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, Kähler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the Kähler 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 Kähler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by Alekseevskǐı (and the corresponding special Kähler 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 Alekseevskǐı 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. 1 On leave of absence from the Institute for Theoretical Physics, Utrecht, The Netherlands; E-mail BDEWIT at CRNVMA.CERN.CH or FYS.RUU.NL 2 Onderzoeksleider, N.F.W.O. Belgium; Bitnet FGBDA19 at BLEKUL11 CERN-TH.6302/91 KUL-TF-91/43 THU-91/22 December, 1991.