Abstract
We study the moduli stabilization from the viewpoint of modular flavor symmetries. We systematically analyze stabilized moduli values in possible configurations of flux compactifications, investigating probabilities of moduli values and showing which moduli values are favorable from our moduli stabilization. Then, we examine their implications on modular symmetric flavor models. It is found that distributions of complex structure modulus τ determining the flavor structure are clustered at a fixed point with the residual ℤ3 symmetry in the SL(2, ℤ) fundamental region. Also, they are clustered at other specific points such as intersecting points between |τ|2 = k/2 and Re τ = 0,±1/4,±1/2, although their probabilities are less than the ℤ3 fixed point. In general, CP-breaking vacua in the complex structure modulus are statistically disfavored in the string landscape. Among CP-breaking vacua, the values Re τ = ±1/4 are most favorable in particular when the axio-dilaton S is stabilized at Re S = ±1/4. That shows a strong correlation between CP phases originated from string moduli.
Highlights
The modular group transforms non-trivially zero-modes corresponding to quarks and leptons in four-dimensional (4D) effective field theory
We systematically analyze stabilized moduli values in possible configurations of flux compactifications, investigating probabilities of moduli values and showing which moduli values are favorable from our moduli stabilization
Unification of the modular symmetry and the 4D CP is developed in the heterotic string theory and in Type IIB string theory, where the 4D CP is identified with an outer automorphism of the SL(2, Z) modular group on toroidal background [19, 20] and Sp(2n, Z) modular group on Calabi-Yau threefolds [21]
Summary
We briefly review the phenomenologically attractive modular symmetry appearing in the string-derived effective action, with an emphasis on Type IIB supergravity on the factorizable T 6 torus and its orientifold. The dynamics of the moduli fields is discussed in the context of flux compactifications as reviewed in section 2.2 on a simple T 6/(Z2 × Z2) orientifold background. The reader who are interested in the distributions of the moduli fields may skip to section 3
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