×

Mordell-Weil torsion and the global structure of gauge groups in F-theory. (English) Zbl 1333.81264

Summary: We study the global structure of the gauge group \(G\) of F-theory compactified on an elliptic fibration \(Y\). The global properties of \(G\) are encoded in the torsion subgroup of the Mordell-Weil group of rational sections of \(Y\). Generalising the Shioda map to torsional sections we construct a specific integer divisor class on \(Y\) as a fractional linear combination of the resolution divisors associated with the Cartan subalgebra of \(G\). This divisor class can be interpreted as an element of the refined coweight lattice of the gauge group. As a result, the spectrum of admissible matter representations is strongly constrained and the gauge group is non-simply connected. We exemplify our results by a detailed analysis of the general elliptic fibration with Mordell-Weil group \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\) as well as a further specialization to \(\mathbb{Z} \oplus \mathbb{Z}_2\). Our analysis exploits the representation of these fibrations as hypersurfaces in toric geometry.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations

Keywords:

F-theory; D-branes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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]. · Zbl 0925.14005 · doi:10.1016/0550-3213(96)00242-8
[2] 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]. · Zbl 0925.14007 · doi:10.1016/0550-3213(96)00369-0
[3] A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [INSPIRE]. · Zbl 0976.81503
[4] 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].
[5] 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]. · Zbl 1246.81223 · doi:10.1016/j.nuclphysb.2011.10.034
[6] S. Krause, C. Mayrhofer and T. Weigand, G4flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys.B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE]. · Zbl 1246.81271 · doi:10.1016/j.nuclphysb.2011.12.013
[7] T.W. Grimm and H. Hayashi, F-theory fluxes, chirality and Chern-Simons theories, JHEP03 (2012) 027 [arXiv:1111.1232] [INSPIRE]. · Zbl 1309.81218 · doi:10.1007/JHEP03(2012)027
[8] S. Krause, C. Mayrhofer and T. Weigand, Gauge fluxes in F-theory and type IIB orientifolds, JHEP08 (2012) 119 [arXiv:1202.3138] [INSPIRE]. · Zbl 1397.81160 · doi:10.1007/JHEP08(2012)119
[9] D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil group of elliptically-fibered Calabi-Yau threefolds, JHEP10 (2012) 128 [arXiv:1208.2695] [INSPIRE]. · Zbl 1397.81389 · doi:10.1007/JHEP10(2012)128
[10] M. Cvetič, T.W. Grimm and D. Klevers, Anomaly cancellation and abelian gauge symmetries in F-theory, JHEP02 (2013) 101 [arXiv:1210.6034] [INSPIRE]. · Zbl 1342.81525 · doi:10.1007/JHEP02(2013)101
[11] C. Mayrhofer, E. Palti and T. Weigand, U(1) symmetries in F-theory GUTs with multiple sections, JHEP03 (2013) 098 [arXiv:1211.6742] [INSPIRE]. · Zbl 1342.81733 · doi:10.1007/JHEP03(2013)098
[12] V. Braun, T.W. Grimm and J. Keitel, New global F-theory GUTs with U(1) symmetries, JHEP09 (2013) 154 [arXiv:1302.1854] [INSPIRE].
[13] 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].
[14] M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections, JHEP06 (2013) 067 [arXiv:1303.6970] [INSPIRE]. · Zbl 1342.81414 · doi:10.1007/JHEP06(2013)067
[15] V. Braun, T.W. Grimm and J. Keitel, Geometric engineering in toric F-theory and GUTs with U(1) gauge factors, JHEP12 (2013) 069 [arXiv:1306.0577] [INSPIRE]. · doi:10.1007/JHEP12(2013)069
[16] M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors, JHEP04 (2014) 010 [arXiv:1306.3987] [INSPIRE]. · doi:10.1007/JHEP04(2014)010
[17] 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]. · Zbl 1285.81053 · doi:10.1016/j.nuclphysb.2014.02.006
[18] M. Cvetič, D. Klevers and H. Piragua, F-theory compactifications with multiple U(1)-factors: addendum, JHEP12 (2013) 056 [arXiv:1307.6425] [INSPIRE]. · doi:10.1007/JHEP12(2013)056
[19] 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, JHEP03 (2014) 021 [arXiv:1310.0463] [INSPIRE]. · doi:10.1007/JHEP03(2014)021
[20] D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE]. · Zbl 1348.83091
[21] G. Martini and W. Taylor, 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces, arXiv:1404.6300 [INSPIRE]. · Zbl 1388.83862
[22] N.C. Bizet, A. Klemm and D.V. Lopes, Landscaping with fluxes and the E8 Yukawa Point in F-theory, arXiv:1404.7645 [INSPIRE].
[23] J. Marsano, N. Saulina and S. Schäfer-Nameki, Compact F-theory GUTs with U(1)(PQ), JHEP04 (2010) 095 [arXiv:0912.0272] [INSPIRE]. · Zbl 1272.81163 · doi:10.1007/JHEP04(2010)095
[24] H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, More on dimension-4 proton decay problem in F-theory — Spectral surface, discriminant locus and monodromy, Nucl. Phys.B 840 (2010) 304 [arXiv:1004.3870] [INSPIRE]. · Zbl 1206.81125 · doi:10.1016/j.nuclphysb.2010.07.011
[25] 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].
[26] M.J. Dolan, J. Marsano and S. Schäfer-Nameki, Unification and phenomenology of F-theory GUTs with U(1)PQ, JHEP12 (2011) 032 [arXiv:1109.4958] [INSPIRE]. · Zbl 1306.81215 · doi:10.1007/JHEP12(2011)032
[27] J. Marsano, N. Saulina and S. Schäfer-Nameki, On G-flux, M5 instantons and U(1)s in F-theory, arXiv:1107.1718 [INSPIRE]. · Zbl 1272.81163
[28] A. Maharana and E. Palti, Models of particle physics from type IIB string theory and F-theory: a review, Int. J. Mod. Phys.A 28 (2013) 1330005 [arXiv:1212.0555] [INSPIRE]. · doi:10.1142/S0217751X13300056
[29] O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP08 (2013) 115 [arXiv:1305.0318] [INSPIRE]. · Zbl 1342.81248 · doi:10.1007/JHEP08(2013)115
[30] G. McCabe, The structure and interpretation of the standard model, Elsevier, THe Netherlands (2007).
[31] J.C. Baez and J. Huerta, The algebra of grand unified theories, Bull. Am. Math. Soc.47 (2010) 483 [arXiv:0904.1556] [INSPIRE]. · Zbl 1196.81252 · doi:10.1090/S0273-0979-10-01294-2
[32] P.S. Aspinwall and D.R. Morrison, Nonsimply connected gauge groups and rational points on elliptic curves, JHEP07 (1998) 012 [hep-th/9805206] [INSPIRE]. · doi:10.1088/1126-6708/1998/07/012
[33] 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]. · Zbl 0992.81060
[34] M. Fukae, Y. Yamada and S.-K. Yang, Mordell-Weil lattice via string junctions, Nucl. Phys.B 572 (2000) 71 [hep-th/9909122] [INSPIRE]. · Zbl 1068.81588 · doi:10.1016/S0550-3213(00)00013-4
[35] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul40 (1991) 83. · Zbl 0757.14011
[36] I. Shimada, On elliptic K3 surfaces, Michigan Math. J.47 (2000) 423, [math.AG/0505140]. · Zbl 1085.14509 · doi:10.1307/mmj/1030132587
[37] Z. Guralnik, String junctions and nonsimply connected gauge groups, JHEP07 (2001) 002 [hep-th/0102031] [INSPIRE]. · doi:10.1088/1126-6708/2001/07/002
[38] T. Shioda, Mordell-Weil Lattices and Galois Representation. I, Proc. Japan Acad.A 65 (1989) 268. · Zbl 0715.14015 · doi:10.3792/pjaa.65.268
[39] R. Wazir, Arithmetic on elliptic threefolds, Compos. Math.140 (2001) 567 [math/0112259]. · Zbl 1060.11039 · doi:10.1112/S0010437X03000381
[40] D.S. Park, Anomaly equations and intersection theory, JHEP01 (2012) 093 [arXiv:1111.2351] [INSPIRE]. · Zbl 1306.81268 · doi:10.1007/JHEP01(2012)093
[41] P. Berglund, A. Klemm, P. Mayr and S. Theisen, On type IIB vacua with varying coupling constant, Nucl. Phys.B 558 (1999) 178 [hep-th/9805189] [INSPIRE]. · Zbl 1068.81610 · doi:10.1016/S0550-3213(99)00420-4
[42] P. Candelas, E. Perevalov and G. Rajesh, Comments on A, B, C chains of heterotic and type-II vacua, Nucl. Phys.B 502 (1997) 594 [hep-th/9703148] [INSPIRE]. · Zbl 0934.81046 · doi:10.1016/S0550-3213(97)00374-X
[43] V. Bouchard and H. Skarke, Affine Kac-Moody algebras, CHL strings and the classification of tops, Adv. Theor. Math. Phys.7 (2003) 205 [hep-th/0303218] [INSPIRE]. · doi:10.4310/ATMP.2003.v7.n2.a1
[44] J.H. Silverman, The arithmetic of elliptic curves, Springer, Germany (2008).
[45] S. Lang, Fundamentals of diophantine geometry, Springer, Germany (1983). · Zbl 0528.14013 · doi:10.1007/978-1-4757-1810-2
[46] S. Lang and A. Néron, Rational points of abelian varieties over function fields, Amer. J. Math.81 (1959) 95. · Zbl 0099.16103 · doi:10.2307/2372851
[47] R. Kloosterman, On the classification of degree 1 elliptic threefolds with constant j-invariant, Illinois J. Math.55 (2011) 771. · Zbl 1283.14014
[48] A. Grassi and V. Perduca, Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts, Adv. Theor. Math. Phys.17 (2013) 741 [arXiv:1201.0930] [INSPIRE]. · Zbl 1291.81313 · doi:10.4310/ATMP.2013.v17.n4.a2
[49] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan24 (1972) 20.. · Zbl 0226.14013 · doi:10.2969/jmsj/02410020
[50] D. A. Cox and S. Zucker, Intersection numbers of sections of elliptic surfaces, Inv. Math.53 (1979) 1. · Zbl 0444.14004 · doi:10.1007/BF01403189
[51] T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive abelian gauge symmetries and fluxes in F-theory, JHEP12 (2011) 004 [arXiv:1107.3842] [INSPIRE]. · Zbl 1306.81105 · doi:10.1007/JHEP12(2011)004
[52] A.P. Braun, A. Collinucci and R. Valandro, The fate of U(1)’s at strong coupling in F-theory, JHEP07 (2014) 028 [arXiv:1402.4054] [INSPIRE]. · doi:10.1007/JHEP07(2014)028
[53] M.R. Douglas, D.S. Park and C. Schnell, The Cremmer-Scherk mechanism in F-theory compactifications on K3 manifolds, JHEP05 (2014) 135 [arXiv:1403.1595] [INSPIRE]. · doi:10.1007/JHEP05(2014)135
[54] H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys.B 718 (2005) 203 [hep-th/0502059] [INSPIRE]. · Zbl 1207.81126 · doi:10.1016/j.nuclphysb.2005.04.011
[55] M. Buican, D. Malyshev, D.R. Morrison, H. Verlinde and M. Wijnholt, D-branes at singularities, compactification and hypercharge, JHEP01 (2007) 107 [hep-th/0610007] [INSPIRE]. · doi:10.1088/1126-6708/2007/01/107
[56] D. Bump, Lie groups, 2nd edition, Springer, Germany (2013). · Zbl 1279.22001 · doi:10.1007/978-1-4614-8024-2
[57] P. DiFrancesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, Germany (1997). · doi:10.1007/978-1-4612-2256-9
[58] M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys.B 481 (1996) 215 [hep-th/9605200] [INSPIRE]. · Zbl 1049.81581 · doi:10.1016/S0550-3213(96)90131-5
[59] 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]. · Zbl 1270.81174 · doi:10.4310/CNTP.2012.v6.n1.a2
[60] S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP08 (2011) 094 [arXiv:1106.3854] [INSPIRE]. · Zbl 1298.81307 · doi:10.1007/JHEP08(2011)094
[61] 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]. · Zbl 0947.81054 · doi:10.1016/S0550-3213(96)00410-5
[62] O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N =1 six-dimensional E8theory, Nucl. Phys.B 487 (1997) 93 [hep-th/9610251] [INSPIRE]. · Zbl 0925.14015 · doi:10.1016/S0550-3213(96)00690-6
[63] A. Sen, Orientifold limit of F-theory vacua, Phys. Rev.D 55 (1997) 7345 [hep-th/9702165] [INSPIRE].
[64] R. Donagi and M. Wijnholt, Higgs bundles and UV completion in F-theory, Commun. Math. Phys.326 (2014) 287 [arXiv:0904.1218] [INSPIRE]. · Zbl 1285.81070 · doi:10.1007/s00220-013-1878-8
[65] V. Braun and D.R. Morrison, F-theory on genus-one fibrations, JHEP08 (2014) 132 [arXiv:1401.7844] [INSPIRE]. · Zbl 1333.81200 · doi:10.1007/JHEP08(2014)132
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.