Maximal Higgs bundles for adjoint forms via Cayley correspondence

For a fixed compact Riemann surface X, of genus at least 2, we count the number of connected components of the moduli space of maximal Higgs bundles over X for the hermitian groups $PSp(2n,R)$, $PSO^*(2n)$, $PSO_0(2,n)$ and $E_6^{-14}$. Hence the same result follows for the number of connected components of the moduli space of maximal representations of $\pi_1X$ in these groups. We use the Cayley correspondence proved in [3] as our main tool.