Exotic components of $\mathrm{SO}(p,q)$ surface group representations, and their Higgs bundle avatars

TL;DR

This paper reveals new exotic components in the moduli spaces of Higgs bundles for surface group representations into the Lie groups SO(p,q), expanding the understanding of topological invariants and connected components beyond previously known cases.

Contribution

It introduces previously unknown exotic components in the moduli spaces for SO(p,q) groups, which are outside the classes of groups with known exotic components.

Findings

Existence of new exotic components in SO(p,q) moduli spaces.

These components are not distinguished by known topological invariants.

The results extend the classification of surface group representations.

Abstract

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups $\mathrm{SO}(p,q)$ with $2<p<q$. These groups lie outside formerly know classes of groups associated with exotic components.