The Bekenstein-Hawking area-entropy relation S BH = A/4 is derived for a class of five-dimensional extremal black holes in string theory by counting the degeneracy of BPS soliton bound states.

A renormalizable theory of quantum gravity coupled to a dilaton and conformal matter in two spacetime dimensions is analyzed. The theory is shown to be exactly solvable classically. Included among the exact classical solutions are configurations describing the formation of a black hole by collapsing matter. The problem of Hawking radiation and back reaction of the metric is analyzed to leading order in a I/N expansion, where N is the number of matter fields. The results suggest that the collapsing matter radiates away all of its energy before an event horizon has a chance to form, and black holes thereby disappear from the quantum-mechanical spectrum. It is argued that the matter asymptotically approaches a zero-energy "bound state" which can carry global quantum numbers and that a unitary S matrix including such states should exist.

A holographic duality is proposed relating quantum gravity on dS D (D-dimensional de Sitter space) to conformal field theory on a single S D−1 ((D-1)-sphere), in which bulk de Sitter correlators with points on the boundary are related to CFT correlators on the sphere, and points on I + (the future boundary of dS D ) are mapped to the antipodal points on S D−1 relative to those on I − . For the case of dS 3 , which is analyzed in some detail, the central charge of the CFT 2 is computed in an analysis of the asymptotic symmetry group at I ± . This dS/CFT proposal is supported by the computation of correlation functions of a massive scalar field. In general the dual CFT may be non-unitary and (if for example there are sufficently massive stable scalars) contain complex conformal weights. We also consider the physical region O − of dS 3 corresponding to the causal past of a timelike observer, whose holographic dual lives on a plane rather than a sphere. O − can be foliated by asymptotically flat spacelike slices. Time evolution along these slices is generated by L 0 +L 0 , and is dual to scale transformations in the boundary CFT 2 .

It is shown that extremal magnetic black hole solutions of N = 2 supergravity coupled to vector multiplets XI with a generic holomorphic prepotential F(x.') can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the X" are real, and it is seen that the logarithm of the conformal factor of the spatial metric equals the Kahler potential on the vector multiplet moduli space. Several examples are discussed in detail.

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Black holes whose near-horizon geometries are locally, but not necessarily globally, AdS 3 (three-dimensional anti-de Sitter space) are considered. Using the fact that quantum gravity on AdS 3 is a conformal field theory, we microscopically compute the black hole entropy from the asymptotic growth of states. Precise numerical agreement with the Bekenstein-Hawking area formula for the entropy is found. The result pertains to any consistent quantum theory of gravity, and does not use string theory or supersymmetry.

Extremal black holes in M-theory compactification on M × S 1 are microscopically represented by fivebranes wrapping P × S 1 , where M is a Calabi-Yau threefold and P is a four-cycle in M . Additional spacetime charges arise from momentum around the S 1 and expectation values for the self-dual three-form field strength in the fivebrane. The microscopic entropy of the fivebrane as a function of all the charges is determined from a two-dimensional (0, 4) sigma model whose target space includes the fivebrane moduli space. This entropy is compared to the macroscopic formula. Precise agreement is found for both the tree-level and one-loop expressions.

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