supersymmetry

# 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 $\mathbb{K}^n$, for $\mathbb{K}$ 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
$\phantom{AA}$spin groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq SL(2, \mathbb{H})$$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}$$\phantom{A}$ $\mathbb{O}$ 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)