Superintegrability of $(q,t)$-deformed matrix models and the quantum toroidal algebra

We study solutions of $q$-Virasoro constraints in refined or $(q,t)$-deformed matrix models with various potentials. Our goal is to establish an analogue of the W (or cut-and-joint) representation and its relation to “superintegrability” — a special form of averages of Macdonald polynomials. The examples considered come from localization of 3D SUSY theories, including the refined Chern-Simons model or gauge theories with adjoint and fundamental matter. We show that these models are governed by certain recursion relations given by a quantum toroidal algebra, also known as the Ding-Iohara-Miki (DIM) algebra or the elliptic Hall algebra. For the Chern-Simons model at $q=t$ these recursions seem to reproduce the recently obtained skein recursion relations for the unknot.