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

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 $\Delta=1$ scalars. For correlators involving scalars with $\Delta\ne 1$, 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.