Ray-Singer Torsion for a Hyperbolic 3-Manifold and Asymptotics of Chern-Simons-Witten Invariant

TL;DR

This paper expresses the Ray-Singer torsion of hyperbolic 3-manifolds using Selberg zeta-functions and explores its implications for the semiclassical analysis of Witten's invariant in Chern-Simons theory and quantum gravity.

Contribution

It provides a novel expression of Ray-Singer torsion in terms of Selberg zeta-functions and applies this to analyze the asymptotics of Witten's invariant in 3D quantum field theory.

Findings

Explicit formula for Ray-Singer torsion using Selberg zeta-functions

Semiclassical asymptotics of Witten's invariant derived

Insights into sum over topologies in 3D quantum gravity

Abstract

The Ray-Singer torsion for a compact smooth hyperbolic 3-dimensional manifold ${\cal H}^3$ is expressed in terms of Selberg zeta-functions, making use of the associated Selberg trace formulae. Applications to the evaluation of the semiclassical asymptotics of the Witten's invariant for the Chern-Simons theory with gauge group SU(2) as well as to the sum over topologies in 3-dimensional quantum gravity are presented.