We study non-abelian discrete flavor symmetries, which can appear in magnetized brane models. For example, D 4 , ∆(27) and ∆(54) can appear and matter fields with several representations can appear. We also study the orbifold background, where non-abelian flavor symmetries are broken in a certain way. *

We study three generation models in the four-dimensional spacetime, which can be derived from the ten-dimensional N = 1 super Yang-Mills theory on the orbifold background with a non-vanishing magnetic flux. We classify the flavor structures and show possible patterns of Yukawa matrices. Some examples of numerical studies are also shown. *

We compute three-point and higher order couplings in magnetized brane models. We show that higher order couplings are written as products of three-point couplings. This behavior is the same as higher order amplitudes by conformal field theory calculations e.g. in intersecting D-brane models. *

We show that, without any extra physical degree introduced, the standard model can be readily reformulated as a double field theory. Consequently, the standard model can couple to an arbitrary stringy gravitational background in an O(4,4) T-duality covariant manner and manifest two independent local Lorentz symmetries, Spin(1,3)×Spin(3,1). While the diagonal gauge fixing of the twofold spin groups leads to the conventional formulation on the flat Minkowskian background, the enhanced symmetry makes the standard model more rigid, and also stringy, than it appeared. The CP violating θ term may no longer be allowed by the symmetry, and hence the strong CP problem can be solved. There are now stronger constraints imposed on the possible higher order corrections. We speculate that the quarks and the leptons may belong to the two different spin classes.

It is pointed out that there exist a few problems to be overcome toward an observable sub-eV QCD axion in superstring compactification. We give a general expression for the axion decay constant. For a large domain wall number N DW , the axion decay constant can be substantially lowered from a generic value of a scalar singlet VEV. The Yukawa coupling structure in the recent Z 12−I model is studied completely, including the needed nonrenormalizable terms toward realistic quark and lepton masses. In this model we find an approximate global symmetry and vacuum so that a QCD axion results but its decay constant is at the GUT scale. The axion-photon-photon coupling is calculated for a realistic vacuum satisfying the quark and lepton mass matrix conditions. It is the first time calculation of c aγγ in realistic string compactifications: c aγγ = 5 3 − 1.93 ≃ −0.26.