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Factorization of 5D super Yang-Mills on $Y^{p,q}$ spaces

We continue our study on the partition function for 5D supersymmetric Yang-Mills theory on toric Sasaki-Einstein $Y^{p,q}$ manifolds. Previously, using the localisation technique we have computed the perturbative part of the partition function. In this work we show how the perturbative part factorises into four pieces, each corresponding to the perturbative answer of the same theory on $\mathbb{R}^4 \times S^1$. This allows us to identify the equivariant parameters and to conjecture the full partition functions (including the instanton contributions) for $Y^{p,q}$ spaces. The conjectured partition function receives contributions only from singular contact instantons supported along the closed Reeb orbits. At the moment we are not able to prove this fact from the first principles.