Eigenvalues and eigenforms on Calabi-Yau threefolds

TL;DR

This paper introduces a numerical algorithm to compute the spectrum of the Laplace-de Rham operator on Calabi-Yau threefolds, specifically on the Fermat quintic, extending previous scalar Laplace methods.

Contribution

The authors develop a new numerical method for calculating eigenvalues and eigenforms of the Laplace operator on Calabi-Yau manifolds, including $(p,q)$-forms, using approximate metrics.

Findings

Successfully computed the spectrum on the Fermat quintic threefold.

Validated the algorithm by computing the spectrum on projective space.

Extended previous scalar Laplace computations to $(p,q)$-forms.

Abstract

We present a numerical algorithm for computing the spectrum of the Laplace-de Rham operator on Calabi-Yau manifolds, extending previous work on the scalar Laplace operator. Using an approximate Calabi-Yau metric as input, we compute the eigenvalues and eigenforms of the Laplace operator acting on $(p,q)$-forms for the example of the Fermat quintic threefold. We provide a check of our algorithm by computing the spectrum of $(p,q)$-eigenforms on $\mathbb{P}^{3}$.