Cohomological Hall algebras and character varieties

TL;DR

This paper explores the connection between twisted and untwisted character varieties through Cohomological Hall algebras, proposing a conjectural extension of known Donaldson-Thomas theory results.

Contribution

It introduces a conjectural lift of the relationship between character varieties into the Cohomological Hall algebra framework, building on prior Donaldson-Thomas theory calculations.

Findings

Hausel and Villegas' calculations of E polynomials for character varieties

Proposal of a conjectural extension to Cohomological Hall algebras

Insight into the algebraic structure of character varieties

Abstract

In this paper we investigate the relationship between twisted and untwisted character varieties via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi-Yau categories introduced by Kontsevich and Soibelman. In terms of Donaldson--Thomas theory, this relationship is completely understood via the calculations of Hausel and Villegas of the E polynomials of twisted character varieties and untwisted character stacks. We present a conjectural lift of this relationship to the cohomological Hall algebra setting.