Selberg zeta functions and Wilson spools in 3D Chern–Simons quantum gravity

Abstract

In quantum gravity, the Euclidean path integral encodes the full quantum dynamics of bothfields and background geometry. In general, it cannot be evaluated exactly and thus requires expansion around classical saddle-point solutions. We focus on the one-loop determinant, which captures these leading quantum corrections through the functional determinant of the kinetic operator governing the theory. In the context of 3D gravity, we study these functional determinants using two distinct methods: the Selberg zeta function and the Wilson spool. The Selberg zeta function, a generalization of the Riemann zeta function in terms of the lengths of closed geodesics on hyperbolic quotient geometries, provides a simple method for computing functional determinants. It has since been applied in quantum gravity to study the functional determinant of the Euclidean BTZ black hole, a three-dimensional hyperbolic quotient. The Wilson spool, arising from the Chern–Simons formulation of three-dimensional gravity, is of particular interest because it couples matter to quantum gravity and incorporates off-shell fluctuations, unlike standard saddle-point methods. We begin by establishing a formal connection between the two objects in the context of the BTZ black hole. We then turn to Euclidean de Sitter space, where we show that, despite the absence of a Selberg zeta function, one can construct an associated trace formula. We show that, with an appropriate choice of test function, the trace formula gives rise to the Wilson spool. Finally, we comment on how this procedure can be extended to construct Wilson spool candidates for more general spaces, such as lens spaces and higher-dimensional spheres.