# Mordell-Weil torsion in the mirror of multi-sections

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## Abstract

We give further evidence that genus-one fibers with multi-sections are mirror dual to fibers with Mordell-Weil torsion. In the physics of F-theory compactifications this implies a relation between models with a non-simply connected gauge group and those with discrete symmetries. We provide a combinatorial explanation of this phenomenon for toric hypersurfaces. In particular this leads to a criterion to deduce Mordell-Weil torsion directly from the polytope. For all 3134 complete intersection genus-one curves in three-dimensional toric ambient spaces we confirm the conjecture by explicit calculation. We comment on several new features of these models: the Weierstrass forms of many models can be identified by relabeling the coefficient sections. This reduces the number of models to 1024 inequivalent ones. We give an example of a fiber which contains only non-toric sections one of which becomes toric when the fiber is realized in a different ambient space. Similarly a singularity in codimension one can have a toric resolution in one representation while it is non-toric in another. Finally we give a list of 24 inequivalent genus-one fibers that simultaneously exhibit multi-sections and Mordell-Weil torsion in the Jacobian. We discuss a self-mirror example from this list in detail.

## Keywords

Differential and Algebraic Geometry F-Theory Global Symmetries String Duality## Notes

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## References

- [1]C. Vafa,
*Evidence for F-theory*,*Nucl. Phys.***B 469**(1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [2]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].ADSMathSciNetCrossRefMATHGoogle Scholar - [3]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].ADSMathSciNetCrossRefGoogle Scholar - [4]P.S. Aspinwall and D.R. Morrison,
*Nonsimply connected gauge groups and rational points on elliptic curves*,*JHEP***07**(1998) 012 [hep-th/9805206] [INSPIRE].ADSCrossRefGoogle Scholar - [5]C. Mayrhofer, D.R. Morrison, O. Till and T. Weigand,
*Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory*,*JHEP***10**(2014) 016 [arXiv:1405.3656] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [6]V. Braun and D.R. Morrison,
*F-theory on Genus-One Fibrations*,*JHEP***08**(2014) 132 [arXiv:1401.7844] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [7]D.R. Morrison and W. Taylor,
*Sections, multisections and*U(1)*fields in F-theory*, arXiv:1404.1527 [INSPIRE]. - [8]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].ADSCrossRefGoogle Scholar - [9]T. Banks and N. Seiberg,
*Symmetries and Strings in Field Theory and Gravity*,*Phys. Rev.***D 83**(2011) 084019 [arXiv:1011.5120] [INSPIRE].ADSGoogle Scholar - [10]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].ADSCrossRefGoogle Scholar - [11]T.W. Grimm, A. Kapfer and D. Klevers,
*The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle*,*JHEP***06**(2016) 112 [arXiv:1510.04281] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [12]M. Cvetič, R. Donagi, D. Klevers, H. Piragua and M. Poretschkin,
*F-theory vacua with ℤ*_{3}*gauge symmetry*,*Nucl. Phys.***B 898**(2015) 736 [arXiv:1502.06953] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [13]L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa,
*F-theory and the Classification of Little Strings*,*Phys. Rev.***D 93**(2016) 086002 [arXiv:1511.05565] [INSPIRE].ADSMathSciNetGoogle Scholar - [14]W. Stein et al.,
*Sage Mathematics Software (Version 7.1)*, The Sage Development Team (2016) http://www.sagemath.org. - [15]A. Novoseltsev,
*Lattice polytope module for Sage*, The Sage Development Team (2010) http://www.sagemath.org/doc/reference/geometry/sage/geometry/lattice polytope.html. - [16]V. Braun and A. Novoseltsev,
*Toric geometry module for Sage*, The Sage Development Team (2013) http://www.sagemath.org/doc/reference/schemes/sage/schemes/toric/variety.html. - [17]L. Lin and T. Weigand,
*Towards the Standard Model in F-theory*,*Fortsch. Phys.***63**(2015) 55 [arXiv:1406.6071] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [18]M. Cvetič, D. Klevers, D.K.M. Peña, P.-K. Oehlmann and J. Reuter,
*Three-Family Particle Physics Models from Global F-theory Compactifications*,*JHEP***08**(2015) 087 [arXiv:1503.02068] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [19]V. Braun, T.W. Grimm and J. Keitel,
*Complete Intersection Fibers in F-theory*,*JHEP***03**(2015) 125 [arXiv:1411.2615] [INSPIRE]. - [20]W. Fulton,
*Introduction to toric varieties*, volume 131, Annals of Mathematics Studies, Princeton University Press (1993).Google Scholar - [21]D.A. Cox and S. Katz,
*Mirror symmetry and algebraic geometry*, volume 68, Mathematical surveys and monographs, American Mathematical Society (1999) [INSPIRE]. - [22]D. Cox, J. Little and H. Schenck,
*Toric Varieties*, Graduate studies in mathematics, American Mathematical Society (2011).Google Scholar - [23]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].ADSMathSciNetGoogle Scholar - [24]M. Kreuzer and H. Skarke,
*PALP: A Package for analyzing lattice polytopes with applications to toric geometry*,*Comput. Phys. Commun.***157**(2004) 87 [math.NA/0204356] [INSPIRE]. - [25]J. Reuter,
*Non-perturbative aspects of string theory from elliptic curves*, Ph.D. Thesis, Universität Bonn, Bonn Germany (2015) [INSPIRE] and online pdf version at http://hss.ulb.uni-bonn.de/2015/4107/4107.htm. - [26]L. Borisov,
*Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties*, alg-geom/9310001. - [27]V.V. Batyrev and L.A. Borisov,
*On Calabi-Yau complete intersections in toric varieties*, alg-geom/9412017 [INSPIRE]. - [28]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].ADSMathSciNetCrossRefMATHGoogle Scholar - [29]R. Blumenhagen, B. Jurke, T. Rahn and H. Roschy,
*Cohomology of Line Bundles: A Computational Algorithm*,*J. Math. Phys.***51**(2010) 103525 [arXiv:1003.5217] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [30]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].ADSCrossRefGoogle Scholar - [31]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].ADSMathSciNetCrossRefMATHGoogle Scholar