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

TL;DR

This paper presents a fast algorithm to compute Weil-Petersson metrics on moduli spaces of Calabi-Yau manifolds, using Donaldson's quantization, and demonstrates convergence and practical examples.

Contribution

It introduces a new efficient algorithm for approximating Weil-Petersson metrics and proves its convergence via Donaldson's quantization framework.

Findings

Algorithm converges to Weil-Petersson metric

Explicit computations on Calabi-Yau Quintics

Broader applicability to Dirac operator metrics

Abstract

We introduce a simple and very fast algorithm that computes Weil-Petersson metrics on moduli spaces of polarized Calabi-Yau manifolds. Also, by using Donaldson's quantization link between the infinite and finite dimensional G.I.T quotients that describe moduli spaces of varieties, we define a natural sequence of Kaehler metrics. We prove that the sequence converges to the Weil-Petersson metric. We also develop an algorithm that numerically approximates such metrics, and hence the Weil-Petersson metric itself. Explicit examples are provided on a family of Calabi-Yau Quintic hypersurfaces in CP^4. The scope of our second algorithm is much broader; the same techniques can be used to approximate metrics on null spaces of Dirac operators coupled to Hermite Yang-Mills connections.