nLab
division algebra and supersymmetry

Idea

There is a close relationship between

• the four normed division algebras

• spin geometry,

• the Lie algebra cohomology of the super Poincare Lie algebra,

• and supersymmetry in quantum field theory and string theory.

This is based on the fact that in certain dimensions, spin group representations are naturally identified with a ${𝕂}^n$, for $𝕂$ one of the normed division algebras.

…

Applications

The structure of the normed division algebras governs the existence of 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.

References

Normed division algebras are used to construct Lie 2- and 3-superalgebras extending the Poincaré Lie superalgebra, and then Lie 2- and 3-supergroups extending the Poincaré Lie supergroups, here:

• John Baez, John Huerta, Division algebras and supersymmetry I (arXiv:0909.0551)

• John Baez, John Huerta, Division algebras and supersymmetry II (arXiv:1003.34360)

• John Huerta, Division Algebras, Supersymmetry and Higher Gauge Theory, (arXiv:1106.3385)

• John Huerta, Division Algebras and supersymmetry III, (arXiv:1109.3574)

The relationship in string theory via octonion algebra between the NRS spinning string and the Green-Schwarz superstring sigma-models is discussed in

• Rafael I. Nepomechie, Nonabelian bosonization, triality, and superstring theory Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210

• I. 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)