Generalized indices for N=1 theories in four-dimensions

We use localization techniques to calculate the Euclidean partition functions for N = 1 theories on four-dimensional manifolds M of the form S-1 x M-3, where M-3 is a circle bundle over a Riemann surface. These are generalizations of the N = 1 indices in four-dimensions including the lens space index. We show that these generalized indices are holomorphic functions of the complex structure moduli on M. We exhibit the deformation by background flat connection.