On S-Duality in Abelian Gauge Theory

TL;DR

This paper explores how U(1) gauge theory's electric-magnetic duality extends to general four-manifolds, revealing the partition function's modular form transformation and implications for supersymmetric Yang-Mills theory and Donaldson invariants.

Contribution

It demonstrates that the partition function transforms as a modular form on general four-manifolds, impacting low-energy interactions in supersymmetric gauge theories and the computation of Donaldson invariants.

Findings

Partition function transforms as a modular form, not invariant.

Duality influences low-energy interactions in N=2 supersymmetric Yang-Mills.

Analysis of abelian duality's role in Donaldson invariants.

Abstract

U(1) gauge theory on ${\bf R}^4$ is known to possess an electric-magnetic duality symmetry that inverts the coupling constant and extends to an action of $SL(2,{\bf Z})$. In this paper, the duality is studied on a general four-manifold and it is shown that the partition function is not a modular-invariant function but transforms as a modular form. This result plays an essential role in determining a new low-energy interaction that arises when N=2 supersymmetric Yang-Mills theory is formulated on a four-manifold; the determination of this interaction gives a new test of the solution of the model and would enter in computations of the Donaldson invariants of four-manifolds with $b_2^+\leq 1$. Certain other aspects of abelian duality, relevant to matters such as the dependence of Donaldson invariants on the second Stieffel-Whitney class, are also analyzed.