Numerical Weil–Petersson metrics on moduli spaces of Calabi–Yau manifolds

Abstract

We introduce a simple and very fast algorithm to compute Weil–Petersson metrics on moduli spaces of Calabi–Yau varieties. Additionally, we introduce a second algorithm to approximate the same metric using Donaldson’s quantization link between infinite and finite dimensional Geometric Invariant Theoretical (GIT) quotients that describe moduli spaces of varieties. Although this second algorithm is slower and more sophisticated, it can also be used to compute similar metrics on other moduli spaces (e.g. moduli spaces of vector bundles on Calabi–Yau varieties). We study the convergence properties of both algorithms and provide explicit computer implementations using a family of Calabi–Yau quintic hypersurfaces in P 4 . Also, we include discussions on: the existing methods that are used to compute this class of metrics, the background material that we use to build our algorithms, and how to extend the second algorithm to the vector bundle case.