Killing Spinors and SYM on Curved Spaces

TL;DR

This work constructs two families of globally supersymmetric Yang–Mills theories on curved spacetimes that admit Killing spinors, using generalized Killing spinor equations to ensure supersymmetry in nontrivial geometries. Family A (with internal $\Gamma^{[1]}$) and Family B (with $\Gamma^{[3]}$) differ by curvature-induced mass terms and, in the latter, cubic scalar couplings, yielding distinct SUSY algebras and vacuum structures. A key finding is that curvature generically lifts flat directions, so there is no continuous Coulomb branch in many cases, though AdS backgrounds can host half-BPS Coulomb branches that reduce to the standard flat-space branch in the Ricci-flat limit. The analysis also extends to Euclidean signatures and outlines partial superalgebras, with implications for curved D-brane worldvolume theories and AdS/CFT applications, while highlighting open questions about the underlying algebraic structures and quantum behavior. The approach centers on the Killing spinor framework with equations of the form $\nabla_{\mu}\varepsilon = \alpha\Gamma_{\mu}\Gamma\varepsilon$, enabling explicit construction and comparison of curved-space SUSY theories across dimensions $n<6$ and $n<8$ for the two families.

Abstract

We construct two families of globally supersymmetric counterparts of standard Poincaré supersymmetric SYM theories on curved space-times admitting Killing spinors, in all dimensions less than six and eight respectively. The former differs from the standard theory only by mass terms for the fermions and scalars and modified supersymmetry transformation rules, the latter in addition has cubic Chern-Simons like couplings for the scalar fields. We partially calculate the supersymmetry algebra of these models, finding R-symmetry extensions proportional to the curvature. We also show that generically these theories have no continuous Coulomb branch of maximally supersymmetric vacua, but that there exists a half-BPS Coulomb branch approaching the standard Coulomb branch in the Ricciflat limit.