E-polynomials of character varieties for real curves

TL;DR

This paper computes the E-polynomials of certain real character varieties associated with Riemann surfaces with involution, using point counting over finite fields and generating functions expressed via plethystic logarithms.

Contribution

It provides a new explicit formula for the E-polynomials of these real character varieties, extending previous work on complex cases.

Findings

Derived a generating function for E-polynomials as a plethystic logarithm.

Connected point counting over finite fields with topological invariants.

Extended methods to real curves with involution.

Abstract

We calculate the E-polynomial for a class of the (complex) character varieties $\mathcal{M}_n^{\tau}$ associated to a genus $g$ Riemann surface $\Sigma$ equipped with an orientation reversing involution $\tau$. Our formula expresses the generating function $\sum_{n=1}^{\infty} E(\mathcal{M}_n^{\tau}) T^n$ as the plethystic logarithm of a product of sums indexed by Young diagrams. The proof uses point counting over finite fields, emulating Hausel and Rodriguez-Villegas.