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Calabi-Yau threefolds with large \(h^{2,1}\). (English) Zbl 1333.81384
Summary: We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section (‘EFS’) and have a large Hodge number \(h^{2,1}\). EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have \(h^{2,1}\geq 350\) by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefolds with \(h^{2,1} \geq 350\), as well as three apparently new Calabi-Yau threefolds. We speculate that there are no other Calabi-Yau threefolds (elliptically fibered or not) with Hodge numbers that exceed this bound. We summarize the theoretical and practical obstacles to a complete enumeration of all possible EFS Calabi-Yau threefolds and fourfolds, including those with small Hodge numbers, using this approach.

MSC:
81T45 Topological field theories in quantum mechanics
14J81 Relationships with physics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J30 \(3\)-folds
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References:
[1] Candelas, P.; Horowitz, GT; Strominger, A.; Witten, E., Vacuum configurations for superstrings, Nucl. Phys., B 258, 46, (1985)
[2] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 1, Cambridge University Press, Cambridge U.K. (1987) [INSPIRE].
[3] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 2, Cambridge University Press, Cambridge U.K. (1987) [INSPIRE].
[4] J. Polchinski, String theory, Cambridge University Press, Cambridge U.K. (1998).
[5] T. Hübsch, Calabi-Yau manifolds: a bestiary for physicists, World Scientific, Singapore (1992).
[6] M. Gross, D. Huybrechts and D. Joyce, Calabi-Yau manifolds and related geometries, Springer-Verlag, Berlin Germany (2003).
[7] Davies, R., The expanding zoo of Calabi-Yau threefolds, Adv. High Energy Phys., 2011, 901898, (2011)
[8] He, Y-H, Calabi-Yau geometries: algorithms, databases and physics, Int. J. Mod. Phys., A 28, 1330032, (2013)
[9] Gross, M., A finiteness theorem for elliptic Calabi-Yau threefolds, Duke Math. J., 74, 271, (1994)
[10] N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra, volume II, Kinokuniya, Tokyo Japan (1988), pg. 405.
[11] Vafa, C., Evidence for F-theory, Nucl. Phys., B 469, 403, (1996)
[12] Morrison, DR; Vafa, C., Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys., B 473, 74, (1996)
[13] Seiberg, N.; Witten, E., Comments on string dynamics in six-dimensions, Nucl. Phys., B 471, 121, (1996)
[14] Morrison, DR; Vafa, C., Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys., B 476, 437, (1996)
[15] Grassi, A., On minimal models of elliptic threefolds, Math. Ann., 290, 287, (1991)
[16] W.P. Barth, K. Hulek, C.A.M. Peters and A. Van de Ven, Compact complex surfaces, Springer, Berlin Germany (2004).
[17] M. Reid, Chapters on algebraic surfaces, in Complex algebraic geometry Park City U.S.A. (1993), IAS/Park City Math. Ser.3 (1997) 3 [alg-geom/9602006].
[18] Kumar, V.; Morrison, DR; Taylor, W., Global aspects of the space of 6D \( \mathcal{N}=1 \) supergravities, JHEP, 11, 118, (2010)
[19] Morrison, DR; Taylor, W., Classifying bases for 6D F-theory models, Central Eur. J. Phys., 10, 1072, (2012)
[20] Morrison, DR; Taylor, W., Toric bases for 6D F-theory models, Fortsch. Phys., 60, 1187, (2012)
[21] G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, arXiv:1404.6300 [INSPIRE].
[22] Taylor, W., On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP, 08, 032, (2012)
[23] Kreuzer, M.; Skarke, H., Complete classification of reflexive polyhedra in four-dimensions, Adv. Theor. Math. Phys., 4, 1209, (2002)
[24] D.R. Morrison, TASI lectures on compactification and duality, hep-th/0411120 [INSPIRE].
[25] F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [INSPIRE].
[26] W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [INSPIRE].
[27] Kodaira, K., On compact analytic surfaces. II, Ann. Math., 77, 563, (1963)
[28] Kodaira, K., On compact analytic surfaces. III, Ann. Math., 78, 1, (1963)
[29] Bershadsky, M.; etal., Geometric singularities and enhanced gauge symmetries, Nucl. Phys., B 481, 215, (1996)
[30] Katz, S.; Morrison, DR; Schäfer-Nameki, S.; Sully, J., Tate’s algorithm and F-theory, JHEP, 08, 094, (2011)
[31] Grassi, A.; Morrison, DR, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys., 6, 51, (2012)
[32] Wazir, R., Arithmetic on elliptic threefolds, Composit. Math., 140, 567, (2004)
[33] Grassi, A.; Morrison, DR, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, J. Alg. Geom., 12, 321, (2003)
[34] C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand, Mordell-Weil torsion and the global structure of gauge groups in F-theory, arXiv:1405.3656 [INSPIRE].
[35] L. Badescu, Algebraic surfaces, Springer Verlag, Berlin Germany (2001).
[36] Witten, E., Phase transitions in M-theory and F-theory, Nucl. Phys., B 471, 195, (1996)
[37] Kumar, V.; Park, DS; Taylor, W., 6D supergravity without tensor multiplets, JHEP, 04, 080, (2011)
[38] Braun, V., Toric elliptic fibrations and F-theory compactifications, JHEP, 01, 016, (2013)
[39] Green, MB; Schwarz, JH; West, PC, Anomaly free chiral theories in six-dimensions, Nucl. Phys., B 254, 327, (1985)
[40] Sagnotti, A., A note on the Green-Schwarz mechanism in open string theories, Phys. Lett., B 294, 196, (1992)
[41] Erler, J., Anomaly cancellation in six-dimensions, J. Math. Phys., 35, 1819, (1994)
[42] Sadov, V., Generalized Green-Schwarz mechanism in F-theory, Phys. Lett., B 388, 45, (1996)
[43] Morrison, DR; Taylor, W., Matter and singularities, JHEP, 01, 022, (2012)
[44] Anderson, LB; Taylor, W., Geometric constraints in dual F-theory and heterotic string compactifications, JHEP, 08, 025, (2014)
[45] W. Fulton, Introduction to toric varieties, Ann. Math. Study131, Princeton University Press, Princeton U.S.A. (1993).
[46] Batyrev, V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J., 69, 349, (1993)
[47] D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
[48] Heckman, JJ; Morrison, DR; Vafa, C., On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP, 05, 028, (2014)
[49] Kumar, V.; Morrison, DR; Taylor, W., Mapping 6D \( \mathcal{N}=1 \) supergravities to F-theory, JHEP, 02, 099, (2010)
[50] Katz, SH; Vafa, C., Matter from geometry, Nucl. Phys., B 497, 146, (1997)
[51] Esole, M.; Yau, S-T, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math. Phys., 17, 1195, (2013)
[52] M. Esole, J. Fullwood and S.-T. Yau, \(D\)_{5}elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory, arXiv:1110.6177 [INSPIRE].
[53] Lawrie, C.; Schäfer-Nameki, S., The Tate form on steroids: resolution and higher codimension fibers, JHEP, 04, 061, (2013)
[54] Grassi, A.; Halverson, J.; Shaneson, JL, Matter from geometry without resolution, JHEP, 10, 205, (2013)
[55] Hayashi, H.; Lawrie, C.; Morrison, DR; Schäfer-Nameki, S., Box graphs and singular fibers, JHEP, 05, 048, (2014)
[56] A. Grassi, J. Halverson and J.L. Shaneson, Non-Abelian gauge symmetry and the Higgs mechanism in F-theory, arXiv:1402.5962 [INSPIRE].
[57] M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases, arXiv:1402.6331 [INSPIRE].
[58] Grimm, TW; Weigand, T., On abelian gauge symmetries and proton decay in global F-theory guts, Phys. Rev., D 82, 086009, (2010)
[59] Grimm, TW; Kerstan, M.; Palti, E.; Weigand, T., Massive abelian gauge symmetries and fluxes in F-theory, JHEP, 12, 004, (2011)
[60] Park, DS; Taylor, W., Constraints on 6D supergravity theories with abelian gauge symmetry, JHEP, 01, 141, (2012)
[61] Park, DS, Anomaly equations and intersection theory, JHEP, 01, 093, (2012)
[62] Morrison, DR; Park, DS, F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds, JHEP, 10, 128, (2012)
[63] Cvetič, M.; Grimm, TW; Klevers, D., Anomaly cancellation and abelian gauge symmetries in F-theory, JHEP, 02, 101, (2013)
[64] Mayrhofer, C.; Palti, E.; Weigand, T., U(1) symmetries in F-theory GUTs with multiple sections, JHEP, 03, 098, (2013)
[65] Braun, V.; Grimm, TW; Keitel, J., New global F-theory GUTs with U(1) symmetries, JHEP, 09, 154, (2013)
[66] Borchmann, J.; Mayrhofer, C.; Palti, E.; Weigand, T., Elliptic fibrations for SU(5) × U(1) × U(1) F-theory vacua, Phys. Rev., D 88, 046005, (2013)
[67] Cvetič, M.; Klevers, D.; Piragua, H., F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections, JHEP, 06, 067, (2013)
[68] Cvetič, M.; Klevers, D.; Piragua, H., F-theory compactifications with multiple U(1)-factors: addendum, JHEP, 12, 056, (2013)
[69] Braun, V.; Grimm, TW; Keitel, J., Geometric engineering in toric F-theory and GUTs with U(1) gauge factors, JHEP, 12, 069, (2013)
[70] Cvetič, M.; Grassi, A.; Klevers, D.; Piragua, H., Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors, JHEP, 04, 010, (2014)
[71] Borchmann, J.; Mayrhofer, C.; Palti, E.; Weigand, T., SU(5) tops with multiple U(1)s in F-theory, Nucl. Phys., B 882, 1, (2014)
[72] Cvetič, M.; Klevers, D.; Piragua, H.; Song, P., Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1) × U(1) × U(1) gauge symmetry, JHEP, 03, 021, (2014)
[73] Braun, AP; Collinucci, A.; Valandro, R., The fate of U(1)’s at strong coupling in F-theory, JHEP, 07, 028, (2014)
[74] Douglas, MR; Park, DS; Schnell, C., The Cremmer-Scherk mechanism in F-theory compactifications on K3 manifolds, JHEP, 05, 135, (2014)
[75] Klemm, A.; Lian, B.; Roan, SS; Yau, S-T, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys., B 518, 515, (1998)
[76] Aluffi, P.; Esole, M., New orientifold weak coupling limits in F-theory, JHEP, 02, 020, (2010)
[77] K. Matsuki, Introduction to the Mori program, Springer-Verlag, Berlin Germany (2002).
[78] Grassi, A., Divisors on elliptic Calabi-Yau four folds and the superpotential in F-theory. 1, J. Geom. Phys., 28, 289, (1998)
[79] Grimm, TW; Taylor, W., Structure in 6D and 4D \( \mathcal{N}=1 \) supergravity theories from F-theory, JHEP, 10, 105, (2012)
[80] Mohri, K., F-theory vacua in four-dimensions and toric threefolds, Int. J. Mod. Phys., A 14, 845, (1999)
[81] Kreuzer, M.; Skarke, H., Calabi-Yau four folds and toric fibrations, J. Geom. Phys., 26, 272, (1998)
[82] Knapp, J.; Kreuzer, M.; Mayrhofer, C.; Walliser, N-O, Toric construction of global F-theory guts, JHEP, 03, 138, (2011)
[83] N.C. Bizet, A. Klemm and D.V. Lopes, Landscaping with fluxes and the E_{8}Yukawa point in F-theory, arXiv:1404.7645 [INSPIRE].
[84] Candelas, P.; Constantin, A.; Skarke, H., An abundance of K3 fibrations from polyhedra with interchangeable parts, Commun. Math. Phys., 324, 937, (2013)
[85] Candelas, P.; Font, A., Duality between the webs of heterotic and type-II vacua, Nucl. Phys., B 511, 295, (1998)
[86] Gray, J.; Haupt, AS; Lukas, A., All complete intersection Calabi-Yau four-folds, JHEP, 07, 070, (2013)
[87] J. Gray, A.S. Haupt and A. Lukas, Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds, arXiv:1405.2073 [INSPIRE].
[88] J. Gray, private communication.
[89] Keller, CA; Ooguri, H., Modular constraints on Calabi-Yau compactifications, Commun. Math. Phys., 324, 107, (2013)
[90] Friedan, D.; Keller, CA, Constraints on 2D CFT partition functions, JHEP, 10, 180, (2013)
[91] Wall, CTC, Classification problems in differential topology, V: on certain 6-manifolds, Invent. Math., 1, 355, (1966)
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