Branes and moduli spaces of Higgs bundles on smooth projective varieties

Given a smooth complex projective variety M and a smooth closed curve X subset of M such that the homomorphism of fundamental groups pi(1)(X) -> pi(1)(M) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.