Chen--Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

TL;DR

This paper studies the topology of moduli spaces of stable parabolic Higgs bundles on complex curves, computing orbifold Euler characteristics and Chen--Ruan cohomology, revealing new geometric and topological invariants.

Contribution

It provides explicit descriptions of fixed point loci, computes orbifold Euler characteristics, and analyzes Chen--Ruan cohomology for moduli spaces of parabolic Higgs bundles, extending understanding of their orbifold structure.

Findings

Connected components of fixed point loci described

Orbifold Euler characteristic computed for certain cases

Chen--Ruan cohomology groups and product structure analyzed

Abstract

We consider the moduli space of stable parabolic Higgs bundles of rank $r$ and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth complex projective curve $X$ of genus $g$, with $g\,\geq\, 2$. The group $\Gamma$ of $r$-torsion points of the Jacobian of $X$ acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from $\Gamma$. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen--Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen--Ruan product structure in special cases.