Rigidly Supersymmetric Gauge Theories on Curved Superspace

TL;DR

The paper develops rigid N=1 gauged sigma models and gauge theories on curved Einstein four-manifolds, uncovering constraints analogous to those in supergravity, such as exact Kahler forms on quotients and vanishing Fayet-Iliopoulos parameters. It demonstrates that the physical data organize through the combination K minus a superpotential dependent term, with the superpotential behaving as a section of an affine bundle, and analyzes two spacetime classes arising from auxiliary fields. In AdS4-like backgrounds the Kahler class must be cohomologically trivial and FI parameters vanish, while a second class with nonzero curvature in other directions yields different but consistent structures; the work also situates these theories in the broader language of stacks and gerbes to capture universality and nonperturbative aspects, and discusses anomaly considerations via background principles. The results lay groundwork for exploring instanton effects and other nonperturbative phenomena in curved spacetime SUSY theories and point to a rich interplay between geometry and field theory with potential applications in string compactifications and quantum field theory in curved backgrounds.

Abstract

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 Kähler 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 Kähler 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.