Journal of High Energy Physics

, 2016:31 | Cite as

Mordell-Weil torsion in the mirror of multi-sections

  • Paul-Konstantin Oehlmann
  • Jonas Reuter
  • Thorsten Schimannek
Open Access
Regular Article - Theoretical Physics


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.


Differential and Algebraic Geometry F-Theory Global Symmetries String Duality 


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.


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Paul-Konstantin Oehlmann
    • 1
    • 2
  • Jonas Reuter
    • 2
  • Thorsten Schimannek
    • 2
  1. 1.Physics Department, Virginia TechBlacksburgU.S.A.
  2. 2.Bethe Center for Theoretical PhysicsPhysikalisches Institut der Universität BonnBonnGermany

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