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arXiv · 2211.10136

3 Definitions of BF Theory on Homology 3-Spheres

Matthias Blau·Mbambu Kakona·George Thompson
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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.

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Matthias Blau, Mbambu Kakona, George Thompson. 2022-11-18. 3 Definitions of BF Theory on Homology 3-Spheres. https://doi.org/10.1007/jhep04(2023)146

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