$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

TL;DR

This paper develops a framework for lifting Wess-Zumino-Witten models to quantum toroidal algebras, deriving algebraic correlators satisfying the $(q,t)$-KZ equations, and connects these to Nekrasov functions and 6d gauge theories.

Contribution

It introduces algebraic correlators for quantum toroidal algebras without screenings and constructs intertwiners related to refined topological vertices.

Findings

Correlation functions satisfy the $(q,t)$-KZ equation with ${ m R}$-matrix.

Explicit matching with Nekrasov functions for instanton counting on ALE spaces.

Application to 6d gauge theories via elliptic $(q,t)$-KZ equations.

Abstract

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type $A_n$. The correlation functions of these operators satisfy the $(q,t)$-Knizhnik-Zamolodchikov (KZ) equation, which features the ${\cal R}$-matrix. Matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly. We also present an important application of the DIM formalism to the study of $6d$ gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic $(q,t)$-KZ equations.