Two dimensional Sen connections in general relativity

The two dimensional version of the Sen connection for spinors and tensors on spacelike 2-surfaces is constructed. A complex metric ${\gamma }_{AB}$ on the spin spaces is found which characterizes both the algebraic and extrinsic geometrical properties of the 2-surface $\mathrm{\$}$. The curvature of the two dimensional Sen operator ${\mathrm{\Delta }}_e$ is the pull back to $\mathrm{\$}$ 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 of the 2 dimensional Sen and the induced spin connections is the anti-self-dual part of the torsion’. The irreducible parts of ${\mathrm{\Delta }}_e$ 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 2 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.