superalgebra

and

supergeometry

# 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.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\mathbb{R}$ the real numbers
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\mathbb{C}$ the complex numbers
$6 = 5+1$$Spin(5,1) \simeq SL(2, \mathbb{H})$$\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) \underset{some\,sense}{\simeq} SL(2,\mathbb{O})$$\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 was apparently first made explicit in

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

A pedagogical review 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 September 20, 2016 00:34:50 by Urs Schreiber (89.204.137.119)