  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.
