Generic HKT geometries in the harmonic superspace approach

TL;DR

This paper demonstrates how generic HKT geometries can be systematically derived using N=4 supersymmetric quantum mechanics and harmonic superspace, providing a classification framework and explicit examples like the Taub-NUT manifold.

Contribution

It introduces a method to derive all HKT geometries from two harmonic potentials within the harmonic superspace approach, extending previous conjectures.

Findings

Derived the metric for the Taub-NUT manifold as an example.

Showed that generic HKT geometries can be obtained from two harmonic potentials.

Provided a simple proof that the generic case yields HKT geometry.

Abstract

We explain how a generic HKT geometry can be derived using the language of N = 4 supersymmetric quantum mechanics. To this end, one should consider a Lagrangian involving several (4,4,0) multiplets defined in harmonic superspace and subject to nontrivial harmonic constraints. Conjecturally, this general construction worked out earlier by Delduc and Ivanov gives a complete classification of all HKT geometries. Each such geometry is generated by two different functions (potentials) of a special type that depend on harmonic superfields and on harmonics. Given these two potentials, one can derive the vielbeins, metric, connections and curvatures, but this is not so simple: one should solve rather complicated differential equations. We illustrate the general construction by giving a detailed derivation of the metric for the hyper-Kaehler Taub-NUT manifold. In the generic case, we arrive at an HKT geometry. In this paper, we give a simple proof of this assertion.