On the cohomology of character stacks for non-orientable surfaces

TL;DR

This paper challenges a proposed formula for the mixed Poincaré series of character stacks on non-orientable surfaces by providing a specific counterexample involving real elliptic curves.

Contribution

It presents a counterexample to a conjectured formula, using explicit descriptions of character stacks for real elliptic curves, thus refining understanding in the field.

Findings

The proposed formula does not hold universally for non-orientable surfaces.

Explicit descriptions of character stacks for real elliptic curves are provided.

A specific counterexample invalidates the previous conjecture.

Abstract

We give a counterexample to a formula suggested by the work of Letellier and Rodriguez-Villegas (arXiv:2008.13435) for the mixed Poincar\'e series of character stacks for non-orientable surfaces. The counterexample is obtained by an explicit description of these character stacks for (real) elliptic curves.