Fourier transform from momentum space to twistor space

TL;DR

This paper introduces a Fourier transform from momentum space to twistor space within twistor string theory, clarifying its application and properties, especially for complex projective twistor spaces related to Minkowski spacetime.

Contribution

It defines a new Fourier transform for momentum-space functions in twistor theory, including inverse transforms, and relates these to cohomology group representations.

Findings

Fourier transform is explicitly defined via complex integrals.

Transform and inverse transform are shown to relate to cohomology groups.

Twistor operator representations are compatible with the Fourier transform.

Abstract

A Fourier transform from momentum space to twistor space is introduced in twistor string theory, for the first time, for the case where the twistor space is a three-dimensional real projective space, corresponding to ultra-hyperbolic spacetime. In this case, the Fourier transform is the same as in the standard analysis. However, when the twistor space is a three-dimensional complex projective space, corresponding to the complexified Minkowski space, some aspects of the Fourier transform have yet to be clarified. For example, no concrete method is known for calculating the complex integral, and nor is it known which functions it should be applied to or what results are obtained. In this paper, we define a Fourier transform for momentum-space functions in terms of a certain complex integral, assuming that the functions can be expanded as power series. We also define the inverse Fourier transform for twistor-space functions in terms of a certain complex integral. Then, we show that the momentum-space functions to which the Fourier transform can be applied and the twistor-space functions that are obtained can be represented in terms of cohomology groups. Furthermore, we show that the twistor operator representations popular in the twistor theory literature are valid for the Fourier transform and its inverse transform. We also show that the functions over which the application of the operators is closed can be represented in terms of cohomology groups.