Elliptic Quantum Toroidal Algebra $U_{q,t,p}(g{l}_{1,tor})$ and Affine Quiver Gauge Theories

Abstract

We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(g{l}_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(g{l}_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized by using the elliptic Ruijsenaars difference operator. Intertwining operators of $U_{q,t,p}(g{l}_{1,tor})$-modules w.r.t. the Drinfeld comultiplication are also constructed. We show that $U_{q,t,p}(g{l}_{1,tor})$ gives a realization of the affine quiver $W$-algebra $W_{q,t}(\Gamma ({\hat{A}}_0))$ proposed by Kimura-Pestun. This realization turns out to be useful to derive the Nekrasov instanton partition functions, i.e. the ${\chi }_{y^{-}}$ and elliptic genus, of the 5d and 6d lifts of the 4d $\mathcal{N}=2^{\ast }$ theories and provide a new Alday-Gaiotto-Tachikawa correspondence.

arXiv comment63 pages, 3 figures, to be published in Letters in Mathematical Physics