Pure Spinors are Higher-Dimensional Twistors

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 corresponding pure spinors have been interpreted as four and six-dimensional twistor variables. In this paper, we argue that the identification of pure spinors and twistors holds in any even dimension, and we use pure spinors to construct massless solutions in higher dimensions.