nLab tractor bundle



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

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 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.


Last revised on March 19, 2017 at 00:41:19. See the history of this page for a list of all contributions to it.