synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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
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
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).
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]
Discussion relating to Killing vectors in supergeometry (superisometries) is in
and later in
See also
Volume 154, Number 3 (1993), 509-521 (Euclid)
Discussion regarding the conceptualization of Killing spinors in super-Cartan geometry is in
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.
General discussion of Killing with an eye towards applications in supersymmetry is around page 907 in volume II of
specifically
in 11-dimensional supergravity:
for G₂-structures in M-theory on G₂-manifolds:
Generalization to “differential spinors”:
Last revised on July 18, 2024 at 11:11:05. See the history of this page for a list of all contributions to it.