Abstract
We study elliptic fibrations for F-theory compactifications realizing 4d and 6d supersymmetric gauge theories with abelian gauge factors. In the fibration these U(1) symmetries are realized in terms of additional rational section. We obtain a universal characterization of all the possible U(1) charges of matter fields by determining the corresponding codimension two fibers with rational sections. In view of modelling supersymmetric Grand Unified Theories, one of the main examples that we analyze are U(1) symmetries for SU(5) gauge theories with \( \overline{\mathbf{5}} \) and 10 matter. We use a combination of constraints on the normal bundle of rational curves in Calabi-Yau three- and four-folds, as well as the splitting of rational curves in the fibers in codimension two, to determine the possible configurations of smooth rational sections. This analysis straightforwardly generalizes to multiple U(1)s. We study the flops of such fibers, as well as some of the Yukawa couplings in codimension three. Furthermore, we carry out a universal study of the U(1)-charged GUT singlets, including their KK-charges, and determine all realizations of singlet fibers. By giving vacuum expectation values to these singlets, we propose a systematic way to analyze the Higgsing of U(1)s to discrete gauge symmetries in F-theory.
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References
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
K. Kodaira, On compact analytic surfaces: II, Ann. Math. 77 (1963) 563.
A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964) 128.
H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box Graphs and Singular Fibers, JHEP 05 (2014) 048 [arXiv:1402.2653] [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].
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 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].
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 and H. Piragua, F-Theory Compactifications with Multiple U(1)-Factors: Addendum, JHEP 12 (2013) 056 [arXiv:1307.6425] [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].
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].
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].
V. Braun, T.W. Grimm and J. Keitel, Complete Intersection Fibers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [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].
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].
A.P. Braun, A. Collinucci and R. Valandro, G-flux in F-theory and algebraic cycles, Nucl. Phys. B 856 (2012) 129 [arXiv:1107.5337] [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].
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
M.J. Dolan, J. Marsano, N. Saulina and S. Schäfer-Nameki, F-theory GUTs with U(1) Symmetries: Generalities and Survey, Phys. Rev. D 84 (2011) 066008 [arXiv:1102.0290] [INSPIRE].
F. Baume, E. Palti and S. Schwieger, On E 8 and F-theory GUTs, JHEP 06 (2015) 039 [arXiv:1502.03878] [INSPIRE].
M.J. Dolan, J. Marsano and S. Schäfer-Nameki, Unification and Phenomenology of F-theory GUTs with U(1) P Q , JHEP 12 (2011) 032 [arXiv:1109.4958] [INSPIRE].
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].
M. Kuntzler and S. Schäfer-Nameki, Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group, arXiv:1406.5174 [INSPIRE].
C. Lawrie and D. Sacco, Tate’s algorithm for F-theory GUTs with two U(1)s, JHEP 03 (2015) 055 [arXiv:1412.4125] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].
D.-E. Diaconescu and S. Gukov, Three-dimensional N = 2 gauge theories and degenerations of Calabi-Yau four folds, Nucl. Phys. B 535 (1998) 171 [hep-th/9804059] [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [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].
T. Shioda, Mordell-Weil lattices and Galois representation. I, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989) 268.
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].
C. Mayrhofer, E. Palti, O. Till and T. Weigand, On Discrete Symmetries and Torsion Homology in F-theory, JHEP 06 (2015) 029 [arXiv:1410.7814] [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].
N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra. Vol. II, Kinokuniya, Tokyo, (1988), pg. 405-431.
O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, (2001).
C. Lawrie and S. Schafer-Nameki, in progress.
T.W. Grimm, A. Kapfer and J. Keitel, Effective action of 6D F-theory with U(1) factors: Rational sections make Chern-Simons terms jump, JHEP 07 (2013) 115 [arXiv:1305.1929] [INSPIRE].
T.W. Grimm and A. Kapfer, Anomaly Cancelation in Field Theory and F-theory on a Circle, arXiv:1502.05398 [INSPIRE].
S. Katz, Rational curves on Calabi-Yau threefolds, in Essays on mirror manifolds, Int. Press, Hong Kong, (1992), pg. 168-180.
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Am. J. Math. 79 (1957) 121.
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964) 259.
D. Eisenbud and J. Harris, 3264 & All That Intersection Theory in Algebraic Geometry, (2013).
R. Miranda, The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica, ETS Editrice, Pisa, (1989).
M. Reid, Minimal models of canonical 3-folds, in Algebraic varieties and analytic varieties, Tokyo, 1981, Adv. Stud. Pure Math. 1 (1983) 131, North-Holland, Amsterdam, Netherlands.
H.B. Laufer, On CP 1 as an exceptional set, in Recent developments in several complex variables, Proc. Conf. Princeton University, Princeton, N.J., U.S.A (1979), Ann. Math. Stud. 100261, Princeton Univ. Press, Princeton, N.J., U.S.A. (1981).
A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions I, arXiv:1407.3520 [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. Schafer-Nameki, Box Graphs and Resolutions II, to appear.
K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, (2002).
K. Matsuki, Weyl groups and birational transformations among minimal models, Mem. Am. Math. Soc. 116 (1995) 557.
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].
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].
V. Braun and D.R. Morrison, F-theory on Genus-One Fibrations, JHEP 08 (2014) 132 [arXiv:1401.7844] [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].
I. García-Etxebarria, T.W. Grimm and J. Keitel, Yukawas and discrete symmetries in F-theory compactifications without section, JHEP 11 (2014) 125 [arXiv:1408.6448] [INSPIRE].
S. Krippendorf, D.K. Mayorga Pena, P.-K. Oehlmann and F. Ruehle, Rational F-theory GUTs without exotics, JHEP 07 (2014) 013 [arXiv:1401.5084] [INSPIRE].
L.E. Ibáñez and G.G. Ross, Discrete gauge symmetries and the origin of baryon and lepton number conservation in supersymmetric versions of the standard model, Nucl. Phys. B 368 (1992) 3 [INSPIRE].
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Lawrie, C., Schäfer-Nameki, S. & Wong, JM. F-theory and all things rational: surveying U(1) symmetries with rational sections. J. High Energ. Phys. 2015, 144 (2015). https://doi.org/10.1007/JHEP09(2015)144
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DOI: https://doi.org/10.1007/JHEP09(2015)144