Topological invariants of parabolic $G$-Higgs bundles

TL;DR

This paper investigates the topological features of moduli spaces of parabolic G-Higgs bundles over Riemann surfaces, computing dimensions and invariants, and counting connected components using orbifold cohomology.

Contribution

It introduces new topological invariants for maximal parabolic G-Higgs bundles and provides methods to count moduli space components, extending previous results.

Findings

Computed dimensions of parabolic Teichmüller components for split real forms.

Introduced new topological invariants for maximal parabolic G-Higgs bundles.

Counted connected components of moduli spaces using orbifold cohomology.

Abstract

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{\"u}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure.