Phase and Scaling Properties of Determinants Arising in Topological Field Theories

TL;DR

This paper generalizes zeta-regularization to analyze determinants in topological field theories, revealing their phase and scaling properties, and reproduces a non-perturbative feature of Chern-Simons theory via path integrals.

Contribution

It introduces a generalized zeta-regularization method for determinants in topological theories, providing explicit formulas for phase and scaling dependence on manifold cohomology.

Findings

Derived formulas for determinants' phase and scale dependence

Reproduced non-perturbative features of Chern-Simons theory

Extended regularization techniques to odd-dimensional manifolds

Abstract

In topological field theories determinants of maps with negative as well as positive eigenvalues arise. We give a generalisation of the zeta-regularisation technique to derive expressions for the phase and scaling-dependence of these determinants. For theories on odd-dimensional manifolds a simple formula for the scaling dependence is obtained in terms of the dimensions of certain cohomology spaces. This enables a non-perturbative feature of Chern-Simons gauge theory to be reproduced by path-integral methods.