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Article
Report number	arXiv:1510.04281 ; CERN-PH-TH-2015-230 ; CERN-PH-TH-2015-230
Title	The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle
Author(s)	Grimm, Thomas W. (Munich, Max Planck Inst. ; Utrecht U.) ; Kapfer, Andreas (Munich, Max Planck Inst.) ; Klevers, Denis (CERN)
Publication	2016-06-20
Imprint	14 Oct 2015
Number of pages	44
Note	43 pages, 3 figures
In:	JHEP 06 (2016) 112
DOI	10.1007/JHEP06(2016)112
Subject category	Particle Physics - Theory
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.
Copyright/License	arXiv nonexclusive-distrib. 1.0 publication: © 2016-2025 The Author(s) (License: CC-BY-4.0), sponsored by SCOAP³ preprint: © 2015-2025 CERN (License: CC-BY-4.0)

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