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\begin{document}

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\section*{Killing spinor}

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

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

[[!include Riemannian geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_supersymmetry}{Relation to supersymmetry}\dotfill \pageref*{relation_to_supersymmetry} \linebreak


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

A \emph{Killing spinor} on a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] $X$ is a \emph{[[spinor]]} -- a [[section]] of some [[spinor bundle]] $v \in \Gamma(S)$ -- that is taken by the [[covariant derivative]] of the corresponding [[Levi-Civita connection]] to a multiple of itself

\begin{displaymath}
\nabla_v \psi = \kappa \gamma_v \psi
\end{displaymath}
for some constant $\kappa$.

If that constant is 0, hence if the spinor is \emph{covariant constant}, then one also speaks of a \emph{covariant constant spinor} or \emph{parallel spinor} (with respect to the given metric structure).

More generally, a \emph{twistor spinor} or \emph{conformal Killing spinor} is a $\psi$ such that

\begin{displaymath}
\nabla_v \psi = \frac{1}{dim(X)} \gamma_v D \psi
  \,,
\end{displaymath}
where $D$ is the given [[Dirac operator]] (e.g. \hyperlink{Baum00}{Baum 00}).

A Killing spinor with non-vanishing $\kappa$ may be understood as a genuine covariantly constant spinor, but with respect to a [[super-Cartan geometry]] modeled not on [[super-Euclidean space]]/[[super-Minkowski spacetime]], but on its spherical/hyperbolic or deSitter/anti-deSitter versions (\hyperlink{EgeilehChami13}{Egeileh-Chami 13, p. 60 (8/8)}).

Similarly a \emph{[[Killing vector]]} is a covariantly constant [[vector field]].

Pairing two covariant constant spinors to a vector yields a Killing vector.

In [[supergravity]], [[super spacetimes]] which solves the [[equations of motion]] and admit Killing spinors are \emph{[[BPS states]]} (at least if they are asymptotically flat and of finite mass).

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

\begin{itemize}%
\item [[Killing vector]]


\item [[Killing tensor]]


\item [[Killing-Yano tensor]]


\item [[superisometry]]


\item [[supersymmetry and Calabi-Yau manifolds]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Lecture notes include

\begin{itemize}%
\item \emph{Parallel and Killing spinor fields} ([[KillingSpinors.pdf:file]])


\item [[Helga Baum]], \emph{Twistor and Killing spinors in Lorentzian geometry}, S\'e{}minaires \& Congr\`e{}s, 4, 2000 (\href{http://www.emis.de/journals/SC/2000/4/pdf/smf_sem-cong_4_35-52.pdf}{pdf})


\item [[Helga Baum]], \emph{Conformal Killing spinors and the holonomy problem in Lorentzian geometry} (\href{http://www.mathematik.hu-berlin.de/~baum/publikationen-fr/Publikationen-ps-dvi/IMA06-neu.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item \"O{}zg\"u{}r A\c{c}k, \emph{Field equations from Killing spinors} (\href{https://arxiv.org/abs/1705.04685}{arXiv:1705.04685})

\end{itemize}
Discussion relating to [[Killing vectors]] in [[supergeometry]] ([[superisometries]]) is in

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], pages 1198-1199 of \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991)

\end{itemize}
and later in

\begin{itemize}%
\item [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], \emph{Killing spinors are Killing vector fields in Riemannian Supergeometry} (\href{http://arxiv.org/abs/dg-ga/9704002}{arXiv:dg-ga/9704002})

\end{itemize}
See also

\begin{itemize}%
\item [[Christian Bär]], \emph{Real Killing spinors and holonomy}, Comm. Math. Phys. Volume 154, Number 3 (1993), 509-521 (\href{https://projecteuclid.org/euclid.cmp/1104253076}{Euclid})

\end{itemize}
Discussion regarding the conceptualization of Killing spinors in [[super-Cartan geometry]] is in

\begin{itemize}%
\item Michel Egeileh, Fida El Chami, \emph{Some remarks on the geometry of superspace supergravity}, J.Geom.Phys. 62 (2012) 53-60 (\href{http://inspirehep.net/record/1333125}{spire})

\end{itemize}
Discussion relating to [[special holonomy]] includes

\begin{itemize}%
\item Andrei Moroianu, [[Uwe Semmelmann]], \emph{Parallel spinors and holonomy groups}, Journal of Mathematical Physics 41 (2000), 2395-2402 (\href{http://arxiv.org/abs/math/9903062}{arXiv:math/9903062})

\end{itemize}
Discussion of classification includes

\begin{itemize}%
\item [[Thomas Friedrich]], \emph{Zur Existenz paralleler Spinorfelder \"u{}ber Riemannschen Mannigfaltigkeiten} Czechoslavakian-GDR-Polish scientific school on differential geometry Boszkowo/ Poland 1978, Sci. Comm., Part 1,2; 104-124 (1979)


\item [[Thomas Friedrich]], \emph{Zur Existenz paralleler Spinorfelder \"u{}ber Riemannschen Mannigfaltigkeiten}, Colloquium Mathematicum vol. XLIV, Fasc. 2 (1981), 277-290.



\end{itemize}
\hypertarget{relation_to_supersymmetry}{}\subsubsection*{{Relation to supersymmetry}}\label{relation_to_supersymmetry}

General discussion of Killing with an eye towards applications in [[supersymmetry]] is around page 907 in volume II of

\begin{itemize}%
\item [[Pierre Deligne]], P. Etingof, [[Dan Freed]], L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and [[Edward Witten]], (eds.) \emph{Quantum Fields and Strings, A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999. (\href{http://www.math.ias.edu/qft}{web version})

\end{itemize}
specifically

in [[heterotic supergravity]]:

\begin{itemize}%
\item [[Ulf Gran]], [[George Papadopoulos]], Diederik Roest, \emph{Supersymmetric heterotic string backgrounds}, Phys.Lett.B656:119-126, 2007 (\href{https://arxiv.org/abs/0706.4407}{arXiv:0706.4407})

\end{itemize}
in [[11-dimensional supergravity]]:

\begin{itemize}%
\item [[Jerome Gauntlett]], Stathis Pakis, \emph{The Geometry of $D=11$ Killing Spinors}, JHEP 0304 (2003) 039 (\href{http://arxiv.org/abs/hep-th/0212008}{arXiv:hep-th/0212008})

\end{itemize}
for [[G2-structures]] in [[M-theory on G2-manifolds]]:

\begin{itemize}%
\item [[Thomas Friedrich]], Stefan Ivanov, \emph{Parallel spinors and connections with skew-symmetric torsion in string theory}, AsianJ.Math.6:303-336,2002 (\href{http://arxiv.org/abs/math/0102142}{arXiv:math/0102142})

\end{itemize}
[[!redirects Killing spinors]]

[[!redirects covariantly constant spinor]] [[!redirects covariantly constant spinors]]

[[!redirects Killing spinor field]] [[!redirects Killing spinor fields]]

[[!redirects parallel spinor]] [[!redirects parallel spinors]]

[[!redirects twistor spinor]] [[!redirects twistor spinors]]

[[!redirects parallel spinor field]] [[!redirects parallel spinor fields]]



\end{document}