The H/Q-correspondence and a generalization of the supergravity c-map (Q6589770)

Given a hyperkähler manifold with a so-called rotating circle action, the HK/QK correspondence of \textit{A. Haydys} [J. Geom. Phys. 58, No. 3, 293--306 (2008; Zbl 1143.53043)] produces a quaternionic Kähler manifold with circle symmetry of the same dimension. Haydys has also introduced an inverse construction called the QK/HK correspondence.\N\NThe HK/QK correspondence has subsequently been studied from different points of view, and, in particular, was generalized to the case of indefinite metrics [\textit{D. V. Alekseevsky} et al., Commun. Math. Phys. 324, No. 2, 637--655 (2013; Zbl 1300.53047)]. It produces many examples of complete quaternionic Kähler manifolds and gives a natural context for the quaternionic Kähler manifolds arising from projective special Kähler manifolds via the supergravity \(c\)-map construction [\textit{D. V. Alekseevsky} et al., J. Geom. Phys. 92, 271--287 (2015; Zbl 1326.53061)].\N\NIn this paper, the authors provide a version, called the \(H/Q\)-correspondence, of the above construction in the absence of metrics by associating a quaternionic manifold \(\bar M\) with a hypercomplex manifold \(M\) together with a rotating vector field. The resulting quaternionic manifold \(\bar M\) depends on the choice of some additional data, in particular a closed integral two-form on \(M\). They then use the \(H/Q\)-correspondence to give a generalized version of the supergravity \(c\)-map construction which associates a quaternionic manifold to a conical special complex manifold. This is done by first generalizing the rigid \(c\)-map construction, i.e., constructing a hypercomplex structure with Ricci-flat Obata connection on the tangent bundle of a special complex manifold \(N\). If \(N\) is conical, then the authors exhibit a canonical rotating vector field on \(TN\), so that the \(H/Q\)-correspondence can be applied.\N\NThe authors provide examples of the \(H/Q\)-correspondence where the input is a hypercomplex Hopf manifold which does not admit a compatible hyperkähler metric, thus showing that the \(H/Q\)-correspondence is a strict generalization of the \(HK/QK\)-correspondence.\N\NIn earlier work, the authors have given a construction of a hypercomplex manifold from a quaternionic manifold, called the \(Q/H\)-correspondence, which generalizes the \(QK/HK\)-correspondence [\textit{V. Cortés} and \textit{K. Hasegawa}, Osaka J. Math. 58, No. 1, 213--238 (2021; Zbl 1490.53037)]. The question whether there exists a canonical choice of two-form on the resulting hypercomplex manifold, so that the \(H/Q\)-correspondence gives an inverse to the \(Q/H\)-correspondence, is left for future investigation.