Global supersymmetry on curved spaces in various dimensions

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 Hn+1 and compact spheres Sn or - by turning on appropriate Wilson lines corresponding to R-symmetry vector fields - on S1×Sn, with n<6. By group theory arguments we show that Euclidean field theories with rigid supersymmetry cannot be consistently defined on round spheres Sd 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.