Spinorial Superspaces and Super Yang-Mills Theories

In physics literature about supersymmetry, many authors refer to “super Minkowski spaces”. These spaces are affine supermanifolds with certain distinguished spin structures. In these notes, we make the notion of such spin structures precise and generalise the setup to curved supermanifolds. This leads to the more general notion of spinorial superspaces. By working in a suitable geometric and coordinate-free setting, many explicit coordinate computations appearing in physics literature can be replaced by more conceptual methods. As an application of the rather general framework of spinorial superspaces, we formulate $\mathcal{N}=1$ super Yang-Mills theories on curved superspaces of spacetime dimensions $d=3$ and $d=4$ and show how to reduce the theory to field theories defined on an underlying ordinary spacetime manifold.