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Estimating Calabi-Yau hypersurface and triangulation counts with equation learners. (English) Zbl 1414.83080
Summary: We provide the first estimate of the number of fine, regular, star triangulations of the four-dimensional reflexive polytopes, as classified by Kreuzer and Skarke (KS). This provides an upper bound on the number of Calabi-Yau threefold hypersurfaces in toric varieties. The estimate is performed with deep learning, specifically the novel equation learner (EQL) architecture. We demonstrate that EQL networks accurately predict numbers of triangulations far beyond the \(h^{1,1}\) training region, allowing for reliable extrapolation. We estimate that number of triangulations in the KS dataset is \(10^{10,505}\), dominated by the polytope with the highest \(h^{1,1}\) value.

83E30 String and superstring theories in gravitational theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[1] Candelas, P.; Horowitz, GT; Strominger, A.; Witten, E., Vacuum Configurations for Superstrings, Nucl. Phys., B 258, 46, (1985)
[2] Aspinwall, PS; Greene, BR; Morrison, DR, Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory, Nucl. Phys., B 416, 414, (1994) · Zbl 0899.32006
[3] Strominger, A., Massless black holes and conifolds in string theory, Nucl. Phys., B 451, 96, (1995) · Zbl 0925.83071
[4] Dasgupta, K.; Rajesh, G.; Sethi, S., M theory, orientifolds and G-flux, JHEP, 08, 023, (1999) · Zbl 1060.81575
[5] Giddings, SB; Kachru, S.; Polchinski, J., Hierarchies from fluxes in string compactifications, Phys. Rev., D 66, 106006, (2002)
[6] Bousso, R.; Polchinski, J., Quantization of four form fluxes and dynamical neutralization of the cosmological constant, JHEP, 06, 006, (2000) · Zbl 0990.83543
[7] Douglas, MR, The Statistics of string/M theory vacua, JHEP, 05, 046, (2003)
[8] Carifio, J.; Cunningham, WJ; Halverson, J.; Krioukov, D.; Long, C.; Nelson, BD, Vacuum Selection from Cosmology on Networks of String Geometries, Phys. Rev. Lett., 121, 101602, (2018)
[9] Halverson, J.; Long, C.; Sung, B., Algorithmic universality in F-theory compactifications, Phys. Rev., D 96, 126006, (2017)
[10] Garriga, J.; Schwartz-Perlov, D.; Vilenkin, A.; Winitzki, S., Probabilities in the inflationary multiverse, JCAP, 01, 017, (2006) · Zbl 1236.83021
[11] Ashok, S.; Douglas, MR, Counting flux vacua, JHEP, 01, 060, (2004) · Zbl 1243.83060
[12] Denef, F.; Douglas, MR, Distributions of flux vacua, JHEP, 05, 072, (2004)
[13] Taylor, W.; Wang, Y-N, The F-theory geometry with most flux vacua, JHEP, 12, 164, (2015) · Zbl 1388.81367
[14] Taylor, W.; Wang, Y-N, A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua, JHEP, 01, 137, (2016) · Zbl 1388.81013
[15] Taylor, W.; Wang, Y-N, Scanning the skeleton of the 4D F-theory landscape, JHEP, 01, 111, (2018) · Zbl 1384.83066
[16] F. Denef and M.R. Douglas, Computational complexity of the landscape. I., Annals Phys.322 (2007) 1096 [hep-th/0602072] [INSPIRE].
[17] Cvetič, M.; Garcia-Etxebarria, I.; Halverson, J., On the computation of non-perturbative effective potentials in the string theory landscape: IIB/F-theory perspective, Fortsch. Phys., 59, 243, (2011) · Zbl 1209.81162
[18] Bao, N.; Bousso, R.; Jordan, S.; Lackey, B., Fast optimization algorithms and the cosmological constant, Phys. Rev., D 96, 103512, (2017)
[19] Denef, F.; Douglas, MR; Greene, B.; Zukowski, C., Computational complexity of the landscape II — Cosmological considerations, Annals Phys., 392, 93, (2018) · Zbl 1390.83337
[20] Halverson, J.; Ruehle, F., Computational Complexity of Vacua and Near-Vacua in Field and String Theory, Phys. Rev., D 99, (2019)
[21] Y.-H. He, Deep-Learning the Landscape, arXiv:1706.02714 [INSPIRE].
[22] Ruehle, F., Evolving neural networks with genetic algorithms to study the String Landscape, JHEP, 08, 038, (2017) · Zbl 1381.83128
[23] Krefl, D.; Seong, R-K, Machine Learning of Calabi-Yau Volumes, Phys. Rev., D 96, (2017)
[24] Klaewer, D.; Schlechter, L., Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties, Phys. Lett., B 789, 438, (2019) · Zbl 1406.14001
[25] Bull, K.; He, Y-H; Jejjala, V.; Mishra, C., Machine Learning CICY Threefolds, Phys. Lett., B 785, 65, (2018)
[26] Wang, Y-N; Zhang, Z., Learning non-Higgsable gauge groups in 4D F-theory, JHEP, 08, 009, (2018)
[27] Hashimoto, K.; Sugishita, S.; Tanaka, A.; Tomiya, A., Deep learning and the AdS/CFT correspondence, Phys. Rev., D 98, (2018)
[28] Liu, J., Artificial Neural Network in Cosmic Landscape, JHEP, 12, 149, (2017) · Zbl 1383.85016
[29] R. Jinno, Machine learning for bounce calculation, arXiv:1805.12153 [INSPIRE]. · Zbl 1372.94122
[30] Kreuzer, M.; Skarke, H., Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys., 4, 1209, (2002) · Zbl 1017.52007
[31] G. Martius and C.H. Lampert, Extrapolation and learning equations, arXiv:1610.02995.
[32] Carifio, J.; Halverson, J.; Krioukov, D.; Nelson, BD, Machine Learning in the String Landscape, JHEP, 09, 157, (2017) · Zbl 1382.81155
[33] Batyrev, VV, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom., 3, 493, (1994) · Zbl 0829.14023
[34] M. Kreuzer and H. Skarke, PALP: A Package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun.157 (2004) 87 [math/0204356] [INSPIRE]. · Zbl 1196.14007
[35] J. Rambau, TOPCOM: Triangulations of point configurations and oriented matroids, in Mathematical SoftwareICMS 2002, A.M. Cohen, X.-S. Gao and N. Takayama eds., pp. 330-340, World Scientific (2002) [http://www.zib.de/PaperWeb/abstracts/ZR-02-17]. · Zbl 1057.68150
[36] Altman, R.; Gray, J.; He, Y-H; Jejjala, V.; Nelson, BD, A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List, JHEP, 02, 158, (2015) · Zbl 1388.53071
[37] Long, C.; McAllister, L.; McGuirk, P., Heavy Tails in Calabi-Yau Moduli Spaces, JHEP, 10, 187, (2014) · Zbl 1333.81223
[38] M. Demirtas, C. Long, L. McAllister and M. Stillman, The Kreuzer-Skarke Axiverse, arXiv:1808.01282 [INSPIRE].
[39] Halverson, J.; Tian, J., Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications, Phys. Rev., D 95, (2017)
[40] R. Grinis and A. Kasprzyk, Normal forms of convex lattice polytopes, arXiv:1301.6641.
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