Abstract
We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.
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References
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
S.T. Yau, ‘Open problems in geometry, Proc. Symp. Pure Math 54 (1993) 1.
M. Gross, A finiteness theorem for elliptic Calabi-Yau threefolds, Duke Math. J. 74 (1994) 271.
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds — I, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds —II, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].
W. Taylor and Y.-N. Wang, Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua, Adv. Theor. Math. Phys. 21 (2017) 1063 [arXiv:1504.07689] [INSPIRE].
S.B. Johnson and W. Taylor, Calabi-Yau threefolds with large h2,1 , JHEP 10 (2014) 023 [arXiv:1406.0514] [INSPIRE].
S.B. Johnson and W. Taylor, Enhanced gauge symmetry in 6D F-theory models and tuned elliptic Calabi-Yau threefolds, Fortsch. Phys. 64 (2016) 581 [arXiv:1605.08052] [INSPIRE].
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds, Nucl. Phys. B 298 (1988) 493 [INSPIRE].
L.B. Anderson et al., A new construction of Calabi-Yau manifolds: generalized CICYs, Nucl. Phys. B 906 (2016) 441 [arXiv:1507.03235] [INSPIRE].
M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys. 4 (2002) 1209 [hep-th/0002240] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Multiple fibrations in Calabi-Yau geometry and string dualities, JHEP 10 (2016) 105 [arXiv:1608.07555] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
L.B. Anderson, J. Gray and B. Hammack, Fibrations in non-simply connected Calabi-Yau quotients, JHEP 08 (2018) 128 [arXiv:1805.05497] [INSPIRE].
P. Candelas, A. Constantin and H. Skarke, An abundance of K 3 fibrations from polyhedra with interchangeable parts, Commun. Math. Phys. 324 (2013) 937 [arXiv:1207.4792] [INSPIRE].
M. Kreuzer and H. Skarke, http://hep.itp.tuwien.ac.at/∼kreuzer/CY.html.
Y.-C. Huang and W. Taylor, Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers, JHEP 02 (2019) 087 [arXiv:1805.05907] [INSPIRE].
Y.-C. Huang and W. Taylor, On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds, JHEP 03 (2019) 014 [arXiv:1809.05160] [INSPIRE].
V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993) 349.
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
F. Rohsiepe, Fibration structures in toric Calabi-Yau fourfolds, hep-th/0502138 [INSPIRE].
V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP 01 (2013) 016 [arXiv:1110.4883] [INSPIRE].
V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification of tops, Adv. Theor. Math. Phys. 7 (2003) 205 [hep-th/0303218] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, Geometric engineering in toric F-theory and GUTs with U(1) gauge factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE].
D. Klevers et al., F-theory on all toric hypersurface fibrations and its Higgs branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE].
Y.-C. Huang and W. Taylor, Mirror symmetry and elliptic Calabi-Yau manifolds, JHEP 04 (2019) 083 [arXiv:1811.04947] [INSPIRE].
P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].
P. Berglund, Y.C. Huang, W. Taylor and Y.N. Wang, to appear.
J. Halverson, C. Long and W. Taylor, to appear.
P. Candelas et al., Codimension three bundle singularities in F-theory, JHEP 06 (2002) 014 [hep-th/0009228] [INSPIRE].
C. Lawrie and S. Schäfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].
J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, Elliptic fibrations for SU(5) × U(1) × U(1) F-theory vacua, Phys. Rev. D 88 (2013) 046005 [arXiv:1303.5054] [INSPIRE].
M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors, JHEP 04 (2014) 010 [arXiv:1306.3987] [INSPIRE].
M. Dierigl, P.-K. Oehlmann and F. Ruehle, Global tensor-matter transitions in F-theory, Fortsch. Phys. 66 (2018) 1800037 [arXiv:1804.07386] [INSPIRE].
I. Achmed-Zade, I. Garćıa-Etxebarria and C. Mayrhofer, A note on non-flat points in the SU(5) × U(1)P Q F-theory model, JHEP 05 (2019) 013 [arXiv:1806.05612] [INSPIRE].
J. Tian and Y.-N. Wang, E-string spectrum and typical F-theory geometry, arXiv:1811.02837 [INSPIRE].
F. Apruzzi, L. Lin and C. Mayrhofer, Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations, JHEP 05 (2019) 187 [arXiv:1811.12400] [INSPIRE].
F. Apruzzi et al., Fibers add flavor. Part I: classification of 5d SCFTs, flavor symmetries and BPS states, JHEP 11 (2019) 068 [arXiv:1907.05404] [INSPIRE].
P. Berglund and T. Hubsch, A generalized construction of Calabi-Yau models and mirror symmetry, SciPost Phys. 4 (2018) 009 [arXiv:1611.10300] [INSPIRE].
D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys. B 518 (1998) 515 [hep-th/9701023] [INSPIRE].
W. Taylor and Y.-N. Wang, A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua, JHEP 01 (2016) 137 [arXiv:1510.04978] [INSPIRE].
M. Demirtas, C. Long, L. McAllister and M. Stillman, The Kreuzer-Skarke axiverse, arXiv:1808.01282 [INSPIRE].
R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [INSPIRE].
R. Davies, The expanding zoo of Calabi-Yau threefolds, Adv. High Energy Phys. 2011 (2011) 901898 [arXiv:1103.3156] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, Complete intersection fibers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [INSPIRE].
M.-x. Huang, S. Katz and A. Klemm, Topological string on elliptic CY 3-folds and the ring of Jacobi forms, JHEP 10 (2015) 125 [arXiv:1501.04891] [INSPIRE].
J. Gray, A.S. Haupt and A. Lukas, Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds, JHEP 09 (2014) 093 [arXiv:1405.2073] [INSPIRE].
F. Schöller and H. Skarke, All weight systems for Calabi-Yau fourfolds from reflexive polyhedra, Commun. Math. Phys. 372 (2019) 657 [arXiv:1808.02422] [INSPIRE].
J. Halverson, C. Long and B. Sung, Algorithmic universality in F-theory compactifications, Phys. Rev. D 96 (2017) 126006 [arXiv:1706.02299] [INSPIRE].
W. Taylor and Y.-N. Wang, Scanning the skeleton of the 4D F-theory landscape, JHEP 01 (2018) 111 [arXiv:1710.11235] [INSPIRE].
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Huang, YC., Taylor, W. Fibration structure in toric hypersurface Calabi-Yau threefolds. J. High Energ. Phys. 2020, 172 (2020). https://doi.org/10.1007/JHEP03(2020)172
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DOI: https://doi.org/10.1007/JHEP03(2020)172