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\section*{division algebra and supersymmetry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{exceptional_structures}{}\paragraph*{{Exceptional structures}}\label{exceptional_structures}

[[!include exceptional structures -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

There is a close relationship between

\begin{itemize}%
\item the four real [[normed division algebras]]


\item [[spin geometry]] and real (Majorana) [[spin representations]];


\item the [[Lie algebra cohomology]] of the [[super Poincare Lie algebra]],


\item and [[supersymmetry]] in [[quantum field theory]] and [[string theory]].



\end{itemize}
This is based on the fact that in certain dimensions, [[spin group]] [[representation]]s are naturally identified with a $\mathbb{K}^n$, for $\mathbb{K}$ one of the [[normed division algebras]], see at [[spin group]] \emph{\href{spin%20group#ExceptionalIsomorphisms}{The exceptional isomorphisms}}.

For detailed exposition see

\begin{itemize}%
\item \emph{[[geometry of physics -- supersymmetry]]} the section \emph{\href{geometry+of+physics+--+supersymmetry##RealSpinRepresentationViaNormedDivisionAlgebra}{Real spin representations via Real alternative division algebra}}


\item \emph{[[geometry of physics -- fundamental super p-branes]]}



\end{itemize}
[[!include exceptional spinors and division algebras -- table]]

The structure of the [[normed division algebras]] also governs the existence of the \emph{[[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]]''.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hopf invariant one]]

\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

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

\begin{itemize}%
\item [[Taichiro Kugo]], [[Paul Townsend]], \emph{Supersymmetry and the division algebras}, Nuclear Physics B, Volume 221, Issue 2 (1982) p. 357-380. (\href{http://inspirehep.net/record/181889}{spires}, \href{http://cds.cern.ch/record/140183/files/198301032.pdf}{pdf})


\item A. Sudbery, \emph{Division algebras, (pseudo)orthogonal groups and spinors}, Jour. Phys. A17 (1984), 939--955.


\item [[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


\item K.-W. Chung, A. Sudbery, \emph{Octonions and the Lorentz and conformal groups of ten-dimensional space-time}, Phys. Lett. B 198 (1987), 161--164.


\item [[Corinne Manogue]], A. Sudbery, \emph{General solutions of covariant superstring equations of motion}, Phys. Rev. D 12 (1989), 4073--4077


\item J\"o{}rg Schray, \emph{The general classical solution of the superparticle}, Class. Quant. Grav. 13 (1996), 27--38. (\href{https://arxiv.org/abs/hep-th/9407045}{arXiv:hep-th/9407045})


\item [[Tevian Dray]], J. Janesky, [[Corinne Manogue]], Octonionic hermitian matrices with non-real eigenvalues, Adv. Appl. Clifford Algebras 10 (2000), 193--216 (\href{https://arxiv.org/abs/math/0006069}{arXiv:math/0006069})



\end{itemize}
Streamlined proof and exposition regarding is in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry I}, in R. Doran, G. Friedman and [[Jonathan Rosenberg]] (eds.), \emph{Superstrings, Geometry, Topology, and $C*$-algebras}, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (\href{http://arxiv.org/abs/0909.0551}{arXiv:0909.0551})


\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry II}, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.34360})



\end{itemize}
Discussion with an emphasis on [[super Yang-Mills theory]] and [[U-duality]] in [[supergravity]] and the [[Freudenthal magic square]] is in

\begin{itemize}%
\item [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, \emph{A magic square from Yang-Mills squared} (\href{http://arxiv.org/abs/1301.4176}{arXiv:1301.4176})


\item A. Anastasiou, [[Leron Borsten]], [[Mike Duff]], L. J. Hughes, S. Nagy, \emph{Super Yang-Mills, division algebras and triality} (\href{http://arxiv.org/abs/1309.0546}{arXiv:1309.0546})


\item A. Anastasiou, [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, \emph{A magic pyramid of supergravities}, \href{http://arxiv.org/abs/1312.6523}{arXiv:1312.6523}



\end{itemize}
The relationship in [[string theory]] via [[octonion]] algebra between the NRS [[spinning string]] and the [[Green-Schwarz superstring]] [[sigma-models]] is discussed in

\begin{itemize}%
\item Rafael I. Nepomechie, \emph{Nonabelian bosonization, triality, and superstring theory} Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pages 207-210


\item [[Itzhak Bars]], D. Nemschansky and S. Yankielowicz, SLACPub-3758.


\item H. Tachibana, K. Imeda, \emph{Octonions, superstrings and ten-dimensional spinors} , Il nuovo cimento, Vol 104 B N.1



\end{itemize}
The relation of the division algebras to ordinary (Lie algebraic) extensions of the [[super Poincare Lie algebra]] is discussed in

\begin{itemize}%
\item Jerzy Lukierski, Francesco Toppan, \emph{Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory} (\href{https://arxiv.org/abs/hep-th/0203149}{arXiv:hep-th/0203149}, \href{http://cbpfindex.cbpf.br/publication_pdfs/NF00102.2010_08_03_10_47_48.pdf}{pdf})


\item A. Anastasiou, L. Borsten, [[Mike Duff]], L. J. Hughes, S. Nagy, \emph{An octonionic formulation of the M-theory algebra} (\href{http://arxiv.org/abs/1402.4649}{arXiv:1402.4649})



\end{itemize}
Normed division algebras are used to describe the construction of [[Lie 2-algebra]] extensions of the [[super Poincare Lie algebra]] in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry II}, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.34360})


\item [[John Huerta]], \emph{Division Algebras, Supersymmetry and Higher Gauge Theory}, (\href{http://arxiv.org/abs/1106.3385}{arXiv:1106.3385})


\item [[John Huerta]], \emph{Division Algebras and supersymmetry III}, (\href{http://arxiv.org/abs/1109.3574}{arXiv:1109.3574})



\end{itemize}
This is made use of in the [[homotopy theory|homotopy theoretic]] description of [[M-theory]]-structure in

\begin{itemize}%
\item [[John Huerta]], [[Urs Schreiber]], \emph{[[schreiber:M-Theory from the Superpoint]]}, Letters of Mathematical Physics, 2018 (\href{https://arxiv.org/abs/1702.01774}{arXiv:1702.01774})


\item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]} (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987})



\end{itemize}
[[!redirects division algebras and supersymmetry]]

[[!redirects supersymmetry and division algebra]] [[!redirects supersymmetry and division algebras]]

[[!redirects real spin representations and division algebras]] [[!redirects division algebras and real spin representations]]



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