Global Supersymmetry on Curved Spaces in Various Dimensions

TL;DR

This work classifies Euclidean curved spaces across dimensions that admit rigid supersymmetry by combining Nahm’s group-theoretic constraints with on-shell supergravity background analyses. It shows that supersymmetry can exist on conformally flat spaces such as $\mathbb{H}^{n+1}$, $\mathbb{S}^n$, and on product spaces like $\mathbb{S}^1\times\mathbb{S}^n$ via R‑symmetry Wilson lines (for $n<6$), while round spheres with $d>5$ are excluded for unitary theories. The study covers four-, five-, and six-dimensional cases using $N=1$ and $N=2$ supergravity, and Romans $F(4)$ gauged supergravity, highlighting how background fields and Wilson lines control supersymmetry preservation. It also discusses ellipsoids and orbifolds as additional allowed geometries and outlines a general on-shell method to construct supersymmetric Lagrangians on curved spaces. Collectively, the results provide a framework for placing SUSY theories on curved manifolds and clarify fundamental dimensional limits for rigid and superconformal supersymmetry in Euclidean settings.

Abstract

We propose methods towards a systematic determination of d dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity point of view. In particular, by using appropriate gauged supergravities in various dimensions we show that supersymmetry can be defined in conformally flat spaces, such as non-compact hyperboloids $H^{n+1}$ and compact spheres $S^n$ or --by turning on appropriate Wilson lines corresponding to R-symmetry vector fields-- on $S^1 x S^n$, with n<6. By group theory arguments we show that Euclidean field theories with rigid supersymmetry cannot be consistently defined on round spheres $S^d$ if d>5 (despite the existence of Killing spinors). We also show that distorted spheres and certain orbifolds are also allowed by the group theory classification.