Abstract
Discrete gauge groups naturally arise in F-theory compactifications on genus-one fibered Calabi–Yau manifolds. Such geometries appear in families that are parameterized by the Tate–Shafarevich group of the genus-one fibration. While the F-theory compactification on any element of this family gives rise to the same physics, the corresponding M-theory compactifications on these geometries differ and are obtained by a fluxed circle reduction of the former. In this note, we focus on an element of order three in the Tate–Shafarevich group of the general cubic. We discuss how the different M-theory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles in I2-fibers that appear at certain codimension two loci in the base. We explicitly identify all three curves that give rise to the corresponding Higgs fields. In this analysis the investigation of different resolved phases of the underlying geometry plays a crucial rôle.
Highlights
Discrete symmetries play a key rôle for constructing extensions of the standard model of particle physics
Discrete symmetries are used to forbid terms in the MSSM superpotential that would allow for fast proton decay or other processes which are highly suppressed in the standard model
In this note we have considered the physics of F-theory compactifications with Z3 discrete gauge group using M-/F-theory duality
Summary
Discrete symmetries play a key rôle for constructing extensions of the standard model of particle physics. For each choice of Wilson line, there is one Kaluza-Klein mode within each tower that becomes massless and serves as fivedimensional Higgs field which - once they acquire a vacuum expectation value - lead to n different five-dimensional vacua. These are to be compared with the n different M-theory compactifications on the respective elements of the TS group. For this analysis we discuss two different resolution phases of the dP1 geometry
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