We study sigma models in AdS_4 with global N=1 supersymmetry and find that
they differ significantly from their flat-space cousins -- the target space is
constrained to be a Kahler manifold with an exact Kahler form, the
superpotential transforms under Kahler transformations, the space of
supersymmetric vacua is generically a set of isolated points even when the
superpotential vanishes, and the R-symmetry is classically broken by the
cosmological constant. Remarkably, the exactness of the Kahler class is also
required for the sigma model to arise as a decoupling limit of N=1
supergravity, and ensures the vanishing of gravitational anomalies. As simple
applications of these results, we argue that fields with AdS_4 scale masses are
ubiquitous in, for example, type IIB N=1 AdS_4 vacua stabilized near large
volume; we also show that the Affleck-Dine-Seiberg runaway of N_f < N_c SQCD is
regulated by considering the theory in AdS_4.Comment: 32 pages; v2: minor changes and references added; v3: discussion in
sect. 5 extended, version published in JHE
