Twistors, Generalizations and Exceptional Structures

TL;DR

This paper explores the algebraic and geometric structures of twistors using Clifford algebras, relating them to conformal maps, pure spinors, and exceptional Lie groups, with applications to string theory and quaternionic-Kahler manifolds.

Contribution

It introduces a novel algebraic spinor approach to twistors in Minkowski spacetime using lower-dimensional Clifford algebras and relates twistorial structures to exceptional Lie groups and string theory.

Findings

Twistors can be described via Clifford algebras in lower dimensions.

Identification of twistor fibers with coset spaces in various dimensions.

Connections established between twistors, pure spinors, and quaternionic-Kahler manifolds.

Abstract

This paper is intended to describe twistors via the paravector model of Clifford algebras and to relate such description to conformal maps in the Clifford algebra over R(4,1), besides pointing out some applications of the pure spinor formalism. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac-Clifford algebra, using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose the Clifford algebra over R{4,1} is also used to describe conformal maps, instead of R{2,4}. It is possible to identify the twistor fiber in four, six and eight dimensions, respectively, with the coset spaces SO(4)/(SU(2) x U(1)/Z_2) = CP1, SO(6)/(SU(3)x U(1)/Z_2) = CP3 and SO(8)/(Spin(6)x Spin(2)/Z_2). The last homogeneous space is closely related to the SO(8) spinor decomposition reserving SO(8) symmetry in type IIB superstring theory. Indeed, aside the IIB theory, there is no SO(8) spinor decomposition preserving SO(8) symmetry and, in this case, one can introduce distinct coordinates and conjugate momenta only if the Spin(8) symmetry is broken by a Spin(6) x Spin(2) subgroup of Spin(8). Also, it is reviewed how to generalize the Penrose flagpole, constructing a flagpole that is more general than the Penrose one, which arises in a particular case (Benn & Tucker description). We investigate the well-known relation between this flagpole and the SO(2n)/U(n) twistorial structure, which emerges when one considers the set X of all totally isotropic subspaces of C{2n}, and an isomorphism from the set of pure spinors to X. Finally we point out some relations between twistors fibrations and the classification of compact homogeneous quaternionic-Kahler manifolds (the so-called Wolf spaces), and exceptional Lie structures.