×

zbMATH — the first resource for mathematics

Patterns in Calabi-Yau distributions. (English) Zbl 1376.81059
Summary: 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.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Candelas, P.; Horowitz, G.T.; Strominger, A.; Witten, E., Vacuum configurations for superstrings, Nucl. Phys. B, 258, 46, (1985)
[2] Candelas, P.; Dale, A.M.; Lutken, C.A.; Schimmrigk, R., Complete intersection Calabi-Yau manifolds, Nucl. Phys. B, 298, 493, (1988)
[3] Candelas, P.; Lutken, C.A.; Schimmrigk, R., Complete intersection Calabi-Yau manifolds. 2. three generation manifolds, Nucl. Phys. B, 306, 113, (1988)
[4] Gagnon, M., Ho-Kim, Q.: An exhaustive list of complete intersection Calabi-Yau manifolds. Mod. Phys. Lett. A 9, 2235 (1994) · Zbl 1020.14501
[5] Hitchin, N.: Generalized Calabi-Yau manifolds. Quart. J. Math. 54, 281. arXiv:math.DG/0209099 · Zbl 1076.32019
[6] Douglas, M.R., The statistics of string/M theory vacua, JHEP, 0305, 046, (2003)
[7] Candelas, P.; Lynker, M.; Schimmrigk, R., Calabi-Yau manifolds in weighted P(4), Nucl. Phys. B, 341, 383, (1990) · Zbl 0962.14029
[8] Batyrev, V.: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties. arXiv:alg-geom/9310003 · Zbl 1342.81425
[9] 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 · Zbl 0908.14015
[10] Kreuzer, M.; Skarke, H., On the classification of reflexive polyhedra, Commun. Math. Phys., 185, 495, (1997) · Zbl 0894.14026
[11] Avram, A.C.; Kreuzer, M.; Mandelberg, M.; Skarke, H., The web of Calabi-Yau hypersurfaces in toric varieties, Nucl. Phys. B, 505, 625, (1997) · Zbl 0896.14026
[12] Kreuzer, M.; Skarke, H., Classification of reflexive polyhedra in three-dimensions, Adv. Theor. Math. Phys., 2, 847, (1998) · Zbl 0934.52006
[13] Kreuzer, M.; Skarke, H., Reflexive polyhedra, weights and toric Calabi-Yau fibrations, Rev. Math. Phys., 14, 343, (2002) · Zbl 1079.14534
[14] Kreuzer, M.; Skarke, H., Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys., 4, 1209, (2002) · Zbl 1017.52007
[15] Kreuzer, Maximilian; Skarke, Harald, Calabi-Yau 4-folds and toric fibrations, J. Geom. Phys., 26, 272-290, (1998) · Zbl 0956.14030
[16] Gray, J.; Haupt, A.; Lukas, A., Calabi-Yau fourfolds in products of projective space, Proc. Symp. Pure Math., 88, 281, (2014) · Zbl 1325.14058
[17] Gray, J., Haupt, A., Lukas, A.: All complete intersection Calabi-Yau four-folds. JHEP 1307, 070 (2013) arXiv:1303.1832 [hep-th] · Zbl 1342.14086
[18] 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] · Zbl 1334.14023
[19] 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] · Zbl 1388.53071
[20] Davies, R., The expanding zoo of Calabi-Yau threefolds, Adv. High Energy Phys., 2011, 901898, (2011) · Zbl 1234.81110
[21] Candelas, P.; Davies, R., New Calabi-Yau manifolds with small Hodge numbers, Fortsch. Phys., 58, 383, (2010) · Zbl 1194.14062
[22] He, Y.H., Calabi-Yau geometries: algorithms, databases, and physics, Int. J. Mod. Phys. A, 28, 1330032, (2013) · Zbl 1277.81059
[23] Anderson, L.B.; He, Y.H.; Lukas, A., Heterotic compactification, an algorithmic approach, JHEP, 0707, 049, (2007)
[24] 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] · Zbl 1329.81313
[25] Gao, P.; He, Y.H.; Yau, S.T., Extremal bundles on calabiyau threefolds, Commun. Math. Phys., 336, 1167, (2015) · Zbl 1328.14068
[26] Anderson, L.B.; Gray, J.; Lukas, A.; Palti, E., Heterotic line bundle standard models, JHEP, 1206, 113, (2012) · Zbl 1397.81406
[27] Braun, V.; He, Y.H.; Ovrut, B.A.; Pantev, T., The exact MSSM spectrum from string theory, JHEP, 0605, 043, (2006)
[28] Taylor, W., On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP, 1208, 032, (2012) · Zbl 1397.14048
[29] 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]
[30] Gao, X.; Shukla, P., On classifying the divisor involutions in Calabi-Yau threefolds, JHEP, 11, 170, (2013) · Zbl 1342.81425
[31] Blumenhagen, R.; Jurke, B.; Rahn, T., Computational tools for cohomology of toric varieties, Adv. High Energy Phys., 2011, 152749, (2011) · Zbl 1234.81107
[32] 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)
[33] 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] · Zbl 1284.14051
[34] Braun, V., On free quotients of complete intersection Calabi-Yau manifolds, JHEP, 1104, 005, (2011) · Zbl 1250.14026
[35] Candelas, P.; Ossa, X.; He, Y.H.; Szendroi, B., Triadophilia: a special corner in the landscape, Adv. Theor. Math. Phys., 12, 429, (2008) · Zbl 1144.81499
[36] Kreuzer, M.; Skarke, H., PALP: a package for analyzing lattice polytopes with applications to toric geometry, Comput. Phys. Commun., 157, 87, (2004) · Zbl 1196.14007
[37] Braun, A.P., Knapp, J., Scheidegger, E., Skarke, H., Walliser, N.O.: PALP—a User Manual. arXiv:1205.4147 [math.AG]
[38] The On-Line Encyclopedia of Integer Sequences. http://oeis.org, Number A090045 · Zbl 1044.11108
[39] He, Y.H.; Lee, S.J.; Lukas, A., Heterotic models from vector bundles on toric Calabi-Yau manifolds, JHEP, 1005, 071, (2010) · Zbl 1287.81094
[40] 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) · Zbl 1063.14504
[41] Stamatis, D.H.: Six Sigma and Beyond: Statistics and Probability, vol. 3, 1st edn. CRC Press (2002) · Zbl 1342.81405
[42] Braun, V., Toric elliptic fibrations and F-theory compactifications, JHEP, 1301, 016, (2013) · Zbl 1342.81405
[43] Johnson, S.B.; Taylor, W., Calabi-Yau threefolds with large \(h\)\^{2,1}, JHEP, 1410, 23, (2014) · Zbl 1333.81384
[44] Taylor, W., Wang, Y.N.: Non-toric Bases for Elliptic Calabi-Yau Threefolds and 6D F-Theory Vacua. arXiv:1504.07689 [hep-th]
[45] Anderson, L.B.; Gao, X.; Gray, J.; Lee, S.J., Multiple fibrations in Calabi-Yau geometry and string dualities, JHEP, 1610, 105, (2016) · Zbl 1390.81403
[46] Candelas, P., Constantin, A., Mishra, C.: Calabi-Yau Threefolds With Small Hodge Numbers. arXiv:1602.06303 [hep-th]
[47] Bianchi, M.; Ferrara, S., Enriques and octonionic magic supergravity models, JHEP, 0802, 054, (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.