Abstract
We explore the distribution of topological numbers in Calabi–Yau manifolds, using the Kreuzer–Skarke dataset of hypersurfaces in toric varieties as a testing ground. While the Hodge numbers are well-known to exhibit mirror symmetry, patterns in frequencies of combination thereof exhibit striking new patterns. We find pseudo-Voigt and Planckian distributions with high confidence and exact fit for many substructures. The patterns indicate typicality within the landscape of Calabi–Yau manifolds of various dimension.
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Candelas P., Horowitz G.T., Strominger A., Witten E.: Vacuum configurations for superstrings. Nucl. Phys. B 258, 46 (1985)
Candelas P., Dale A.M., Lutken C.A., Schimmrigk R.: Complete intersection Calabi–Yau manifolds. Nucl. Phys. B 298, 493 (1988)
Candelas P., Lutken C.A., Schimmrigk R.: Complete intersection Calabi–Yau manifolds. 2. Three generation manifolds. Nucl. Phys. B 306, 113 (1988)
Gagnon, M., Ho-Kim, Q.: An exhaustive list of complete intersection Calabi–Yau manifolds. Mod. Phys. Lett. A 9, 2235 (1994)
Hitchin, N.: Generalized Calabi–Yau manifolds. Quart. J. Math. 54, 281. arXiv:math.DG/0209099
Douglas M.R.: The statistics of string/M theory vacua. JHEP 0305, 046 (2003) arXiv:hep-th/0303194
Candelas P., Lynker M., Schimmrigk R.: Calabi–Yau manifolds in weighted P(4). Nucl. Phys. B 341, 383 (1990)
Batyrev, V.: Dual Polyhedra and Mirror Symmetry for Calabi–Yau Hypersurfaces in Toric Varieties. arXiv:alg-geom/9310003
Batyrev, Victor V., Borisov, Lev A.: On Calabi–Yau complete intersections in toric varieties. In: Andreatta, M., Peternell, T. (eds.) Higher Dimensional Complex Varieties, Proceedings of the International Conference, pp. 39–65. Waller de Gruyter, Trento, Italy, Berlin (1996). arXiv:alg-geom/9412017
Kreuzer M., Skarke H.: On the classification of reflexive polyhedra. Commun. Math. Phys. 185, 495 (1997) arXiv:hep-th/9512204
Avram A.C., Kreuzer M., Mandelberg M., Skarke H.: The web of Calabi–Yau hypersurfaces in toric varieties. Nucl. Phys. B 505, 625 (1997) arXiv:hep-th/9703003
Kreuzer M., Skarke H.: Classification of reflexive polyhedra in three-dimensions. Adv. Theor. Math. Phys. 2, 847 (1998) arXiv:hep-th/9805190
Kreuzer M., Skarke H.: Reflexive polyhedra, weights and toric Calabi–Yau fibrations. Rev. Math. Phys. 14, 343 (2002) arXiv:math/0001106 [math-ag]
Kreuzer M., Skarke H.: Complete classification of reflexive polyhedra in four-dimensions. Adv. Theor. Math. Phys. 4, 1209 (2002) arXiv:hep-th/0002240
Kreuzer Maximilian, Skarke Harald: Calabi–Yau 4-folds and toric fibrations. J. Geom. Phys. 26, 272–290 (1998) arXiv:hep-th/9701175v1
Gray J., Haupt A., Lukas A.: Calabi–Yau fourfolds in products of projective space. Proc. Symp. Pure Math. 88, 281 (2014)
Gray, J., Haupt, A., Lukas, A.: All complete intersection Calabi–Yau four-folds. JHEP 1307, 070 (2013) arXiv:1303.1832 [hep-th]
Anderson, L.B., Apruzzi, F., Gao, X., Gray, J., Lee, S.J.: A new construction of Calabi–Yau manifolds: generalized CICYs. Nucl. Phys. B 906, 441–496 (2016) arXiv:1507.03235 [hep-th]
Altman, R., Gray, J., He, Y.H., Jejjala, V., Nelson, B.D.: A Calabi–Yau database: threefolds constructed from the Kreuzer–Skarke list. JHEP 1502, 158 (2015) arXiv:1411.1418 [hep-th]
Davies R.: The expanding zoo of Calabi–Yau threefolds. Adv. High Energy Phys. 2011, 901898 (2011) arXiv:1103.3156 [hep-th]
Candelas P., Davies R.: New Calabi–Yau manifolds with small Hodge numbers. Fortsch. Phys. 58, 383 (2010) arXiv:0809.4681 [hep-th]
He Y.H.: Calabi–Yau geometries: algorithms, databases, and physics. Int. J. Mod. Phys. A 28, 1330032 (2013) arXiv:1308.0186 [hep-th]
Anderson L.B., He Y.H., Lukas A.: Heterotic compactification, an algorithmic approach. JHEP 0707, 049 (2007). doi:10.1088/1126-6708/2007/07/049. arXiv:hep-th/0702210 [hep-th]
Gabella, M., He, Y.H., Lukas, A.: An abundance of heterotic vacua. JHEP 0812, 027 (2008). doi:10.1088/1126-6708/2008/12/027. arXiv:0808.2142 [hep-th]
Gao P., He Y.H., Yau S.T.: Extremal Bundles on CalabiYau Threefolds. Commun. Math. Phys. 336(3), 1167 (2015). doi:10.1007/s00220-014-2271-y. arXiv:1403.1268 [hep-th]
Anderson L.B., Gray J., Lukas A., Palti E.: Heterotic line bundle standard models. JHEP 1206, 113 (2012). doi:10.1007/JHEP06(2012)113. arXiv:1202.1757 [hep-th]
Braun V., He Y.H., Ovrut B.A., Pantev T.: The exact MSSM spectrum from string theory. JHEP 0605, 043 (2006). doi:10.1088/1126-6708/2006/05/043. arXiv:hep-th/0512177
Taylor W.: On the Hodge structure of elliptically fibered Calabi–Yau threefolds. JHEP 1208, 032 (2012) arXiv:1205.0952 [hep-th]
Taylor, W., Wang, Y.N.: A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua. JHEP 01, 137 (2016) arXiv:1510.04978 [hep-th]
Gao X., Shukla P.: On classifying the divisor involutions in Calabi–Yau threefolds. JHEP 11, 170 (2013) arXiv:1307.1139 [hep-th]
Blumenhagen R., Jurke B., Rahn T.: Computational tools for cohomology of toric varieties. Adv. High Energy Phys. 2011, 152749 (2011) arXiv:1104.1187 [hep-th]
Gray J., He Y.-H., Jejjala V., Jurke B., Nelson B.D., Simon J.: Calabi–Yau manifolds with large volume vacua. Phys. Rev. D 86, 101901 (2012) arXiv:1207.5801 [hep-th]
Candelas, P., Constantin, A., Skarke, H.: An abundance of K3 fibrations from polyhedra with interchangeable parts. Commun. Math. Phys. 324(3), 937–959 (2013) arXiv:1207.4792 [hep-th]
Braun V.: On free quotients of complete intersection Calabi–Yau manifolds. JHEP 1104, 005 (2011) arXiv:1003.3235 [hep-th]
Candelas P., de la Ossa X., He Y.H., Szendroi B.: Triadophilia: a special corner in the landscape. Adv. Theor. Math. Phys. 12, 429 (2008) arXiv:0706.3134 [hep-th]
Kreuzer M., Skarke H.: PALP: a package for analyzing lattice polytopes with applications to toric geometry. Comput. Phys. Commun. 157, 87 (2004) arXiv:math/0204356 [math-sc]
Braun, A.P., Knapp, J., Scheidegger, E., Skarke, H., Walliser, N.O.: PALP—a User Manual. arXiv:1205.4147 [math.AG]
The On-Line Encyclopedia of Integer Sequences. http://oeis.org, Number A090045
He Y.H., Lee S.J., Lukas A.: Heterotic models from vector bundles on toric Calabi–Yau manifolds. JHEP 1005, 071 (2010) arXiv:0911.0865 [hep-th]
Lynker M., Schimmrigk R., Wisskirchen A.: Landau–Ginzburg vacua of string, M theory and F theory at c = 12. Nucl. Phys. B 550, 123 (1999) arXiv:hep-th/9812195
Stamatis, D.H.: Six Sigma and Beyond: Statistics and Probability, vol. 3, 1st edn. CRC Press (2002)
Braun V.: Toric elliptic fibrations and F-theory compactifications. JHEP 1301, 016 (2013). doi:10.1007/JHEP01(2013)016. arXiv:1110.4883 [hep-th]
Johnson S.B., Taylor W.: Calabi–Yau threefolds with large h 2,1. JHEP 1410, 23 (2014). doi:10.1007/JHEP10(2014)023. arXiv:1406.0514 [hep-th]
Taylor, W., Wang, Y.N.: Non-toric Bases for Elliptic Calabi–Yau Threefolds and 6D F-Theory Vacua. arXiv:1504.07689 [hep-th]
Anderson L.B., Gao X., Gray J., Lee S.J.: Multiple fibrations in Calabi–Yau geometry and string dualities. JHEP 1610, 105 (2016). doi:10.1007/JHEP10(2016)105. arXiv:1608.07555 [hep-th]
Candelas, P., Constantin, A., Mishra, C.: Calabi–Yau Threefolds With Small Hodge Numbers. arXiv:1602.06303 [hep-th]
Bianchi M., Ferrara S.: Enriques and octonionic magic supergravity models. JHEP 0802, 054 (2008). doi:10.1088/1126-6708/2008/02/054. arXiv:0712.2976 [hep-th]
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He, YH., Jejjala, V. & Pontiggia, L. Patterns in Calabi–Yau Distributions. Commun. Math. Phys. 354, 477–524 (2017). https://doi.org/10.1007/s00220-017-2907-9
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DOI: https://doi.org/10.1007/s00220-017-2907-9