Cusps in 3d gravity

TL;DR

This paper investigates the role of cusped hyperbolic manifolds in 3D gravity, proposing a renormalization method for divergent sums over topologies and exploring implications for supergravity and black hole spectra.

Contribution

It introduces a renormalization approach using cusped manifolds as counterterms in 3D gravity sums, extending the zeta-function regularization to include these geometries.

Findings

Renormalization of volume sums via cusped manifolds.

Application to supergravity with even and odd spin structures.

Reinterpretation of black hole spectrum regularization.

Abstract

Three dimensional hyperbolic manifolds have accumulation points in the spectrum of their volumes, leading to a divergence in the sum over topologies. The limit points are cusped hyperbolic manifolds, and we propose to renormalize the sum by including the cusped manifold as a counterterm. This gives a reinterpretation of the zeta-function regularization procedure used by Maloney and Witten in the sum over SL(2,Z) black holes. For pure N = 1 supergravity, cusps with even spin structure can be used in a similar way. Cusps with odd spin structure are not needed to cancel any divergence, but they find an application by making the index nonzero.