Non-orientable surfaces and electric-magnetic duality

TL;DR

This paper explores the effects of non-orientable surfaces on electric-magnetic duality in gauge theories, revealing how boundary conditions and fluxes relate through sigma-models and mirror symmetry.

Contribution

It extends the understanding of S-duality and mirror symmetry to gauge theories on non-orientable surfaces, introducing modified flux notions and boundary conditions.

Findings

Verification of mirror symmetry of branes under S-duality.

Matching of fluxes with homotopy classes in sigma-model.

Extension of 't Hooft's flux concepts to non-orientable contexts.

Abstract

We consider the reduction along two compact directions of a twisted N=4 gauge theory on a 4-dimensional orientable manifold which is not a global product of two surfaces but contains a non-orientable surface. The low energy theory is a sigma-model on a 2-dimensional worldsheet with a boundary which lives on branes constructed from the Hitchin moduli space of the non-orientable surface. We modify 't Hooft's notion of discrete electric and magnetic fluxes in gauge theory due to the breaking of discrete symmetry and we match these fluxes with the homotopy classes of maps in sigma-model. We verify the mirror symmetry of branes as predicted by S-duality in gauge theory.