Some Remarks about Duality, Analytic Torsion and Gaussian Integration in Antisymmetric Field Theories

TL;DR

This paper explores the mathematical foundations of duality in antisymmetric quantum field theories, emphasizing Fourier transforms, measure theory, and the role of analytic torsion, with implications for understanding partition functions and degeneracies.

Contribution

It provides a measure-theoretic perspective on duality, utilizing Schwarz's Ansatz to relate analytic torsion to partition functions in antisymmetric field theories.

Findings

Duality relies on Fourier transformations of infinite-dimensional measures.

Analytic torsion factorizes in terms of partition functions for degenerate actions.

Identification of partition functions in even-dimensional cases.

Abstract

From a path integral point of view (e.g. \cite{Q98}) physicists have shown how {\it duality} in antisymmetric quantum field theories on a closed space-time manifold $M$ relies in a fundamental way on Fourier Transformations of formal infinite-dimensional volume measures. We first review these facts from a measure theoretical point of view, setting the importance of the Hodge decomposition theorem in the underlying geometric picture, ignoring the local symmetry which lead to degeneracies of the action. To handle these degeneracies we then apply Schwarz's Ansatz showing how duality leads to a factorization of the analytic torsion of $M$ in terms of the partition functions associated to degenerate "dual" actions, which in the even dimensional case corresponds to the identification of these partition functions.