On the negative spectrum of the 2+1 black hole

In (2+1)-dimensional gravity with negative cosmological constant, the states in the negative energy range, between AdS (M=-1) and the so-called BTZ black hole ($M\geq 0$), correspond to topological defects with angular deficit $0<\alpha <2\pi$. These defects are produced by (static or spinning) 0-branes which, in the extreme case $M\ell =-|J|$, admit globally-defined covariantly constant spinors. Thus, these branes correspond to BPS solitons and are stable ground state candidates for the corresponding supersymmetric extension of 2+1 AdS gravity. These branes constitute external currents that couple in a gauge-invariant way to three-dimensional AdS gravity.