Cusps in 3d gravity

TL;DR

The paper tackles divergences in the 3D gravity path integral caused by accumulation points at cusped hyperbolic manifolds. It introduces a geometric renormalization in which cusps are treated as counterterms with a tunable fugacity, reproducing Maloney–Witten zeta-regularization while enabling generalization to N=1 supergravity. The authors compute the cusp contribution at one loop and show it can cancel the MW divergence, arguing that the cancellation extends to general manifings by locality and the universal origin of accumulation points. In the supersymmetric case, even-spin cusps cancel divergences whereas odd-spin cusps can render the index nonzero, offering a mechanism for a τ-independent index. Overall, the work links a geometric cusp picture to modular-invariant regularization and raises questions about higher-loop effects and the role of off-shell geometries.

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.