F-theory for Engineering Phenomenology Models Sample Clauses

F-theory for Engineering Phenomenology Models. ‌ This section fleshes out some of the ideas of section 1.5. It also justifies the promises made there about the phenomenological attractiveness of F-theory compactifications, by presenting an implementation of the SU (5) GUT model from section 1.3 in F-theory. First and foremost, an embedding of the SU (5) GUT requires an SU (5) gauge group in the compactification. We have seen that this gauge group can be realized on a 7-brane in F-theory, by considering an elliptic fibration over some base manifold B with a singularity of Kodaira type I5. If B has more than one complex dimension – and, as GUTs are to model the physics of the universe, B should be complex three-dimensional – one expects this singularity to enhance in codimension two. The two loci over which this happens have been discussed in section 3.3, and the matter multiplets arising over them transform in the fundamental and antifundamental 5 and 5, as well as the antisymmetric 10/10 representation of SU (5). These are precisely the representations required to engineer the Standard Model fermions and Xxxxx bosons. In order to construct the three generations of quarks and leptons, one has to find a G- flux configuration such that the chiral index of section 3.5 over the 10 matter curve is χc10 = +3. Also, one needs χc5 = −3, since the right-handed down-type quarks and the leptons transform in the 5 representation in an SU (5) GUT. Recall that the two Yukawa couplings generically arising in an I5 singularity at codimen- sion three are 10 5 5 and 10 10 5. After the breaking of SU (5) to GSM , the former can give the down-type Yukawa couplings of the Standard Model, while the latter is suited to replicate the up-type Yukawa couplings, most notably the top Yukawa coupling. Unlike in compactifications of type IIA/IIB string theory, the top Yukawa coupling is presented generically in F-theory models, and does not have to be generated by e.g. instanton effects with small coupling constants. This is rooted in the fact that the singular fiber at the top Yukawa point enhances to Kodaira type IV ∗ with associated Lie group E6. Type IIA/IIB compactifications do not contain any exceptional gauge symmetries, and their appearance in F-theory can be understood as a consequence of the theory being genuinely non-perturbative. Lastly, there are various options to break the SU (5) GUT to the Standard Model gauge group in F-theory. A way of doing so is via hypercharge flux, i.e., gauge flux associated to the generator o...
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