Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

Abstract

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 Kähler 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.

arXiv comment52 pages, 6 figures; v2 - corrected typo in quintic equation

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