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

  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 21 July 2023
  • volume 2023, Article number: 164 (2023)
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Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces
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  • A. Ashmore  ORCID: orcid.org/0000-0001-6178-75381,
  • Y-H. He2,3,4,5,
  • E. Heyes5 &
  • …
  • B. A. Ovrut6 

  • 59 Accesses

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A preprint version of the article is available at arXiv.

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, ℙ3 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.

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Acknowledgments

It is a pleasure to thank Clay Córdova and Edward Mazenc for useful discussions. AA is supported in part by NSF Grant No. PHY2014195 and in part by the Kadanoff Center for Theoretical Physics. AA also acknowledges the support of the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 838776. YHH would like to thank STFC for grant ST/J00037X/2. EH would like to thank SMCSE at City, University of London for the PhD studentship, as well as the Jersey Government for a postgraduate grant. BAO is supported in part by both the research grant DOE No.DESC0007901 and SAS Account 020-0188-2-010202-6603-0338. This work was completed in part with resources provided by the University of Chicago Research Computing Center.

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Authors and Affiliations

  1. Enrico Fermi Institute & Kadanoff Center for Theoretical Physics, University of Chicago, 933 E 56th St., Chicago, IL, 60637, U.S.A.

    A. Ashmore

  2. London Institute for Mathematical Sciences, Royal Institution, 21 Albemarle St., London, W1S 4BS, U.K.

    Y-H. He

  3. Merton College, University of Oxford, Merton St., Oxford, OX1 4JD, U.K.

    Y-H. He

  4. School of Physics, NanKai University, No. 94 Weijin Road, Tianjin, 300071, P.R. China

    Y-H. He

  5. Department of Mathematics, City, University of London, Northampton Square, London, EC1V 0HB, U.K.

    Y-H. He & E. Heyes

  6. Department of Physics, University of Pennsylvania, 209 S 33rd St, Philadelphia, PA, 19104, U.S.A.

    B. A. Ovrut

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Ashmore, A., He, YH., Heyes, E. et al. Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces. J. High Energ. Phys. 2023, 164 (2023). https://doi.org/10.1007/JHEP07(2023)164

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  • Received: 06 June 2023

  • Accepted: 16 July 2023

  • Published: 21 July 2023

  • DOI: https://doi.org/10.1007/JHEP07(2023)164

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Keywords

  • Differential and Algebraic Geometry
  • Superstring Vacua
  • Superstrings and Heterotic Strings
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