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We describe the general strategy for lifting the Wess-Zumino-Witten modelfrom the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to genericquantum toroidal algebras. A nearly exhaustive presentation is given for thetwo series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and$U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not existand thus all the correlators are purely algebraic, i.e. do not includeadditional hypergeometric type integrations/summations. Generalizing theconstruction of the intertwiner (refined topological vertex) of theDing-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of theFock representations of the quantum toroidal algebra of type $A_n$. Thecorrelation functions of these operators satisfy the$(q,t)$-Knizhnik-Zamolodchikov (KZ) equation, which features the ${\calR}$-matrix. Matching with the Nekrasov function for the instanton counting onthe ALE space is worked out explicitly. We also present an importantapplication of the DIM formalism to the study of $6d$ gauge theories describedby the double elliptic integrable systems. We show that the modular andperiodicity properties of the gauge theories are neatly explained by thenetwork matrix models providing solutions to the elliptic $(q,t)$-KZ equations.

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