Higgs bundles and surface group representations in the real symplectic group

TL;DR

This paper investigates the structure of the moduli space of surface group representations into the real symplectic group, focusing on maximal representations and their connected components using Higgs bundle theory and new discrete invariants.

Contribution

It counts the connected components of maximal representations in the moduli space, introducing new discrete invariants derived from twisted Higgs bundles for GL(n,R).

Findings

Number of connected components of maximal representations determined.

New discrete invariants for maximal representations discovered.

Identification of Higgs bundle moduli space with twisted Higgs bundles for GL(n,R).

Abstract

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R).