In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle).
The term tractor is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan conformal connection,[1] and later rediscovered within the formalism of local twistors and generalized to projective connections by Michael Eastwood et al. in [2] Tractor bundles can be defined for arbitrary parabolic geometries.
[3] The tractor bundle for a
vector bundle
is a linear isomorphism at each point from the tangent bundle of
denotes the orthogonal complement of
Given a tractor bundle, the metrics in the conformal class are given by fixing a local section
To go the other way, and construct a tractor bundle from a conformal structure, requires more work.
The tractor bundle is then an associated bundle of the Cartan geometry determined by the conformal structure.
The conformal group for a manifold of signature
, and one obtains the tractor bundle (with connection) as the connection induced by the Cartan conformal connection on the bundle associated to the standard representation of the conformal group.
Because the fibre of the Cartan conformal bundle is the stabilizer of a null ray, this singles out the line bundle
The tractor bundle is the space of 2-jets of solutions
A little work then shows that the sections of the tractor bundle (in a fixed Weyl gauge) can be represented by
+ σ τ + ρ α
The preferred line bundle
Given a change in Weyl gauge
, the components of the tractor bundle change according to the rule
has been used in one place to raise the index.
is invariant under the change in gauge, and the connection can be shown to be invariant using the conformal change in the Levi-Civita connection and Schouten tensor.
be a projective manifold of dimension
Then the tractor bundle is a rank
equipped with the additional data of a line subbundle
such that, for any non-vanishing local section
is a linear isomorphism of the tangent space to
[2] One recovers an affine connection in the projective class from a section
Explicitly, the tractor bundle can be represented in a given affine chart by pairs
is the projective Schouten tensor.
Here the projective Schouten tensor of an affine connection is defined as follows.
Define the Riemann tensor in the usual way (indices are abstract)