We give the first numerical calculation of the spectrum of the Laplacian acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show how to compute the approximate eigenvalues and eigenmodes of the Dolbeault Laplacian acting on bundle-valued $(p,q)$-forms on K\"ahler manifolds. We restrict our attention to line bundles over complex projective space and Calabi-Yau hypersurfaces therein. We give three examples. For two of these, $\mathbb{P}^3$ and a Calabi-Yau one-fold (a torus), we compare our numerics with exact results available in the literature and find complete agreement. For the third example, the Fermat quintic three-fold, there are no known analytic results, so our numerical calculations are the first of their kind. The resulting spectra pass a number of non-trivial checks that arise from Serre duality and the Hodge decomposition. The outputs of our algorithm include all the ingredients one needs to compute physical Yukawa couplings in string compactifications.

翻译：我们首次给出了Calabi-Yau三维流形上作用于丛值形式的拉普拉斯算子谱的数值计算结果。具体而言，我们展示了如何计算Kähler流形上作用于丛值$(p,q)$-形式的Dolbeault拉普拉斯算子的近似特征值与特征模。我们将注意力限制在复射影空间及其中的Calabi-Yau超曲面上的线丛。我们给出了三个算例。对于其中两个（$\mathbb{P}^3$和Calabi-Yau一维流形——环面），我们将数值结果与文献中已有的精确结果进行了比较，发现完全一致。对于第三个算例——费马五次三维流形，目前尚无已知的解析结果，因此我们的数值计算属于首创。所得谱通过了由Serre对偶和Hodge分解导出的若干非平凡检验。我们的算法输出包含了计算弦紧化中物理Yukawa耦合所需的所有要素。