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.
Highlights
That there is no notion of an actual twelve-dimensional background geometry
A similar replacement should apply for unHiggsing a higher rank Mordell-Weil group by induction on its rank, as discussed in [17]. This is expected to establish the existence of the group law postulated in subsection 6.1 on the Cartan divisors of any non-Abelian gauge group in F-theory that can be Higgsed in an adjoint Higgsing to a purely Abelian gauge group
We have systematically studied the relationship between the F-theory effective theory on a circle and the geometry of resolved elliptic fibrations
Summary
We discuss a number of symmetries that are encountered when considering six- and four-dimensional gauge theories on a circle. In subsection 2.1 we first introduce our notation and summarize some basic facts about circle compactifications of gauge theories. In subsection 2.2 we study the manifestation of large gauge transformations with gauge parameters supported on the circle. We comment on the rearrangement of the Kaluza-Klein spectrum and the significance of one-loop induced Chern-Simons terms on the Coulomb branch of the gauge theory
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