Eigenvalues and eigenforms on Calabi-Yau threefolds

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 $\mathbb{P}^{3}$.

Figures (17)

• Figure 1: A plot of the numerical eigenvalues of the Laplacian on $\mathbb{P}^{3}$ for $(0,0)$-forms with $k_{\phi}=3$ and varying $N_{\phi}$. The dashed horizontal lines show the values of the exact eigenvalues $\hat{\lambda}_{m}$ from Table \ref{['tab:P3_exact']}. We also indicate the multiplicity of each eigenvalue, calculated by counting the number of eigenvalues in each grouping, which are in agreement with the exact results.
• Figure 2: A plot of the numerical eigenvalues of the Laplacian on $\mathbb{P}^{3}$ for $(1,0)$-forms with $k_{\phi}=3$ and varying $N_{\phi}$. The dashed horizontal lines show the values of the exact eigenvalues $\hat{\lambda}_{m}$ from Table \ref{['tab:P3_exact']}. We also indicate the multiplicity of each eigenvalue, calculated by counting the number of eigenvalues in each grouping, which are in agreement with the exact results.
• Figure 3: A plot of the numerical eigenvalues of the Laplacian on $\mathbb{P}^{3}$ for $(0,0)$-forms with $N_{\phi}=250{,}000$ and varying $k_{\phi}$. The dashed horizontal lines show the values of the exact eigenvalues $\hat{\lambda}_{m}$ from Table \ref{['tab:P3_exact']}. We also indicate the multiplicity of each eigenvalue, calculated by counting the number of eigenvalues in each grouping. Notice that the multiplicities do not change as $k_{\phi}$ is increased.
• Figure 4: A plot of the numerical eigenvalues of the Laplacian on $\mathbb{P}^{3}$ for $(1,1)$-forms with $N_{\phi}=250{,}000$ and varying $k_{\phi}$. The dashed horizontal lines show the values of the exact eigenvalues $\hat{\lambda}_{m}$ from Table \ref{['tab:P3_exact']}. We also indicate the multiplicity of each eigenvalue, calculated by counting the number of eigenvalues in each grouping. Notice that the multiplicities can change as $k_{\phi}$ is increased for the reason mentioned in the main text.
• Figure 5: A plot of the numerical eigenvalues of the Laplacian on $\mathbb{P}^{3}$ for $(p,q)$-forms with $N_{\phi}={10}^{6}$ and $k_{\phi}=3$ ($(3,0)$ at $k_{\phi}=5$). The dashed horizontal lines show the values of the exact eigenvalues $\hat{\lambda}_{m}$ from Table \ref{['tab:P3_exact']}. We also indicate the multiplicity of each eigenvalue, calculated by counting the number of eigenvalues in each grouping. An asterisk on a multiplicity indicates that this does not match the exact value in Table \ref{['tab:P3_exact']} for the reason discussed in the main text.