Abstract
We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau n-folds in ℙn+1. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the distance from degeneration points in complex structure moduli space.
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P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
E. Calabi, On kähler manifolds with vanishing canonical class, in Algebraic Geometry and Topology, Princeton University Press (1957), p. 78–89 [https://doi.org/10.1515/9781400879915-006].
S.-T. Yau, Calabi’s Conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. 74 (1977) 1798 [INSPIRE].
P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [INSPIRE].
G.W. Moore, Arithmetic and attractors, hep-th/9807087 [INSPIRE].
S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) R5412 [hep-th/9508072] [INSPIRE].
Y. André, G-functions and geometry, Aspects of Mathematics E13, Vieweg+Teubner Verlag (1989) [https://doi.org/10.1007/978-3-663-14108-2_2].
F. Oort, Canonical liftings and dense sets of CM points, Arithmetic Geometry, Cambridge University Press (1997).
J. Pila et al., Canonical Heights on Shimura Varieties and the André-Oort Conjecture, arXiv:2109.08788 [https://doi.org/10.48550/arXiv.2109.08788].
Y.H.J. Lam and A. Tripathy, Attractors are not algebraic, arXiv:2009.12650 [INSPIRE].
W.V.D. Hodge, The topological invariants of algebraic varieties, Proceedings of the International Congress of Mathematicians, Cambridge U.S.A., August 30–September 6 (1950).
P. Candelas, X. de la Ossa, M. Elmi and D. Van Straten, A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two, JHEP 10 (2020) 202 [arXiv:1912.06146] [INSPIRE].
Y.-H. He, Machine-learning the string landscape, Phys. Lett. B 774 (2017) 564 [INSPIRE].
D. Krefl and R.-K. Seong, Machine Learning of Calabi-Yau Volumes, Phys. Rev. D 96 (2017) 066014 [arXiv:1706.03346] [INSPIRE].
F. Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape, JHEP 08 (2017) 038 [arXiv:1706.07024] [INSPIRE].
J. Carifio, J. Halverson, D. Krioukov and B.D. Nelson, Machine Learning in the String Landscape, JHEP 09 (2017) 157 [arXiv:1707.00655] [INSPIRE].
F. Ruehle, Data science applications to string theory, Phys. Rept. 839 (2020) 1 [INSPIRE].
S.K. Donaldson, Some numerical results in complex differential geometry, math/0512625.
M.R. Douglas, R.L. Karp, S. Lukic and R. Reinbacher, Numerical Calabi-Yau metrics, J. Math. Phys. 49 (2008) 032302 [hep-th/0612075] [INSPIRE].
V. Braun, T. Brelidze, M.R. Douglas and B.A. Ovrut, Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds, JHEP 07 (2008) 120 [arXiv:0805.3689] [INSPIRE].
A. Ashmore, Y.-H. He and B.A. Ovrut, Machine Learning Calabi-Yau Metrics, Fortsch. Phys. 68 (2020) 2000068 [arXiv:1910.08605] [INSPIRE].
L.B. Anderson et al., Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning, JHEP 05 (2021) 013 [arXiv:2012.04656] [INSPIRE].
M.R. Douglas, S. Lakshminarasimhan and Y. Qi, Numerical Calabi-Yau metrics from holomorphic networks, arXiv:2012.04797 [INSPIRE].
V. Jejjala, D.K. Mayorga Pena and C. Mishra, Neural network approximations for Calabi-Yau metrics, JHEP 08 (2022) 105 [arXiv:2012.15821] [INSPIRE].
A. Ashmore, L. Calmon, Y.-H. He and B.A. Ovrut, Calabi-Yau Metrics, Energy Functionals and Machine-Learning, International Journal of Data Science in the Mathematical Sciences 1 (2023) 49 [arXiv:2112.10872] [INSPIRE].
M. Larfors, A. Lukas, F. Ruehle and R. Schneider, Learning Size and Shape of Calabi-Yau Spaces, arXiv:2111.01436 [INSPIRE].
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].
M. Gerdes and S. Krippendorf, CYJAX: A package for Calabi-Yau metrics with JAX, Mach. Learn. Sci. Tech. 4 (2023) 025031 arXiv:2211.12520 [INSPIRE].
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B 766 (2007) 21 [hep-th/0605264] [INSPIRE].
A. Ashmore and F. Ruehle, Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold, Phys. Rev. D 103 (2021) 106028 [arXiv:2103.07472] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [INSPIRE].
T. Nagell, Sur les propriétés arithmétiques des cubiques planes du premier genre, Acta Math. 52 (1929) 93.
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.0), https://www.sagemath.org.
D.M. Ghilencea, D. Hoover, C.P. Burgess and F. Quevedo, Casimir energies for 6D supergravities compactified on T2/ZN with Wilson lines, JHEP 09 (2005) 050 [hep-th/0506164] [INSPIRE].
A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].
Calabi-yau threefolds and complex multiplication, in Mirror Symmetry I, American Mathematical Society (1998), pp. 431–444 [https://doi.org/10.1090/amsip/009/20].
S. Gukov and C. Vafa, Rational conformal field theories and complex multiplication, Commun. Math. Phys. 246 (2004) 181 [hep-th/0203213] [INSPIRE].
M. Chen, Complex Multiplication, Rationality and Mirror Symmetry for Abelian Varieties and K3 Surfaces, Ph.D. thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany (2007).
T. Shioda and H. Inose, On Singular K3 Surfaces, in Complex Analysis and Algebraic Geometry, W. Baily & T. Shioda eds., Cambridge University Press (1977), pp. 119–136 [https://doi.org/10.1017/cbo9780511569197.010].
D. Huybrechts, Lectures on K3 Surfaces, Cambridge University Press (2016) [https://doi.org/10.1017/cbo9781316594193] [INSPIRE].
L. Dembélé, A. Panchishkin, J. Voight and W. Zudilin, Special Hypergeometric Motives and Their L-Functions: Asai Recognition, Exper. Math. 31 (2020) 1278 [arXiv:1906.07384] [INSPIRE].
S. Ferrara, G.W. Gibbons and R. Kallosh, Black holes and critical points in moduli space, Nucl. Phys. B 500 (1997) 75 [hep-th/9702103] [INSPIRE].
F. Denef, B.R. Greene and M. Raugas, Split attractor flows and the spectrum of BPS D-branes on the quintic, JHEP 05 (2001) 012 [hep-th/0101135] [INSPIRE].
Acknowledgments
We thank Anthony Ashmore, Sergei Gukov, Sarah Harrison, Andre Lukas and Paul K. Oehlmann for useful discussions. The work of FR is supported by the NSF grants PHY-2210333 and PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions). The work of HA and FR is also supported by startup funding from Northeastern University.
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Ahmed, H., Ruehle, F. Level crossings, attractor points and complex multiplication. J. High Energ. Phys. 2023, 164 (2023). https://doi.org/10.1007/JHEP06(2023)164
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DOI: https://doi.org/10.1007/JHEP06(2023)164