A spool for every quotient: One-loop partition functions in AdS$_3$ gravity

TL;DR

This paper introduces a topological operator called the Wilson spool to compute one-loop determinants for massive spinning fields in Euclidean AdS$_3$ gravity, extending previous methods to more general solutions.

Contribution

It generalizes the Wilson spool prescription to include massive spinning fields on all smooth hyperbolic quotients in AdS$_3$ gravity, providing a gauge-invariant, off-shell operator.

Findings

Reproduces known one-loop determinants on hyperbolic quotients.

Extends the Wilson spool framework to massive spinning fields.

Connects the construction to Selberg trace formula, worldline quantum mechanics, and quasinormal modes.

Abstract

The Wilson spool is a prescription for expressing one-loop determinants as topological line operators in three-dimensional gravity. We extend this program to describe massive spinning fields on all smooth, cusp-free, solutions of Euclidean gravity with a negative cosmological constant. Our prescription makes use of the expression of such solutions as a quotients of hyperbolic space. The result is a gauge-invariant topological operator, which can be promoted to an off-shell operator in the gravitational path integral about a given saddle-point. When evaluated on-shell, the Wilson spool reproduces and extends the known results of one-loop determinants on hyperbolic quotients. We motivate our construction of the Wilson spool from multiple perspectives: the Selberg trace formula, worldline quantum mechanics, and the quasinormal mode method.