Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds

TL;DR

This paper introduces a numerical method to compute the spectrum of the Laplace-Beltrami operator on Calabi-Yau threefolds, providing explicit eigenvalues and eigenfunctions for various examples, which aids in understanding their geometric properties.

Contribution

The paper presents a new numerical algorithm for calculating the Laplacian spectrum on Calabi-Yau manifolds, including Ricci-flat metrics and eigenfunctions, with detailed analysis of symmetry-related eigenvalue multiplicities.

Findings

Eigenvalues and eigenfunctions computed for multiple Calabi-Yau examples

Eigenvalue multiplicities explained via symmetry group representations

Method demonstrates practical computation of Laplacian spectra on complex manifolds

Abstract

A numerical algorithm for explicitly computing the spectrum of the Laplace-Beltrami operator on Calabi-Yau threefolds is presented. The requisite Ricci-flat metrics are calculated using a method introduced in previous papers. To illustrate our algorithm, the eigenvalues and eigenfunctions of the Laplacian are computed numerically on two different quintic hypersurfaces, some Z_5 x Z_5 quotients of quintics, and the Calabi-Yau threefold with Z_3 x Z_3 fundamental group of the heterotic standard model. The multiplicities of the eigenvalues are explained in detail in terms of the irreducible representations of the finite isometry groups of the threefolds.