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

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 $Δ$ 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 $Δ$.