3d Expansions of 5d Instanton Partition Functions

We propose a set of novel expansions of Nekrasov’s instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on ${\mathbb{C}}_{q,t^{-1}}^2\times {\mathbb{S}}^1$, 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.