nLab Killing spinor

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).

Related concepts

• Killing vector

• Killing tensor

• Killing-Yano tensor

• superisometry

• supersymmetry and Calabi-Yau manifolds

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)

See also

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

• Ángel Murcia, Carlos S. Shahbazi, Parallel spinors on globally hyperbolic Lorentzian four-manifolds, Ann Glob Anal Geom 61 (2022) 253–292 [arXiv:2011.02423, doi:10.1007/s10455-021-09808-y]

  (on globally hyperbolic Lorentzian manifolds)

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

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, pages 1198-1199 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

and later in

• Dmitri Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Uwe Semmelmann, Killing spinors are Killing vector fields in Riemannian Supergeometry (arXiv:dg-ga/9704002)

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

• Andrei Moroianu, Uwe Semmelmann, Parallel spinors and holonomy groups, Journal of Mathematical Physics 41 (2000), 2395-2402 (arXiv:math/9903062)

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

• Pierre Deligne, P. Etingof, Dan Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and Edward Witten, (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

specifically

in heterotic supergravity:

• Ulf Gran, George Papadopoulos, Diederik Roest, Supersymmetric heterotic string backgrounds, Phys.Lett.B656:119-126, 2007 (arXiv:0706.4407)

in 11-dimensional supergravity:

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

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

• Thomas Friedrich, Stefan Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, AsianJ.Math.6:303-336,2002 (arXiv:math/0102142)

Generalization to “differential spinors”:

• Carlos S. Shahbazi, Differential spinors and Kundt three-manifolds with skew-torsion [arXiv:2405.03756]