6 - The Penrose Transform

Summary

Introduction

This article is a survey of developments in the Penrose transform since [8]. Recall that in [8] the transform was precisely the homomorphism

for

• U = an open subset of M (= compactified complexified Minkowski space)

• V = the corresponding open subset of P (= projective twistor space)

• O(−n − 2) = the sheaf of germs of holomorphic functions homogeneous of degree −n − 2

• Ƶ_n = the sheaf of germs of holomorphic solutions of the zero-rest-mass free field equations of helicity.

The transform was shown to be an isomorphism under mild topological conditions on U which hold, in particular, for the important special case of U = M⁺ and V = P⁺ (see [14] for standard twistor notation). Thus, one obtains a twistor description of positive frequency massless fields. The method of proof in [8] was to set up a general spectral sequence which specialized to give the different results for different values of n. In particular, one can see how the transform automatically falls into three cases:

Thus, one should view [8] as providing a cohomological ‘machine’ in the form of a spectral sequence which turns cohomology into differential equations.