A Categorical Quantum Toroidal Action on Hilbert Schemes

TL;DR

This paper develops a geometric categorification of the quantum toroidal algebra's action on Hilbert schemes, connecting algebraic structures with geometric and categorical frameworks.

Contribution

It introduces a new geometric categorical action of the quantum toroidal algebra on Hilbert schemes, using detailed geometric analysis of nested Hilbert schemes.

Findings

Categorification of Nakajima's Heisenberg operators

Geometric categorical $U_{q_1,q_2}(\

Hilbert schemes are shown to have a geometric categorical quantum toroidal algebra action.

Abstract

We categorify the commutation of Nakajima's Heisenberg operators $P_{\pm 1}$ and their infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot{gl_1})$ acting on the Grothendieck groups of Hilbert schemes. By combining our result with arxiv:1804.03645 , one obtains a geometric categorical $U_{q_1,q_2}(\ddot{gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities.