Poincar\'e polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)

TL;DR

This paper proves conjectures relating to the Poincaré polynomials of moduli spaces of Higgs bundles and character varieties by simplifying counting formulas over finite fields, confirming key predictions in geometric representation theory.

Contribution

It reduces complex counting formulas to simpler conjectured formulas, confirming important conjectures on Poincaré polynomials and E-polynomials in Higgs bundle and character variety theory.

Findings

Confirmed the conjecture of Hausel and Rodriguez-Villegas on Poincaré polynomials.

Proved the conjecture of Hausel and Thaddeus on E-polynomial independence.

Simplified counting formulas for Higgs bundles over finite fields.

Abstract

Using our earlier results on polynomiality properties of plethystic logarithms of generating series of certain type we show that Schiffmann's formulas for various counts of Higgs bundles over finite fields can be reduced to much simpler formulas conjectured by Mozgovoy. In particular, our result implies the conjecture of Hausel and Rodriguez-Villegas on the Poincar\'e polynomials of twisted character varieties and the conjecture of Hausel and Thaddeus on independence of $E$-polynomials on the degree.