Eigenvalues and eigenfunctions of the scalar Laplace operator on Calabi-Yau manifolds

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 a 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.