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.

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

References in corpus (21)

• A Mini-Landscape of Exact MSSM Spectra in Heterotic Orbifolds
• Heterotic GUT and Standard Model Vacua from simply connected Calabi-Yau Manifolds
• Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua
• Numerical Calabi-Yau metrics
• Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua
• Heterotic MSSM on a Resolved Orbifold
• Machine learning Calabi-Yau metrics
• Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic
• Moduli Space of Torsional Manifolds
• Stability of the Minimal Heterotic Standard Model Bundle
• CYJAX: A package for Calabi-Yau metrics with JAX
• Heterotic backgrounds via generalised geometry: moment maps and moduli
• Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold
• A Statistical Analysis of the Minimal SUSY B-L Theory
• Numerical Metrics, Curvature Expansions and Calabi-Yau Manifolds
• Heterotic Instanton Superpotentials from Complete Intersection Calabi-Yau Manifolds
• Non-vanishing Superpotentials in Heterotic String Theory and Discrete Torsion
• Heterotic Instantons for Monad and Extension Bundles
• Line Bundle Hidden Sectors for Strongly Coupled Heterotic Standard Models
• Level Crossings, Attractor Points and Complex Multiplication
• Hidden Sectors from Multiple Line Bundles for the $B-L$ MSSM

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