Spinorial Superspaces and Super Yang-Mills Theories

TL;DR

This paper introduces the concept of spinorial superspaces, generalizing super Minkowski spaces to curved supermanifolds, and applies this framework to formulate and reduce super Yang-Mills theories on curved superspaces.

Contribution

It formalizes spin structures on supermanifolds, extends the notion to curved cases, and develops a geometric approach to super Yang-Mills theories in this setting.

Findings

Defined spinorial superspaces in a geometric, coordinate-free manner

Formulated $ abla=1$ super Yang-Mills theories on curved superspaces

Reduced super Yang-Mills theories to ordinary spacetime field theories

Abstract

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.