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F-theory fluxes, Chirality and Chern-Simons theories
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We study the charged chiral matter spectrum of four-dimensional F-theory compactifications on elliptically fibered Calabi-Yau fourfolds by using the dual M-theory description. A chiral spectrum can be induced by M-theory four-form flux on the fully resolved Calabi-Yau fourfold. In M-theory this flux yields three-dimensional Chern-Simons couplings in the Coulomb branch of the gauge theory. In the F-theory compactification on an additional circle these couplings are only generated by one-loop corrections with charged fermions running in the loop. This identification allows us to infer the net number of chiral matter fields of the four-dimensional effective theory. The chirality formulas can be evaluated by using the intersection numbers and the cones of effective curves of the resolved fourfolds. We argue that a study of the effective curves also allows to follow the resolution process at each co-dimension. To write simple chirality formulas we suggest to use the effective curves involved in the resolution process to determine the matter surfaces and to connect with the group theory at co-dimension two in the base. We exemplify our methods on examples with SU(5) and SU(5)xU(1) gauge group.
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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.
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