Orthosymplectic modules of cohomological Hall algebras

TL;DR

This paper explores modules over cohomological Hall algebras with classical type stabilizers, revealing their structure as twisted Yetter-Drinfeld modules and connecting to the AGT conjecture via orthosymplectic sheaves.

Contribution

It introduces a new framework for modules of CoHAs with classical stabilizers, demonstrating their compatibility with vertex coproducts and localised coactions, and relates these to the AGT conjecture.

Findings

Modules form twisted Yetter-Drinfeld modules over CoHAs.

Compatibility of CoHA actions with localised vertex coactions.

Connection between dimension zero sheaves and the AGT conjecture.

Abstract

We study modules and comodules for cohomological Hall algebras equipped with their vertex coproducts arising as objects with classical type stabilizer groups. Specifically we consider how classical type parabolic induction gives rise to actions of CoHAs of quivers with potential, of preprojective algberas, and of dimension zero sheaves on a smooth proper surface. In all cases the CoHA action is compatible with a localised (and vertex) coaction making the module a twisted Yetter-Drinfeld module over the CoHA with its localised braided bialgebra structure. In the case of dimension zero sheaves on a surface the action is related to an approach to the AGT conjecture in classical type using moduli stacks of orthosymplectic perverse coherent sheaves, a compactification of the stack of classical type bundles on a surface.