Solving Topological Field Theories on Mapping Tori

TL;DR

This paper derives explicit partition functions for topological field theories like BF and Chern-Simons on mapping tori, revealing their dependence on the underlying diffeomorphism and providing new insights into their topological invariants.

Contribution

It provides concrete expressions for partition functions of BF, U(1|1), and non-abelian models on mapping tori, linking gauge theory, topology, and explicit torsion calculations.

Findings

Explicit partition functions for BF and U(1|1) models on mapping tori

Derived non-abelian generalization and Ray-Singer torsion expressions

Showed dependence of 2D theories on diffeomorphisms, not just isomorphism classes

Abstract

Using gauge theory and functional integral methods, we derive concrete expressions for the partition functions of BF theory and the U(1|1) model of Rozansky and Saleur on $\Sigma x S^{1}$, both directly and using equivalent two-dimensional theories. We also derive the partition function of a certain non-abelian generalization of the U(1|1) model on mapping tori and hence obtain explicit expressions for the Ray-Singer torsion on these manifolds. Extensions of these results to BF and Chern-Simons theories on mapping tori are also discussed. The topological field theory actions of the equivalent two-dimensional theories we find have the interesting property of depending explicitly on the diffeomorphism defining the mapping torus while the quantum field theory is sensitive only to its isomorphism class defining the mapping torus as a smooth manifold.