Abstract
Many four-dimensional supersymmetric compactifications of F-theory contain gauge groups that cannot be spontaneously broken through geometric deformations. These “non-Higgsable clusters” include realizations of SU(3), SU(2), and SU(3) × SU(2), but no SU(n) gauge groups or factors with n > 3. We study possible realizations of the standard model in F-theory that utilize non-Higgsable clusters containing SU(3) factors and show that there are three distinct possibilities. In one, fields with the non-abelian gauge charges of the standard model matter fields are localized at a single locus where non-perturbative SU(3) and SU(2) seven-branes intersect; cancellation of gauge anomalies implies that the simplest four-dimensional chiral SU(3) × SU(2) × U(1) model that may arise in this context exhibits standard model families. We identify specific geometries that realize non-Higgsable SU(3) and SU(3) × SU(2) sectors. This kind of scenario provides a natural mechanism that could explain the existence of an unbroken QCD sector, or more generally the appearance of light particles and symmetries at low energy scales.
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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].
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].
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 and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
L. Kofman et al., Beauty is attractive: moduli trapping at enhanced symmetry points, JHEP 05 (2004) 030 [hep-th/0403001] [INSPIRE].
N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra, vol. II, Kinokuniya, Tokyo Japan (1988), pg. 405.
K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.
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 [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, 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, N = (1, 0) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [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].
L.B. Anderson and W. Taylor, Geometric constraints in dual F-theory and heterotic string compactifications, JHEP 08 (2014) 025 [arXiv:1405.2074] [INSPIRE].
D.R. Morrison and W. Taylor, Non-Higgsable clusters for 4D F-theory models, arXiv:1412.6112 [INSPIRE].
J.H. Schwarz, An SL(2, Z) multiplet of type IIB superstrings, Phys. Lett. B 360 (1995) 13 [Erratum ibid. B 364 (1995) 252] [hep-th/9508143] [INSPIRE].
M.R. Gaberdiel and B. Zwiebach, Exceptional groups from open strings, Nucl. Phys. B 518 (1998) 151 [hep-th/9709013] [INSPIRE].
O. DeWolfe and B. Zwiebach, String junctions for arbitrary Lie algebra representations, Nucl. Phys. B 541 (1999) 509 [hep-th/9804210] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Matter from geometry without resolution, JHEP 10 (2013) 205 [arXiv:1306.1832] [INSPIRE].
A. Grassi, J. Halverson and J.L. Shaneson, Non-Abelian gauge symmetry and the Higgs mechanism in F-theory, arXiv:1402.5962 [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, math/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].
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].
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, 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].
S.B. Johnson and W. Taylor, Calabi-Yau threefolds with large h 2,1, JHEP 1410 (2014) 23 [arXiv:1406.0514] [INSPIRE].
V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori, Duke Math. J. 69 (1993) 349.
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].
M. Kreuzer and H. Skarke, Calabi-Yau four folds and toric fibrations, J. Geom. Phys. 26 (1998) 272 [hep-th/9701175] [INSPIRE].
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].
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].
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, Heterotic string/F-theory duality from mirror symmetry, Adv. Theor. Math. Phys. 2 (1999) 1307 [hep-th/9811217] [INSPIRE].
K. Mohri, F-theory vacua in four-dimensions and toric threefolds, Int. J. Mod. Phys. A 14 (1999) 845 [hep-th/9701147] [INSPIRE].
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 N = 1 supergravity theories from F-theory, JHEP 10 (2012) 105 [arXiv:1204.3092] [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].
J. Halverson and W. Taylor, to appear.
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].
K. Matsuki, Introduction to the Mori program, Springer-Verlag, Berlin Germany (2002).
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
J. Knapp and M. Kreuzer, Toric methods in F-theory model building, Adv. High Energy Phys. 2011 (2011) 513436 [arXiv:1103.3358] [INSPIRE].
W. Fulton, Introduction to toric varieties, Annals of Mathematics Study 131, Princeton University Press, Princeton U.S.A. (1993).
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. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory compactifications for supersymmetric GUTs, JHEP 08 (2009) 030 [arXiv:0904.3932] [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, I. Garcia-Etxebarria and J. Halverson, Global F-theory models: instantons and gauge dynamics, JHEP 01 (2011) 073 [arXiv:1003.5337] [INSPIRE].
C.-M. Chen, J. Knapp, M. Kreuzer and C. Mayrhofer, Global SO(10) F-theory GUTs, JHEP 10 (2010) 057 [arXiv:1005.5735] [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].
L. Lin and T. Weigand, Towards the standard model in F-theory, arXiv:1406.6071 [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, arXiv:1408.4808 [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.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].
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [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].
V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [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].
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].
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].
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].
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].
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].
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].
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
C. Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988) 177.
D.R. Morrison, D. Park and W. Taylor, to appear.
E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [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].
J. Halverson, Anomaly nucleation constrains SU(2) gauge theories, Phys. Rev. Lett. 111 (2013) 261601 [arXiv:1310.1091] [INSPIRE].
E. Witten, An SU(2) anomaly, Phys. Lett. B 117 (1982) 324 [INSPIRE].
S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].
V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactifications, JHEP 03 (2005) 007 [hep-th/0502058] [INSPIRE].
L. Susskind, The anthropic landscape of string theory, hep-th/0302219 [INSPIRE].
M. Graña, Flux compactifications in string theory: a comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].
F. Denef, M.R. Douglas and S. Kachru, Physics of string flux compactifications, Ann. Rev. Nucl. Part. Sci. 57 (2007) 119 [hep-th/0701050] [INSPIRE].
A. Giryavets, S. Kachru and P.K. Tripathy, On the taxonomy of flux vacua, JHEP 08 (2004) 002 [hep-th/0404243] [INSPIRE].
O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Enumerating flux vacua with enhanced symmetries, JHEP 02 (2005) 037 [hep-th/0411061] [INSPIRE].
W. Taylor, On the Hodge structure of elliptically fibered Calabi-Yau threefolds, JHEP 08 (2012) 032 [arXiv:1205.0952] [INSPIRE].
P. Candelas, E. Perevalov and G. Rajesh, Toric geometry and enhanced gauge symmetry of F-theory/heterotic vacua, Nucl. Phys. B 507 (1997) 445 [hep-th/9704097] [INSPIRE].
T. Gherghetta, C.F. Kolda and S.P. Martin, Flat directions in the scalar potential of the supersymmetric standard model, Nucl. Phys. B 468 (1996) 37 [hep-ph/9510370] [INSPIRE].
Y.-H. He, V. Jejjala, C. Matti and B.D. Nelson, Veronese geometry and the electroweak vacuum moduli space, Phys. Lett. B 736 (2014) 20 [arXiv:1402.3312] [INSPIRE].
J.A. Casas, Charge and color breaking, Adv. Ser. Direct. High Energy Phys. 21 (2010) 469 [hep-ph/9707475] [INSPIRE].
S. Abel and T. Falk, Charge and color breaking in the constrained MSSM, Phys. Lett. B 444 (1998) 427 [hep-ph/9810297] [INSPIRE].
T. Cohen and J.G. Wacker, Here be dragons: the unexplored continents of the CMSSM, JHEP 09 (2013) 061 [arXiv:1305.2914] [INSPIRE].
M. Cvetič, J. Halverson and P. Langacker, Implications of string constraints for exotic matter and Z′s beyond the standard model, JHEP 11 (2011) 058 [arXiv:1108.5187] [INSPIRE].
N. Seiberg and W. Taylor, Charge lattices and consistency of 6D supergravity, JHEP 06 (2011) 001 [arXiv:1103.0019] [INSPIRE].
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Grassi, A., Halverson, J., Shaneson, J. et al. Non-Higgsable QCD and the standard model spectrum in F-theory. J. High Energ. Phys. 2015, 86 (2015). https://doi.org/10.1007/JHEP01(2015)086
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DOI: https://doi.org/10.1007/JHEP01(2015)086