Chen-Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

TL;DR

This paper computes the Chen-Ruan cohomology of the orbifold formed by the action of r-torsion points on the moduli space of semi-stable parabolic vector bundles over a Riemann surface with fixed determinant.

Contribution

It provides the first explicit computation of Chen-Ruan cohomology for moduli spaces of parabolic bundles under a specific group action.

Findings

Explicit Chen-Ruan cohomology formulas derived

Identification of orbifold structure related to r-torsion points

Insights into the topology of moduli spaces with parabolic structures

Abstract

Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $\alpha$. Let $P \mathcal{M}_{\xi}$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $\xi$ over $X$ with weight system $\alpha$, where $r$ is a prime number and $\xi$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_{\xi}$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.