Untangling Selberg from the Wilson spool: 1-loop determinants and trace formulae in (A)dS$_{3}$

TL;DR

This work unifies two established methods for computing 1-loop determinants in three-dimensional perturbative gravity—the Selberg zeta function and the Wilson spool—by showing their equivalence for arbitrary-spin fields on Euclidean BTZ and by deriving a novel trace-formula-based spool for Euclidean de Sitter on $S^3$. It demonstrates that the Wilson spool reproduces the representation-theoretic form of the BTZ Selberg zeta function across spins and provides a new, off-shell spool construction for $S^3$ via a Fredholm-determinant trace formula, independent of Chern–Simons arguments. The results illuminate the topological and representation-theoretic structures underlying 1-loop determinants, and they point to generalizations to lens spaces and higher-dimensional spheres, as well as potential extensions to fermions and higher dimensions. Collectively, the paper offers a coherent framework linking geometric, spectral, and topological data in 3D quantum gravity with practical computational tools for quantum corrections. These insights may inform quantum gravity thermodynamics, holographic correspondences, and the treatment of discrete quotients in Euclidean (A)dS spacetimes.

Abstract

Leading quantum effects in perturbative quantum gravity are captured by functional determinants of kinetic operators. We study such 1-loop determinants in three-dimensional Euclidean (anti-) de Sitter gravity evaluated using two seemingly disparate tools, the Selberg zeta function and the Wilson spool. For the Euclidean BTZ black hole, we demonstrate the Wilson spool for massive bosons of arbitrary spin directly equates to a representation-theoretic version of the Selberg zeta function. In the case of Euclidean de Sitter, we show a new trace formula associated with the Fredholm determinant for the scalar Laplacian on the three-sphere reproduces the Wilson spool. Generalizing the trace formula, we comment on how to extend this Wilson spool construction to lens space quotients and higher-dimensional spheres.

Figures (3)

• Figure 1: (a) The contour when split into lowest weight representations, $\mathcal{C}_+$, and highest weight representations, $\mathcal{C}_-$. (b) The contour expressed in terms of only highest weight representations, $\mathcal{C}=2\mathcal{C}_+$. (c) The contour $\mathcal{C}$ deformed to enclose poles that lie on the real axis.
• Figure 2: The contour $\Gamma$ wrapping around the real axis enclosing poles at $z=n$ for $n\in\mathbb{N}$.
• Figure 3: (a) The contour $\mathcal{C}$ for the Euclidean de-Sitter Wilson spool. (b) The deformed contour $\mathcal{C}$.