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\begin{document}

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\section*{supergravity Lie 6-algebra}

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

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

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

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

[[!include string theory - contents]]

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

[[!include supergeometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{RelationToSupergravity}{Relation to $D = 11$ supergravity}\dotfill \pageref*{RelationToSupergravity} \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}

The \emph{supergravity Lie 6-algebra} is a [[super L-∞ algebra]] such that [[connection on an ∞-bundle|∞-connections]] with values in it encode

\begin{itemize}%
\item a [[vielbein]] field and a [[spin connection]], hence the [[first order formulation of gravity]] for a [[graviton]] field in 11-[[dimension]]s;


\item a [[gravitino]] field;


\item the [[supergravity C-field]] in degree 3 \emph{and} its [[electric-magnetic duality|magnetic dual]].



\end{itemize}
This is such that the [[field strength]]s and [[Bianchi identities]] of these fields are governed by certain fermionic [[L-∞ algebra cohomology|super L-∞ algebraic cocycles]] as suitable for [[11-dimensional supergravity]].

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

\begin{prop}
\label{TheSevenCocycle}\hypertarget{TheSevenCocycle}{}
The [[supergravity Lie 3-algebra]] $\mathfrak{sugra}_3(10,1)$ carries an [[L-∞ algebra cohomology|L-∞ algebra cocycle]] $\mu_7 \in CE(\mathfrak{sugra}_3(10,1))$ of degree 7, given in the standard generators $\{e^a\}$ ([[vielbein]]), $\{\omega^{a b}\}$ ([[spin connection]]) $\{\psi^\alpha\}$ ([[gravitino]]) and $\{c_3\}$ ([[supergravity C-field]]) by

\begin{displaymath}
\mu_7
   \coloneqq
   \frac{i}{2}\bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
   +
  15 \mu_4 \wedge c_3
  \,,
\end{displaymath}
where

\begin{displaymath}
\mu_4 = \frac{i}{2}\bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2}
\end{displaymath}
is the 4-cocycle which defines $\mathfrak{sugra}_3(10,1)$ as an extension of $\mathbb{R}^{10,1\vert \mathbf{32}}$m and where $c_3$ is the generator that cancels the class of this cocycle, $d_{CE} c_3 \propto \mu_4$.

\end{prop}
This appears in (\hyperlink{DAuriaFre}{DAuria-Fre, page 18}) and \hyperlink{CastellaniDAuriaFre}{Castellani-DAuria-Fre, III.8.3}.

\begin{proof}
One computes

\begin{displaymath}
\begin{aligned}
    d_{CE} \mu_7 = 
    & 
     - \frac{5}{4} \bar \psi \wedge \Gamma^{a_1 \cdots a_4 b} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_4} \wedge \bar \psi \wedge \Gamma_b \psi
  \\
  & - i 15 \wedge \Gamma^{a b} e_a \wedge \bar \psi \wedge \Gamma_b \psi \wedge c_3
   \\
   & +
    \frac{15}{4} \bar \psi \wedge \Gamma_{a b} \psi \wedge e^a \wedge e^b \wedge \bar \psi \wedge \Gamma_{c d} \psi \wedge e^c \wedge e^d 
  \end{aligned}
  \,.
\end{displaymath}
This expression vanishes due to the [[Fierz identities]]

\begin{displaymath}
\bar \psi \wedge 
  \Gamma^{a_1 \cdots a_4 b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 
  3 \bar \psi \wedge \Gamma^{[a_1 a_2} \psi \wedge \bar \psi \wedge \Gamma^{a_3 a_4 ]} \psi
\end{displaymath}
and

\begin{displaymath}
\bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 0 
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Hence if we write

\begin{displaymath}
g_4 \coloneqq \mu_4 = \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2}
\end{displaymath}
and

\begin{displaymath}
g_7 
  \coloneqq
  \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
\end{displaymath}
then

\begin{displaymath}
d g_7 \propto g_4 \wedge g_4
  \,.
\end{displaymath}
This is the structure of the [[equations of motion]] of the [[field strength]] $G_4$ of the [[supergravity C-field]] and its [[Hodge dual]] $G_7 = \ast G_4$ in [[11-dimensional supergravity]].

\end{remark}
\begin{def}
\label{TheDefinition}\hypertarget{TheDefinition}{}
The \emph{supergravity Lie 6-algebra} $\mathfrak{sugra}_{7}(10,1)$ is the [[super L-∞ algebra|super Lie 7-algebra]] that is the $b^6 \mathbb{R}$-[[L-∞ algebra cohomology|extension]] of $\mathfrak{sugra}_3(10,1)$ classified by the cocycle $\mu_7$ from def. \ref{TheSevenCocycle}.

\begin{displaymath}
b^5 \mathbb{R} \to \mathfrak{sugra}_6 \to \mathfrak{sugra}_3
  \,.
\end{displaymath}
\end{def}
\begin{remark}
\label{InTermsOfGenerators}\hypertarget{InTermsOfGenerators}{}
This means that the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}_6)$ is generated from

\begin{itemize}%
\item $\{e^a\}$ ([[vielbein]]) in degree $(1,even)$


\item $\{\omega^{a b}\}$ ([[spin connection]]) in degree $(1,even)$;


\item $\{\psi^\alpha\}$ ([[gravitino]]) in degree $(1,odd)$


\item $\{c_3\}$ ([[supergravity C-field]]) in degree $(3,even)$


\item $\{c_6\}$ (magnetic dual [[supergravity C-field]]) in degree $(6,even)$



\end{itemize}
with [[differential]] defined by

\begin{displaymath}
d_{CE}  : \omega^{a b} \mapsto \omega^{a c} \wedge \omega_c{}^b
\end{displaymath}
\begin{displaymath}
d_{CE} : e^a = -\omega^{a b} e_b  - \frac{1}{2}i \bar \psi \wedge  \Gamma^a \psi
\end{displaymath}
\begin{displaymath}
d_{CE} : \psi \mapsto - \frac{1}{4}\omega^{a b} \Gamma^{a b}
\end{displaymath}
\begin{displaymath}
d_{CE} : c_3 \mapsto \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
\end{displaymath}
\begin{displaymath}
d_{CE} 
   \colon
  c_6 
  \mapsto
  - \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
  -
  \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3
  \,.
\end{displaymath}
\end{remark}
This appears as (\hyperlink{CastellaniDAuriaFre}{Castellani-DAuria-Fre, (III.8.18)}).

\begin{remark}
\label{}\hypertarget{}{}
According to (\hyperlink{CastellaniDAuriaFre}{Castellani-DAuria-Fre, comment below (III.8.18)}): ``no further extension is possible''.

\end{remark}
\hypertarget{RelationToSupergravity}{}\subsection*{{Relation to $D = 11$ supergravity}}\label{RelationToSupergravity}

The supergravity Lie 6-algebra is something like the gauge $L_\infty$-algebra of [[11-dimensional supergravity]], in the sense discussed at \emph{[[D'Auria-Fre formulation of supergravity]]} .

\begin{defn}
\label{ModifiedWeilAlgebra}\hypertarget{ModifiedWeilAlgebra}{}
Write $W(\mathfrak{sugra}_6(10,1))$ for the [[Weil algebra]] of the supergravity Lie 6-algebra.

Write $g_4$ and $g_7$ for the shifted generators of the Weil algebra corresponding to $c_3$ and $c_6$, respectively.

Define a modified Weil algebra $\tilde W(\mathfrak{sugra}_6(10,1))$ by declaring it to have the same generators and differential as before, except that the differential for $c_6$ is modified to

\begin{displaymath}
d_{\tilde W} c_6 := d_{W} c_6  + 15 g_4 \wedge c_3
\end{displaymath}
and hence the differential of $g_7$ is accordingly modified in the unique way that ensures $d_{\tilde W}^2 = 0$ (yielding the modified [[Bianchi identity]] for $g_7$).

\end{defn}
This ansatz appears as (\hyperlink{CastellaniDAuriaFre}{CastellaniDAuriaFre, (III.8.24)}).

Note that this amounts simply to a redefinition of [[curvature]]s

\begin{displaymath}
\tilde g_7 := g_7 + 15 g_4 \wedge c_3
  \,.
\end{displaymath}
\begin{note}
\label{SugraEOM}\hypertarget{SugraEOM}{}
A field configuration of [[11-dimensional supergravity]] is given by [[∞-groupoid of ∞-Lie algebra valued forms|L-∞ algebra valued differential forms]] with values in $\mathfrak{sugra}_6$. Among all of these the solutions to the equations of motion (the points in the [[covariant phase space]]) can be characterized as follows:

A field configuration

\begin{displaymath}
\Omega^\bullet(X) \leftarrow \tilde W(\mathfrak{sugra}_6) : \Phi
\end{displaymath}
solves the equations of motion precisely if

\begin{enumerate}%
\item all [[curvature]]s sit in the ideal of differential forms spanned by the 1-form fields $E^a$ ([[vielbein]]) and $\Psi$ ([[gravitino]]);

more precisely if we have

\begin{itemize}%
\item $\tau = 0$

(super-[[torsion]]);


\item $G_4 = (G_4)_{a_1, \cdots a_4} E^{a_1} \wedge \cdots E^{a_4}$

([[field strength]] of [[supergravity C-field]])


\item $G_7 = (G_7)_{a_1, \cdots a_7} E^{a_1} \wedge \cdots E^{a_7}$

([[electric-magnetic duality|dual]] field strength)


\item $\rho = \rho_{a b} E^a \wedge E^b  + H_a \Psi \wedge E^a$

([[Dirac operator]] applied to [[gravitino]])


\item $R^{a b} = R^{a b}_{c d} E^c \wedge E^d + \bar \Theta^{a b}{}_c \Psi \wedge E^c + \bar \Psi \wedge K^{a b} \Psi$

([[Riemann tensor]]: [[field strength]] of [[gravity]])



\end{itemize}

\item such that the coefficients of terms containing $\Psi$s are polynomials in the coefficients of the terms containing no $\Psi$s. (``rheonomy'').



\end{enumerate}
\end{note}
This is the content of (\hyperlink{CastellaniDAuriaFre}{CastellaniDAuriaFre, section III.8.5}).

In particular this implies that on-shell the 4- and 7-field strength are indeed dual of each other

\begin{displaymath}
G_7 \propto \star G_4
  \,.
\end{displaymath}
This is the content of (\hyperlink{CastellaniDAuriaFre}{CastellaniDAuriaFre, equation (III.8.52)}).

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

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

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


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


\item [[M-theory super Lie algebra]]



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

The supergravity Lie 6-algebra appears first on page 18 of

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

\end{itemize}
and is recalled in section 4 of

\begin{itemize}%
\item [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, K. Pilch, [[Peter van Nieuwenhuizen]], \emph{Gauging of $d = 11$ supergravity?}, Annals of Physics Volume 146, Issue 1, March 1983, Pages 35--77

\end{itemize}
A textbook discussion is in section III.8.3 of

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fre]],

\emph{[[Supergravity and Superstrings - A Geometric Perspective]]}



\end{itemize}
The same is being recalled for instance in section 3 of

\begin{itemize}%
\item [[Pietro Fré]], Pietro Antonio Grassi, \emph{Free Differential Algebras, Rheonomy, and Pure Spinors} (\href{http://arxiv.org/abs/0801.3076}{arXiv:0801.3076})

\end{itemize}
Then it is rediscovered around equation (8.8) in

\begin{itemize}%
\item C. Chryssomalakos, [[José de Azcárraga]], J. M. Izquierdo and C. P\'e{}rez Bueno, \emph{The geometry of branes and extended superspaces} (\href{http://arxiv.org/abs/hep-th/9904137}{arXiv:hep-th/9904137})

\end{itemize}
which gives a detailed and comprehensive discussion.

A discussion in the context of [[smooth super ∞-groupoids]] is 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]]}

\end{itemize}
in the last section of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}


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