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

TL;DR

This work develops a robust numerical pipeline to compute the scalar Laplace spectrum on Calabi–Yau manifolds by constructing Ricci-flat metrics via Donaldson’s balanced metrics, building finite function bases, and solving generalized eigenvalue problems for the Laplacian. The method is validated on CP^3 with exact analytic results and extended to Calabi–Yau quintics, their Z5×Z5 quotients, and a heterotic standard-model manifold, with eigenvalue degeneracies explained by finite groups and covering-space representation theory. Weyl’s law is checked to ensure correct volume normalization and asymptotic growth, while detailed moduli-dependence and symmetry-breaking patterns illustrate how geometry and complex structure influence the KK spectrum. The results illuminate the connection between Calabi–Yau geometry, discrete symmetries, and observable physics such as the gravitational potential in four dimensions, providing concrete tools for extracting phenomenological data from string compactifications.

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.

Figures (16)

• Figure 1: Spectrum of the scalar Laplacian on $\mathop{ {\mathbb{P}}}\nolimits^3$ with the rescaled Fubini-Study metric. Here we fix the space of functions by choosing degree ${k_\phi}=3$, and evaluate the Laplace operator at a varying number of points ${n_\phi}$.
• Figure 2: Spectrum of the scalar Laplacian on $\mathop{ {\mathbb{P}}}\nolimits^3$ with the rescaled Fubini-Study metric. Here we evaluate the spectrum of the Laplace operator as a function of ${k_\phi}$, while keeping the number of points fixed at ${n_\phi}= 00000.$. Note that ${k_\phi}$ determines the dimension of the matrix approximation to the Laplace operator.
• Figure 3: Check of Weyl's formula for the spectrum of the scalar Laplacian on $\mathop{ {\mathbb{P}}}\nolimits^3$ with the rescaled Fubini-Study metric. We fix the space of functions by taking ${k_\phi}=3$ and evaluate $\frac{\lambda_n^3}{n}$ as a function of $n$ at a varying number of points ${n_\phi}$. Note that the data used for the eigenvalues is the same as for ${k_\phi}=3$ in \ref{['fig:SpecCP3Np']}.
• Figure 4: Eigenvalues of the scalar Laplace operator on the same "random quintic" defined in eq. \ref{['eq:RandomQt']}. The metric is computed at degree ${k_h}=8$, using ${n_h}= 166000.$ points. The Laplace operator is evaluated at degree ${k_\phi}=3$ on a varying number ${n_\phi}$ of points.
• Figure 5: Eigenvalues of the scalar Laplace operator on a random quintic plotted against ${k_\phi}$. The metric is computed at degree ${k_h}=8$, using ${n_h}= 166000.$ points. The Laplace operator is then evaluated at ${n_\phi}= 00000.$ points.