# A class of cubic hypersurfaces and quaternionic K\"ahler manifolds of co-homogeneity one

## TL;DR

This paper classifies certain special real manifolds with reducible cubic potentials and explores their connection to quaternionic K"ahler manifolds via the supergravity q-map, revealing new inhomogeneous examples of co-homogeneity one.

## Contribution

It provides a complete classification of projective special real manifolds with reducible cubic potentials and constructs new inhomogeneous quaternionic K"ahler manifolds using the q-map.

## Key findings

- Four series of projective special real manifolds identified
- Two series are homogeneous, two have co-homogeneity one
- Constructed inhomogeneous quaternionic K"ahler manifolds of co-homogeneity one

## Abstract

We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with co-homogeneity one. Complete projective special real manifolds give rise to complete quaternionic K\"ahler manifolds via the supergravity q-map, which is the composition of the supergravity c-map and r-map. We develop curvature formulas for manifolds in the image of the q-map. Applying the q-map to one of the above series of projective special real manifolds, we obtain a series of complete quaternionic K\"ahler manifolds, which are shown to be inhomogeneous (of co-homogeneity one) based on our curvature formulas.