Central singularity of the BTZ geometries

The nature of the central singularity of the BTZ geometries---stationary vacuum solutions of $2+1$ gravity with negative cosmological constant Λ=-^-2 and SO(2)×R isometry---is discussed. The essential tool for this analysis is the holonomy operator on a closed path (i.e., Wilson loop) around the central singularity. The study considers the holonomies for the Lorentz and AdS₃ connections. The analysis is carried out for all values of the mass M and angular momentum J, namely, for black holes (M≥|J|) and naked singularities (M<|J|). In general, both Lorentz and AdS₃ holonomies are nontrivial in the zero-radius limit revealing the presence of deltalike singularity at the origin in the curvature and torsion two-forms. However, in the cases M±J/=-n_±², with n_±∈N, recently identified by Giribet et al. [BPS defects in AdS₃ supergravity, arXiv:2402.00171] as Bogomol'nyi-Prasad-Sommerfield configurations, the AdS₃ holonomy reduces to the identity. Nevertheless, except for the AdS₃ spacetime (M=-1, $J=0$), all BTZ geometries have a central singularity which is not revealed by local operations.