Quaternionic K\"ahler metrics associated with special K\"ahler manifolds

TL;DR

This paper provides explicit formulas for quaternionic K"ahler metrics derived from special K"ahler manifolds using the HK/QK correspondence, including proofs of known metrics and applications to conical K"ahler manifolds.

Contribution

It introduces explicit formulas for quaternionic K"ahler metrics from the HK/QK correspondence and applies them to various cases, including the Ferrara-Sabharwal metric and conical K"ahler manifolds.

Findings

Explicit formula for quaternionic K"ahler metrics from HK/QK correspondence

New proof that Ferrara-Sabharwal metric is quaternionic K"ahler

Application to conical K"ahler manifolds

Abstract

We give an explicit formula for the quaternionic K\"ahler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic K\"ahler. A similar explicit formula is given for the analogous (K/K) correspondence between K\"ahler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical K\"ahler manifold.