The H/Q-correspondence and a generalization of the supergravity c-map

TL;DR

This paper generalizes the supergravity c-map by constructing a quaternionic manifold from a hypercomplex one with a rotating vector field, and explores related structures like the hypercomplex tangent bundle and special complex manifolds.

Contribution

It introduces a new construction linking hypercomplex and quaternionic manifolds, extending the supergravity c-map and related geometric correspondences.

Findings

Constructs conical hypercomplex manifolds from hypercomplex manifolds with rotating vector fields.

Establishes a quaternionic manifold association to conical special complex manifolds.

Shows the tangent bundle of any special complex manifold has a canonical Ricci-flat hypercomplex structure.

Abstract

Given a hypercomplex manifold with a rotating vector field (and additional data), we construct a conical hypercomplex manifold. As a consequence, we associate a quaternionic manifold to a hypercomplex manifold of the same dimension with a rotating vector field. This is a generalization of the HK/QK-correspondence. As an application, we show that a quaternionic manifold can be associated to a conical special complex manifold of half its dimension. Furthermore, a projective special complex manifold (with a canonical c-projective structure) associates with a quaternionic manifold. The latter is a generalization of the supergravity c-map. We do also show that the tangent bundle of any special complex manifold carries a canonical Ricci-flat hypercomplex structure, thereby generalizing the rigid c-map.