Categorifying Quiver Linking/Unlinking using CoHA Modules

TL;DR

This paper provides a representation-theoretic categorification of quiver linking and unlinking operations, connecting moduli space cohomologies with Cohomological Hall Algebra actions in the context of knot-quiver relations.

Contribution

It introduces a new interpretation of linking/unlinking as relations between moduli spaces, enhancing the understanding of the knots-quivers correspondence through CoHA modules.

Findings

Moduli spaces of linked/unlinked quivers are naturally related.

Cohomologies of these moduli spaces are connected via CoHA actions.

The work categorifies quiver linking/unlinking at the CoHA module level.

Abstract

The knots-quivers correspondence is a relation between knot invariants and enumerative invariants of quivers, which in particular translates the knot operations of linking and unlinking to a certain mutation operation on quivers. In this paper we show that the moduli spaces of a quiver and its linking/unlinking are naturally related, giving a purely representation theoretic interpretation of these operations. We obtain a relation between the cohomologies of these spaces which is moreover compatible with a natural action of the Cohomological Hall Algebra. The result is a categorification of quiver linking/unlinking at the level of CoHA modules.