The geometry of supersymmetric partition functions

We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z(M). Our primary focus is the dependence of Z(M) on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N = 1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z(M) is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z(M) is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N = 2 theories with a U(1) R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that ZM is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S-3 x S-1 or S-2 x S-1, which are related to supersymmetric indices, and manifolds diffeomorphic to S-3 (squashed spheres). In examples where Z(M) has been calculated explicitly, our results explain many of its observed properties.