Boundary dynamics and the statistical mechanics of the 2 + 1-dimensional black hole

Abstract

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.

Anti–de Sitter–CFT correspondence in three-dimensional supergravity

“…An earlier statistical description of the 2 + 1 black hole entropy was given by Carlip [39] in terms of horizon degrees of freedom. For further work in these directions, see [16,[40][41][42][43][44][45][46]. In view of the relevance of the 2 + 1 black hole to higher dimensional ones [47,48], this question clearly deserves further study.…”

“…(See [15,16] for a different approach to the problem of boundary conditions in the connection representation.) The bosonic boundary conditions must be supplemented by appropriate boundary conditions on the fermionic fields.…”

“…After this gauge transformation has been performed, all the asymptotically relevant components of the fields occur at the same order O(1). Furthermore, because r has dropped out, the analysis could be carried out in the same way at any finite value of r. A Virasoro algebra for all values of r has been investigated in [16].…”

“…Although the OSp(1|2)-WZW theory is not superconformal, the resulting theory is. What happens is that the other component (along e) of the fermionic Kac-Moody supercurrent is "transmuted" into the super-Virasoro generator since its transformation law becomes (20) once the gauge parameters are restricted by (14) and (16). From the WZW point of view, supersymmetry on the worldsheet arises therefore in a non trivial way.…”

A note on Einstein gravity on AdS3 and boundary conformal field theory

“…We have introduced explicitly the first subleading terms in the asymptotic expansion of the metric field and we want to investigate how they are related to the conformal field theory on the boundary. The infinitesimal diffeomorphisms ζ a (r, t, θ)preserving (3)(4)(5)(6) are of the form [1,2] …”