Counting stringy points on a family of character varieties

TL;DR

This paper derives explicit formulas for counting stringy points on certain character varieties over finite fields, confirming mirror symmetry conjectures and providing new insights into their topological invariants.

Contribution

It introduces a novel explicit counting formula for stringy points on parabolic type A character varieties with generic monodromy, linking algebraic and topological properties.

Findings

Confirmed the Betti Topological Mirror Symmetry Conjecture for these varieties

Derived formulas for stringy E-polynomials of the varieties

Established a new Frobenius-type formula for counting points

Abstract

We provided explicit formulas for the number of stringy points over finite fields of parabolic type A character varieties with generic semisimple monodromy. This leads to formulas for their stringy E-polynomials. In particular, they satisfy the Betti Topological Mirror Symmetry Conjecture of T. Hausel and M. Thaddeus, as well as a refinement regarding isotypic components. Our proof is based on a Frobenius' type formula for Clifford's type settings and an analysis of it in a specific set-up related to regular wreath products with cyclic groups.