# nLab Killing spinor

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# 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

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)

Discussion relating to Killing vectors in supergeometry (superisometries) originates around

• 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 $D=11$ Killing Spinors, JHEP 0304 (2003) 039 (arXiv:hep-th/0212008)

Revised on May 8, 2015 13:36:13 by Urs Schreiber (195.113.30.252)