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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Tractor bundles are certain associated bundles considered in conformal geometry and more generally in parabolic geometry.
Even more generally, given a subgroup inclusion $H \hookrightarrow G$ of Lie groups and given a linear representation of $G$ on some vector space $V$, with restricted representation by $H$, then the operation of forming associated bundles
to $H$-principal bundles $P$ constitutes a construction of natural vector bundles on the category of $(H\to G)$-Cartan geometries. For the case that $H \hookrightarrow G$ is a parabolic subgroup, hence for the case of parabolic geometry, and the case that $P \to X$ is the $H$-frame bundle of an $(H \to G)$ Cartan geometry on a manifold $X$, then these associated bundles $P \underset{H}{\times} V \to X$ are called tractor bundles (e.g Čap-Souček 07, p. 11), often denoted
These tractor bundles carry special connections and as such are called tractor connections (Čap-Gover 02).
Some $G$-representations are special and hence some tractor bundles are singled out (e.g. Čap-Slovák 09, 1.5.7):
For the case that $V = \mathfrak{g}$ is the Lie algebra of $G$ equipped with its adjoint action, then
is called the adjoint tractor bundle. A key point is that for parabolic geometries the Lie algebra structure is retained by this bundle in that $\mathcal{A}X$ is a bundle of Lie algebras, hence its space of sections $\Gamma(\mathcal{A}X)$ carries a Lie bracket $\{-,-\}$ which makes each fiber isomorphic to $\mathfrak{g}$ not just as a vector space, but as Lie algebra. In particular therefore for any representation $V$ then the sections of the associated tractor bundle $V X$ carry an action by sections of $\mathcal{A}X$.
By the axioms on Cartan geometries (see also at Cartan connection) the tractor bundle corresponding to the quotient $\mathfrak{g}/\mathfrak{h}$ is isomorphic to the tangent bundle $T X$.
The inclusion $\mathfrak{p}\hookrightarrow \mathfrak{g}$ of the Lie algebra of the parabolic subgroup induces a tractor subbundle $P \underset{H}{\times} \mathfrak{p} \hookrightarrow P \underset{H}{\times}\mathfrak{g}$ denoted
(e.g Čap-Souček 07, p. 5). The corresponding quotient is again the tangent bundle
(e.g Čap-Souček 07, p. 12).
Dually, the tractor bundle associated to $(\mathfrak{g}/\mathfrak{p})^\ast \simeq \mathfrak{p}_+$ is $\mathcal{A}^1 X$ which is isomorphic to the cotangent bundle $T^\ast X$.
From this one gets that differential forms with values in the tractor bundle of $V$ are equivalently sections of the tractor bundle given by the tensor product representation $\wedge^k \mathfrak{p}_+ \otimes V$
The differential in the Lie algebra homology of $\mathfrak{p}_+$ with coefficients in $V$ is a natural transformation
(often called the Kostant codifferential $\partial^\ast$ is this parabolic context) and hence gives homomorphisms of natural bundles
This yields a complex of bundles whose chain homology is fiberwise the Lie algebra homology, hence $ker(\partial^\ast)/im(\partial^\ast)$ is the bundle associated to the Lie algebra homology group of $\mathfrak{p}_+$ regarded as an $H$-representation:
(e.g Čap-Souček 07, p. 17)
For homogeneous Cartan geometries $X = G/H$ this gives a geometric construction of BGG resolutions of representations, for general Cartan geometries it gives a curved generalization of that.
Andreas Čap, A. Rod Gover, Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), 1511–1548.
Andreas Čap, Vladimír Souček, Curved Casimir operators and the BGG machinery, SIGMA 3, 2007, 111 (arxiv:0708.3180 doi)
Andreas Čap, Jan Slovák, Parabolic Geometries I – Background and General Theory, AMS 2009
Sean Curry, A. Rod Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, 2014 (arXiv:1412.7559)
Wikipedia, Tractor bundle
Last revised on March 18, 2017 at 20:41:19. See the history of this page for a list of all contributions to it.