A class of cubic hypersurfaces and quaternionic Kähler 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 Kähler 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 Kähler manifolds, which are shown to be inhomogeneous (of co-homogeneity one) based on our curvature formulas.

Are all supergravity theories Yang–Mills squared?

“…e.g. [58]) from real special homogeneous manifolds. By contrast, for the addition of hyper multiplets to the minimally coupled model of section II A 4 there is no real special homogeneous starting point (i.e.…”

Special geometry, Hessian structures and applications

“…Already in dimension two there are continuous families of non-isomorphic PSR spaces. Complete PSR manifolds based on reducible cubic polynomials have been classified in [145] and belong to four infinite series, two of which consist of homogenous spaces while the other two consist of spaces of co-homogeneity one.…”

“…Homogeneous (pseudo-)PSK manifolds of the form G/K, where G is a semisimple Lie group and K a compact subgroup are automatically symmetric spaces [146]. Examples of PSK manifolds with co-homogeneity one have been constructed by applying the r-map to non-homogneous PSR manifolds [145]. A general criterion of the geodesic completeness of PSK manifolds has been proved…”

“…Moreover, since it preserves completeness, one can use complete, non-homogeneous PSK manifolds to obtain complete, non-homogeneous QK manifolds. Two infinite families of complete QK manifold of co-homogeneity 1 have been constructed in [145].…”

Symmetries of quaternionic Kähler manifolds with $S^1$-symmetry

“…Since there are many examples of complete, simply connected PSK manifolds [CHM12], including homogeneous spaces [dWvP92, AC00] and manifolds of cohomogeneity one [CDJL17], this result yields many examples of quaternionic Kähler manifolds with non-trivial fundamental group.…”