Klein and Conformal Superspaces, Split Algebras and Spinor Orbits

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.

1+31 pages; v2: one Ref. added, and other minor changes. To be published in Reviews in Mathematical Physics (2017)

References in corpus (12)

• ICTP Lectures on Covariant Quantization of the Superstring
• Yang-Mills origin of gravitational symmetries
• Tractors, Mass and Weyl Invariance
• Global symmetries of Yang-Mills squared in various dimensions
• On the Magical Supergravities in Six Dimensions
• The Minkowski and conformal superspaces
• Geometry of Killing spinors in neutral signature
• Pseudo-Riemannian VSI spaces II
• The magic square of Lie groups: the $2\times 2$ case
• Quadratic deformation of Minkowski space
• Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square
• The symplectic origin of conformal and Minkowski superspaces

Cited by in corpus (4)

• Orbits in Non-Supersymmetric Magic Theories
• Quantum Klein Space and Superspace
• The Segre embedding of the quantum conformal superspace
• The $q$-linked complex Minkowski space, its real forms and deformed isometry groups