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 rela
tion between 2 + 1-dimensional quantum gravity and quantised Chem-Simons th
eory. In the micro-canonical ensemble, we find the Bekenstein-Hawking entro
py by relating a Kac-Moody algebra of global gauge charges to a Virasoro al
gebra with a classical central charge via a twisted Sugawara construction.
This construction is valid at all values of the black hole radius. At infin
ity it gives the asymptotic isometries of the black hole, and at the horizo
n 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 de
fine the partition function by using a surface term at infinity that is com
patible 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 inc
lude a source term at the horizon that induces a non-trivial spin-structure
on the WZW partition function, (C) 1999 Published by Elsevier Science B.V.