division algebra and supersymmetry
∞-Lie theory (higher geometry)
Formal Lie groupoids
There is a close relationship between
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.
|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 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
- John Baez, John Huerta, Division algebras and supersymmetry I, in R. Doran, G. Friedman, Jonathan Rosenberg(eds.) Superstrings, Geometry, Topology, and -algebras, , Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (arXiv:0909.0551)
Discussion with an emphasis on super Yang-Mills theory and U-duality in supergravity and the Freudenthal magic square is in
Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, A magic square from Yang-Mills squared (arXiv:1301.4176)
A. Anastasiou, Leron Borsten, Mike Duff, L. J. Hughes, S. Nagy, Super Yang-Mills, division algebras and triality (arXiv:1309.0546)
A. Anastasiou, Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, A magic pyramid of supergravities, arXiv:1312.6523
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 November 2, 2016 08:57:10
by Urs Schreiber