Elliptic Quantum Toroidal Algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ and Elliptic Stable Envelopes for the $A^{(1)}_{N-1}$ Quiver Varieties

TL;DR

This paper introduces a new construction of vertex operators for the elliptic quantum toroidal algebra using elliptic stable envelopes, leading to novel algebraic relations and connections to K-theoretic vertex functions.

Contribution

It develops a novel vertex operator construction for the elliptic quantum toroidal algebra combining algebraic representations with elliptic stable envelopes, and establishes their algebraic relations.

Findings

Constructed vertex operators consistent with elliptic stable envelopes.

Derived exchange relations and $L$-operators satisfying elliptic dynamical $R$-matrix relations.

Connected vertex operators to K-theoretic vertex functions for quiver varieties.

Abstract

We propose a new construction of vertex operators of the elliptic quantum toroidal algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ by combining representations of the algebra and formulas of the elliptic stable envelopes for the $A^{(1)}_{N-1}$ quiver variety ${\cal M}(v,w)$. Compositions of the vertex operators turn out consistent to the shuffle product formula of the elliptic stable envelopes. Their highest to highest expectation values provide K-theoretic vertex functions for ${\cal M}(v,w)$. We also derive exchange relation of the vertex operators and construct a $L$-operator satisfying the $RLL=LLR^*$ relation with $R$ and $R^*$ being elliptic dynamical $R$-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of $L$ and defining a comultiplication $\Delta$ in terms of it, we show that our vertex operators are intertwining operators of the $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$-modules w.r.t $\Delta$.