Sasaki manifolds are the odd dimensional analogues of Kähler manifolds. In this talk, I will discuss the equivariant index of the horizontal Dolbeault complex on toric Sasaki manifolds. We show that the index localises to certain closed Reeb orbits and can be expressed as a sum over lattice points of the moment cone. This kind of index problems have appeared recently in the evaluation of partition functions of certain supersymmetric gauge theories.