Topological mirror symmetry for rank two character varieties of surface groups

TL;DR

This paper proves the equality of Hodge numbers for the intersection cohomology of rank two character varieties, confirming a mirror symmetry conjecture and advancing understanding of their topological and geometric properties.

Contribution

It establishes topological mirror symmetry for rank two surface group character varieties by proving the equality of their stringy Hodge numbers.

Findings

Hodge numbers of SL2 and PGL2 moduli spaces are equal

Confirmed mirror symmetry predictions for rank two character varieties

Answered a longstanding question by Tamás Hausel

Abstract

The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tam\'as Hausel in Remark 3.30 of "Global topology of the Hitchin system".