Conformal field theories in six-dimensional twistor space

Abstract

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 6-dimensional case in which twistor space is the six-quadric Q in CP 7 with a view to applications to the self-dual (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 give 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 (H 2 and H 3 ) in which the H 3 s arise as obstructions to extending the H 2 s off Q into CP 7 . We also develop the theory of Sparling's 'Ξ-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 Ξ-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 6-dimensional analogue of the half-Fourier transform between functions on twistor space and momentum space. We give a treatment of the elementary conformally invariant Φ 3 amplitude on twistor space and finish with a discussion of conformal field theories in twistor space.

Twistor methods for AdS5

“…Reality conditions on the X AB -and hence the homogeneous coordinates X ABcorrespond to a reality structure on these spinor bundles [22,23]. Introduce a quaternionic conjugation acting on Z A ∈ S A by…”

“…The existence of this alternative description for a scalar Φ is related to a certain obstruction problem in twistor space [22,23,32]. In particular, allowing g to extend off the quadric…”

“…Equation (3.9) can be used to produce an integral formula for Φ in terms of g (this is adapted from a similar argument for the indirect transform for flat 6-dimensional spacetime [22]). Extending off the quadric is accomplished at the level of the incidence relations by imposing …”

“…The existence of an indirect transform is again related to an obstruction problem in twistor space [22,32], with any indirect representative χ related to a direct representative ψ bȳ…”

“…A basic example relevant for scattering amplitude calculations is a momentum eigenstate: an on-shell space-time field modelled on e ik·x is encoded in terms of certain distributional cohomology classes on twistor space [22,33]. In AdS, the S-matrix is replaced by correlation functions of specified boundary data for the space-time fields [18,19].…”