Penrose transform

In mathematical physics, the Penrose transform, introduced by Roger Penrose (1967, 1968, 1969), is a complex analogue of the Radon transform that relates massless fields on spacetime to cohomology of sheaves on complex projective space. The projective space in question is the twistor space, a geometrical space naturally associated to the original spacetime, and the twistor transform is also geometrically natural in the sense of integral geometry. The Penrose transform is a major component of classical twistor theory.

Abstractly, the Penrose transform operates on a double fibration of a space Y, over two spaces X and Z

In the classical Penrose transform, Y is the spin bundle, X is a compactified and complexified form of Minkowski space and Z is the twistor space. More generally examples come from double fibrations of the form

where G is a complex semisimple Lie group and H₁ and H₂ are parabolic subgroups.

The Penrose transform operates in two stages. First, one pulls back the sheaf cohomology groups H^r(Z,F) to the sheaf cohomology H^r(Y,η⁻¹F) on Y; in many cases where the Penrose transform is of interest, this pullback turns out to be an isomorphism. One then pushes the resulting cohomology classes down to X; that is, one investigates the direct image of a cohomology class by means of the Leray spectral sequence. The resulting direct image is then interpreted in terms of differential equations. In the case of the classical Penrose transform, the resulting differential equations are precisely the massless field equations for a given spin.

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