Comments on N = (2,2) supersymmetry on two-manifolds

We study curved-space rigid supersymmetry for two-dimensional N = (2, 2) supersymmetric fields theories with a vector-like R-symmetry by coupling such theories to background supergravity. The associated Killing spinors can be viewed as holomorphic sections of particular complex line bundles over Euclidean space-time, which severely restricts the allowed supersymmetric couplings on compact orientable Riemann surfaces without boundaries. For genus g > 1, the only consistent non-singular couplings are the ones dictated by the topological A-twist. On spaces with S (2) topology, there exist additional supersymmetric backgrounds with m = 0 or +/- 1 unit of flux for the R-symmetry gauge field. The m = -1 case includes the Omega-background on the sphere. We also systematically work out the curved-space supersymmetry multiplets and supersymmetric Lagrangians.