Higher-Dimensional Twistor Transforms using Pure Spinors

TL;DR

This paper extends the twistor transform using pure spinors to higher even dimensions beyond six, providing explicit formulas for massless field solutions in these spaces.

Contribution

It introduces a method to construct twistor transforms in dimensions greater than six with nonlinear pure spinor constraints, expanding the scope of twistor theory.

Findings

Constructed twistor transforms for d>6 with nonlinear constraints

Provided explicit formulas for massless field solutions in higher dimensions

Generalized known transforms from 4 and 6 dimensions to higher even dimensions

Abstract

Hughston has shown that projective pure spinors can be used to construct massless solutions in higher dimensions, generalizing the four-dimensional twistor transform of Penrose. In any even (Euclidean) dimension d=2n, projective pure spinors parameterize the coset space SO(2n)/U(n), which is the space of all complex structures on R^{2n}. For d=4 and d=6, these spaces are CP^1 and CP^3, and the appropriate twistor transforms can easily be constructed. In this paper, we show how to construct the twistor transform for d>6 when the pure spinor satisfies nonlinear constraints, and present explicit formulas for solutions of the massless field equations.