Integrality of HLV kernels

TL;DR

This paper proves that the generating functions related to character varieties and Higgs bundle moduli spaces have polynomial coefficients with integer values, confirming conjectures about their algebraic structure.

Contribution

It establishes the integrality of the coefficients of these generating functions, advancing understanding of their algebraic and geometric properties.

Findings

Coefficients are polynomials in q and t with integer coefficients.

Supports conjectures relating to mixed Hodge numbers of character varieties.

Provides a foundation for further algebraic and geometric analysis of these moduli spaces.

Abstract

We prove that the coefficients of the generating function of Hausel, Letellier, Villegas, and its recent generalization by Carlsson and Villegas, which according to various conjectures should compute mixed Hodge numbers of character varieties and moduli spaces of Higgs bundles of curves of genus $g$ with $n$ punctures, are polynomials in $q$ and $t$ with integer coefficients for any $g,n\geq 0$.