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

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