Gauge Theory, Sigma Models and Generalised Geometry

When the target space of a supersymmetric sigma model is a generalised Kahler manifold, there are two topological twists, generalising the A-model and B-model on a Kahler manifold. Kapustin and Witten considered a reduction of a 4-dimensional N = 4 gauge theory to such a sigma model and explained geometric Langlands programme by electric-magnetic duality. The target space is Hitchin's moduli space, which is hyper-Kahler, and the sigma model at low energies is either a B-model or a B-field transform of an A-model, all of which are anomaly-free. In this paper, we consider the reduction of the N = 4 gauge theory on an orientable 4-manifold containing embedded non-orientable surfaces. The resulting theory is a sigma model on a worldsheet whose boundary lives on branes from Hitchin's moduli space for non-orientable surfaces. We show that these branes are supported on submanifolds preserved by the generalised complex structures and that the low energy theory remains anomaly-free at the quantum level. We match the topological sectors and discrete symmetries of the high and low energy theories in a way that is manifestly covariant on the worldsheet.