3D Conformal Field Theory in Twistor Space
Aswini Bala, Sachin Jain, Dhruva K. S., Deep Mazumdar, Vibhor Singh
TL;DR
This work develops a twistor-space bootstrap for three-dimensional Lorentzian CFTs by formulating and solving conformal Ward identities in twistor and momentum-twistor spaces. It shows that helicity operators augment conformal symmetry, yielding first-order Euler equations whose solutions include both standard polynomial and distributional delta-type terms, which correctly reproduce two- and three-point Wightman functions, including parity-odd structures. The results are cross-validated by translating to momentum-space spinor-helicity variables via half-Fourier transforms and by analytic continuation from Euclidean correlators, establishing a consistent, compact framework for conserved-current correlators in CFT$_3$. The approach clarifies the role of distributional solutions, parity-odd sectors, and mixed representations, and points to promising extensions to higher-point functions, supersymmetric twistor space, and connections to higher-spin and AdS/CFT contexts.
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.