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 S5, T1,1, Y7,3, Y2,1, Y2,0, and Y4,0. We find complete agreement with previous results obtained by Qiu and Zabzine using equivariant indices except for the orbifold limits Yp,0 with p > 1.