We calculate the density of states of the 2+1 dimensional BTZ black hole in the micro-and grand-canonical ensembles. Our starting point is the relation between 2+1 dimensional quantum gravity and quantised Chern-Simons theory. In the microcanonical ensemble, we find the Bekenstein-Hawking entropy by relating a Kac-Moody algebra of global gauge charges to a Virasoro algebra with a classical central charge via a twisted Sugawara construction. This construction is valid at all values of the black hole radius. At infinity it gives the asymptotic isometries of the black hole, and at the horizon it gives an explicit form for a set of deformations of the horizon whose algebra is the same Virasoro algebra. In the grand-canonical ensemble we define the partition function by using a surface term at infinity that is compatible with fixing the temperature and angular velocity of the black hole. We then compute the partition function directly in a boundary Wess-Zumino-Witten theory, and find that we obtain the correct result only after we include a source term at the horizon that induces a non-trivial spin-structure on the WZW partition function.

We compute the canonical partition function of 2+1 dimensional de Sitter space using the Euclidean SU (2) × SU (2) Chern-Simons formulation of 3d gravity with a positive cosmological constant. Firstly, we point out that one can work with a ChernSimons theory with level k = l/4G, and its representations are therefore unitary for integer values of k. We then compute explicitly the partition function using the standard character formulae for SU (2) WZW theory and find agreement, in the large k limit, with the semiclassical result. Finally, we note that the de Sitter entropy can also be obtained as the degeneracy of states of representations of a Virasoro algebra with c = 3l/2G.

We discuss semiclassical states in quantum gravity corresponding to Schwarzschild as well as Reissner-Nordström black holes. We show that reduced quantization of these models is equivalent to Wheeler-DeWitt quantization with a particular factor ordering. We then demonstrate how the entropy of black holes can be consistently calculated from these states. While this leads to the Bekenstein-Hawking entropy in the Schwarzschild and non-extreme Reissner-Nordström cases, the entropy for the extreme Reissner-Nordström case turns out to be zero. ͓S0556-2821͑97͒01904-8͔

We discuss black hole quantization in the Wheeler-DeWitt approach. Our consideration is based on a detailed investigation of the canonical formulation of gravity with special considerations of surface terms. Since the phase space of gravity for noncompact spacetimes or spacetimes with boundaries is ill-defined unless one takes boundary degrees of freedom into account, we give a Hamiltonian formulation of the Einstein-Hilbert action as well as a Hamiltonian formulation of the surface terms. It then is shown how application to black hole spacetimes connects the boundary degrees of freedom with thermodynamical properties of black hole physics. Our treatment of the surface terms thereby naturally leads to the Nernst theorem. Moreover, it will produce insights into correlations between the Lorentzian and the Euclidean theory. Next we discuss quantization, which we perform in a standard manner. It is shown how the thermodynamical properties can be rediscovered from the quantum equations by a WKB-like approximation scheme. Back reaction is treated by going beyond the first order approximation. We end our discussion by a rigorous investigation of the so-called BTZ solution in (2ϩ1)-dimensional gravity. ͓S0556-2821͑98͒01504-5͔