nLab division algebra and supersymmetry

Idea

There is a close relationship between

• the four real normed division algebras

• spin geometry and real (Majorana) spin representations;

• 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 $\mathbb{K}^n$, for $\mathbb{K}$ one of the normed division algebras, see at spin group The exceptional isomorphisms.

For detailed exposition see

• geometry of physics – supersymmetry the section Real spin representations via Real alternative division algebra

• geometry of physics – fundamental super p-branes

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ 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”.

Related concepts

• SL(2,H)

• Hopf invariant one

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. (spire:181889, doi:10.1016/0550-3213(83)90584-9, pdf)

• Anthony Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17 (1984),

  939–955 (doi:10.1088/0305-4470/17/5/018)

• Jonathan Evans, Supersymmetric Yang–Mills theories and division algebras, Nucl. Phys. B298 (1988), 92–108 (https://doi.org/10.1016/0550-3213(88)90305-7)

• K.-W. Chung, Anthony Sudbery, Octonions and the Lorentz and conformal groups of ten-dimensional space-time, Phys. Lett. B 198 (1987), 161–164 (doi:10.1016/0370-2693(87)91489-4)

• Corinne Manogue, Anthony Sudbery, General solutions of covariant superstring equations of motion, Phys. Rev. D 12 (1989), 4073–4077 (doi:10.1103/PhysRevD.40.4073)

• 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)

Corresponding discussion of twistor space:

• Ingemar Bengtsson, Martin Cederwall, Particles, Twistors and the Division Algebras, Nucl.Phys. B302 (1988) 81-103 (spire)

• Martin Cederwall, Introduction to Division Algebras, Sphere Algebras and Twistors (arXiv:hep-th/9310115)

and its AdS version

• Alex Arvanitakis, Alec E. Barns-Graham, Paul Townsend, Twistor description of spinning particles in AdS, JHEP 01 (2018) 059 (arXiv:1710.09557)

Streamlined proof and exposition is in

• John Baez, John Huerta: Division algebras and supersymmetry I, in: Superstrings, Geometry, Topology, and $C^*\ast$-algebras, Proc. Symp. Pure Math. 81, AMS (2010) 65-80 [arXiv:0909.0551, ISBN:978-0-8218-4887-6]

• John Baez, John Huerta: Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011) 1373-1410 [arXiv:1003.34360, doi:10.4310/ATMP.2011.v15.n5.a4]

The case of 2-component quaternionic spinors in 6d (see at SL(2,H)) is discussed in more detail in:

• Joás Venâncio, Carlos Batista, Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes, Advances in Applied Clifford Algebras 31 71 (2021) [arXiv:2007.04296, doi:10.1007/s00006-021-01172-1]

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, Mia J. Hughes, Silvia Nagy, A magic square from Yang-Mills squared (arXiv:1301.4176)

• Alexandros Anastasiou, Leron Borsten, Mike Duff, Mia J. Hughes, Silvia 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 (arXiv:hep-th/0203149, 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

• John Baez, John Huerta, Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (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)

This is made use of in the homotopy theoretic description of M-theory in:

• John Huerta, Urs Schreiber, M-Theory from the Superpoint, Letters of Mathematical Physics, 2018 (arXiv:1702.01774)

• John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes (arXiv:1805.05987)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super-exceptional M5-brane model – Emergence of SU(2)-flavor sector (arXiv:2006.00012)