Maximal Higgs bundles for adjoint forms via Cayley correspondence

TL;DR

This paper counts the connected components of the moduli space of maximal Higgs bundles for certain hermitian groups over a fixed Riemann surface, using Cayley correspondence as the main tool.

Contribution

It provides the first explicit count of connected components for maximal Higgs bundles for these specific groups, extending the understanding of their moduli spaces.

Findings

Number of connected components for PSp(2n,R), PSO^*(2n), PSO_0(2,n), and E_6^{-14}

Corresponding counts for maximal representations of π_1X

Application of Cayley correspondence to these counts

Abstract

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.