S-Duality in Gauge Theories as a Canonical Transformation

TL;DR

This paper demonstrates that S-duality in four-dimensional abelian gauge theories can be understood as a canonical transformation, revealing the electric-magnetic interchange and modular anomaly emergence, with extensions to higher dimensions and non-abelian cases.

Contribution

It formulates S-duality as a canonical transformation in phase space and analyzes its implications for abelian and non-abelian gauge theories, including constraints and dual models.

Findings

S-duality acts as a canonical transformation exchanging electric and magnetic variables.

Modular anomaly arises from integrating out momenta in phase space.

Non-abelian dual theories are of Freedman-Townsend type, not simple inversions of coupling.

Abstract

We show that S-duality in four dimensional non-supersymmetric abelian gauge theories can be formulated as a canonical transformation in the phase space of the theory. This transformation is the usual interchange between electric and magnetic degrees of freedom. It is shown that in phase space the modular anomaly emerges as the result of integrating out the momenta degrees of freedom. The generalization to d dimensional abelian gauge theories of p forms is also considered. In the case of non-abelian gauge theories a careful analysis of the constraints implied by the canonical transformation shows that it does not relate Yang-Mills theories with inverted couplings. In fact the dual theory is shown to be of Freedman- Townsend's type, also with ${\tilde \tau}=-1/\tau$, $\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^2}$.