SPECIAL GEOMETRY, CUBIC POLYNOMIALS AND HOMOGENEOUS QUATERNIONIC SPACES

The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certain N = 2 supergravity theories, where dimensional reduction induces a mapping between special real, Kahler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the Kahler and quaternionic manifolds in homogeneous holomorphic functions of second degree. We classify all cubic polynomials that have an invariance group that acts transitively on the real manifold. The corresponding Kahler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by Alekseevskii (and the corresponding special Kahler spaces given by Cecotti), but there is a new class of rank-3 spaces of quaternionic dimension larger than 3. We also point out that some of the rank-4 Alekseevskii spaces were not fully specified and correspond to a finite variety of inequivalent spaces. A simpler version of the equation that underlies the classification of this paper also emerges in the context of W3 algebras.