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 localizes to certain closed Reeb orbits and can be expressed as a sum over lattice points of the moment cone. This index problem arises for instance in the calculation of partition functions of cohomologically twisted supersymmetric gauge theories. This is joint work with Marcos Orseli.
