3d Expansions of 5d Instanton Partition Functions

We propose a set of novel expansions of Nekrasovs instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on $mathbb{C}^2_{q,t^{-1}} times mathbb{S}^1$, we show that the instanton partition function admits expansions in terms of partition functions of unitary gauge theories living on the 3d subspaces $mathbb{C}_{q} times mathbb{S}^1$, $mathbb{C}_{t^{-1}} times mathbb{S}^1$ and their intersection along $mathbb{S}^1$. These new expansions are natural from the BPS/CFT viewpoint, as they can be matched with $W_{q,t}$ correlators involving an arbitrary number of screening charges of two kinds. Our constructions generalize and interpolate existing results in the literature.