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\section*{M-theory supersymmetry algebra}

[[!redirects M-theory super Lie algebra]]

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

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

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

[[!include supergeometry - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_the_algebra_of_conserved_currents_of_the_mbranes}{As the algebra of conserved currents of the M-branes}\dotfill \pageref*{as_the_algebra_of_conserved_currents_of_the_mbranes} \linebreak
\noindent\hyperlink{as_the_lie_algebra_of_derivations_of_the_sugra_lie_3algebra}{As the Lie algebra of derivations of the SuGra Lie 3-algebra}\dotfill \pageref*{as_the_lie_algebra_of_derivations_of_the_sugra_lie_3algebra} \linebreak
\noindent\hyperlink{AsAn11DimensionalBoundaryCondition}{As an 11-dimensional boundary condition for the M2-brane}\dotfill \pageref*{AsAn11DimensionalBoundaryCondition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesFromTheM2Cocycle}{From the M2-Cocyle}\dotfill \pageref*{ReferencesFromTheM2Cocycle} \linebreak
\noindent\hyperlink{alternative}{Alternative}\dotfill \pageref*{alternative} \linebreak


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

A [[super Lie algebra]] which is a \href{super+Poincare+Lie+algebra#PolyvectorExtensions}{polyvector} [[Lie algebra extension|extension]] of the [[super Poincaré Lie algebra]] ([[supersymmetry]]) in $D = 10+1$ for $N=1$ [[supersymmetry]] by [[charges]] corresponding to the [[M2-brane]] and the [[M5-brane]] (``[[extended supersymmetry]]'').

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_the_algebra_of_conserved_currents_of_the_mbranes}{}\subsubsection*{{As the algebra of conserved currents of the M-branes}}\label{as_the_algebra_of_conserved_currents_of_the_mbranes}

In (\hyperlink{AGIT89}{AGIT 89}) it is shows that the M-algebra-like polyvector extensions arise as the algebras of [[conserved currents]] of the [[Green-Schwarz super p-brane sigma-models]].

By the discussion at \href{conserved+current#InHigherPrequantumGeometry}{conserved current -- In higher prequantum geometry} this means that this is the degree-0 piece in the [[Heisenberg Lie n-algebra]] which is induced by regarding the WZW-curvature terms as super [[n-plectic forms]] on $\mathbb{R}^{10,1|32}$.

\hypertarget{as_the_lie_algebra_of_derivations_of_the_sugra_lie_3algebra}{}\subsubsection*{{As the Lie algebra of derivations of the SuGra Lie 3-algebra}}\label{as_the_lie_algebra_of_derivations_of_the_sugra_lie_3algebra}

In (\hyperlink{Castellani05}{Castellani 05}) it is implicitly shown, (\hyperlink{FSS13}{FSS 13}), that the M-extension arises as the [[derivations]]/automorphisms of the [[supergravity Lie 3-algebra]]/[[supergravity Lie 6-algebra]] (see there for the details).

\hypertarget{AsAn11DimensionalBoundaryCondition}{}\subsubsection*{{As an 11-dimensional boundary condition for the M2-brane}}\label{AsAn11DimensionalBoundaryCondition}

The original construction in (\hyperlink{DAuriaFre82}{D'Auria-Fré 82}) asks for a [[super Lie algebra]] [[Lie algebra extension|extension]] $\mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g}$ of [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert 32}$ such that the 4-[[cocycle]] $\mu_4 = \overline{\psi} \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$ for the M2-brane trivializes when pulled back to this:

\begin{displaymath}
\itexarray{
    & & \mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g}
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\simeq} && \mathbb{R}^{10,1\vert 32}
    \\
    & \searrow && \swarrow_{\mathrlap{\mu_4}}
    \\
    && B^{3}\mathbb{R}
  }
  \,.
\end{displaymath}
(In the language of [[local prequantum field theory]] this identifies a [[boundary condition]] for the WZW term of the M2-brane.)

They find, see also (\hyperlink{BandosAzcarragaIzquierdoPiconVarela04}{Bandos-Azcarraga-Izquierdo-PiconVarela 04}) that a solution for $\mathfrak{g}$ includes a fermionic extension of the M-theory super Lie algebra.

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

\begin{itemize}%
\item [[type II super Lie algebra]]


\item [[BPS state]]


\item [[supergravity Lie 3-algebra]]


\item [[supergravity Lie 6-algebra]]


\item [[M2-brane]], [[M5-brane]]


\item [[M-theory]]



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

\hypertarget{ReferencesFromTheM2Cocycle}{}\subsubsection*{{From the M2-Cocyle}}\label{ReferencesFromTheM2Cocycle}

Two versions of a fermionic extension of the Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ on which the [[M2-brane]] [[Lie algebra cohomology|4-cocycle]] trivializes were first found in

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]], last pages of \emph{[[GeometricSupergravity.pdf:file]]}, Nuclear Physics B201 (1982) 101-140

\end{itemize}
and then a 1-parameter family of such was discovered in

\begin{itemize}%
\item [[Igor Bandos]], [[José de Azcárraga]], J.M. Izquierdo, M. Picon, O. Varela, \emph{On the underlying gauge group structure of D=11 supergravity}, Phys.Lett.B596:145-155,2004 (\href{http://arxiv.org/abs/hep-th/0406020}{arXiv:hep-th/0406020})


\item [[Igor Bandos]], [[José de Azcárraga]], Moises Picon, Oscar Varela, \emph{On the formulation of $D=11$ supergravity and the composite nature of its three-from field}, Annals Phys. 317 (2005) 238-279 (\href{https://arxiv.org/abs/hep-th/0409100}{arXiv:hep-th/0409100})



\end{itemize}
That a limiting case of this is given by the [[orthosymplectic super Lie algebra]] $\mathfrak{osp}(1\vert 32)$ is due to

\begin{itemize}%
\item J.J. Fernandez, J.M. Izquierdo, M.A. del Olmo, \emph{Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators}, Nuclear Physics B Volume 897, August 2015, Pages 87--97 (\href{http://arxiv.org/abs/1504.05946}{arXiv:1504.05946})

\end{itemize}
Further discussion is in

\begin{itemize}%
\item [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], \emph{Hidden Gauge Structure of Supersymmetric Free Differential Algebras}, JHEP 1608 (2016) 095 (\href{https://arxiv.org/abs/1606.07328}{arXiv:1606.07328})


\item [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], \emph{More on the Hidden Symmetries of 11D Supergravity} (\href{https://arxiv.org/abs/1705.06251}{arXiv:1705.06251})



\end{itemize}
where the algebra is referred to as the \emph{DF-algebra}, in honor of \hyperlink{DAuriaFre82}{D'Auria-Fré 82}.

All this is reviewed in

\begin{itemize}%
\item [[José de Azcárraga]], section 5 of \emph{Superbranes, D=11 CJS supergravity and enlarged superspace coordinates/fields correspondence}, AIP Conf. Proc. 767:243-267, 2005 (\href{https://arxiv.org/abs/hep-th/0501198}{arXiv:hep-th/0501198})

\end{itemize}
Another, alternative ``weak [[decomposable differential form|decomposition]]'' of the M2-brane [[extended super-Minkowski spacetime]] was found in

\begin{itemize}%
\item [[Lucrezia Ravera]], section 3 of \emph{Hidden Role of Maxwell Superalgebras in the Free Differential Algebras of D=4 and D=11 Supergravity} (\href{https://arxiv.org/abs/1801.08860}{arXiv:1801.08860})

\end{itemize}
with the interesting difference that for this splitting [[super Lie algebra]] is non-abelian, in fact an extension of the Lie algebra of the [[Spin group]] $Spin(10,1)$ (\hyperlink{Ravera18}{Ravera 18, (3.5)-(3-6)}).

See also

\begin{itemize}%
\item [[Lucrezia Ravera]], \emph{Group Theoretical Hidden Structure of Supergravity Theories in Higher Dimensions} (\href{https://arxiv.org/abs/1802.06602}{arXiv:1802.06602})

\end{itemize}
That the underlying bosonic [[body]] of this super Lie algebra happens to be the [[typical fiber]] of what would be the 11-d [[exceptional generalized tangent bundle]], namely the \href{E11#FundamentalRepresentationAndBraneCharges}{level-2 truncation of the l1-representation} of [[E11]] according to (\href{E11#West04}{West 04}) was highlighted in the review

\begin{itemize}%
\item [[Silvia Vaula]], \emph{On the underlying $E_{11}$ symmetry of the $D= 11$ Free Differential Algebra}, JHEP 0703:010, 2007 (\href{http://arxiv.org/abs/hep-th/0612130}{arXiv:hep-th/0612130})

\end{itemize}
For analogous discussion in [[7d supergravity]] and [[4d supergravity]], see the references there.

\hypertarget{alternative}{}\subsubsection*{{Alternative}}\label{alternative}

From a different perspective the M-theory algebra extensions were (apparently independently) introduced in

\begin{itemize}%
\item [[Jan-Willem van Holten]], [[Antoine Van Proeyen]], \emph{$N=1$ supersymmetry algebras in $d=2,3,4 \,mod\, 8$} J.Phys. A15, 3763 (1982).


\item [[Paul Townsend]], \emph{p-Brane Democracy} (\href{http://arxiv.org/abs/hep-th/9507048}{arXiv:hep-th/9507048})



\end{itemize}
with further amplification including

\begin{itemize}%
\item [[Paul Townsend]], \emph{M(embrane) theory on $T^0$}, Nucl.Phys.Proc.Suppl.68:11-16,1998 (\href{http://arxiv.org/abs/hep-th/9708034}{arXiv:hep-th/9708034})


\item [[Paul Townsend]], \emph{M-theory from its superalgebra}, Cargese lectures 1997 (\href{http://arxiv.org/abs/hep-th/9712004}{arXiv:hep-th/9712004})



\end{itemize}
In their global form, where differential forms are replaced by their de Rham cohomology classes on curved superspacetimes, these algebras were identified (for the case including the 2-form piece but not the 5-form piece) as the algebras of [[conserved currents]] of the [[Green-Schwarz super p-brane sigma-models]] in

\begin{itemize}%
\item [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], \emph{Topological Extensions of the Supersymmetry Algebra for Extended Objects}, Phys.Rev.Lett. 63 (1989) 2443 (\href{https://inspirehep.net/record/26393?ln=en}{spire})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[José de Azcárraga]], Jos\'e{} M. Izquierdo, section 8.8 of \emph{[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]}, Cambridge monographs of mathematical physics, (1995)

\end{itemize}
The generalization of this including also the contribution of the [[M5-brane]] was considered in

\begin{itemize}%
\item [[Dmitri Sorokin]], [[Paul Townsend]], \emph{M-theory superalgebra from the M-5-brane}, Phys.Lett. B412 (1997) 265-273 (\href{http://arxiv.org/abs/hep-th/9708003}{arXiv:hep-th/9708003})

\end{itemize}
Further detailed discussion along these lines producing also the [[type II supersymmetry algebras]] is in

\begin{itemize}%
\item Hanno Hammer, \emph{Topological Extensions of Noether Charge Algebras carried by D-p-branes}, Nucl.Phys. B521 (1998) 503-546 (\href{http://arxiv.org/abs/hep-th/9711009}{arXiv:hep-th/9711009})

\end{itemize}
The full extension was named ``M-algebra'' in

\begin{itemize}%
\item [[Ergin Sezgin]], \emph{The M-Algebra}, Phys.Lett. B392 (1997) 323-331 (\href{http://arxiv.org/abs/hep-th/9609086}{arXiv:hep-th/9609086})

\end{itemize}
In (\hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}) the motivation is from the formulation of the fields of [[11-dimensional supergravity]] as connections with values in the [[supergravity Lie 3-algebra]], see at \emph{[[D'Auria-Fré formulation of supergravity]]}. Realization of the M-theory super Lie algebra as the algebra of [[derivations]] of the [[supergravity Lie 3-algebra]] is in

\begin{itemize}%
\item [[Leonardo Castellani]], \emph{Lie derivatives along antisymmetric tensors, and the M-theory superalgebra}, J. Phys. Math. Volume 3 (2011), 1-7. (\href{http://arxiv.org/abs/hep-th/0508213}{arXiv:hep-th/0508213}, \href{http://projecteuclid.org/euclid.jpm/1359468398}{Euclid})

\end{itemize}
with amplification in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} (2013)


\item [[nLab:Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Lie n-algebras of BPS charges]]}



\end{itemize}
Discussion of a formulation in terms of [[octonions]] (see also at \emph{[[division algebra and supersymmetry]]}) includes

\begin{itemize}%
\item A. Anastasiou, [[Leron Borsten]], [[Michael 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}
Arguments that the charges of the M-theory super Lie algebra may be identified inside [[E11]] are given in

\begin{itemize}%
\item [[Peter West]], \emph{$E_{11}$, $SL(32)$ and Central Charges}, Phys.Lett.B575:333-342,2003 (\href{http://arxiv.org/abs/hep-th/0307098v2}{arXiv:hep-th/0307098v2})


\item [[Paul Cook]], around p. 75 of \emph{Connections between Kac-Moody algebras and M-theory} (\href{http://arxiv.org/abs/0711.3498}{arXiv:0711.3498})



\end{itemize}
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[[!redirects D'Auiria-Fre algebra]] [[!redirects D'Auiria-Fre algebras]]



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