Elliptic Quantum Toroidal Algebra U_{t_1,t_2,p}(gl_{1,tor}), Vertex Operators and L-operators

TL;DR

This paper introduces new vertex operators for the elliptic quantum toroidal algebra, linking them to elliptic stable envelopes, and constructs an L-operator satisfying the RLL relation, advancing the algebra's representation theory.

Contribution

The paper develops novel vertex operators for the elliptic quantum toroidal algebra and connects them with elliptic stable envelopes and R-matrix relations, providing new tools for its representation theory.

Findings

Vertex operators reproduce elliptic stable envelope formulas.

Vacuum expectation values yield K-theoretic vertex functions.

Constructed L-operator satisfies RLL=LLR^* relation.

Abstract

We propose new vertex operators, both the type I and the type II dual, of the elliptic quantum toroidal algebra U_{t_1,t_2,p}(gl_{1,tor}) by combining representations of U_{t_1,t_2,p}(gl_{1,tor}) and the notions of the elliptic stable envelopes for the instanton moduli space M(n,r). The vertex operators reproduce the shuffle product formula of the elliptic stable envelopes by their composition. We also show that the vacuum expectation value of a composition of the vertex operators gives a correct formula of the K-theoretic vertex function for M(n,r). We then derive exchange relations among the vertex operators and construct a L-operator satisfying the RLL=LLR^* relation with R and R^* being elliptic dynamical instanton R-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of L, we define a comultiplication \Delta in terms of it. It turns out that the new vertex operators are intertwining operators of the U_{t_1,t_2,p}(gl_{1,tor})-modules w.r.t \Delta.