Hodge polynomials of the SL(2,C)-character variety of an elliptic curve with two marked points

TL;DR

This paper calculates the Hodge polynomials of the SL(2,C)-character variety of an elliptic curve with two marked points, linking it to Higgs bundles and instantons, and providing new insights into its geometric structure.

Contribution

It introduces explicit computations of Hodge polynomials for these character varieties and extends results to doubly periodic instantons, connecting different moduli spaces.

Findings

Hodge polynomials of the character variety are computed.

The character variety is shown to be diffeomorphic to a moduli space of Higgs bundles.

Results extend to the moduli space of doubly periodic instantons.

Abstract

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.