Localisation on Sasaki-Einstein manifolds from holomorphic 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,T-1, Y-7,Y-3, Y-2,Y-1, Y-2,Y-0, and Y-4,Y-0. We find complete agreement with previous results obtained by Qiu and Zabzine using equivariant indices except for the orbifold limits Y-p,Y-0 with p > 1.