S-duality in Abelian gauge theory revisited

TL;DR

This paper extends the definition of the U(1) gauge theory partition function to a broader class of four-manifolds, analyzes its modular properties, and discusses the impact of curvature and non-compactness on S-duality.

Contribution

It provides a generalized framework for the partition function on various four-manifolds and explores the conditions under which S-duality is preserved or restored.

Findings

Partition function extended to compact and ALF four-manifolds.

Curvature corrections affect modular weights in the compact case.

S-duality can be restored with gravitational counter terms.

Abstract

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its modular properties. In the compact case, compared with the purely topological result of Witten, we find a non-trivial curvature correction to the modular weights of the partition function. But S-duality can be restored by adding gravitational counter terms to the Lagrangian in the usual way. In the ALF case however we encounter non-trivial difficulties stemming from original non-compact ALF phenomena. Fortunately our careful definition of the partition function makes it possible to circumnavigate them and conclude that the partition function has the same modular properties as in the compact case.