Abstract
We analyze non-Higgsable clusters of gauge groups and matter that can arise at the level of geometry in 4D F-theory models. Non-Higgsable clusters seem to be generic features of F-theory compactifications, and give rise naturally to structures that include the nonabelian part of the standard model gauge group and certain specific types of potential dark matter candidates. In particular, there are nine distinct single nonabelian gauge group factors, and only five distinct products of two nonabelian gauge group factors with matter, including SU(3) × SU(2), that can be realized through 4D non-Higgsable clusters. There are also more complicated configurations involving more than two gauge factors; in particular, the collection of gauge group factors with jointly charged matter can exhibit branchings, loops, and long linear chains.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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. 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].
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. 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].
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].
F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [INSPIRE].
W. Taylor, Lectures on D-branes, gauge theory and M(atrices), in High energy physics and cosmology, Proceedings of 1997 Summer School, ICTP, Trieste Italy (1997) [hep-th/9801182] [INSPIRE].
W. Taylor and M. Van Raamsdonk, Multiple D0-branes in weakly curved backgrounds, Nucl. Phys. B 558 (1999) 63 [hep-th/9904095] [INSPIRE].
W. Taylor and M. Van Raamsdonk, Multiple Dp-branes in weak background fields, Nucl. Phys. B 573 (2000) 703 [hep-th/9910052] [INSPIRE].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
R. Donagi, S. Katz and E. Sharpe, Spectra of D-branes with Higgs vevs, Adv. Theor. Math. Phys. 8 (2005) 813 [hep-th/0309270] [INSPIRE].
S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-branes and monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].
R. Donagi and M. Wijnholt, Gluing branes — I, JHEP 05 (2013) 068 [arXiv:1104.2610] [INSPIRE].
L.B. Anderson, J.J. Heckman and S. Katz, T-branes and geometry, JHEP 05 (2014) 080 [arXiv:1310.1931] [INSPIRE].
M.R. Douglas, D-branes and matrix theory in curved space, Nucl. Phys. Proc. Suppl. 68 (1998) 381 [hep-th/9707228] [INSPIRE].
M.R. Douglas, A. Kato and H. Ooguri, D-brane actions on Kähler manifolds, Adv. Theor. Math. Phys. 1 (1998) 237 [hep-th/9708012] [INSPIRE].
F. Ferrari, On matrix geometry and effective actions, Nucl. Phys. B 871 (2013) 181 [arXiv:1301.3722] [INSPIRE].
A. Collinucci and R. Savelli, T-branes as branes within branes, arXiv:1410.4178 [INSPIRE].
A. Collinucci and R. Savelli, F-theory on singular spaces, arXiv:1410.4867 [INSPIRE].
J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, fluxes and compact three-generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [INSPIRE].
R. Blumenhagen, T.W. Grimm, B. Jurke and T. Weigand, Global F-theory GUTs, Nucl. Phys. B 829 (2010) 325 [arXiv:0908.1784] [INSPIRE].
T.W. Grimm, S. Krause and T. Weigand, F-theory GUT vacua on compact Calabi-Yau fourfolds, JHEP 07 (2010) 037 [arXiv:0912.3524] [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].
J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yau’s, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [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].
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].
N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra, volume II, Kinokuniya (1988), pp. 405-431.
K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
P.S. Aspinwall, S.H. Katz and D.R. Morrison, Lie groups, Calabi-Yau threefolds and F-theory, Adv. Theor. Math. Phys. 4 (2000) 95 [hep-th/0002012] [INSPIRE].
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].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, J. Algebraic Geom. 12 (2003) 321 [math.AG/0005196] [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].
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, Commun. Math. Phys. 336 (2015) 1231 [arXiv:1402.5962] [INSPIRE].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases, arXiv:1402.6331 [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 [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, Prog. Theor. Exp. Phys. 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Intriligator, 6d, \( \mathcal{N} \) = (1, 0) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].
B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of minimal 6d SCFTs, arXiv:1412.3152 [INSPIRE].
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].
S.D. Cutkosky, Zariski decomposition of divisors on algebraic varieties, Duke Math. J. 53 (1986) 149.
W. Fulton, Introduction to toric varieties, Annals of Mathematics Study 131, Princeton University Press, Princeton U.S.A. (1993).
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortschr. Phys. 60 (2012) 1187 [arXiv:1204.0283] [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].
S.H. Katz, D.R. Morrison and M.R. Plesser, Enhanced gauge symmetry in type-II string theory, Nucl. Phys. B 477 (1996) 105 [hep-th/9601108] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [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. Berglund and P. Mayr, Stability of vector bundles from F-theory, JHEP 12 (1999) 009 [hep-th/9904114] [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].
J. Halverson and W. Taylor, to appear.
M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [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, arXiv:1404.6300 [INSPIRE].
S.B. Johnson and W. Taylor, Calabi-Yau threefolds with large h 2,1, JHEP 10 (2014) 023 [arXiv:1406.0514] [INSPIRE].
Y. Wang and W. Taylor, Non-toric bases for 6D F-theory models and elliptic Calabi-Yau threefolds, to appear.
W.P. Barth, K. Hulek, C.A.M. Peters and A. Van de Ven, Compact complex surfaces, Springer (2004).
N. Seiberg and W. Taylor, Charge lattices and consistency of 6D supergravity, JHEP 06 (2011) 001 [arXiv:1103.0019] [INSPIRE].
V.V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993) 349.
P. Candelas, A.M. Dale, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds, Nucl. Phys. B 298 (1988) 493 [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. 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].
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, 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].
V. Braun, T.W. Grimm and J. Keitel, Complete intersection fibers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [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, JHEP 09 (2014) 093 [arXiv:1405.2073] [INSPIRE].
O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Type IIA moduli stabilization, JHEP 07 (2005) 066 [hep-th/0505160] [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].
J.J. Heckman, Particle physics implications of F-theory, Ann. Rev. Nucl. Part. Sci. 60 (2010) 237 [arXiv:1001.0577] [INSPIRE].
T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].
D.R. Morrison, D. Park and W. Taylor, to appear.
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: 1412.6112
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
Morrison, D.R., Taylor, W. Non-Higgsable clusters for 4D F-theory models. J. High Energ. Phys. 2015, 80 (2015). https://doi.org/10.1007/JHEP05(2015)080
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2015)080