Rigidly supersymmetric gauge theories on curved superspace

In this note we construct rigidly supersymmetric gauged sigma models and gauge theories on certain Einstein four-manifolds, and discuss constraints on these theories. In work elsewhere, it was recently shown that on some nontrivial Einstein four-manifolds such as AdS(4), N = 1 rigidly supersymmetric sigma models are constrained to have target spaces with exact Kahler forms. Similarly, in gauged sigma models and gauge theories, we find that supersymmetry imposes constraints on Fayet-Iliopoulos parameters, which have the effect of enforcing that Kahler forms on quotient spaces be exact. We discuss the 'background principle' in this context. We also discuss general aspects of universality classes of gauged sigma models, as encoded by stacks, and also discuss affine bundle structures implicit in these constructions. In an appendix, we discuss how anomalies in four-dimensional gauge theories, such as those which play an important role in our analysis, can be recast in the language of stacks.