Two-dimensional Sen connections in general relativity

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Abstract
The two-dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface . The curvature of the two-dimensional Sen operator is the pullback to of the anti-self-dual part of the spacetime curvature, while its `torsion' is a boost-gauge invariant expression of the extrinsic curvatures of . The difference between the two-dimensional Sen and the induced spin connections is the anti-self-dual part of the `torsion'. The irreducible parts of are shown to be the familiar 2-surface twistor and the Weyl--Sen--Witten operators. Two Sen--Witten type identities are derived; the first is an identity between the two-dimensional twistor and the Weyl--Sen--Witten operators and the integrand of Penrose's charge integral, while the second contains the `torsion' as well. For spinor fields satisfying the 2-surface twistor equation the first reduces to Tod's formula for the kinematical twistor.
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