Rigidly Supersymmetric Gauge Theories on Curved Superspace

TL;DR

This paper constructs rigid supersymmetric gauge theories on curved four-manifolds, revealing how supersymmetry constrains target space geometry and Fayet-Iliopoulos parameters, with implications for the structure of these models.

Contribution

It introduces methods to build supersymmetric gauge theories on curved spaces and analyzes geometric constraints imposed by supersymmetry, including effects on Kähler forms and Fayet-Iliopoulos parameters.

Findings

Supersymmetry constrains target spaces to have exact Kähler forms on certain manifolds.

Fayet-Iliopoulos parameters are constrained, affecting the geometry of quotient spaces.

Discussion of universality classes and affine bundle structures in gauged sigma models.

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 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.