in two independent ways, the number of BPS quantum states arising from supergravity degrees of freedom in a system with fixed total D4D0 charge. First, we count states generated by quantizing the spacetime degrees of freedom of "entropyless" multicentered solutions consisting of D0-branes bound to a D6D6 pair. Second, we determine the number of free supergravity excitations of the corresponding AdS 3 geometry with the same total charge. We find that, although these two approaches yield a priori different sets of states, the leading degeneracies in a large charge expansion are equal to each other and that, furthermore, the number of such states is parametrically smaller than that arising from the D4D0 black hole's entropy. This strongly suggests that supergravity alone is not sufficient to capture all degrees of freedom of large supersymmetric black holes. Comparing the free supergravity calculation to that of the D6D6D0 system we find that the bound on the free spectrum imposed by the stringy exclusion principle (a unitarity bound in the dual CFT) seems to be captured in the dynamics of the fully interacting but classcial supergravity equations of motion.

In this paper we construct a detailed map from pure and mixed half-BPS states of the D1-D5 system to half-BPS solutions of type IIB supergravity. Using this map, we can see how gravity arises through coarse graining microstates, and we can explicitly confirm the microscopic description of conical defect metrics, the M = 0 BTZ black hole and of small black rings. We find that the entropy associated to the natural geometric stretched horizon typically exceeds that of the mixed state from which the geometry was obtained.