Numerical metrics, curvature expansions and Calabi-Yau manifolds

Wei Cui,James Gray

doi:10.1007/jhep05(2020)044

Abstract

We discuss the extent to which numerical techniques for computing approximations to Ricci-flat metrics can be used to investigate hierarchies of curvature scales on Calabi-Yau manifolds. Control of such hierarchies is integral to the validity of curvature expansions in string effective theories. Nevertheless, for seemingly generic points in moduli space it can be difficult to analytically determine if there might be a highly curved region localized somewhere on the Calabi-Yau manifold. We show that numerical techniques are rather efficient at deciding this issue.

Highlights

One might suspect that if the coefficients appearing in a defining relation are very large or very small that one might have an issue
In this paper we have studied to what extent numerical methods can be utilized to detect hierarchies of curvature scales appearing in Ricci-flat metrics on Calabi-Yau manifolds at different locations in moduli space
By illustrating that we can reproduce the expected behavior of such quantities as the system approaches singular points in complex structure moduli space, we have demonstrated that such techniques are rather effective in deciding this issue

Summary

Numerical computation

We briefly review some of the existing methods for obtaining numerical approximations to Ricci-flat metrics on Calabi-Yau manifolds [9,10,11,12,13,14,15,16,17,18]. Given an ansatz for the Kahler potential on a Calabi-Yau manifold of the form discussed in the last paragraph, the step is to introduce a procedure for adjusting the parameters that appear in the ansatz to some optimal value where the associated metric is as close as possible to Ricci-flat. Instead we obtain an approximation to the Ricci-flat Kahler potential at each finite k, called the optimal metric [15], and increase the value of k until a desired precision is reached At this stage it is useful to mention that, as noted in [15], one might expect an ansatz of the form (2.1) to be most useful for Calabi-Yau manifolds that are smooth and which do not exhibit a hierarchy of scales of curvature. One of the burdens of this paper will be to show that (2.1) is practically useful in mapping out where in moduli space such large curvature regions are occuring and that the above concerns do not prevent us from achieving useful results

Examples

Finding high curvature regions

Examples 1: one parameter families

Examples 2: a two parameter family

Findings

Conclusions and future directions