nLab Killing spinor

Contents

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

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

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 $X$ is a 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

$\nabla_v \psi = \kappa \gamma_v \psi$

for some constant $\kappa$.

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 $\psi$ such that

$\nabla_v \psi = \frac{1}{dim(X)} \gamma_v D \psi \,,$

where $D$ is the given Dirac operator (e.g. 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 (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

General

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)

• Özgür Açık, Field equations from Killing spinors (arXiv:1705.04685)

• Ángel Murcia, C. S. Shahbazi, Parallel spinors on globally hyperbolic Lorentzian four-manifolds (arXiv:2011.02423)

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

and later in

• 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

specifically

Last revised on February 1, 2021 at 22:14:30. See the history of this page for a list of all contributions to it.