Moduli Space of Sheaves and Categorified Commutator of Functors

TL;DR

This paper develops a categorification of quantum toroidal algebra actions on the Grothendieck groups of moduli spaces of stable sheaves, introducing new intersection-theoretic descriptions of these moduli spaces.

Contribution

It provides a novel weak categorification framework for quantum toroidal algebra actions using intersection theory on moduli spaces of sheaves.

Findings

Constructed a weak categorification of the algebra action.

Introduced two new intersection-theoretic descriptions of moduli spaces.

Extended previous work by Schiffmann-Vasserot and Neguț.

Abstract

We construct a weak categorification of the quantum toroidal algebra action on the Grothendieck group of moduli space of stable (or framed) sheaves over an algebraic surface, which is constructed by Schiffmann-Vasserot and Negu\c{t}. The new ingredient is two intersection-theoretic descriptions of the quadruple moduli space of stable sheaves.