Conformal field theories in six-dimensional twistor space

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space–time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the six-dimensional case in which twistor space is the 6-quadric QQ in CP7CP7 with a view to applications to the self-dual (0,2)(0,2)-theory. We show how spinor-helicity momentum eigenstates have canonically defined distributional representatives on twistor space (a story that we extend to arbitrary dimension). These yield an elementary proof of the surjectivity of the Penrose transform. We give a direct construction of the twistor transform between the two different representations of massless fields on twistor space (H2H2 and H3H3) in which the H3H3s arise as obstructions to extending the H2H2s off QQ into CP7CP7.We also develop the theory of Sparling’s ‘ΞΞ-transform’, the analogous totally real split signature story based now on real integral geometry where cohomology no longer plays a role. We extend Sparling’s ΞΞ-transform to all helicities and homogeneities on twistor space and show that it maps kernels and cokernels of conformally invariant powers of the ultrahyperbolic wave operator on twistor space to conformally invariant massless fields on space–time. This is proved by developing the six-dimensional analogue of the half-Fourier transform between functions on twistor space and momentum space. We give a treatment of the elementary conformally invariant Φ3Φ3 amplitude on twistor space and finish with a discussion of conformal field theories in twistor space.