3D Conformal Field Theory in Twistor Space

TL;DR

This paper develops a twistor space framework for 3D Lorentzian conformal field theories, solving Ward identities and analyzing the structure of two and three-point functions, including distributional and parity-odd solutions.

Contribution

It formulates conformal Ward identities in twistor space and reveals the role of helicity operators, introducing distributional solutions and extending analysis to parity-odd correlators.

Findings

Distributional solutions match CFT correlators

Parity-odd functions have unique twistor space forms

Results verified via momentum space and spinor helicity analysis

Abstract

The aim of this paper is to study three dimensional Lorentzian conformal field theories in twistor space. We formulate the conformal Ward identities and solve for two and three point Lorentzian Wightman functions. We found that the Helicity operators apart from the conformal generators play an important role in fixing their functional form. The equations take the form of first order Euler equations which in addition to the usual solutions that are polynomials, also possess weak solutions which are distributional in nature. All of these play an important role in our analysis. For instance, in the case of three point functions, the distributional solutions are indeed the ones realized by the CFT correlators. We also extend our analysis to parity odd Wightman functions which take an interesting form in twistor space. We verify our results by systematically analyzing the corresponding Wightman functions in momentum space and spinor helicity variables and matching with the twistor results via a half-Fourier transform.