Branes and moduli spaces of Higgs bundles on smooth projective varieties

Given a smooth complex projective variety $M$ and a smooth closed curve $X \subset M$ such that the homomorphism of fundamental groups $\pi_1(X) \rightarrow \pi_1(M)$ is surjective, we study the restriction map of Higgs bundles, namely from 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 set-up where a finite group is acting on $M$ via holomorphic automorphisms or anti-holomorphic involutions, and the curve $X$ is preserved by the action. Branes are studied in this context.