Non-orientable surfaces and electric-magnetic duality

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 the sigma-model. We verify the mirror symmetry of branes as predicted by S-duality in gauge theory.