Constructing Calabi–Yau metrics from hyperkähler spaces

Abstract

Recently, a metric construction for Calabi–Yau threefolds from a four-dimensional hyperkähler 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 hyperkähler base. We find that a discrete choice arises: the U(1) factor can either depend solely on the fiber coordinates or vanish. In each case, the metric is determined by a differential equation for the modified Kähler potential. As explicit examples, we obtain the generalized resolutions (up to orbifold singularity) of the cone of the Einstein–Sasaki spaces Yp, q. We also obtain a large class of new singular CY3 metrics with SU(2) × U(1) or SU(2) × U(1)2 isometries.