  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.
