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.
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, A Superstring Inspired Standard Model, Phys. Lett. B 180 (1986) 69 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, A Three Generation Superstring Model. 1. Compactification and Discrete Symmetries, Nucl. Phys. B 278 (1986) 667 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, A Three Generation Superstring Model. 2. Symmetry Breaking and the Low-Energy Theory, Nucl. Phys. B 292 (1987) 606 [INSPIRE].
T. Matsuoka and D. Suematsu, Realistic Models From the E(8) X E(8)-prime Superstring Theory, Prog. Theor. Phys. 76 (1986) 886 [INSPIRE].
B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, 273 Yukawa Couplings for a Three Generation Superstring Model, Phys. Lett. B 192 (1987) 111 [INSPIRE].
S.K. Donaldson, Some numerical results in complex differential geometry, math/0512625.
M. Larfors, A. Lukas, F. Ruehle and R. Schneider, Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds, Mach. Learn. Sci. Tech. 3 (2022) 035014 [arXiv:2205.13408] [INSPIRE].
A. Strominger and E. Witten, New Manifolds for Superstring Compactification, Commun. Math. Phys. 101 (1985) 341 [INSPIRE].
A. Strominger, Yukawa Couplings in Superstring Compactification, Phys. Rev. Lett. 55 (1985) 2547 [INSPIRE].
W.R. Inc., Mathematica, version 13.2, https://www.wolfram.com/mathematica.
R. Kuwabara, Spectrum of the Schrödinger operator on a line bundle over complex projective spaces, Tohoku Math. J. 40 (1988) 199.
C.T. Prieto, Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces, Differ. Geom. Appl. 24 (2006) 288.
P. Candelas, Yukawa Couplings Between (2,1) Forms, Nucl. Phys. B 298 (1988) 458 [INSPIRE].
P. Candelas and X. de la Ossa, Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Vol. 1: introduction, Cambridge Monographs on Mathematical Physics (1988) [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Vol. 2: loop amplitudes, anomalies and phenomenology, Cambridge Monographs on Mathematical Physics (1988) [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton, NJ, U.S.A. (1992) [INSPIRE].
S.K. Donaldson, Anti self-dual yang-mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1 [INSPIRE].
K. Uhlenbeck and S.T. Yau, On the existence of hermitian-yang-mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) S257.
S. Bochner, Curvature and Betti Numbers, Annals Math. 49 (1948) 379.
K. Kodaira, On a Differential-Geometric Method in the Theory of Analytic Stacks, Proceedings of the National Academy of Sciences 39 (1953) 1268.
S. Nakano, On complex analytic vector bundles., J. Math. Soc. Japan 7 (1955) 1.
J.-P. Demailly, Sur l’identite de Bochner-Kodaira-Nakano en geometrie hermitienne, in P. Lelong, P. Dolbeault and H. Skoda eds., Séminaire d’Analyse, Lect. Notes Math. 1198 (1986) 88.
D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on Sn and Pn(C), Osaka J. Math. 15 (1978) 515.
P. Bérard and B. Helffer, Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle, Lett. Math. Phys. 106 (2016) 1729.
J. Milnor, Eigenvalues of the laplace operator on certain manifolds, Proceedings of the National Academy of Sciences 51 (1964) 542.
G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Diff. Geom. 32 (1990) 99.
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2305.08901
About this article
Cite this article
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
- Differential and Algebraic Geometry
- Superstring Vacua
- Superstrings and Heterotic Strings