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\begin{document}

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\section*{11d supergravity Lie 3-algebra}

[[!redirects supergravity Lie 3-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{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_chevalleyeilenberg_algebra}{The Chevalley-Eilenberg algebra}\dotfill \pageref*{the_chevalleyeilenberg_algebra} \linebreak
\noindent\hyperlink{hidden_super_lie_1algebra}{Hidden super Lie 1-algebra}\dotfill \pageref*{hidden_super_lie_1algebra} \linebreak
\noindent\hyperlink{relation_to_m5brane_action_functional}{Relation to M5-brane action functional}\dotfill \pageref*{relation_to_m5brane_action_functional} \linebreak
\noindent\hyperlink{Polyvector}{Relation to the 11-dimensional polyvector super Poincar\'e{}-algebra}\dotfill \pageref*{Polyvector} \linebreak
\noindent\hyperlink{via_derivations}{Via derivations}\dotfill \pageref*{via_derivations} \linebreak
\noindent\hyperlink{via_the_heisenberg_lie_3algebras}{Via the Heisenberg Lie 3-algebras}\dotfill \pageref*{via_the_heisenberg_lie_3algebras} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The \emph{supergravity Lie 3-algebra} $\mathfrak{sugra}(10,1)$ or [[M2-brane]] extension $\mathfrak{m}2\mathfrak{brane}$ is a [[super L-∞ algebra]] that is a shifted [[∞-Lie algebra cohomology|extension]]

\begin{displaymath}
0 \to b^2 \mathbb{R} \to 
  \mathfrak{sugra}(10,1) \to
  \mathfrak{siso}(10,1)
  \to
  0
\end{displaymath}
of the [[super Poincare Lie algebra]] $\mathfrak{siso}(10,1)$ in 10+1 dimensions induced by the exceptional degree 4-[[Lie algebra cohomology|super Lie algebra cocycle]]

\begin{displaymath}
\mu_4 = \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
  \;\;
  \in CE(\mathfrak{siso}(10,1))
  \,.
\end{displaymath}
This is the same mechanism by which the [[String Lie 2-algebra]] is a shifted central extension of $\mathfrak{so}(n)$.

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

\hypertarget{the_chevalleyeilenberg_algebra}{}\subsubsection*{{The Chevalley-Eilenberg algebra}}\label{the_chevalleyeilenberg_algebra}

\begin{prop}
\label{TheCEAlgebra}\hypertarget{TheCEAlgebra}{}
The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}(10,1))$ is generated on

\begin{itemize}%
\item elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$


\item a single element $c$ of degree $(3,even)$


\item and elements $\{\psi^\alpha\}$ of degree $(1,odd)$



\end{itemize}
with the differential defined by

\begin{displaymath}
d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
\end{displaymath}
\begin{displaymath}
d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
\end{displaymath}
\begin{displaymath}
d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
\end{displaymath}
\begin{displaymath}
d_{CE} \, c = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b
  \,.
\end{displaymath}
\end{prop}
\begin{quote}%
(fill in details)


\end{quote}
\hypertarget{hidden_super_lie_1algebra}{}\subsubsection*{{Hidden super Lie 1-algebra}}\label{hidden_super_lie_1algebra}

At the end of (\hyperlink{DAuriaFr82}{D'Auria-Fre 82}) the authors ask for a super Lie 1-algebra $\mathfrak{g}$, equipped with a degree-3 element $A$ in its [[Chevalley-Eilenberg algebra]], and equipped with a homomorphism $p\colon \mathfrak{g}\longrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}$ such that the pullback of the 4-cocycle $\mu_4$ along $p$ is trivialized by $A$:

\begin{displaymath}
p^\ast \mu_4 = d_{CE}A
  \,.
\end{displaymath}
In the [[model structure on L-infinity algebras|homotopy theory of L-infinity algebra]] this means that

\begin{displaymath}
\itexarray{
     \mathfrak{g} &\longrightarrow& \ast
     \\
     {}^{\mathllap{p}}
     \downarrow
       &\swArrow_{\mathrlap{A}}&
     \downarrow
     \\
     \mathbb{R}^{10,1\vert \mathbf{32}}
       &\underset{\mu_4}{\longrightarrow}&
     \mathbf{B}^3 \mathbb{R}
  }
  \,.
\end{displaymath}
Compare this to the characterization of $\mathfrak{sugra}(10,1)$ as the [[homotopy fiber]] of $\mu_4$, hence as the \emph{universal} solution to this situation

\begin{displaymath}
\itexarray{
     \mathfrak{sugra}(10,1) &\longrightarrow& \ast
     \\
     \downarrow
       &\swArrow_{}&
     \downarrow
     \\
     \mathbb{R}^{10,1\vert \mathbf{32}}
       &\underset{\mu_4}{\longrightarrow}&
     \mathbf{B}^3 \mathbb{R}
  }
  \,.
\end{displaymath}
In any case, in (\hyperlink{DAuriaFr82}{D'Auria-Fre 82}) possible choices for $p \colon \mathfrak{g} \to \mathbb{R}^{10,1\vert\mathbf{32}}$ are found.

Curiously, the bosonic [[body]] of $\mathfrak{g}$ is such that when adapted to a compactification to 4d, then it is the [[exceptional tangent bundle]] on which the [[U-duality]] group [[E7]] has a canonical action.

In (\hyperlink{BAIPV04}{BAIPV 04}) these solutions are shown to extend to a 1-parameter family of solutions. Further comments are in (\hyperlink{AndrianopoliDAuriaRavera16}{Andrianopoli-D'Auria-Ravera 16}).

\hypertarget{relation_to_m5brane_action_functional}{}\subsubsection*{{Relation to M5-brane action functional}}\label{relation_to_m5brane_action_functional}

The supergravity Lie 3-algebra carries a 7-cocycle (the one that induces the [[supergravity Lie 6-algebra]]-extension of it). The corresponding WZW term is that of the [[M5-brane]] in its [[Green-Schwarz action functional]]-like formulation.

[[!include brane scan]]

\hypertarget{Polyvector}{}\subsubsection*{{Relation to the 11-dimensional polyvector super Poincar\'e{}-algebra}}\label{Polyvector}

\hypertarget{via_derivations}{}\paragraph*{{Via derivations}}\label{via_derivations}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathfrak{der}(\mathfrak{sugra}(10,1))$ be the [[automorphism ∞-Lie algebra]] of $\mathfrak{sugra}(10,1)$. This is a [[dg-Lie algebra]]. Write $\mathfrak{der}(\mathfrak{sugra}(10,1))_0$ for the ordinary [[Lie algebra]] in degree 0.

This is [[isomorphic]] to the polyvector-extension of the [[super Poincaré Lie algebra]] (see there) in $d = 10+1$ -- the ``[[M-theory super Lie algebra]]'' -- with ``2-brane central charge'': the Lie algebra spanned by generators $\{P_a, Q_\alpha, J_{a b}, Z^{a b}\}$ and graded Lie brackets those of the [[super Poincaré Lie algebra]] as well as

\begin{displaymath}
[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
\end{displaymath}
\begin{displaymath}
[Q_\alpha, Z^{a b}] = 2 i (C \Gamma^{[a})_{\alpha \beta}Q^{b]\beta}
\end{displaymath}
etc.

\end{prop}
This observation appears implicitly in (\hyperlink{Castellani05}{Castellani 05, section 3.1}), see (\hyperlink{FSS13}{FSS 13}).

\begin{proof}
With the presentation of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}(10,1))$ as in prop. \ref{TheCEAlgebra} above, the generators are identified with [[derivation]]s on $CE(\mathfrak{sugra}(10,1))$ as

\begin{displaymath}
P_a = [d_{CE}, \frac{\partial}{\partial e^a} ]
\end{displaymath}
and

\begin{displaymath}
Q_\alpha = [d_{CE}, \frac{\partial}{\partial \psi^\alpha} ]
\end{displaymath}
and

\begin{displaymath}
J_{a b} = [d_{CE}, \frac{\partial}{\partial \omega^{a b}} ]
\end{displaymath}
and

\begin{displaymath}
Z^{a b} = [d_{CE}, e^a \wedge e^b \wedge \frac{\partial}{\partial c}]
\end{displaymath}
etc. With this it is straightforward to compute the commutators. Notably the last term in

\begin{displaymath}
[Q_\alpha, Q_\beta] = i (C \Gamma^a)_{\alpha \beta} P_a + (C \Gamma_{a b})Z^{a b}
\end{displaymath}
arises from the contraction of the 4-cocycle $\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b$ with $\frac{\partial}{\partial \psi^\alpha}\wedge \frac{\partial}{\partial \psi^\beta}$.

\end{proof}
\hypertarget{via_the_heisenberg_lie_3algebras}{}\paragraph*{{Via the Heisenberg Lie 3-algebras}}\label{via_the_heisenberg_lie_3algebras}

(\ldots{})

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

The field configurations of 11-dimensional [[supergravity]] may be identified with [[∞-Lie algebra-valued forms]] with values in $\mathfrak{sugra}(10,1)$. See [[D'Auria-Fre formulation of supergravity]].

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

[[supergravity Lie 6-algebra]] $\to$ \textbf{supergravity Lie 3-algebra} $\to$ [[super Poincaré Lie algebra]]

\begin{itemize}%
\item [[4d supergravity Lie 2-algebra]]


\item [[extended supersymmetry]]


\item [[type II supergravity Lie 2-algebra]]


\item [[type II supersymmetry algebra]]


\item [[M-theory supersymmetry algebra]]



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

The [[Chevalley-Eilenberg algebra]] of $\mathfrak{sugra}(10,1)$ first appears in (3.15) of

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

\end{itemize}
and later in the textbook

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}

\end{itemize}
The manifest interpretation of this as a [[Lie 3-algebra]] and the supergravity field content as [[∞-Lie algebra valued forms]] with values in this is mentioned in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:L-∞ algebra connections]]}


\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]]}



\end{itemize}
A systematic study of the super-[[Lie algebra cohomology]] involved is in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry I} (\href{http://arxiv.org/abs/0909.0551}{arXiv:0909.0551})


\item [[John Baez]], [[John Huerta]], \emph{Division algebras and supersymmetry II} (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.34360})



\end{itemize}
See also [[division algebra and supersymmetry]].

Further discussion of its ``hidden'' super Lie algebra includes

\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]], [[Jose de Azcarraga]], 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})


\item L. Andrianopoli, [[Riccardo D'Auria]], L. Ravera, \emph{Hidden Gauge Structure of Supersymmetric Free Differential Algebras} (\href{https://arxiv.org/abs/1606.07328}{arXiv:1606.07328})



\end{itemize}
The computation of the automorphism Lie algebra of $\mathfrak{sugra}(10,1)$ 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})

\end{itemize}
A similar argument with more explicit use of the Lie 3-algebra as underlying the [[Green-Schwarz action functional|Green-Schwarz-like action functional]] for the [[M5-brane]] is 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}
[[!redirects m2brane]]



\end{document}