3 Definitions of BF Theory on Homology 3-Spheres

TL;DR

This paper evaluates the partition function of 3D BF theory on certain homology spheres, proposing three different definitions of the resulting finite-dimensional integrals to better understand its topological invariants.

Contribution

It introduces three novel definitions of the BF theory partition function on homology 3-spheres, connecting path integral evaluation with residue calculus and large k limits of Chern-Simons matrix integrals.

Findings

Three consistent definitions of the BF partition function are proposed.

The methods relate BF theory to Chern-Simons matrix models.

Results shed light on the sum over flat connections in topological gauge theories.

Abstract

3-dimensional BF theory with gauge group $G$ (= Chern-Simons theory with non-compact gauge group $TG$) is a deceptively simple yet subtle topological gauge theory. Formally, its partition function is a sum/integral over the moduli space $\mathcal{M}$ of flat connections, weighted by the Ray-Singer torsion. In practice, however, this formal expression is almost invariably singular and ill-defined. In order to improve upon this, we perform a direct evaluation of the path integral for certain classes of 3-manifolds (namely integral and rational Seifert homology spheres). By a suitable choice of gauge, we sidestep the issue of having to integrate over $\mathcal{M}$ and reduce the partition function to a finite-dimensional Abelian matrix integral which, however, itself requires a definition. We offer 3 definitions of this integral, firstly via residues, and then via a large $k$ limit of the corresponding $G\times G$ or $G_C$ Chern-Simons matrix integrals (obtained previously). We then check and discuss to which extent the results capture the expected sum/integral over all flat connections.