  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 K\"ahler 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 K\"ahler forms on quotient spaces be exact. 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.
