An Ode to the Penrose and Witten transforms in Twistor space for 3D CFT

TL;DR

This work develops a comprehensive twistor-space framework for 3D conformal field theories, anchored by Sp$(4)$ invariants and Penrose/Witten transforms. It shows that incorporating the infinity twistor is essential to describe general scaling dimensions, parity-odd sectors, and non-conserved spinning operators, while preserving conformal invariance in a distributional sense. The authors construct explicit Wightman functions for conserved currents and $\Delta=1$ scalars, extend the construction to general $\Delta$ via Legendre and epsilon transforms, and establish a robust supersymmetric (\N=1) extension using super-twistors and $OSp(\mathcal{N}|4)$ invariants. This framework enables a twistor-space bootstrap program for spinning correlators in 3D, with potential links to dual conformal and Yangian symmetries and a path toward higher-point SUSY correlators. Overall, the paper clarifies how infinity-twistor structures and projective-delta invariants encode the full representation content of 3D CFT operators in twistor space and sets up practical tools for SUSY generalizations.

Abstract

Here we discuss the construction of Sp$(4;\mathbb{R})$ invariant objects in the twistor space for three dimensional conformal field theories. The Sp$(4;\mathbb{R})$ invariant projective delta function, alongside the Twistor symplectic dot product invariants form the basis for conformal Wightman functions involving conserved currents and $Δ=1$ scalars. For correlators involving scalars with $Δ\ne 1$, generic spinning primaries and parity odd correlators we show that the infinity twistor of $\mathbb{R}^{2,1}$ must be incorporated into the analysis. We show that this feature can be traced to the Penrose and Witten transforms of these operators that we derive. We then discuss the super-twistor space construction and derive the supersymmetric Penrose transform for $\mathcal{N}=1$ theories using the Fourier transform and the supersymmetric Witten transform. We construct OSp$(\mathcal{N}|4;\mathbb{R})$ invariants and its application to several super-Wightman functions. Similar to the non supersymmetric case, we find an important role played by the (super) infinity twistor which we exemplify through parity odd super-correlators and a supersymmetric contact term.

Figures (3)

• Figure 1: The trio of Twistors, Dual-Twistors and Spinor-Helicity variables
• Figure 2: CPT and the Twistor Fourier transform. CPT exchanges positive and negative helicities and $W$ and $Z$ while keeping the functional form fixed. The Fourier transform does not change the helicity but converts a function of $W$ into a different function of $Z$.
• Figure 3: The interplay between the Penrose transform and Witten transform