nLab
Killing spinor

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          A Killing spinor on a (pseudo-)Riemannian manifold XX is a spinor – a section of some spinor bundle vΓ(S)v \in \Gamma(S) – that is taken by the covariant derivative of the corresponding Levi-Civita connection to a multiple of itself

          vψ=κγ vψ \nabla_v \psi = \kappa \gamma_v \psi

          for some constant κ\kappa.

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

          More generally, a twistor spinor or conformal Killing spinor is a ψ\psi such that

          vψ=1dim(X)γ vDψ, \nabla_v \psi = \frac{1}{dim(X)} \gamma_v D \psi \,,

          where DD is the given Dirac operator (e.g. Baum 00).

          A Killing spinor with non-vanishing κ\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 (Egeileh-Chami 13, p. 60 (8/8)).

          Similarly a 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 BPS states (at least if they are asymptotically flat and of finite mass).

          References

          Lecture notes include

          • Parallel and Killing spinor fields (pdf)

          • Helga Baum, Twistor and Killing spinors in Lorentzian geometry, Séminaires & Congrès, 4, 2000 (pdf)

          • Helga Baum, Conformal Killing spinors and the holonomy problem in Lorentzian geometry (pdf)

          See also

          Discussion relating to Killing vectors in supergeometry (superisometries) is in

          and later in

          See also

          • Christian Bär, Real Killing spinors and holonomy, Comm. Math. Phys.

            Volume 154, Number 3 (1993), 509-521 (Euclid)

          Discussion regarding the conceptualization of Killing spinors in super-Cartan geometry is in

          • Michel Egeileh, Fida El Chami, Some remarks on the geometry of superspace supergravity, J.Geom.Phys. 62 (2012) 53-60 (spire)

          Discussion relating to special holonomy includes

          A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of

          Discussion of classification includes

          • Thomas Friedrich, Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten Czechoslavakian-GDR-Polish scientific school on differential geometry Boszkowo/ Poland 1978, Sci. Comm., Part 1,2; 104-124 (1979)

          • Thomas Friedrich, Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten, Colloquium Mathematicum vol. XLIV, Fasc. 2 (1981), 277-290.

          Discussion relating to G2-structures includes

          Discussion in 11-dimensional supergravity includes

          • Jerome P. Gauntlett, Stathis Pakis, The Geometry of D=11D=11 Killing Spinors, JHEP 0304 (2003) 039 (arXiv:hep-th/0212008)

          Last revised on May 16, 2017 at 05:21:26. See the history of this page for a list of all contributions to it.