A remark on Schwarz's topological field theory

TL;DR

This paper offers an alternative evaluation of Schwarz's topological field theory partition function, showing it equals 1 and providing a new perspective on analytic torsion as a ratio of volumes of differential form spaces.

Contribution

It introduces a novel approach to evaluate the partition function, linking analytic torsion to volume ratios and demonstrating an analogous result for Reidemeister torsion.

Findings

Partition function Z equals 1 under the new evaluation.

Analytic torsion can be viewed as a volume ratio of differential form spaces.

Reidemeister torsion also admits a similar volume ratio interpretation.

Abstract

The standard evaluation of the partition function $Z$ of Schwarz's topological field theory results in the Ray--Singer analytic torsion. Here we present an alternative evaluation which results in Z=1. Mathematically, this amounts to a novel perspective on analytic torsion: it can be formally written as a ratio of volumes of spaces of differential forms which is formally equal to 1 by Hodge duality. An analogous result for Reidemeister combinatorial torsion is also obtained.