Spinning States and Unitarity in 3D Gravity

Abstract

We revisit the proposal to cure the negative density of states in the three-dimensional gravitational path integral by adding spinning states whose spin scales with the central charge. We show that sub-extremal and extremal spinning states below the black hole threshold, together with certain overspinning states above it, can cancel the known negativities in the spectrum. We interpret the states below threshold as bulk spinning defects. The overspinning states were previously interpreted as classical spinning strings; here we instead study their realization as maximal, fixed-point-free BTZ quotients. All of these spinning geometries exhibit causal pathologies in their Lorentzian continuations. The overspinning geometries arise from mixed elliptic--hyperbolic identifications and possess one formally real chiral temperature. We derive the extremal scalar correlator and propose a quotient-motivated image expression for the overspinning background.

v2: 26 pages plus appendices. Substantially revised: clarified the positivity analysis, added a finite-central-charge check of the convergent remainder, and revised the quotient and scalar-correlator analyses. References updated and typos corrected. Main spectral conclusions unchanged