Some index formulae on the moduli space of stable parabolic vector bundles

TL;DR

This paper derives explicit formulas for the Chern characters and classes of index and determinant bundles on the moduli space of stable parabolic vector bundles using a families index theorem for hyperbolic cusp operators.

Contribution

It introduces new index formulae for families of d-bar operators on moduli spaces, linking cusp contributions to natural vector bundles and confirming consistency with existing results.

Findings

Explicit formulas for Chern characters of index bundles.

Expressions for contributions from cusps in terms of natural vector bundles.

Consistency with Takhtajan and Zograf's results on Chern classes.

Abstract

We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.