Spinning up the spool: Massive spinning fields in 3d quantum gravity

We show how to incorporate massive spinning fields into the Euclidean path integral of three-dimensional quantum gravity via its Chern-Simons formulation. The coupling of the spinning fields to gravity is captured by a Wilson spool, a collection of Wilson loops winding around closed paths of the geometry, and generalizes the proposal of [1,2]. We present a robust derivation of the Wilson spool by providing a new group-theoretic perspective of the quasinormal mode method for one-loop determinants. We test our proposal on Euclidean BTZ and $S^3$ backgrounds. We also evaluate explicitly the quantum corrections to the path integral on $S^3$, and report on how $G_N$ and the mass are renormalized to leading order in perturbation theory.