nLab Killing spinor



Riemannian geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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).



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

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.

Relation to supersymmetry

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


in heterotic supergravity:

in 11-dimensional supergravity:

for G2-structures in M-theory on G2-manifolds:

Last revised on April 15, 2023 at 08:47:30. See the history of this page for a list of all contributions to it.