Global Supersymmetry on Curved Spaces in Various Dimensions

TL;DR

This paper systematically identifies curved spaces where Euclidean supersymmetric field theories can be consistently defined, using group theory and supergravity methods, revealing restrictions and possibilities across various dimensions.

Contribution

It provides a comprehensive classification of curved spaces supporting rigid supersymmetry in various dimensions, including new insights into conformally flat spaces and limitations on spheres.

Findings

Supersymmetry can be defined on conformally flat spaces like hyperboloids and spheres.

Rigid supersymmetry cannot be consistently defined on round spheres in dimensions greater than 5.

Distorted spheres and orbifolds are also compatible with supersymmetry.

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.