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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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 (II), 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].
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].
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].
V. Braun and D.R. Morrison, F-theory on Genus-One Fibrations, JHEP 08 (2014) 132 [arXiv:1401.7844] [INSPIRE].
D.R. Morrison and W. Taylor, Sections, multisections and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
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].
T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [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, A. Kapfer and D. Klevers, The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle, JHEP 06 (2016) 112 [arXiv:1510.04281] [INSPIRE].
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].
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].
W. Stein et al., Sage Mathematics Software (Version 7.1), The Sage Development Team (2016) http://www.sagemath.org.
A. Novoseltsev, Lattice polytope module for Sage, The Sage Development Team (2010) http://www.sagemath.org/doc/reference/geometry/sage/geometry/lattice polytope.html.
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.
L. Lin and T. Weigand, Towards the Standard Model in F-theory, Fortsch. Phys. 63 (2015) 55 [arXiv:1406.6071] [INSPIRE].
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].
V. Braun, T.W. Grimm and J. Keitel, Complete Intersection Fibers in F-theory, JHEP 03 (2015) 125 [arXiv:1411.2615] [INSPIRE].
W. Fulton, Introduction to toric varieties, volume 131, Annals of Mathematics Studies, Princeton University Press (1993).
D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, volume 68, Mathematical surveys and monographs, American Mathematical Society (1999) [INSPIRE].
D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate studies in mathematics, American Mathematical Society (2011).
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].
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].
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.
L. Borisov, Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties, alg-geom/9310001.
V.V. Batyrev and L.A. Borisov, On Calabi-Yau complete intersections in toric varieties, alg-geom/9412017 [INSPIRE].
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].
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].
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].
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].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1604.00011
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Oehlmann, PK., Reuter, J. & Schimannek, T. Mordell-Weil torsion in the mirror of multi-sections. J. High Energ. Phys. 2016, 31 (2016). https://doi.org/10.1007/JHEP12(2016)031
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2016)031