nLab
tractor bundle

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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

Tractor bundles are certain associated bundles considered in conformal geometry and more generally in parabolic geometry.

Even more generally, given a subgroup inclusion HGH \hookrightarrow G of Lie groups and given a linear representation of GG on some vector space VV, with restricted representation by HH, then the operation of forming associated bundles

PP×HV P \mapsto P \underset{H}{\times} V

to HH-principal bundles PP constitutes a construction of natural vector bundles on the category of (HG)(H\to G)-Cartan geometries. For the case that HGH \hookrightarrow G is a parabolic subgroup, hence for the case of parabolic geometry, and the case that PXP \to X is the HH-frame bundle of an (HG)(H \to G) Cartan geometry on a manifold XX, then these associated bundles P×HVXP \underset{H}{\times} V \to X are called tractor bundles (e.g Čap-Souček 07, p. 11), often denoted

VXX. V X \to X \,.

These tractor bundles carry special connections and as such are called tractor connections (Čap-Gover 02).

Some GG-representations are special and hence some tractor bundles are singled out (e.g. Čap-Slovák 09, 1.5.7):

  • For the case that V=𝔤V = \mathfrak{g} is the Lie algebra of GG equipped with its adjoint action, then

    𝒜XP×H𝔤 \mathcal{A}X \coloneqq P \underset{H}{\times} \mathfrak{g}

    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 𝒜X\mathcal{A}X is a bundle of Lie algebras, hence its space of sections Γ(𝒜X)\Gamma(\mathcal{A}X) carries a Lie bracket {,}\{-,-\} which makes each fiber isomorphic to 𝔤\mathfrak{g} not just as a vector space, but as Lie algebra. In particular therefore for any representation VV then the sections of the associated tractor bundle VXV X carry an action by sections of 𝒜X\mathcal{A}X.

  • By the axioms on Cartan geometries (see also at Cartan connection) the tractor bundle corresponding to the quotient 𝔤/𝔥\mathfrak{g}/\mathfrak{h} is isomorphic to the tangent bundle TXT X.

  • The inclusion 𝔭𝔤\mathfrak{p}\hookrightarrow \mathfrak{g} of the Lie algebra of the parabolic subgroup induces a tractor subbundle P×H𝔭P×H𝔤P \underset{H}{\times} \mathfrak{p} \hookrightarrow P \underset{H}{\times}\mathfrak{g} denoted

    𝒜 0X𝒜X \mathcal{A}^0 X \hookrightarrow \mathcal{A}X

    (e.g Čap-Souček 07, p. 5). The corresponding quotient is again the tangent bundle

    𝒜X/𝒜 0X=P×H𝔤/𝔭TX \mathcal{A}X/\mathcal{A}^0 X = P\underset{H}{\times} \mathfrak{g}/\mathfrak{p} \simeq T X

    (e.g Čap-Souček 07, p. 12).

  • Dually, the tractor bundle associated to (𝔤/𝔭) *𝔭 +(\mathfrak{g}/\mathfrak{p})^\ast \simeq \mathfrak{p}_+ is 𝒜 1X\mathcal{A}^1 X which is isomorphic to the cotangent bundle T *XT^\ast X.

    𝒜 1XT *X. \mathcal{A}^1 X \simeq T^\ast X \,.
  • From this one gets that differential forms with values in the tractor bundle of VV are equivalently sections of the tractor bundle given by the tensor product representation k𝔭 +V\wedge^k \mathfrak{p}_+ \otimes V

    Ω k(X,VX)Γ(P×H k𝔭 +V). \Omega^k(X, V X) \simeq \Gamma( P \underset{H}{\times} \wedge^k \mathfrak{p}_+ \otimes V ) \,.
  • The differential in the Lie algebra homology of 𝔭 +\mathfrak{p}_+ with coefficients in VV is a natural transformation

    k+1𝔭 +V k𝔭 +V \wedge^{k+1} \mathfrak{p}_+ \otimes V \longrightarrow \wedge^{k} \mathfrak{p}_+ \otimes V

    (often called the Kostant codifferential *\partial^\ast is this parabolic context) and hence gives homomorphisms of natural bundles

    k *:Λ k+1T *X XVXΛ kT *X XVX \partial^\ast_k \colon \Lambda^{k+1} T^\ast X \otimes_X V X \longrightarrow \Lambda^{k} T^\ast X \otimes_X V X

    This yields a complex of bundles whose chain homology is fiberwise the Lie algebra homology, hence ker( *)/im( *)ker(\partial^\ast)/im(\partial^\ast) is the bundle associated to the Lie algebra homology group of 𝔭 +\mathfrak{p}_+ regarded as an HH-representation:

    ker( k1 *)/im( k *)P×HH k(𝔭 +,V) ker(\partial^\ast_{k-1})/im(\partial^\ast_k) \simeq P \underset{H}{\times} H_k(\mathfrak{p}_+, V)

    (e.g Čap-Souček 07, p. 17)

    For homogeneous Cartan geometries X=G/HX = G/H this gives a geometric construction of BGG resolutions of representations, for general Cartan geometries it gives a curved generalization of that.

References

Last revised on March 18, 2017 at 20:41:19. See the history of this page for a list of all contributions to it.