There is a close relationship between
the four real normed division algebras
This is based on the fact that in certain dimensions, spin group representations are naturally identified with a , for one of the normed division algebras, see at spin group The exceptional isomorphisms.
For exposition see geometry of physics -- supersymmetry the section Real spin representations via Real alternative division algebra.
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
|the real numbers|
|the complex numbers|
|the quaternions||little string|
|the octonions||heterotic/type II string|
The structure of the normed division algebras also governs the existence of the brane scan and the super-∞-Lie algebras such as the supergravity Lie 3-algebra. By the D'Auria-Fre formulation of supergravity the ∞-Lie algebra valued forms with values in these constitute the field content of (11-dimensional) supergravity.
Combining this, one finds that supergravity coupled to super Yang-Mills theory (super Einstein-Yang-Mills theories) are parameterized by triples of real normed division algebras, forming a “magic pyramid”.
The relation between supersymmetry and division algebras was gradually established by a variety of authors, including
A. Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17 (1984), 939–955.
Jonathan Evans, Supersymmetric Yang–Mills theories and division algebras, Nucl. Phys. B298 (1988), 92–108. Also available as hhttp://www-lib.kek.jp/cgi-bin/img index?198801412i
K.-W. Chung, A. Sudbery, Octonions and the Lorentz and conformal groups of ten-dimensional space-time, Phys. Lett. B 198 (1987), 161–164.
Corinne Manogue, A. Sudbery, General solutions of covariant superstring equations of motion, Phys. Rev. D 12 (1989), 4073–4077
Jörg Schray, The general classical solution of the superparticle, Class. Quant. Grav. 13 (1996), 27–38. (arXiv:hep-th/9407045)
Streamlined proof and exposition regarding is in
John Baez, John Huerta, Division algebras and supersymmetry I, in R. Doran, G. Friedman and Jonathan Rosenberg (eds.), Superstrings, Geometry, Topology, and -algebras, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (arXiv:0909.0551)
Rafael I. Nepomechie, Nonabelian bosonization, triality, and superstring theory Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210
Itzhak Bars, D. Nemschansky and S. Yankielowicz, SLACPub-3758.
H. Tachibana, K. Imeda, Octonions, superstrings and ten-dimensional spinors , Il nuovo cimento, Vol 104 B N.1
The relation of the division algebras to ordinary (Lie algebraic) extensions of the super Poincare Lie algebra is discussed in
Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory (pdf)