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^3s arise as obstructions to extending the H^2s 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.

arXiv comment37 pages, 3 figures, in v2 a number of inaccuracies are removed and the discussion is improved; in v3, one reference added;v4 extra acknowledgement

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Cited by in corpus (33)

• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Non-Abelian Tensor Multiplet Equations from Twistor Space
• $D=6$, $\mathcal{N}=(2,0)$ and $\mathcal{N}=(4,0)$ theories
• (2,0) Lagrangian Structures
• The polarized scattering equations for 6d superamplitudes
• Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory
• Twistor methods for AdS${}_5$
• Semistrict Higher Gauge Theory
• On Twistors and Conformal Field Theories from Six Dimensions
• Self-interacting chiral p-forms in higher dimensions
• Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations
• Geometrical Aspects of An Abelian (2,0) Action
• $L_{\infty }$-Algebras, the BV Formalism, and Classical Fields
• Twistor actions for gauge theory and gravity
• Ambitwistors, oscillators and massless fields on $Ad{S}_5$
• Spinor description of $D=5$ massless low-spin gauge fields
• Conformal Contact Terms and Semi-Local Terms
• Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes
• Ambitwistor Strings in Six and Five Dimensions
• The supersymmetric Penrose transform in six dimensions
• Higher Gauge Theory
• On Closed Twistor String Theory
• Instanton String and M-Wave in Multiple M5-Branes System
• Instantons on the six-sphere and twistors
• Instantons in six dimensions and twistors
• Massless chiral fields in six dimensions
• From Twistor-Particle Models to Massive Amplitudes
• Conformally Invariant Spinorial Equations in Six Dimensions
• An Alternative Perspective on Ambitwistor String Theory
• Killing Spinors and Related Symmetries in Six Dimensions
• Explicit Non-Abelian Gerbes with Connections
• A Proposed $E_8\times E_8$ Kinematic Scaffolding for the Standard Model with Pre-Gravitation
• The resolution of Euclidean massless field operators of higher spins on ${\mathbb{R}}^6$ and the $L^2$ method