Higgs bundles for the non-compact dual of the special orthogonal group

TL;DR

This paper studies Higgs bundles for the non-compact dual of the special orthogonal group, revealing a connectedness property of their moduli space at maximal and zero Toledo invariants, and counting components of surface group representations.

Contribution

It demonstrates the connectedness of the moduli space of Higgs bundles for SO*(2n) at maximal and zero Toledo invariants, providing new insights into their structure.

Findings

Connectedness of the moduli space at maximal Toledo invariant

Connectedness of the moduli space at zero Toledo invariant

Counting of connected components of surface group representations

Abstract

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO*(2n).