Spinor Algebras

TL;DR

The work provides a comprehensive classification of spacetime supersymmetry algebras in arbitrary dimension and signature by linking Poincaré, conformal, and orthosymplectic structures. It introduces Spin$(s,t)$-algebras as the bosonic cores of minimal conformal superalgebras and shows how orthosymplectic algebras ${ m osp}(N|2p,\mathbb{R})$ realize maximal extensions, with contractions yielding the familiar supertranslation and Poincaré algebras. The analysis hinges on a detailed Spin$(V)$-morphism/Clifford-algebra framework, including real, quaternionic, and complex spinor types, and classifies all consistent embeddings into classical Lie groups (orthogonal, symplectic, and linear) across $D$ and $ ho$ modulo 8. The results recover known Minkowskian cases and provide a unified approach applicable to non-standard signatures, highlighting the role of R-symmetry, central charges, and the Coleman–Mandula/Haag–Lopuszański–Sohnius theorems in restricting allowed algebras. Overall, the paper maps the landscape of spacetime supersymmetry algebras and their maximal realizations across dimensions, signatures, and spinor types, with explicit tables connecting Lorentz, conformal, and orthosymplectic structures.

Abstract

We consider supersymmetry algebras in space-times with arbitrary signature and minimal number of spinor generators. The interrelation between super Poincaré and super conformal algebras is elucidated. Minimal super conformal algebras are seen to have as bosonic part a classical semimisimple algebra naturally associated to the spin group. This algebra, the Spin$(s,t)$-algebra, depends both on the dimension and on the signature of space time. We also consider maximal super conformal algebras, which are classified by the orthosymplectic algebras.