Abstract
Three-dimensional spacetime with a negative cosmological constant has proven
to be a remarkably fertile ground for the study of gravity and higher spin
fields. The theory is topological and, since there are no propagating field
degrees of freedom, the asymptotic symmetries become all the more crucial. For
pure (2+1) gravity they consist of two copies of the Virasoro algebra. There
exists a black hole which may be endowed with all the corresponding charges.
The pure (2+1) gravity theory may be reformulated in terms of two Chern-Simons
connections for sl(2,R). An immediate generalization containing gravity and a
finite number of higher spin fields may be achieved by replacing sl(2,R) by
sl(3,R) or, more generally, by sl(N,R). The asymptotic symmetries are then two
copies of the so-called W\_N algebra, which contains the Virasoro algebra as a
subalgebra. The question then arises as to whether there exists a
generalization of the standard pure gravity (2+1) black hole which would be
endowed with all the W\_N charges. The original pioneering proposal of a black
hole along this line for N=3 turns out, as shown in this paper, to actually
belong to the so called "diagonal embedding" of sl(2,R) in sl(3,R), and it is
therefore endowed with charges of lower rather than higher spins. In
contradistinction, we exhibit herein the most general black hole which belongs
to the "principal embedding". It is endowed with higher spin charges, and
possesses two copies of W\_3 as its asymptotic symmetries. The most general
diagonal embedding black hole is studied in detail as well, in a way in which
its lower spin charges are clearly displayed. The extension to N>3 is also
discussed. A general formula for the entropy of a generalized black hole is
obtained in terms of the on-shell holonomies.
Users
Please
log in to take part in the discussion (add own reviews or comments).