Localisation on Sasaki-Einstein manifolds from holomophic functions on the cone

We study super Yang-Mills theories on five-dimensional Sasaki-Einstein manifolds. Using localisation techniques, we find that the contribution from the vector multiplet to the perturbative partition function can be calculated by counting holomorphic functions on the associated Calabi-Yau cone. This observation allows us to use standard techniques developed in the context of quiver gauge theories to obtain explicit results for a number of examples; namely $S^5$, $T^{1,1}$, $Y^{7,3}$, $Y^{2,1}$, $Y^{2,0}$, and $Y^{4,0}$. We find complete agreement with previous results obtained by Qiu and Zabzine using equivariant indices except for the orbifold limits $Y^{p,0}$ with $p>1$.