Elliptic modular double and 4d partition functions

We consider 4d supersymmetric (special) unitary $\Gamma$ 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 $W_{q,t;q'}(\Gamma)$ 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.