Constructing Calabi-Yau Metrics From Hyperkahler Spaces

TL;DR

This paper extends a method for constructing Calabi-Yau 3-fold metrics from hyperkahler spaces by incorporating a U(1) factor, leading to new explicit metrics and resolutions of Einstein-Sasaki cones.

Contribution

It introduces a generalized formalism with a U(1) factor, providing explicit differential equations and new classes of Calabi-Yau metrics with specific symmetries.

Findings

Derived explicit metrics for generalized resolutions of Einstein-Sasaki cones.

Identified conditions under which the U(1) factor depends on fibre coordinates or vanishes.

Produced new singular Calabi-Yau 3-fold metrics with specific isometries.

Abstract

Recently, a metric construction for the Calabi-Yau 3-folds from a four-dimensional hyperkahler space by adding a complex line bundle was proposed. We extend the construction by adding a U(1) factor to the holomorphic (3,0)-form, and obtain the explicit formalism for a generic hyperkahler base. We find that a discrete choice arises: the U(1) factor can either depend solely on the fibre coordinates or vanish. In each case, the metric is determined by one differential equation for the modified Kahler potential. As explicit examples, we obtain the generalized resolutions (up to orbifold singularity) of the cone of the Einstein-Sasaki spaces Y^{p,q}. We also obtain a large class of new singular CY3 metrics with SU(2)\times U(1) or SU(2)\times U(1)^2 isometries.