Electric-magnetic Duality in Noncommutative Geometry

TL;DR

This paper explores how S-duality in U(1) gauge theory on a 4-manifold can be understood through noncommutative geometry, revealing deep connections with string theory dualities.

Contribution

It constructs a noncommutative space from Wilson-'t Hooft operators and shows S-duality as an inner automorphism arising from Dirac operators.

Findings

S-duality acts as an inner automorphism of the noncommutative algebra

A noncommutative space encodes both geometry and loop space of the manifold

Connections between noncommutative geometry, string theory, and T-duality are discussed

Abstract

The structure of S-duality in U(1) gauge theory on a 4-manifold M is examined using the formalism of noncommutative geometry. A noncommutative space is constructed from the algebra of Wilson-'t Hooft line operators which encodes both the ordinary geometry of M and its infinite-dimensional loop space geometry. S-duality is shown to act as an inner automorphism of the algebra and arises as a consequence of the existence of two independent Dirac operators associated with the spaces of self-dual and anti-selfdual 2-forms on M. The relations with the noncommutative geometry of string theory and T-duality are also discussed.