Elliptic Quantum Toroidal Algebra $U_{t_1,t_2,p}({\mathfrak{gl}}_{N,tor})$ and Elliptic Stable Envelopes for the $A_{N-1}^{(1)}$ 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 $A_{N-1}^{(1)}$ quiver variety $\mathcal{M}(v,w)$. 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 $\mathcal{M}(v,w)$. We also derive exchange relation of the vertex operators and construct a $L$-operator satisfying the $RLL=LL{R}^{\ast }$ relation with $R$ and $R^{\ast }$ 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 $\Delta$.

63 pages