nLab tractor bundle

Idea

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

$$P \mapsto P \underset{H}{\times} V$$

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

$$V X \to X \,.$$

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

  $$\mathcal{A}X \coloneqq P \underset{H}{\times} \mathfrak{g}$$

  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 $\mathcal{A}X$ 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

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

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

  $$\mathcal{A}X/\mathcal{A}^0 X = P\underset{H}{\times} \mathfrak{g}/\mathfrak{p} \simeq T X$$

  (e.g Čap-Souček 07, p. 12).

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

  $$\mathcal{A}^1 X \simeq T^\ast X \,.$$

• 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 $\wedge^k \mathfrak{p}_+ \otimes V$

  $$\Omega^k(X, V X) \simeq \Gamma( P \underset{H}{\times} \wedge^k \mathfrak{p}_+ \otimes V ) \,.$$

• The differential in the Lie algebra homology of $\mathfrak{p}_+$ with coefficients in $V$ is a natural transformation

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

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

  $$\partial^\ast_k \colon \Lambda^{k+1} T^\ast X \otimes_X V X \longrightarrow \Lambda^{k} T^\ast X \otimes_X V X$$

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

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

  (e.g Čap-Souček 07, p. 17)

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

Related concepts

• Fefferman-Graham ambient construction

References

• 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