  Kapustin and Witten showed that a twisted version of N=4 gauge theory in four
dimensions compactifies to a two-dimensional sigma-model whose target space is
the Hitchin moduli space. In this talk, I consider the reduction of the gauge
theory on a four dimensional orientable spacetime manifold which is not a
global product of two surfaces but contains embedded non-orientable surfaces.
The low energy theory is a sigma-model on a two dimensional worldsheet whose
boundary components end on branes constructed from the Hitchin moduli space
associated to a non-orientable surface. I will also compare the discrete
topological fluxes in four and two dimensional theories and verify the mirror
symmetry on branes as predicted by the S-duality in gauge theory. This provides
another non-trivial test of $S$-duality using reduction along possibly
non-orientable surfaces. Finally, I consider the quantisation of the Hitchin
moduli space from a non-orientable surface as an example of quantisation via
branes and mirror symmetry.
