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).
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Construction
Let M be a pseudo-Riemannian conformal manifold with a spin structure and a conformal metric of signature (p,q). The conformal group is the pseudo-orthogonal group
Representation via Weyl spinors
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
Local twistor transport
The connection, sometimes called local twistor transport, is given by
Here
Canonical filtration
In general, the local twistor bundle T is equipped with a short exact sequence of vector bundles
where
where
Curvature
The curvature of the local twistor connection involves both the Weyl curvature and the Cotton tensor. (It is the Cartan conformal curvature.) The curvature preserves the space