Special geometry, cubic polynomials and homogeneous quaternionic spaces

TL;DR

This work classifies homogeneous quaternionic spaces arising from $N=2$ supergravity by reducing five-dimensional real spaces encoded by a cubic form $C(h)=d_{ABC}\,h^A h^B h^C$ to their Kähler and quaternionic images via the $r$ and $c\circ r$ maps. The core result is a Clifford-algebra–driven classification of the $d_{ABC}$ tensors, yielding explicit cubic forms ${\cal Y}(x)$ built from real Clifford generators and organizing solutions into families $L(-1,r)$, $L(0,P,\dot P)$, $L(q,P)$, and $L(4m,P,\dot P)$, with corresponding real, Kähler, and quaternionic spaces. These results identify a well-defined subset of Alekseevskii’s normal quaternionic spaces and reveal a new rank-3 family and a finite set of rank-4 spaces not fully specified previously, suggesting a complete table of homogeneous quaternionic spaces. The analysis also uncovers a link to $W_3$ algebras and offers a coherent dimensional-oxidation narrative connecting five-, four-, and three-dimensional supergravities. Overall, the paper extends the landscape of homogeneous special geometries and clarifies how Clifford-algebra representations control the symmetry structure of these spaces.

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, 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 \Al\ (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 \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.