On bisection loci in a four-section geometry, the F-theory gauge group enlarges from Z2 to U(1) times Z2, and Higgsing can break it down to a discrete Z4 gauge group.
A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations
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abstract
We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantities can be computed in terms of topological invariants in straight manner. Since the general fiber is defined by a cubic curve, basic arithmetic operations on the curve can be done using the chord-tangent group law. We will use this model to determine the spectrum of singular fibers of an elliptic fibration of rank one and compute a generating function for its Euler characteristic. With a view toward string theory, we determine a semi-stable degeneration which is understood as a weak coupling limit in F-theory. We show that it satisfies a non-trivial topological relation at the level of homological Chern classes. This relation ensures that the D3 charge in F-theory is the same as the one in the weak coupling limit.
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2019 1verdicts
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F-theory models with $U(1)\times \mathbb{Z}_2,\, \mathbb{Z}_4$ and transitions in discrete gauge groups
On bisection loci in a four-section geometry, the F-theory gauge group enlarges from Z2 to U(1) times Z2, and Higgsing can break it down to a discrete Z4 gauge group.