Klein and Conformal Superspaces, Split Algebras and Spinor Orbits

TL;DR

This paper explores Klein and Klein-Conformal superspaces in 2+2 dimensions using split composition algebras, revealing new geometric structures and spinor orbit stratifications relevant for supersymmetry.

Contribution

It introduces a novel realization of superspaces via split composition algebras and analyzes spinor orbit stratification in these contexts.

Findings

Realization of superspaces over split composition algebras

Identification of spinor bundle sections as split algebra modules

Analysis of spinor orbit stratification impacts superspace structure

Abstract

We discuss $\mathcal{N}=1$ Klein and Klein-Conformal superspaces in $D=(2,2)$ space-time dimensions, realizing them in terms of their functor of points over the split composition algebra $\mathbb{C}_{s}$. We exploit the observation that certain split form of orthogonal groups can be realized in terms of matrix groups over split composition algebras; this leads to a natural interpretation of the the sections of the spinor bundle in the critical split dimensions $D=4$, $6$ and $10$ as $\mathbb{C}_{s}^{2}$, $\mathbb{H}_{s}^{2}$ and $\mathbb{O}_{s}^{2}$, respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-Conformal superspace structure.