Abstract
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 ≥ 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 ≥ 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 1, Cambridge University Press, Cambridge U.K. (1987) [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 2, Cambridge University Press, Cambridge U.K. (1987) [INSPIRE].
J. Polchinski, String theory, Cambridge University Press, Cambridge U.K. (1998).
T. Hübsch, Calabi-Yau manifolds: a bestiary for physicists, World Scientific, Singapore (1992).
M. Gross, D. Huybrechts and D. Joyce, Calabi-Yau manifolds and related geometries, Springer-Verlag, Berlin Germany (2003).
R. Davies, The expanding zoo of Calabi-Yau threefolds, Adv. High Energy Phys. 2011 (2011) 901898 [arXiv:1103.3156] [INSPIRE].
Y.-H. He, Calabi-Yau geometries: algorithms, databases and physics, Int. J. Mod. Phys. A 28 (2013) 1330032 [arXiv:1308.0186] [INSPIRE].
M. Gross, A finiteness theorem for elliptic Calabi-Yau threefolds, Duke Math. J. 74 (1994) 271 [alg-geom/9305002] [INSPIRE].
N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra, volume II, Kinokuniya, Tokyo Japan (1988), pg. 405.
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. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
W.P. Barth, K. Hulek, C.A.M. Peters and A. Van de Ven, Compact complex surfaces, Springer, Berlin Germany (2004).
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].
V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D \( \mathcal{N}=1 \) supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [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].
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, arXiv:1404.6300 [INSPIRE].
W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [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].
D.R. Morrison, TASI lectures on compactification and duality, hep-th/0411120 [INSPIRE].
F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [INSPIRE].
W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [INSPIRE].
K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.
M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].
A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].
R. Wazir, Arithmetic on elliptic threefolds, Composit. Math. 140 (2004) 567 [math.NT/0112259].
A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, J. Alg. Geom. 12 (2003) 321 [math.AG/0005196] [INSPIRE].
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].
L. Badescu, Algebraic surfaces, Springer Verlag, Berlin Germany (2001).
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [INSPIRE].
V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP 01 (2013) 016 [arXiv:1110.4883] [INSPIRE].
M.B. Green, J.H. Schwarz and P.C. West, Anomaly free chiral theories in six-dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].
A. Sagnotti, A note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B 294 (1992) 196 [hep-th/9210127] [INSPIRE].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [INSPIRE].
L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].
W. Fulton, Introduction to toric varieties, Ann. Math. Study 131, Princeton University Press, Princeton U.S.A. (1993).
V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993) 349.
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [arXiv:1312.5746] [INSPIRE].
V. Kumar, D.R. Morrison and W. Taylor, Mapping 6D \( \mathcal{N}=1 \) supergravities to F-theory, JHEP 02 (2010) 099 [arXiv:0911.3393] [INSPIRE].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math. Phys. 17 (2013) 1195 [arXiv:1107.0733] [INSPIRE].
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].
C. Lawrie and S. Schäfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Matter from geometry without resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].
H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box graphs and singular fibers, JHEP 05 (2014) 048 [arXiv:1402.2653] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Non-Abelian gauge symmetry and the Higgs mechanism in F-theory, arXiv:1402.5962 [INSPIRE].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases, arXiv:1402.6331 [INSPIRE].
T.W. Grimm and T. Weigand, On Abelian gauge symmetries and proton decay in global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [INSPIRE].
T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive Abelian gauge symmetries and fluxes in F-theory, JHEP 12 (2011) 004 [arXiv:1107.3842] [INSPIRE].
D.S. Park and W. Taylor, Constraints on 6D supergravity theories with Abelian gauge symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE].
D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
D.R. Morrison and D.S. Park, F-theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE].
M. Cvetič, T.W. Grimm and D. Klevers, Anomaly cancellation and Abelian gauge symmetries in F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE].
C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP 03 (2013) 098 [arXiv:1211.6742] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, New global F-theory GUTs with U(1) symmetries, JHEP 09 (2013) 154 [arXiv:1302.1854] [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, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections, JHEP 06 (2013) 067 [arXiv:1303.6970] [INSPIRE].
M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: addendum, JHEP 12 (2013) 056 [arXiv:1307.6425] [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].
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].
J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) tops with multiple U(1)s in F-theory, Nucl. Phys. B 882 (2014) 1 [arXiv:1307.2902] [INSPIRE].
M. Cvetič, D. Klevers, H. Piragua and P. Song, Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1) × U(1) × U(1) gauge symmetry, JHEP 03 (2014) 021 [arXiv:1310.0463] [INSPIRE].
A.P. Braun, A. Collinucci and R. Valandro, The fate of U(1)’s at strong coupling in F-theory, JHEP 07 (2014) 028 [arXiv:1402.4054] [INSPIRE].
M.R. Douglas, D.S. Park and C. Schnell, The Cremmer-Scherk mechanism in F-theory compactifications on K3 manifolds, JHEP 05 (2014) 135 [arXiv:1403.1595] [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].
P. Aluffi and M. Esole, New orientifold weak coupling limits in F-theory, JHEP 02 (2010) 020 [arXiv:0908.1572] [INSPIRE].
K. Matsuki, Introduction to the Mori program, Springer-Verlag, Berlin Germany (2002).
A. Grassi, Divisors on elliptic Calabi-Yau four folds and the superpotential in F-theory. 1, J. Geom. Phys. 28 (1998) 289 [INSPIRE].
T.W. Grimm and W. Taylor, Structure in 6D and 4D \( \mathcal{N}=1 \) supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].
K. Mohri, F-theory vacua in four-dimensions and toric threefolds, Int. J. Mod. Phys. A 14 (1999) 845 [hep-th/9701147] [INSPIRE].
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
J. Knapp, M. Kreuzer, C. Mayrhofer and N.-O. Walliser, Toric construction of global F-theory GUTs, JHEP 03 (2011) 138 [arXiv:1101.4908] [INSPIRE].
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].
P. Candelas, A. Constantin and H. Skarke, An abundance of K3 fibrations from polyhedra with interchangeable parts, Commun. Math. Phys. 324 (2013) 937 [arXiv:1207.4792] [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].
J. Gray, A.S. Haupt and A. Lukas, All complete intersection Calabi-Yau four-folds, JHEP 07 (2013) 070 [arXiv:1303.1832] [INSPIRE].
J. Gray, A.S. Haupt and A. Lukas, Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds, arXiv:1405.2073 [INSPIRE].
J. Gray, private communication.
C.A. Keller and H. Ooguri, Modular constraints on Calabi-Yau compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2D CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
C.T.C. Wall, Classification problems in differential topology, V: on certain 6-manifolds, Invent. Math. 1 (1966) 355.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1406.0514
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Johnson, S.B., Taylor, W. Calabi-Yau threefolds with large h 2,1 . J. High Energ. Phys. 2014, 23 (2014). https://doi.org/10.1007/JHEP10(2014)023
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)023