In differential geometry, the local twistor bundle is a specific vector bundle with connection that can be associated to any conformal manifold, at least locally.
Intuitively, a local twistor is an association of a twistor space to each point of space-time, together with a conformally invariant connection that relates the twistor spaces at different points.
This connection can have holonomy that obstructs the existence of "global" twistors (that is, solutions of the twistor equation in open sets).
admits a fundamental representation, the spin representation, and the associated bundle is the local twistor bundle.
Local twistors can be represented as pairs of Weyl spinors on M (in general from different spin representations, determined by the reality conditions specific to the signature).
In the case of a four-dimensional Lorentzian manifold, such as the space-time of general relativity, a local twistor has the form Here we use index conventions from Penrose & Rindler (1986), and
are two-component complex spinors for the Lorentz group
The connection, sometimes called local twistor transport, is given by Here
the Schouten tensor, contracted on one index with the canonical one-form.
An analogous equation holds in other dimensions, with appropriate Clifford algebra multipliers between the two Weyl spin representations (Sparling 1986).
In general, the local twistor bundle T is equipped with a short exact sequence of vector bundles where
is a distinguished sub-bundle, that corresponds to the marked point of contact of the conformal Cartan connection.
That is, there is a canonical marked one-dimensional subspace X in the tractor bundle, and
Under the Plücker embedding, the tractor bundle in four dimensions is isomorphic to the exterior square of the local twistor bundle, and
it involves only the conformally-invariant Weyl curvature.