Massive Ray-Singer torsion and path integrals

Zero modes are an essential part of topological field theories, but they frequently also obstacle to the explicit evaluation associated path integrals. In order address this issue in case Ray-Singer Torsion, which appears various gauge we introduce a massive variant Torsion involves determinants twisted Laplacian with mass without zero modes. This has advantage allowing one explicitly keep track mode dependence theory. We establish number general properties Torsion. For product manifolds $M=N \times S^1$ and mapping tori is able interpret term as flat $\mathbb{R}_{+}$ connection can represent integral Schwarz type Using techniques, judicious choice algebraic fixing condition change variables leaves free action, evaluate torsion closed form. discuss applications, including calculation on $S^1$ for $G=PSL(2,R)$ derivation generalisation formula Fried finite tori.