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The arithmetic of elliptic fibrations in gauge theories on a circle

  • Thomas W. Grimm
  • Andreas Kapfer
  • Denis Klevers
Open Access
Regular Article - Theoretical Physics

Abstract

The geometry of elliptic fibrations translates to the physics of gauge theories in F-theory. We systematically develop the dictionary between arithmetic structures on elliptic curves as well as desingularized elliptic fibrations and symmetries of gauge theories on a circle. We show that the Mordell-Weil group law matches integral large gauge transformations around the circle in Abelian gauge theories and explain the significance of Mordell-Weil torsion in this context. We also use Higgs transitions and circle large gauge transformations to introduce a group law for genus-one fibrations with multi-sections. Finally, we introduce a novel arithmetic structure on elliptic fibrations with non-Abelian gauge groups in F-theory. It is defined on the set of exceptional divisors resolving the singularities and divisor classes of sections of the fibration. This group structure can be matched with certain integral non-Abelian large gauge transformations around the circle when studying the theory on the lower-dimensional Coulomb branch. Its existence is required by consistency with Higgs transitions from the non-Abelian theory to its Abelian phases in which it becomes the Mordell-Weil group. This hints towards the existence of a new underlying geometric symmetry.

Keywords

Effective field theories F-Theory Gauge Symmetry M-Theory 

Notes

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

  • Thomas W. Grimm
    • 1
    • 2
    • 3
  • Andreas Kapfer
    • 1
  • Denis Klevers
    • 4
  1. 1.Max-Planck-Institut für PhysikMunichGermany
  2. 2.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands
  3. 3.Center for Extreme Matter and Emergent PhenomenaUtrecht UniversityUtrechtThe Netherlands
  4. 4.Theory Group, Physics DepartmentCERNGeneva 23Switzerland

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