Complete Calabi-Yau metrics from Kahler metrics in D=4

TL;DR

This paper derives a local form of certain Calabi-Yau metrics with a Hamiltonian Killing vector using a single nonlinear equation, improving previous methods by avoiding reliance on hyperkahler structures and simplifying the construction process.

Contribution

It introduces a new method to describe Calabi-Yau metrics with a Killing vector without needing an initial hyperkahler structure, simplifying the metric construction.

Findings

Explicit example with SU(3) holonomy found.

Method avoids complications of nonlinear operators on hyperkahler spaces.

Connection with previous solution-generating techniques clarified.

Abstract

In the present work the local form of certain Calabi-Yau metrics possessing a local Hamiltonian Killing vector is described in terms of a single non linear equation. The main assumptions are that the complex $(3,0)$-form is of the form $e^{ik}\widetilde{\Psi}$, where $\widetilde{\Psi}$ is preserved by the Killing vector, and that the space of the orbits of the Killing vector is, for fixed value of the momentum map coordinate, a complex 4-manifold, in such a way that the complex structure of the 4-manifold is part of the complex structure of the complex 3-fold. The link with the solution generating techniques of [26]-[28] is made explicit and in particular an example with holonomy exactly SU(3) is found by use of the linearization of [26], which was found in the context of D6 branes wrapping a holomorphic 1-fold in a hyperkahler manifold. But the main improvement of the present method, unlike the ones presented in [26]-[28], does not rely in an initial hyperkahler structure. Additionally the complications when dealing with non linear operators over the curved hyperkahler space are avoided by use of this method.