The Partition Function for Topological Field Theories

TL;DR

This paper calculates the partition function for abelian and non-abelian BF theories in various dimensions, linking it to Ray-Singer torsion and revealing dimension-dependent properties.

Contribution

It provides a unified method to compute the partition function for topological BF theories using Hodge decomposition and relates it to Ray-Singer torsion across dimensions.

Findings

Partition function relates to Ray-Singer torsion in odd dimensions.

Partition function equals one in even dimensions.

Method applies to both abelian and non-abelian theories.

Abstract

We use a Hodge decomposition and its generalization to non-abelian flat vector bundles to calculate the partition function for abelian and non- abelian BF theories in $n$ dimensions. This enables us to provide a simple proof that the partition function is related to the Ray-Singer torsion defined on flat vector bundles for all odd-dimensional manifolds, and is equal to unity for even dimensions.