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F-theory on genus-one fibrations. (English) Zbl 1333.81200

Summary: We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various \({\mathrm{SU}}(5)\times\mathrm{U}(1)^n\) and \(E_6\) models are presented as examples. To each genus-one fibration one can associate a \(\tau\)-function on the base as well as an \({\mathrm{SL}}(2,\mathbb Z)\) representation which together define the IIB axio-dilaton and 7-brane content of the theory. The set of genus-one fibrations with the same \(\tau\)-function and \({\mathrm{SL}}(2,\mathbb Z)\) representation, known as the Tate-Shafarevich group, supplies an important degree of freedom in the corresponding F-theory model which has not been studied carefully until now.
Six-dimensional anomaly cancellation as well as Witten’s zero-mode count on wrapped branes both imply corrections to the usual F-theory dictionary for some of these models. In particular, neutral hypermultiplets which are localized at codimension-two fibers can arise. (All previous known examples of localized hypermultiplets were charged under the gauge group of the theory.) Finally, in the absence of a section some novel monodromies of Kodaira fibers are allowed which lead to new breaking patterns of non-Abelian gauge groups.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
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[1] C. Vafa, Evidence for F-theory, Nucl. Phys.B 469 (1996) 403 [hep-th/9602022] [INSPIRE]. · Zbl 1003.81531 · doi:10.1016/0550-3213(96)00172-1
[2] 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
[3] 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
[4] J.H. Schwarz, An SL(2,ℤ \[\mathbb{Z} )\] multiplet of type IIB superstrings, Phys. Lett.B 360 (1995) 13 [Erratum ibid.B 364 (1995) 252] [hep-th/9508143] [INSPIRE].
[5] P.S. Aspinwall, Some relationships between dualities in string theory, Nucl. Phys. Proc. Suppl.46 (1996) 30 [hep-th/9508154] [INSPIRE]. · Zbl 0957.81599 · doi:10.1016/0920-5632(96)00004-7
[6] K. Kodaira, On compact analytic surfaces. II, Ann. Math.77 (1963) 563. · Zbl 0118.15802 · doi:10.2307/1970131
[7] K. Kodaira, On compact analytic surfaces. III, Ann. Math.78 (1963) 1. · Zbl 0171.19601 · doi:10.2307/1970500
[8] N. Nakayama, On Weierstrass models, in Algebraic geometry and commutative algebra, vol. II, Kinokuniya, Japan (1988), pg. 405. · Zbl 0699.14049
[9] I. Dolgachev and M. Gross, Elliptic threefolds. I. Ogg-Shafarevich theory, J. Algebraic Geom.3 (1994) 39 [alg-geom/9210009]. · Zbl 0803.14021
[10] E. Witten, Toroidal compactification without vector structure, JHEP02 (1998) 006 [hep-th/9712028] [INSPIRE]. · Zbl 0958.81065
[11] J. de Boer et al., Triples, fluxes and strings, Adv. Theor. Math. Phys.4 (2002) 995 [hep-th/0103170] [INSPIRE]. · Zbl 1011.81065
[12] 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
[13] P. Berglund, J.R. Ellis, A.E. Faraggi, D.V. Nanopoulos and Z. Qiu, Elevating the free fermionℤ \[\mathbb{Z} 2\] × ℤ \[\mathbb{Z} 2\] orbifold model to a compactification of F-theory, Int. J. Mod. Phys.A 15 (2000) 1345 [hep-th/9812141] [INSPIRE].
[14] S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math.80 (1958) 659. · Zbl 0097.36203 · doi:10.2307/2372778
[15] I.R. Šafarevič, Principal homogeneous spaces defined over a function field, Trudy Mat. Inst. Steklov.64 (1961) 316 [AMS Transl.37 (1964) 85]. · Zbl 0129.12804
[16] N. Nakayama, Global structure of an elliptic fibration, Publ. Res. Inst. Math. Sci.38 (2002) 451. · Zbl 1039.14004 · doi:10.2977/prims/1145476270
[17] R. Donagi and T. Pantev, Torus fibrations, gerbes and duality, Mem. Amer. Math. Soc.193 (2008) vi+90 [math.AG/0306213] [INSPIRE]. · Zbl 1140.14001
[18] D.R. Morrison, Wilson lines in F-theory, lecture at Harvard University, unpublished, U.S.A. January 8 1999.
[19] P. Deligne, Courbes elliptiques: formulaire (d’après J. Tate), in Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp Belgium 1972), Lect. Notes Math.476, Springer, Berlin Germany (1975), pg. 53.
[20] D. Mumford and K. Suominen, Introduction to the theory of moduli, in Algebraic geometry, Oslo Norway 1970 (Proc. Fifth Nordic Summer-School in Math.), Wolters-Noordhoff, The Netherlands (1972), pg. 171. · Zbl 0242.14004
[21] S.Y. An et al., Jacobians of genus one curves, J. Number Theory90 (2001) 304. · Zbl 1066.14035 · doi:10.1006/jnth.2000.2632
[22] M. Artin, F. Rodriguez-Villegas and J. Tate, On the Jacobians of plane cubics, Adv. Math.198 (2005) 366. · Zbl 1092.14054 · doi:10.1016/j.aim.2005.06.004
[23] A. Weil, Remarques sur un mémoire d’Hermite (in French), Arch. Math. (Basel)5 (1954) 197. · Zbl 0056.03402 · doi:10.1007/BF01899338
[24] J.J. Duistermaat, Discrete integrable systems. QRT maps and elliptic surfaces, Springer Monographs in Mathematics, Springer, Berlin Germany (2010), pg. xxii+627. · Zbl 1219.14001
[25] 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
[26] G. Salmon, A treatise on the analytic geometry of three dimensions, vol. 1, 7th ed., C.H. Rowe ed., Chelsea Publishing Company, New York U.S.A. (1958).
[27] Sage Development Team collaboration, V. Braun and J. Keitel, Weierstrass form for complete intersection of two quadratic equations, http://trac.sagemath.org/14855, (2013).
[28] D.A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math.99 (1977) 447. · Zbl 0373.13006 · doi:10.2307/2373926
[29] T. Fisher, The invariants of a genus one curve, Proc. Lond. Math. Soc.97 (2008) 753 [math.NT/0610318]. · Zbl 1221.11135 · doi:10.1112/plms/pdn021
[30] I. Dolgachev and A. Libgober, On the fundamental group of the complement to a discriminant variety, in Algebraic geometry (Chicago U.S.A. 1980), Lect. Notes Math.862, Springer, Berlin Germany (1981), pg. 1. · Zbl 0475.14011
[31] R. Miranda, Smooth models for elliptic threefolds, in The birational geometry of degenerations (Cambridge U.S.A. 1981), Progr. Math.29, Birkhäuser, Boston U.S.A. (1983), pg. 85. · Zbl 0583.14014
[32] 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
[33] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, in Algebraic geometry, Sendai Japan 1985, Adv. Stud. Pure Math.10, North-Holland, The Netherlands (1987), pg. 283. · Zbl 0672.14006
[34] A. Grothendieck, Le groupe de Brauer, I, II, III (in French), in Dix Exposés sur la Cohomologie des Schémas, North-Holland, The Netherlands (1968), pg. 46.
[35] P.S. Aspinwall, D.R. Morrison and M. Gross, Stable singularities in string theory, Commun. Math. Phys.178 (1996) 115 [hep-th/9503208] [INSPIRE]. · Zbl 0854.53059 · doi:10.1007/BF02104911
[36] C. Vafa and E. Witten, On orbifolds with discrete torsion, J. Geom. Phys.15 (1995) 189 [hep-th/9409188] [INSPIRE]. · Zbl 0816.53053 · doi:10.1016/0393-0440(94)00048-9
[37] F. Denef, Lectures on constructing string vacua, in String theory and the real world: from particle physics to astrophysics, C. Bachas ed., Elsevier, The Netherlands (2008), pg. 483 [arXiv:0803.1194] [INSPIRE].
[38] B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nucl. Phys.B 337 (1990) 1 [INSPIRE]. · Zbl 0744.53045 · doi:10.1016/0550-3213(90)90248-C
[39] M. Gross and P.M.H. Wilson, Large complex structure limits of K3 surfaces, J. Diff. Geom.55 (2000) 475 [math.DG/0008018]. · Zbl 1027.32021
[40] J. Marsano, N. Saulina and S. Schäfer-Nameki, On G-flux, M 5 instantons and U(1)s in F-theory, arXiv:1107.1718 [INSPIRE]. · Zbl 1272.81163
[41] 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]. · Zbl 0829.14023
[42] V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP01 (2013) 016 [arXiv:1110.4883] [INSPIRE]. · Zbl 1342.81405 · doi:10.1007/JHEP01(2013)016
[43] M. Kreuzer and H. Skarke, On the classification of reflexive polyhedra, Commun. Math. Phys.185 (1997) 495 [hep-th/9512204] [INSPIRE]. · Zbl 0894.14026 · doi:10.1007/s002200050100
[44] 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
[45] P. Candelas and H. Skarke, F theory, SO(32) and toric geometry, Phys. Lett.B 413 (1997) 63 [hep-th/9706226] [INSPIRE]. · doi:10.1016/S0370-2693(97)01047-2
[46] P. Candelas, E. Perevalov and G. Rajesh, Matter from toric geometry, Nucl. Phys.B 519 (1998) 225 [hep-th/9707049] [INSPIRE]. · Zbl 0920.14018 · doi:10.1016/S0550-3213(98)00009-1
[47] 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]. · Zbl 0925.14026 · doi:10.1016/S0550-3213(97)00563-4
[48] 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
[49] 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]. · Zbl 1284.14051 · doi:10.1007/s00220-013-1802-2
[50] V. Braun, T.W. Grimm and J. Keitel, New global F-theory GUTs with U(1) symmetries, JHEP09 (2013) 154 [arXiv:1302.1854] [INSPIRE].
[51] 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
[52] Y. Hu, C.-H. Liu and S.-T. Yau, Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces, Adv. Theor. Math. Phys.6 (2003) 457 [math/0010082] [INSPIRE]. · Zbl 1033.81069
[53] Sage Development Team collaboration, V. Braun and A.Y. Novoseltsev, Toric varieties framework for Sage, http://sagemath.org/doc/reference/sage/schemes/toric/variety.html, (2010).
[54] 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
[55] 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
[56] 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
[57] 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
[58] D.S. Park and W. Taylor, Constraints on 6D supergravity theories with Abelian gauge symmetry, JHEP01 (2012) 141 [arXiv:1110.5916] [INSPIRE]. · Zbl 1306.81269 · doi:10.1007/JHEP01(2012)141
[59] A. Grassi and D.R. Morrison, Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds, J. Alg. Geom.12 (2003) 321 [IASSNS-HEP-00/27] [math.AG/0005196 [INSPIRE]. · Zbl 1080.14534 · doi:10.1090/S1056-3911-02-00337-5
[60] 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
[61] E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys.B 471 (1996) 195 [hep-th/9603150] [INSPIRE]. · Zbl 1003.81537 · doi:10.1016/0550-3213(96)00212-X
[62] H. Hayashi, C. Lawrie, D.R. Morrison and S. Schäfer-Nameki, Box graphs and singularities, to appear. · Zbl 1333.81369
[63] 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
[64] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp Belgium 1972), Lect. Notes Math.476, Springer, Berlin Germany (1975), pg. 33. · Zbl 1214.14020
[65] 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
[66] E. Witten, New ‘gauge’ theories in six-dimensions, JHEP01 (1998) 001 [hep-th/9710065] [INSPIRE]. · Zbl 0958.81166 · doi:10.1088/1126-6708/1998/01/001
[67] A. Degeratu and K. Wendland, Friendly giant meets pointlike instantons? On a new conjecture by John McKay, in Moonshine — The First Quarter Century and Beyond, A Workshop on the Moonshine Conjectures and Vertex Algebras, LMS Lect. Notes Ser.372, London Mathematical Society, London U.K. (2010). · Zbl 1238.81166
[68] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan24 (1972) 20. · Zbl 0226.14013 · doi:10.2969/jmsj/02410020
[69] J. Tate, Algebraic cycles and poles of zeta functions, in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., U.S.A. 1963), Harper & Row, U.S.A. (1965), pg. 93. · Zbl 0213.22804
[70] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, in Séminaire Bourbaki, vol. 9, Soc. Math. France, France (1995), pg. 415. · Zbl 0199.55604
[71] R. Wazir, Arithmetic on elliptic threefolds, Composit. Math.140 (2004) 567 [math.NT/0112259]. · Zbl 1060.11039 · doi:10.1112/S0010437X03000381
[72] Sage Development Team collaboration, W.A. Stein et al., Sage Mathematics Software (Version 6.1), http://www.sagemath.org/, (2012).
[73] Sage-Combinat community collaboration, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics, http://combinat.sagemath.org/, (2012).
[74] V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP04 (2011) 080 [arXiv:1011.0726] [INSPIRE]. · Zbl 1250.83058 · doi:10.1007/JHEP04(2011)080
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