Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces

This work is a continuation of the study of higher-dimensional generalizations of the twistor correspondence in conformally compactified Minkowski space begun in Part I [J. Harnad and S. Shnider, J. Math. Phys. 33, 3197-3209 (1992)]. The case of odd-dimensional spaces is treated, with an equivariant diffeomorphism (Cartan map) defined between Grassmannians (or flag manifolds) of totally isotropic subspaces of a complex vector space with respect to a nondegenerate quadratic form, and minimal orbits in the Grassmannians (or flag manifolds) based upon spinor modules. Then the Cartan maps for the real forms of signature (p,q) are considered. The extensions to super-Grassmannians are also discussed, with emphasis upon the cases of dimensions 4, 6, and 10. It is found that in dimensions greater than 4, the natural choice for the superdouble flag space correspondence does not result in the expected supernull line foliation arising in the application to the supersymmetric Yang-Mills equations although in dimension 6 the model constructed is relevant to chiral supergravity.