Approximate Ricci-flat Metrics for Calabi-Yau Manifolds

Yau's theorem guarantees the existence of a unique Ricci-flat K ahler metric with a given K ahler class on a Calabi-Yau (CY) manifold. Such Ricci-flat metrics are of mathematical interest and they play an important role in compactifications of string theory. Unfortunately, for compact Calabi-Yau manifolds of complex dimension three or higher, analytic expressions for Ricci-flat metrics are not known. Over the past few years substantial progress has nevertheless been made in numerically computing Ricci-flat metrics on Calabi-Yau three-folds, starting with Donaldson's algorithm [1] and its applications [2-9] and, more recently, using machine learning methods [10-16]. The Ricci-flat metric in numerical form is already useful, enabling us to compute the spectrum of the Laplacian on a CY manifold [5,17] or the masses of quarks in a CY string compactification [18], to name just two applications. However, it would be all the more helpful and exciting to deal with the metric analytically.