Twisted Fock module of toroidal algebra via DAHA and vertex operators

TL;DR

This paper constructs a twisted Fock module for quantum toroidal algebra using vertex operators and representation theory, confirming conjectures related to Hilbert schemes and duality.

Contribution

It introduces a novel construction of the twisted Fock module of quantum toroidal l_1 algebra via vertex operators and DAHA, linking to geometric and duality conjectures.

Findings

Construction of the twisted Fock module using vertex operators.

Consistency with Gorsky-Negu conjecture on stable envelopes.

Manifestation of (l_1,l_n)-duality.

Abstract

We construct the twisted Fock module of quantum toroidal $\mathfrak{gl}_1$ algebra with a slope $n'/n$ using vertex operators of quantum affine $\mathfrak{gl}_n$. The proof is based on the $q$-wedge construction of an integrable level-one $U_q(\widehat{\mathfrak{gl}}_n)$-module and the representation theory of double affine Hecke algebra. The results are consistent with Gorsky-Negu\c{t} conjecture (Kononov-Smirnov theorem) on stable envelopes for Hilbert schemes of points in the plane and can be viewed as a manifestation of $(\mathfrak{gl}_1,\mathfrak{gl}_n)$-duality.