New non-compact Calabi–Yau metrics in D = 6

Abstract

A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and its complex conjugated. This situation in particular gives a dual description of D6-branes wrapping a complex 1-cycle inside the hyperkahler space [9]. The present work generalize the construction given in that reference. But the explicit solutions we present correspond to the non linear problem. This is a non linear equation with respect to two variables which, with the help of some specific anzatz, is reduced to a non linear equation with a single variable solvable in terms of elliptic functions. In these terms we construct an infinite family of non compact Calabi-Yau metrics. Contents 1. Introduction 1 2. Calabi-Yau metrics with an isometry preserving the SU(3) structure 3 2.1 The general form of the SU(3) structure. .

Constructing Calabi–Yau metrics from hyperkähler spaces

“…Furthermore, we find that the ansatz for the holomorphic (3, 0)-form presented in [40] can be supplemented with a U (1) factor. This allows us to construct a much wider class of solutions including ones that have asymptotic cones over Einstein-Sasaki spaces.…”

“…In this section, we review and then extend significantly the metric construction for the CY3 from D = 4 hyperkähler spaces, proposed in [40]. Let us consider a generic hyperkähler space in four dimensions with the complex coordinates z i ,z i (i = 1, 2) and the Kähler…”

“…Note that comparing with the ansatz in [40], we have introduced a U (1) factor e i κ for the holomorphic (3, 0)-form. This factor turns out to be crucial for constructing metrics with asymptotic cones over Einstein-Sasaki spaces.…”

“…Recently, a new technique was developed in [40] for constructing CY3 metrics, generalizing the construction of the D6-brane wrapping on a two-cycle of a four-dimensional hyperkähler space [41]. The essence of the construction is to build a CY3 metric from a hyperkähler one by adding a complex line bundle.…”

“…Differently, in the new construction, the Kähler potential in four dimensions is allowed to be modified by an arbitrary function G. However, the proof of the existence of such CY3 metrics were presented for the R 4 base only. What is curious is that the metrics with the asymptotic structure of cones over Einstein-Sasaki spaces, such as the R 6 or conifolds, are absent from the construction presented in [40].…”