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\begin{document}

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\section*{super Minkowski spacetime}

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

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

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

[[!include supergeometry - contents]]

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{CanonicalCoordinates}{Canonical coordinates}\dotfill \pageref*{CanonicalCoordinates} \linebreak
\noindent\hyperlink{cohomology_and_super_branes}{Cohomology and super $p$-branes}\dotfill \pageref*{cohomology_and_super_branes} \linebreak
\noindent\hyperlink{AsCentralExtensionOfTheSuperpoint}{As a central extension of the superpoint}\dotfill \pageref*{AsCentralExtensionOfTheSuperpoint} \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}

Super-Minkowski spacetime is a [[super spacetime]] which is an analog in [[supergeometry]] of ordinary [[Minkowski spacetime]]. It is a [[super Cartesian space]] whose odd coordinates form a [[real spin representation]].

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

Ordinary $(d+1)$-dimensional [[Minkowski space]] may be understood as the quotient $Iso(\mathbb{R}^{d-1,1})/(Spin(d-1,1))$ of the [[Poincare group]] by the [[spin group]] cover of [[Lorentz group]] -- the [[translation group]].

Analogously, the for each real irreducible [[spin representation]] $N$ the $dim(N)$-extended [[supermanifold]] \textbf{Minkowski superspace} or \textbf{super Minkowski space} is the quotient of [[supergroups]] of the [[super Poincaré group]] by the corresponding [[spin group]] (a [[super Klein geometry]]).

The \emph{[[super-translation group]]}. See there for more details.

Alternatively, regarded as a [[super Lie algebra]] this is the quotient of the [[super Poincaré Lie algebra]] by the relevant [[Lorentz Lie algebra]].

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

\hypertarget{CanonicalCoordinates}{}\subsubsection*{{Canonical coordinates}}\label{CanonicalCoordinates}

We briefly review some basics of the canonical [[coordinates]] and the super [[Lie algebra cohomology]] of the [[super Poincaré Lie algebra]] and [[super Minkowski space]] (see also at \emph{[[super Cartesian space]]} and at \emph{[[signs in supergeometry]]}).

By the general discussion at [[Chevalley-Eilenberg algebra]], we may characterize the [[super Poincaré Lie algebra]] $\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ by its CE-algebra $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ ``of [[left-invariant 1-forms]]'' on its group manifold.

Let $d \in \mathbb{N}$ and let $N$ be a real [[spin representation]] of $Spin(d-1,1)$. See at \emph{[[Majorana representation]]} for details.

\begin{defn}
\label{CEAlgebraOfSuperPoincare}\hypertarget{CEAlgebraOfSuperPoincare}{}
The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ is generated on

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


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



\end{itemize}
where $a \in \{0,1, \cdots, d-1\}$ is a spacetime index, and where $\alpha$ is an index ranging over a basis of the chosen [[Majorana spinor]] representation $N$.

The CE-differential defined as follows

\begin{displaymath}
d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
\end{displaymath}
and

\begin{displaymath}
d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi
  \,.
\end{displaymath}
(which is the differential for the [[semidirect product]] of the [[Poincaré Lie algebra]] [[action|acting]] on the given [[Majorana spinor]] representation)

and

\begin{displaymath}
d_{CE} \, e^{a } 
    = 
   \omega^a{}_b \wedge e^b + \overline{\psi} \wedge \Gamma^a \psi
\end{displaymath}
where on the right we have the spinor-to-vector pairing in $N$ (\href{Majorana+spinor#SpinorToVectorBilinearPairing}{def.}).

This defines the [[super Poincaré super Lie algebra]]. After discarding the terms involving $\omega$ this becomes the CE algebra of the [[super translation algebra]] underlying super Minkowski spacetime.

\end{defn}
In this way the [[super-Poincaré Lie algebra]] and its extensions is usefully discussed for instance in (\hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}) and in (\href{AzcarragaTownsend89}{Azc\'a{}rraga-Townsend 89}, \hyperlink{CAIB99}{CAIB 99}). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the [[coordinates]] on [[super Minkowski space]].

\begin{remark}
\label{}\hypertarget{}{}
The abstract generators in def. \ref{CEAlgebraOfSuperPoincare} are identified with [[left invariant 1-forms]] on the [[super-translation group]] (= super Minkowski spacetime) as follows.

Let $N$ be a [[real spin representation]] and let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d-1,1\vert N}$ underlying the super-Minkowski super translation group. Then the canonical [[super vielbein]] is the $\mathbb{R}^{d-1,1\vert N}$-valued [[super differential form]] with components

\begin{itemize}%
\item $\psi^\alpha \coloneqq \mathbf{d} \theta^\alpha$.


\item $e^a \coloneqq \mathbf{d} x^a + \overline{\theta} \Gamma^a \mathbf{d} \theta$.



\end{itemize}
Notice that this then gives the above formula for the differential of the [[super-vielbein]] in def. \ref{CEAlgebraOfSuperPoincare} as

\begin{displaymath}
\begin{aligned}
    d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta)
    \\
    & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta
    \\
    & = \frac{i}{2} \overline{\psi}\Gamma^a \psi
  \end{aligned}
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{Supertorsion}\hypertarget{Supertorsion}{}
The term $\frac{i}{2}\bar \psi \Gamma^a \psi$ is sometimes called the \emph{[[supertorsion]]} of the [[super-vielbein]] $e$, because the defining equation

\begin{displaymath}
d_{CE} e^{a } -\omega^a{}_b \wedge e^b = \frac{i}{2}\bar \psi \Gamma^a \psi
\end{displaymath}
may be read as saying that $e$ is [[torsion]]-free except for that term. Notice that this term is the only one that appears when the differential is applied to ``Lorentz scalars'', hence to object in $CE(\mathfrak{siso})$ which have ``all indices contracted''. See also at \emph{[[torsion constraints in supergravity]]}.

Notably we have

\begin{displaymath}
d \left( 
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}
  \right)
  \propto
  \left(
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}}
  \right)
  \wedge
  \left(
    \overline{\Psi} \wedge \Gamma_{a_p} \Psi
  \right)
  \,.
\end{displaymath}
This remaining operation ``$e \mapsto \Psi^2$'' of the differential acting on Loretz scalars is sometimes denoted ``$t_0$'', e.g. in (\hyperlink{BossardHoweStelle09}{Bossard-Howe-Stelle 09, equation (8)}).

This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of $(D,p)$ a [[Fierz identity]] implies that the term

\begin{displaymath}
\left(
    \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}}
  \right)
  \wedge
  \left(
    \overline{\Psi} \wedge \Gamma_{a_p} \Psi
  \right)
\end{displaymath}
vanishes identically, and hence in these dimensions the term

\begin{displaymath}
\overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi
    \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}
\end{displaymath}
is a [[cocycle]]. See also the \hyperlink{BraneScan}{brane scan} table below.

\end{remark}
\hypertarget{cohomology_and_super_branes}{}\subsubsection*{{Cohomology and super $p$-branes}}\label{cohomology_and_super_branes}

As opposed to ordinary [[Minkowski space]], the [[de Rham cohomology]] of [[left invariant forms]] of super-Minkowski space contains nontrivial exceptional [[cocycles]] (the \emph{[[brane scan]]}). These serve as the [[Wess-Zumino-Witten model|WZW terms]] for the [[Green-Schwarz action functional]] (see there for more) of super-$p$-[[branes]] propagating on super-Minkowski space (\hyperlink{FSS13}{FSS 13}).

The corresponding $L_\infty$-extensions are \emph{[[extended superspacetime]]}.

\hypertarget{AsCentralExtensionOfTheSuperpoint}{}\subsubsection*{{As a central extension of the superpoint}}\label{AsCentralExtensionOfTheSuperpoint}

Regarded as a [[super Lie algebra]], super Minkowski spacetime $\mathbb{R}^{d-1,1|N}$ has the single nontrivial super-Lie bracket given by the spinor bilinear pairing

\begin{displaymath}
\overline{(-)}\Gamma (-)
  \colon
  S \otimes S \longrightarrow V
\end{displaymath}
discussed in detail at \emph{[[spin representation]]}.

Notice that this means that if one regards the [[superpoint]] $\mathbb{R}^{0|dim(N)}$ as an [[abelian super Lie algebra]], then super Minkowski spacetime is the [[Lie algebra extension]] of that by this bilinear pairing regarded as a super-[[Lie algebra cohomology|Lie algebra cocycle]] with [[coefficients]] in $\mathbb{R}^{d}$.

\begin{displaymath}
\itexarray{
    \mathbb{R}^{d} &\longrightarrow& \mathbb{R}^{d-1,1|N}
    \\
    && \downarrow 
    \\
    && \mathbb{R}^{0|dim(N)}
  }
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[super anti de Sitter spacetime]]

\end{itemize}
[[!include local and global geometry - table]]

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

The $d = 4$, $N =2$ super Minkowski spacetime was originally introduced in

\begin{itemize}%
\item [[Abdus Salam]] J.A. Strathdee, \emph{Supergauge Transformations}, Nucl.Phys. B76 (1974) 477-482 (\href{http://inspirehep.net/record/89208}{spire})


\item [[Abdus Salam]] J.A. Strathdee, Physical Review D11, 1521-1535 (1975)



\end{itemize}
see at ``[[superspace]] in physics''.

Further discussion includes:

\begin{itemize}%
\item \emph{Super spacetimes and super Poincar\'e{}-group} (\href{http://www.math.ucla.edu/~vsv/papers/ch6.pdf}{pdf})


\item [[Daniel Freed]], lecture 6 of \emph{Classical field theory and Supersymmetry}, IAS/Park City Mathematics Series Volume 11 (2001) (\href{https://www.ma.utexas.edu/users/dafr/pcmi.pdf}{pdf})


\item [[Daniel Freed]], \emph{Lecture 4 of [[Five lectures on supersymmetry]]}


\item [[Veeravalli Varadarajan]], section 7 of \emph{Supersymmetry for mathematicians: An introduction}


\item [[Leonardo Castellani]], [[Riccardo D'Auria]], \emph{[[Pietro Fre]], page 370, part II section II.3.3 of}[[Supergravity and Superstrings - A Geometric Perspective]]\_



\end{itemize}
Discussion of how [[super L-infinity algebra]] [[L-infinity cocycle|extensions]] of super Minkowski spacetime yield all the [[brane scan]] of [[string theory]]/[[M-theory]] is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet]]} (2013)

\end{itemize}
[[!redirects super Minkowski spacetime]] [[!redirects super-Minkowski spacetime]]

[[!redirects super Minkowski spacetimes]] [[!redirects super-Minkowski spacetimes]]

[[!redirects super Minkowski space]] [[!redirects super Minkowski space]] [[!redirects super Minkowski space]]

[[!redirects super Minkowski super Lie algebra]] [[!redirects super Minkowski super Lie algebras]]



\end{document}