Abstract
We use Monte Carlo methods to explore the set of toric threefold bases that support elliptic Calabi-Yau fourfolds for F-theory compactifications to four dimensions, and study the distribution of geometrically non-Higgsable gauge groups, matter, and quiver structure. We estimate the number of distinct threefold bases in the connected set studied to be ∼ 1048. The distribution of bases peaks around h 1,1 ∼ 82. All bases encountered after “thermalization” have some geometric non-Higgsable structure. We find that the number of non-Higgsable gauge group factors grows roughly linearly in h 1,1 of the threefold base. Typical bases have ∼ 6 isolated gauge factors as well as several larger connected clusters of gauge factors with jointly charged matter. Approximately 76% of the bases sampled contain connected two-factor gauge group products of the form SU(3) × SU(2), which may act as the non-Abelian part of the standard model gauge group. SU(3) × SU(2) is the third most common connected two-factor product group, following SU(2) × SU(2) and G 2 × SU(2), which arise more frequently.
Article PDF
Similar content being viewed by others
References
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].
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].
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, 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].
W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].
G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, JHEP 06 (2015) 061 [arXiv:1404.6300] [INSPIRE].
S.B. Johnson and W. Taylor, Calabi-Yau threefolds with large h 2,1, JHEP 10 (2014) 23 [arXiv:1406.0514] [INSPIRE].
W. Taylor and Y.-N. Wang, Non-toric Bases for Elliptic Calabi-Yau Threefolds and 6D F-theory Vacua, arXiv:1504.07689 [INSPIRE].
F. Denef, Les Houches Lectures on Constructing String Vacua, arXiv:0803.1194 [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].
A. Grassi, J. Halverson, J. Shaneson and W. Taylor, Non-Higgsable QCD and the Standard Model Spectrum in F-theory, JHEP 01 (2015) 086 [arXiv:1409.8295] [INSPIRE].
D.R. Morrison and W. Taylor, Non-Higgsable clusters for 4D F-theory models, JHEP 05 (2015) 080 [arXiv:1412.6112] [INSPIRE].
J. Halverson and W. Taylor, \( {\mathrm{\mathbb{P}}}^1 \) -bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models, JHEP 09 (2015) 086 [arXiv:1506.03204] [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].
K. Mohri, F theory vacua in four-dimensions and toric threefolds, Int. J. Mod. Phys. A 14 (1999) 845 [hep-th/9701147] [INSPIRE].
P. Berglund and P. Mayr, Heterotic string / F theory duality from mirror symmetry, Adv. Theor. Math. Phys. 2 (1999) 1307 [hep-th/9811217] [INSPIRE].
T.W. Grimm and W. Taylor, Structure in 6D and 4D N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [INSPIRE].
D.R. Morrison, TASI lectures on compactification and duality, hep-th/0411120 [INSPIRE].
W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, arXiv:1104.2051 [INSPIRE].
K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. 77 (1963) 563, 78 (1963) 1.
J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV, Proceedings of Internat. Summer School, Univ. Antwerp, Antwerp, Belgium, 1972, Lecture Notes in Math. 476 (1975) 33.
M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, 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].
W. Fulton. Introduction to toric varieties. No. 131, Princeton University Press, (1993).
W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven, Compact complex surfaces, Springer, (2004).
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
N. Seiberg, Nontrivial fixed points of the renormalization group in six-dimensions, Phys. Lett. B 390 (1997) 169 [hep-th/9609161] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, D.R. Morrison and D.S. Park, 6D SCFTs and Gravity, JHEP 06 (2015) 158 [arXiv:1412.6526] [INSPIRE].
K. Matsuki, Introduction to the Mori Program, Springer-Verlag, Berlin Germany (2002).
P. Candelas, D.-E. Diaconescu, B. Florea, D.R. Morrison and G. Rajesh, Codimension three bundle singularities in F-theory, JHEP 06 (2002) 014 [hep-th/0009228] [INSPIRE].
R. Wazir, Arithmetic on elliptic threefolds, Compos. Math. 140 (2004) 567.
A. Braun, W. Taylor and Y. Wang, to appear.
M. Lynker, R. Schimmrigk and A. Wisskirchen, Landau-Ginzburg vacua of string, M-theory and F-theory at c = 12, Nucl. Phys. B 550 (1999) 123 [hep-th/9812195] [INSPIRE].
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
H. Skarke, Weight systems for toric Calabi-Yau varieties and reflexivity of Newton polyhedra, Mod. Phys. Lett. A 11 (1996) 1637 [alg-geom/9603007] [INSPIRE].
A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, Searching for K3 fibrations, Nucl. Phys. B 494 (1997) 567 [hep-th/9610154] [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].
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].
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].
L.B. Anderson, F. Apruzzi, X. Gao, J. Gray and S.-J. Lee, A New Construction of Calabi-Yau Manifolds: Generalized CICYs, arXiv:1507.03235 [INSPIRE].
R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory - II: Experimental Predictions, JHEP 01 (2009) 059 [arXiv:0806.0102] [INSPIRE].
T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
D.R. Morrison and W. Taylor, Matter and singularities, JHEP 01 (2012) 022 [arXiv:1106.3563] [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].
C. Lawrie and S. Schäfer-Nameki, The Tate Form on Steroids: Resolution and Higher Codimension Fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].
H. Hayashi, C. Lawrie and S. Schäfer-Nameki, Phases, Flops and F-theory: SU(5) Gauge Theories, JHEP 10 (2013) 046 [arXiv:1304.1678] [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].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and Gauge Theory Phases, arXiv:1402.6331 [INSPIRE].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and Gauge Theory Phases II, arXiv:1407.1867 [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions I, arXiv:1407.3520 [INSPIRE].
P.S. Aspinwall, D.R. Morrison and M. Gross, Stable singularities in string theory, Commun. Math. Phys. 178 (1996) 115 [hep-th/9503208] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Geometry and Topology of String Junctions, arXiv:1410.6817 [INSPIRE].
V. Braun and D.R. Morrison, F-theory on Genus-One Fibrations, JHEP 08 (2014) 132 [arXiv:1401.7844] [INSPIRE].
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
L.B. Anderson, I. García-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE].
C. Mayrhofer, E. Palti, O. Till and T. Weigand, Discrete Gauge Symmetries by Higgsing in four-dimensional F-theory Compactifications, JHEP 12 (2014) 068 [arXiv:1408.6831] [INSPIRE].
D. Klevers, D.K. Mayorga Pena, P.-K. Oehlmann, H. Piragua and J. Reuter, F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE].
M. Cvetič, R. Donagi, D. Klevers, H. Piragua and M. Poretschkin, F-theory vacua with \( {\mathbb{Z}}_3 \) gauge symmetry, Nucl. Phys. B 898 (2015) 736 [arXiv:1502.06953] [INSPIRE].
D. Morrison, D. Park and W. Taylor, to appear.
M. Gross, A finiteness theorem for elliptic Calabi-Yau threefolds, Duke Math. Jour. 74 (1994) 271.
S. Ashok and M.R. Douglas, Counting flux vacua, JHEP 01 (2004) 060 [hep-th/0307049] [INSPIRE].
F. Denef and M.R. Douglas, Distributions of flux vacua, JHEP 05 (2004) 072 [hep-th/0404116] [INSPIRE].
A.P. Braun and T. Watari, Distribution of the Number of Generations in Flux Compactifications, Phys. Rev. D 90 (2014) 121901 [arXiv:1408.6156] [INSPIRE].
A.P. Braun and T. Watari, The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications, JHEP 01 (2015) 047 [arXiv:1408.6167] [INSPIRE].
T. Watari, Statistics of F-theory flux vacua for particle physics, JHEP 11 (2015) 065 [arXiv:1506.08433] [INSPIRE].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].
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: 1510.04978
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
Taylor, W., Wang, YN. A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua. J. High Energ. Phys. 2016, 137 (2016). https://doi.org/10.1007/JHEP01(2016)137
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)137