Spinorial Superspaces and Super Yang-Mills Theories
Johannes Moerland
TL;DR
This work develops a coordinate-free, geometric framework for supersymmetric gauge theories on curved superspaces by introducing spinorial superspaces supported by Spin$(V,q)$-structures. It shows how to formulate $\ olinebreak \mathcal N = 1$ super Yang–Mills theories on spinorial superspaces in $d=3$ and $d=4$ and how to reduce these theories to ordinary spacetimes, yielding familiar Dirac–Yang–Mills Lagrangians with component fields $a$ and $\lambda$. The construction relies on a careful decomposition of the tangent bundle into bosonic and spinorial distributions, a Dolbeault-like chirality decomposition, and a split-superspace formalism that enables odd-direction integration via the Berezinian. The results provide manifest gauge invariance and a natural geometric interpretation of supersymmetric fields, with potential extensions to curved, higher-dimensional, and Euclidean settings. Overall, the paper establishes a robust, algebraically transparent framework for spinorial superspaces and their physical applications to sYM 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.