Existence of parallel spinors on nonsimply connected Riemannian manifolds

Abstract

It is well known, and important for applications, that Ricci-flat Riemannian manifolds of nongeneric holonomy always admit a parallel (covariant constant) spinor if they are simply connected. The nonsimply connected case is much more subtle, however. We show that a parallel spinor can still be found in this case provided that the (real) dimension is not a multiple of four, and provided that the spin structure is carefully chosen.