Special geometry, cubic polynomials and homogeneous quaternionic spaces

TL;DR

This paper classifies homogeneous quaternionic spaces arising from cubic polynomials in supergravity, revealing new classes of spaces and clarifying the structure of known classifications.

Contribution

It provides a complete classification of cubic polynomials with transitive invariance groups, leading to new homogeneous quaternionic spaces and refining existing classifications.

Findings

Identified a subset of normal quaternionic spaces consistent with known classifications.

Discovered a new class of rank-3 quaternionic spaces with larger dimensions.

Clarified the structure of some rank-4 ext{Al} spaces and their inequivalent variants.

Abstract

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 {\em special} real, K\"ahler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the K\"ahler 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 K\"ahler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by \Al\ (and the corresponding special K\"ahler 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 \Al\ 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 $W_3$ algebras.