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Modular transformations of string theory are shown to play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They include CP transformations and provide a unification of CP with traditional flavor symmetries within the framework of the eclectic flavor scheme. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of Local Flavor Unification, where different sectors of the theory (like quarks and leptons) can be subject to different flavor structures.
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It has recently been shown that F-theory based constructions provide a potentially promising avenue for engineering GUT models which descend to the MSSM. In this note we show that in the presence of background fluxes, these models automatically achieve hierarchical Yukawa matrices in the quark and lepton sectors. At leading order, the existence of a U(1) symmetry which is related to phase rotations of the internal holomorphic coordinates at the brane intersection point leads to rank one Yukawa matrices. Subleading corrections to the internal wave functions from variations in the background fluxes generate small violations of this U(1), leading to hierarchical Yukawa structures reminiscent of the Froggatt-Nielsen mechanism. The expansion parameter for this perturbation is in terms of alpha_(GUT)^(1/2). Moreover, we naturally obtain a hierarchical CKM matrix with V_(12) ~ V_(21) ~ epsilon, V_(23) ~ V_(32) ~ epsilon^(2), V_(13) ~ V_(31) ~ epsilon^(3), where epsilon ~ alpha_(GUT)^(1/2), in excellent agreement with observation.
We argue that CP is a gauge symmetry in string theory. As a consequence, CP cannot be explicitly broken either perturbatively or non-pertubatively; there can be no non-perturbative CP-violating parameters. String theory is thus an example of a theory where all $theta$ angles arise due to spontaneous CP violation, and are in principle calculable.
We argue that in large classes of string compactifications with a MSSM-like structure substantial flavor violating SUSY-breaking soft terms are generically induced. We specify to the case of flavor dependent soft-terms in type IIB/F-theory SU(5) unified models, although our results can be easily extended to other settings. The Standard Model (SM) degrees of freedom reside in a local system of 7-branes wrapping a 4-fold S in the extra dimensions. It is known that in the presence of closed string 3-form fluxes SUSY-breaking terms are typically generated. We explore the generation dependence of these soft terms and find that non-universalities arise whenever the flux varies over the 4-fold S. These non-universalities are parametrically suppressed by (M_{GUT}/M_{Pl})^{1/3}. They also arise in the case of varying open string fluxes, in this case parametrically suppressed by alpha_{GUT}^{1/2}. For a standard unification scheme with M_{GUT} = 10^{16} GeV and alpha_{rm GUT} = 1/24 these suppressions are very mild. Although limits from the kaon mass difference Delta m_K are easily obeyed for squark masses above the present LHC limits, constraints from the CP-violation parameter epsilon_K imply squark masses in the multi-TeV region. The constraints from BR(mu rightarrow egamma) turn out to be the strongest ones, with slepton masses of order ~10 TeV or heavier required to obey the experimental limits. These sfermion masses are consistent with the observed large value m_H ~ 126 GeV of the Higgs mass. We discuss under what conditions such strong limits may be relaxed allowing for SUSY particle production at LHC.
Using Z3 asymmetric orbifolds in heterotic string theory, we construct N=1 SUSY three-generation models with the standard model gauge group SU(3)_C times SU(2)_L times U(1)_Y and the left-right symmetric group SU(3)_C times SU(2)_L times SU(2)_R times U(1)_{B-L}. One of the models possesses a gauge flavor symmetry for the Z3 twisted matter.
We study the modular symmetry in magnetized D-brane models on $T^2$. Non-Abelian flavor symmetry $D_4$ in the model with magnetic flux $M=2$ (in a certain unit) is a subgroup of the modular symmetry. We also study the modular symmetry in heterotic orbifold models. The $T^2/Z_4$ orbifold model has the same modular symmetry as the magnetized brane model with $M=2$, and its flavor symmetry $D_4$ is a subgroup of the modular symmetry.
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