Local twistor

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 ${\displaystyle SO(p+1,q+1)}$. There is a conformal Cartan connection on a bundle, the tractor bundle, of M. The spin group of ${\displaystyle SO(p+1,q+1)}$ 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

${\displaystyle Z^{\alpha }={\begin{bmatrix}\omega ^{A}\\\pi _{A'}\end{bmatrix}}.}$

Here we use index conventions from Penrose & Rindler (1986), and ${\displaystyle \omega ^{A}}$ and ${\displaystyle \pi _{A'}}$ are two-component complex spinors for the Lorentz group ${\displaystyle SL(2,\mathbb {C} )}$.

The connection, sometimes called local twistor transport, is given by

${\displaystyle dZ^{\alpha }={\begin{bmatrix}d\omega ^{A}-i\theta ^{AA'}\pi _{A'}\\d\pi _{A'}-iP_{AA'}\omega ^{A}\end{bmatrix}}.}$

Here ${\displaystyle \theta ^{AA'}}$ is the canonical one-form and ${\displaystyle P_{AA'}}$ 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 this formalism, the twistor equation is the requirement that a local twistor be parallel under the connection.

In general, the local twistor bundle T is equipped with a short exact sequence of vector bundles

${\displaystyle 0\to \Pi \to T\to \Omega \to 0}$

where ${\displaystyle \Pi }$ and ${\displaystyle \Omega }$ are two Weyl spin bundles. The bundle ${\displaystyle \Pi }$ 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 ${\displaystyle \Pi }$ is the annihilator of X under Clifford multipliction. In four dimensions, ${\displaystyle \Pi }$ is the space of spinors ${\displaystyle \pi _{A'}}$ and ${\displaystyle \Omega }$ the space of ${\displaystyle \omega ^{A}}$. Under the Plücker embedding, the tractor bundle in four dimensions is isomorphic to the exterior square of the local twistor bundle, and ${\displaystyle \Pi }$ consists of all the twistors incident with

${\displaystyle X^{\alpha \beta }={\begin{bmatrix}0&0\\0&\epsilon _{A'B'}\end{bmatrix}}}$

where ${\displaystyle \epsilon _{A'B'}}$ is the symplectic form on ${\displaystyle \Pi }$.

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 ${\displaystyle \Pi }$, and on ${\displaystyle \Pi }$ it involves only the conformally-invariant Weyl curvature.

• Penrose, R.; Rindler, W. (1986), Spinors and Space-Time: Vol. II, Spinor and Twistor Methods in Space-Time Geometry, Cambridge University Press, ISBN 0-521-25267-9
• Sparling, G (1986), "Towards the geometrization of physics", Nature, 321: 417–419