Narrow Results

We discuss ${𝒩}=1$ 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 ${ℂ}_s$. We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions $D=4,6$ and $10$ as ${ℂ}_s^2$, ${ℍ}_s^2$ and ${𝕆}_s^2$, 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.