In theoretical 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, or more precisely the space of solutions to massless field equations, to sheaf cohomology groups on complex projective space.
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 (which as a complex manifold is
More general examples come from double fibrations of the form where G is a complex semisimple Lie group and H1 and H2 are parabolic subgroups.
First, one pulls back the sheaf cohomology groups Hr(Z,F) to the sheaf cohomology Hr(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.
The classical example is given as follows The maps from Y to X and Z are the natural projections.
Using spinor index notation, the Penrose transform gives a bijection between solutions to the spin
[1] The Penrose–Ward transform is a nonlinear modification of the Penrose transform, introduced by Ward (1977), that (among other things) relates holomorphic vector bundles on 3-dimensional complex projective space CP3 to solutions of the self-dual Yang–Mills equations on S4.
Atiyah & Ward (1977) used this to describe instantons in terms of algebraic vector bundles on complex projective 3-space and Atiyah (1979) explained how this could be used to classify instantons on a 4-sphere.