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 anti-de Sitter (AdS) (M=-1) and the so-called Bañados-Teitelboim-Zanelli black hole (M≥0), correspond to topological defects with angular deficit 0<α<2π. These defects are produced by (static or spinning) 0-branes which, in the extreme case M =-|J|, admit globally-defined covariantly constant spinors. Thus, these branes correspond to Bogomol'nyi-Prasad-Sommerfield 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.