BRST Quantisation and the Product Formula for the Ray-Singer Torsion

TL;DR

This paper presents a quantum field theoretic derivation of the Ray-Singer torsion product formula using BRST formalism, introducing a universal gauge fermion and connecting combinatorial and analytic torsion.

Contribution

It introduces a novel BRST-based approach with an infinite dimensional universal gauge fermion to derive the Ray-Singer torsion formula on product manifolds.

Findings

Derived the Ray-Singer torsion product formula using quantum field theory.

Introduced a new class of Fermionic topological field theories.

Established a link between combinatorial and analytic torsion.

Abstract

We give a quantum field theoretic derivation of the formula obeyed by the Ray-Singer torsion on product manifolds. Such a derivation has proved elusive up to now. We use a BRST formalism which introduces the idea of an infinite dimensional Universal Gauge Fermion, and is of independent interest being applicable to situations other than the ones considered here. We are led to a new class of Fermionic topological field theories. Our methods are also applicable to combinatorially defined manifolds and methods of discrete approximation such as the use of a simplicial lattice or finite elements. The topological field theories discussed provide a natural link between the combinatorial and analytic torsion.