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 Z_M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S ³ × S ¹ or S ² × 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.