Delta functions on twistor space and their sign factors
Jun-ichi Note
TL;DR
The paper tackles the problem that sign factors arising when Fourier transforming Yang-Mills amplitudes to real twistor space break global conformal invariance. It introduces a geometric reinterpretation of these signs via the domain structure of real delta functions and then constructs sign-free delta functions on the complex twistor space using Čech cohomology, demonstrating their conformal invariance. These complex, sign-free delta functions on the dual complex (super) twistor space reproduce three-particle MHV and anti-MHV amplitudes in momentum superspace through inverse Fourier transforms, providing a conformally invariant twistor-space formulation. The work suggests a path toward a field-theoretic operator framework in twistor space and motivates further exploration of BCFW recursion in complex twistor space within the Čech-cohomology context.
Abstract
When performing the Fourier transform of the scattering amplitudes in Yang-Mills theory from momentum space to real twistor space, we encounter sign factors that break global conformal invariance. Previous studies conjectured that the sign factors are intrinsic in the real twistor space corresponding to the split signature space-time; hence, they will not appear in the complex twistor space corresponding to the Lorentzian signature space-time. In this study, we present a new geometrical interpretation of the sign factors by investigating the domain of the delta functions on the real twistor space. In addition, we propose a new definition of delta functions on the complex twistor space in terms of the Cech cohomology group without any sign factors and show that these delta functions have conformal invariance. Moreover, we show that the inverse Fourier transforms of these delta functions are the scattering amplitudes in Yang-Mills theory. Thus, the sign factors do not appear in the complex twistor space.