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\begin{document}

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\section*{kappa-symmetry}

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

\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{as_superdiffeomorphisms_in_the_superembedding_approach}{As super-diffeomorphisms in the super-embedding approach}\dotfill \pageref*{as_superdiffeomorphisms_in_the_superembedding_approach} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

What is called \emph{$\kappa$-symmetry} in [[string theory]]/[[M-theory]] is a certain fermionic [[symmetry]] of [[Green-Schwarz action functionals]] for [[super p-branes]] whose effect is to [[gauge symmetry|gauge away]] half of the [[spinor|spinorial]] [[sigma-model]] [[field (physics)|fields]].

\hypertarget{as_superdiffeomorphisms_in_the_superembedding_approach}{}\subsection*{{As super-diffeomorphisms in the super-embedding approach}}\label{as_superdiffeomorphisms_in_the_superembedding_approach}

In a completely super-covariant formulation of the [[Green-Schwarz action functionals]] -- called the \emph{[[super-embedding formalism]]} -- this $\kappa$-symmetry is simply the odd-graded part of the [[supermanifold|super]]-[[worldvolume]] [[super-diffeomorphism]] symmetry of the [[sigma-model]] (\hyperlink{SorokinTkachVolkov89}{Sorokin-Tkach-Volkov 89}, review includes \hyperlink{Sorokin99}{Sorokin 99, section 4.3}, \hyperlink{HoweSezgin04}{Howe-Sezgin 04, section 4.3}): If

\begin{enumerate}%
\item $X$ denotes a [[superspacetime]] locally modeled on [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert \mathbf{N}}$


\item $\Sigma$ denotes a [[supermanifold|super]]-[[worldvolume]] of a [[super p-brane]] locally modeled on [[super-Minkowski spacetime]] $\mathbb{R}^{p,1\vert \mathbf{N}/2}$


\item so that a [[sigma-model]] [[field (physics)|field]] configuration for a [[super p-brane]] of shape $\Sigma$ to popagate in $X$ is a morphism of [[supermanifolds]] of the form

\begin{displaymath}
\itexarray{
     \Sigma && \text{super-worldvolume}
     \\
     \downarrow^{\mathrlap{\phi}} &&\text{sigma-model super-field}& 
     \\
     X && \text{super-spacetime}
  }
\end{displaymath}


\end{enumerate}
then:

\begin{enumerate}%
\item the postcomposition [[action]] of [[spacetime]] [[super-isometries]] $X \stackrel{\simeq}{\longrightarrow} X$ is in even degree the action of [[spacetime]] [[isometries]] and in odd degree the action of \textbf{[[spacetime]] [[supersymmetry]]} on the sigma-model fields;


\item the precomposition action of [[worldvolume]] [[super-diffeomorphism]] $\Sigma \stackrel{\simeq}{\to} \Sigma$ is in even degree the action of bosonic worldvolume [[diffeomorphism]] and in odd degree the action of \textbf{$\kappa$-symmetry}:



\end{enumerate}
\begin{displaymath}
\itexarray{
      \Sigma  &\underoverset{\simeq}{\kappa\text{-symmetry}}{\longrightarrow}& \Sigma
      \\
      \downarrow^{\mathrlap{\phi}} && \downarrow^{\mathrlap{\phi'}}
      \\
      X &\underoverset{\simeq}{\text{spacetime supersymmetry}}{\longrightarrow}&
    X
  }
  \,.
\end{displaymath}
Notice here the assumption that the number of odd directions on the [[worldvolume]] is half that of the [[target spacetime]]. This is the default assumption for fundamental [[super p-branes]], and it directly reflects the statement that the corrresponding [[black brane]] solutions are $1/2$ [[supergravity]] [[BPS states]].

For example, consider the embedding

\begin{displaymath}
\mathbb{R}^{2,1} \hookrightarrow \mathbb{R}^{10,1}
\end{displaymath}
of 2+1d [[Minkowski spacetime]], thought of as the [[worldvolume]] of a [[membrane]], into 11d Minkowski spacetime, linearly along the coordinate axis. Any such embedding breaks the [[isometry group]] of $\mathbb{R}^{10,1}$ from the 11d [[Poincaré group]] $Iso(10,1)$ to the [[product group]]

\begin{displaymath}
Iso(2,1) \times SO(8)
    \hookrightarrow
  Iso(10,1)
\end{displaymath}
(meaning that this [[subgroup]] is the [[stabilizer subgroup]] of the embedding).

Now consider instead [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$ (with $\mathbf{32}$ the irreducible [[Majorana spinor]] representation in 11), hence the local model [[superspace]] for [[super spacetimes]] in [[11-dimensional supergravity]]. We are to ask what subspace of the [[spin representation]] $\mathbf{32}$ preserves the embedding in that the \href{Majorana+spinor#TheSpinorPairingToVectors}{spinor bilinear pairing} $\overline{\psi}_1 \Gamma \psi_2$ on that subspace lands in $\mathbb{R}^{2,1} \hookrightarrow Iso(2,1) \hookrightarrow Iso(10,1)$ (\hyperlink{Sorokin99}{Sorokin 99, section 5.1}). This is found to be the case for a half-dimensional subspace, and hence we may lift the above to a super-embedding of the form

\begin{equation}
\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} \hookrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}
\label{M2WordvolumeInSpacetime}\end{equation}
(where now $\mathbf{2}$ is the [[irrep|irreducible]] [[Majorana spinor]] representation in 3d, and $8 \otimes \mathbf{2}$ denotes the [[direct sum]] of 8 copies of it) such that the induced [[stabilizer group|stabilizer]] [[supergroup]] inside the [[super Poincaré group ]] now is

\begin{displaymath}
Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}}) \times Spin(8)
    \hookrightarrow
  Iso(\mathbb{R}^{10,1\vert \mathbf{32}})
  \,.
\end{displaymath}
It is in this sense that the membrane ``breaks exactly half the supersymmetry'', namely from $\mathbf{32}$ to $8 \otimes \mathbf{2}$.

If one now thinks of this not as inclusions of global spacetimes, but of their super tangent spaces at the points where the membrane sits in spacetime, then this reflects the local structure of $\kappa$-symmetry: the $\kappa$-symmetries are locally generated by the 16 odd dimensions in $Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} )$, being super-translations along the membrane worldvolume.

This explains why $\kappa$-symmetry in [[Green-Schwarz sigma models]] is taken to quotient out precisely half the spinor components, hence why, in the fully super-covariant formulation, one takes the worldvolume of a super $p$-brane in a superspacetime locally modeled on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ to be $\mathbb{R}^{p,1\vert \mathbf{N}/2}$. But notice that this is not a mathematical necessity. One may consider the worldvolume instead to have fewer odd directions. This then describes [[sigma models]] for ``non-BPS super $p$-branes'' (or rather ``non-half-BPS'' ).

\hypertarget{history}{}\subsection*{{History}}\label{history}

Historically $\kappa$-symmetry was originally observed and considered for [[Green-Schwarz sigma models]] whose [[worldvolume]] is regarded as an ordinary (bosonic) [[smooth manifold]]. Then $\kappa$-symmetry is a ``hidden'' symmetry, with no evident geometric interpretation. As such it was first observed for the [[super-particle]] (\hyperlink{AzcarragLukierski82}{Azcarraga-Lukierski 82}, \hyperlink{Siegel83}{Siegel 83}) and then for the [[super 1-brane in 3d]] (\hyperlink{Siegel84}{Siegel 84}).

Based on this the [[Green-Schwarz action functional]] for the [[superstring]] in 10d ([[heterotic string]], [[type II superstring]]) was found in (\hyperlink{GreenSchwarz84}{Green-Schwarz 84}) by first observing that the plain [[Nambu-Goto action]] for a string on a [[supermanifold]] has twice as many fermionic [[shell|on-shell]] [[degrees of freedom]] as the [[NSR-string]] and then by adding a term to the action (the [[WZW-term]]) to correct this defect, by ensuring that the sum of the NG-action with the WZW term enjoys $\kappa$-symmetry. By the same recipe later $\kappa$-symmetric Green-Schwarz-type [[sigma-model]] actions for all the other [[super p-branes]] were found, for instance for the [[M2-brane]] in (\hyperlink{BergshoeffSezginTownsend87}{Bergshoeff-Sezgin-Townsend 87}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[geometry of physics -- supersymmetry]]

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

Originally, $\kappa$-symmetry was observed for the [[super-particle]] in

\begin{itemize}%
\item [[J. de Azcárraga]], J. Lukierski, \emph{Supersymmetric particles}, Phys. Lett. B113 (1982) 170; Phys. Rev. D28 (1983) 1337.


\item Warren Siegel, \emph{Hidden Local Supersymmetry In The Supersymmetric Particle Action} Phys. Lett. B 128, 397 (1983)



\end{itemize}
and for the [[super 1-brane in 3d]] in

\begin{itemize}%
\item Warren Siegel, \emph{Light Cone Analysis Of Covariant Superstring} , Nucl. Phys. B 236, 311 (1984).

\end{itemize}
Then it was used to define/construct the manifestly spacetime supersymmetric [[Green-Schwarz action functional]] for the [[superstring]] in 10d in

\begin{itemize}%
\item [[Michael Green]], [[John Schwarz]], \emph{Covariant description of superstrings}, Phys. Lett. B136 (1984), 367--370 (\href{http://adsabs.harvard.edu/abs/1984PhLB..136..367G}{web}, \href{http://inspirehep.net/record/193596}{spire:193596}, )

\end{itemize}
and then for the [[M2-brane]] in 11d in

\begin{itemize}%
\item [[Eric Bergshoeff]], [[Ergin Sezgin]], and [[Paul Townsend]], \emph{Supermembranes and eleven-dimensional supergravity}, Phys. Lett. B189 (1987) 75--78. (\href{http://streaming.ictp.trieste.it/preprints/P/87/010.pdf}{pdf})

\end{itemize}
and so forth (see the references at \emph{[[super p-brane]]}).

The [[supergeometry|super-geometric]] interpretation of $\kappa$-symmetry as the odd-graded part of the action of [[super-diffeomorphism]] on the [[super p-brane]] [[worldvolume]], regarded itself as a [[supermanifold]] was first suggested in

\begin{itemize}%
\item [[Dmitri Sorokin]], V. Tkach and [[Dmitrij Volkov]], \emph{Superparticles, twistors and Siegel symmetry}, Mod. Phys. Lett. A4 (1989) 901-908 (\href{http://inspirehep.net/record/271923}{spire:271923}, \href{https://doi.org/10.1142/S0217732389001064}{doi:10.1142/S0217732389001064})

\end{itemize}
Review of this perspective includes

\begin{itemize}%
\item [[Dmitri Sorokin]], \emph{Superbranes and Superembeddings}, Phys.Rept.329:1-101,2000 (\href{http://arxiv.org/abs/hep-th/9906142}{arXiv:hep-th/9906142})


\item [[Paul Howe]], [[Ergin Sezgin]], section 4.3 of \emph{The supermembrane revisited}, Class.Quant.Grav. 22 (2005) 2167-2200 (\href{http://arxiv.org/abs/hep-th/0412245}{arXiv:hep-th/0412245})



\end{itemize}
[[!redirects kappa symmetry]]



\end{document}