Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical β-deformed Hermitean matrix model and the q-deformed matrix models associated to 3d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories on D2×qS1 and $$ {S}_b^3 $$ S b 3 by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the W -algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.

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We show how q-Virasoro constraints can be derived for a large class of (q, t)deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for βensembles. From free field point of view the models considered have zero momentum of the highest weight, which leads to an extra constraint T −1 Z = 0. We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on D 2 × S 1 and S 3 .

Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d N = 2 Yang-Mills-Chern-Simons U (N ) theory on the squashed sphere S 3 b with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level κ 2 and Fayet-Illiopoulos parameter κ 1 . For these values of κ 1 and κ 2 the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent D 2 ×S 1 theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

We consider 4d supersymmetric (special) unitary Γ quiver gauge theories on compact manifolds which are T 2 fibrations over S 2 . We show that their partition functions are correlators of vertex operators and screening charges of the modular double version of elliptic W q,t;q ′ (Γ) algebras. We also consider a generating function of BPS surface defects supported on T 2 and show that it can be identified with a particular coherent state in the Fock module over the elliptic Heisenberg algebra.