nLab
division algebra and supersymmetry

Context

Super-Geometry

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

There is a close relationship between

This is based on the fact that in certain dimensions, spin group representations are naturally identified with a 𝕂 n\mathbb{K}^n, for 𝕂\mathbb{K} 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.

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)"SL(2,𝕆)"Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/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”.

References

The relation between supersymmetry and division algebras was gradually established by a variety of authors, including

  • Taichiro Kugo, Paul Townsend, Supersymmetry and the division algebras, Nuclear Physics B, Volume 221, Issue 2 (1982) p. 357-380. (spires, pdf)

  • 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)

  • Tevian Dray, J. Janesky, Corinne Manogue, Octonionic hermitian matrices with non-real eigenvalues, Adv. Appl. Clifford Algebras 10 (2000), 193–216 (arXiv:math/0006069)

Streamlined proof and exposition regarding is in

Discussion with an emphasis on super Yang-Mills theory and U-duality in supergravity and the Freudenthal magic square is in

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

  • 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)

  • A. Anastasiou, L. Borsten, Mike Duff, L. J. Hughes, S. Nagy, An octonionic formulation of the M-theory algebra (arXiv:1402.4649)

Normed division algebras are used to describe the construction of Lie 2-algebra extensions of the super Poincare Lie algebra in

Revised on January 9, 2017 11:28:27 by Urs Schreiber (188.100.203.189)