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

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

2009-07-09

arxiv.org/abs/0907.1387v1

Mathematics

Differential Geometry

36 pages, 8 figures

Scientific paper

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.

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