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

Abstract 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 Castro et al (2023 J. High Energy Phys. JHEP07(2023)120; Phys. Rev. Lett. 131 171602). 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.