Flavor, CP and Metaplectic Modular Symmetries in Type IIB Chiral Flux Vacua

We examine symmetries of chiral four-dimensional vacua of Type IIB flux compactifications with vanishing superpotential $W=0$. We find that the ${\cal N}=1$ supersymmetric MSSM-like and Pati-Salam vacua possess enhanced discrete symmetries in the effective action below the mass scale of stabilized complex structure moduli and dilaton. Furthermore, a generation number of quarks/leptons is small on these vacua where the flavor, CP and metaplectic modular symmetries are described in the framework of eclectic flavor symmetry.


Introduction
The string theory predicts a huge number of low-energy effective field theories, the so-called string theory landscape.In particular, background fluxes in extra-dimensional spaces lead to the rich and attractive vacuum structure of the string landscape, which will be quantified by a statistical study [1][2][3] as well as the swampland program [4][5][6] 1 .It is known that the statistical study of Type IIB flux vacua is a powerful approach to address the vacuum distribution and selection rules on the moduli spaces.
In Type IIB flux compactifications on T 6 /(Z 2 × Z ′ 2 ) orientifolds, the distribution of complex structure moduli fields was known to be clustered at fixed points of SL(2, Z) modular symmetry of the torus [8,9], where the fixed points in the SL(2, Z) moduli space correspond to τ = i, ω, i∞ with ω = −1+ √ 3i 2 , each with enhanced symmetries.Remarkably, probabilities of moduli values are peaked at a Z 3 fixed point τ = ω, indicating that discrete Z 3 symmetry remains in the low-energy effective action of moduli fields [9].Such a novel feature about the distribution of flux vacua was explored in this simple toroidal orientifold but it is expected to appear in a more generic Calabi-Yau moduli space with symplectic modular symmetry.
In this paper, we further examine semi-realistic four-dimensional (4D) vacua with Standard Model (SM) spectra.Since a generation number of fermions is determined by background fluxes on magnetized D-branes, the generation number and three-form fluxes stabilizing moduli fields will be correlated through tadpole cancellation conditions of D-branes.It is interesting to reveal how much 3-generation models are distributed in the flux landscape.Furthermore, the flavor symmetries of quarks and leptons will also be related to the modular symmetry of the torus, because moduli-dependent Yukawa couplings transform under the moduli symmetry [10].For illustrative purpose, we deal with the simple T 6 /(Z 2 × Z ′ 2 ) orientifolds.By analyzing physically-distinct configurations of background fluxes leading to vanishing superpotential W = 0, we find that the generation number of quarks and leptons is restricted to be small due to the tadpole cancellation condition.Furthermore, flavor symmetry, CP, and modular symmetry in semi-realistic 4D vacua are uniformly described in the context of eclectic flavor symmetry [11,12] as developed in both the top-down and bottom-up approaches [11][12][13][14][15][16].
This paper is organized as follows.In Sec. 2, we first review the flux compactifications on T 6 /(Z 2 × Z ′ 2 ).Next, we incorporate specific magnetized D-brane models without and with a discrete B field in Secs.2.2 and 2.3, respectively.It turned out that the string landscape leads to the small generation number of quarks and leptons.In Sec. 3, we begin with the metaplectic modular symmetry in Sec.3.1, which can be realized in T 2 and T 2 /Z 2 orbifold with magnetic fluxes as discussed in Secs.3.2 and 3.3, respectively.The CP transformation will be unified in the context of generalized modular symmetry in Sec.3.4.Finally, we discuss the unification of flavor, CP, and modular symmetries in Type IIB chiral 4D flux vacua in Sec.3.5.Sec. 4 is devoted to the conclusion.

Moduli distributions in Type IIB flux vacua with SM spectra
In Sec.2.1, we first review the vacuum structure of Type IIB flux compactification on T 6 /(Z 2 × Z ′ 2 ) orientifolds.Next, we introduce semi-realistic magnetized D-brane models in Type IIB flux vacua, taking into account the tadpole cancellation conditions in Secs.2.2 and 2.3.It is found that the generation number of quarks and leptons is restricted to be small due to the tadpole cancellation condition.
It was known that the 4D kinetic terms of the closed string moduli, i.e., three complex structure moduli τ i , the axio-dilaton S and three Kähler moduli T i are derived from the following Kähler potential in units of the reduced Planck mass M Pl = 1: where V denotes the torus volume in units of the string length l s = 2π √ α ′ .The moduli superpotential is induced by background three-form fluxes in Type IIB string theory.Throughout this paper, we focused on the stabilization of complex structure moduli and axio-dilaton.Let us introduce the background Ramond-Ramond (RR) F 3 and Neveu-Schwarz three forms H 3 as follows: where {a 0,1,2,3 , b 0,1,2,3 , c 0,1,2,3 , d 0,1,2,3 } correspond to the integral flux quanta.They lead to the flux-induced superpotential in the 4D effective action [19]: In Ref. [20], the moduli stabilization was performed in the isotropic regime, namely with overall flux quanta: (2.13) The moduli vacuum expectation values (VEVs) are given by for the axio-dilaton and for the overall complex structure modulus, respectively.Here, we redefine the flux quanta Here, we focus on supersymmetric W = 0 minimum.To stabilize Kähler moduli, we will assume non-perturbative dilaton-dependent superpotential W ∼ e −aS to realize constant superpotential below the mass scale of axio-dilaton and complex structure moduli.For more details, see, Ref. [21].
Since the effective action is invariant under the SL(2, Z) τ ≡ SL(2, Z) 1 = SL(2, Z) 2 = SL(2, Z) 3 and SL(2, Z) S modular symmetries2 , one can count finite number of physicallydistinct flux vacua. 3Note that we have to be careful about the tadpole cancellation condition of D-brane charges because we deal with a compact manifold.In particular, we focus on the cancellation condition of the D3-brane charge, and other conditions will be analyzed in the next subsections.Specifically, the flux-induced D3-brane charge In general, it is difficult to stabilize all the moduli fields including twisted moduli localized at orbifold fixed points in addition to untwisted moduli we focused.If Type IIB orientifolds are uplifted to the F-theory in the strong coupling regime, N max flux = O(10 5 ) will be a largest value as discussed in Refs.[23,24].In our analysis, we adopt a phenomenological approach such that we simply ignore the concrete tadpole bound and explore the interplay between moduli stabilization and model building.This approach allows us to understand the vacuum structure of the string landscape more specifically, as will be shown later.Furthermore, each flux quantum is in multiple of 8, that is, {a 0 , a, b, b 0 , c 0 , c, d, d 0 } ∈ 8 Z, and correspondingly N flux ∈ 192 Z. Since the effective action, as well as the tadpole charge, are invariant under the modular symmetry, one can map the moduli VEVs into the fundamental domains.The number of stable vacua is shown in Figure 1, from which there is huge degeneracy at the fixed points in the SL(2, Z) τ moduli space.In particular, the τ = ω vacuum is realized by a high probability such as 62.3 % for N max flux = 192 × 10 and 40.3 % for N max flux = 192 × 1000 [9].It can also be justified in a statistical argument.By taking the flux quanta as the continuous one, the number of supersymmetric W = 0 vacua is analytically estimated as [8]  Here, gcd(l, m, n) = 1 is adopted in the analysis of Ref. [8], but the results are the same with our results as pointed out in Ref. [9].Remarkably, τ = ω corresponding to (l, m, n) = (1, −1, 1) is invariant under the discrete Z 3 symmetry, generated by where S and T are generators of SL(2, Z) τ : with (ST ) 3 = 1.Thus, the effective action in Type IIB flux landscape enjoys the discrete Z 3 symmetry.However, it is unclear whether such a Z 3 symmetry still remains in the effective action with the SM spectra.In the next section, we will engineer the semi-realistic SM-like models on magnetized D-branes and discuss the role of discrete symmetry.

Distribution of g-generation models without discrete B field
In addition to O3-and O7-planes located at fixed loci, we construct semi-realistic models on N a stacks of magnetized D(3 + 2n)-branes wrapping 2n-cycles on T 6 /(Z 2 × Z ′ 2 ) orientifolds.
We turn on the background U (1) a gauge field strength F a on (T 2 ) i , where wrapping numbers of N a D(3 + 2n)-branes on (T 2 ) i are represented by integers m i a with non-vanishing 0, 1, 2 and, 3 values on D3-, D5-, D7-, D9-branes, respectively.Note that {n i a , m i a } for each a and i are assumed to be coprime numbers, and only the wrapping number m i a transforms as ΩR : m i a → −m i a under ΩR.For practical purposes, let us introduce the homology classes of each (T 2 ) i , that is, [0] i and [T 2 ] i for the class of point and of the two-torus with Then, the stack a of D-branes has an associated homology class: Similarly, the 64 O3-and 4 O7 i planes are expressed by RR charges -32 times the following homology classes: Remarkably, these gauge fluxes will lead to semi-realistic D-brane models, that is, gauge groups G SM × G ′ with chiral spectra.In particular, the index theorem tells us that the number of chiral zero-modes between two stacks a and b of D-branes on T 6 = (T 2 ) 1 × (T 2 ) 2 × (T 2 ) 3 is counted by However, some of the couplings of zero-modes are projected out by Z 2 × Z ′ 2 projection (2.1).Indeed, internal fermionic wavefunctions transform as where s i = sign(I i ab ) corresponds to the chirality on each torus and its product (s 1 s 2 s 3 ) corresponds to the 4D chirality.Thus, there exist Z 2 -even and -odd modes on each torus whose explicit form of zero-mode wavefunctions is shown later.Note that the two conditions should be consistent with each other; that is, from which allowed zero-modes are given by a specific combination of Z 2 -even modes (ψ i even ) and Z 2 -odd zero-modes (ψ i odd ) on (T 2 ) i : with i ̸ = j ̸ = k.Since the number of these Z 2 -even and -odd zero-modes is counted by [25] with f i = 1 for odd I i ab and f i = 2 for even I i ab , the total number of zero-modes is still described by Here, we assume I i ab ̸ = 0.If one of the indices is 0, e.g., I 3 ab = 0, the spectrum is not chiral, and the index is counted by [25] On N a stack of D-branes that does not lie on one of the O-planes, the mass spectra consist of U (N a /2) vector multiplets and three adjoint chiral multiplets (called a aa sector) 4 .On the other hand, when 2N a stack of D-branes lies on one of the O-planes, the mass spectra consist of U Sp(N a ) vector multiplets and three antisymmetric chiral multiplets, which we also call a aa sector.In addition, there are chiral multiplets that arise from intersections of two different stacks a and b of D-branes or the stack a and its orientifold image a ′ , as summarized in Table 1.

Sectors
Representations Multiplicities ab + ba ( a , b ) Table 1.Multiplicities of chiral zero-modes in each sector.
Since the magnetic fluxes induce the D3-and D7-brane charges, we have to be careful about their tadpole cancellation conditions: If there exists D9-branes with constant magnetic fluxes, they are mapped to anti D9-branes with the opposite magnetic fluxes under the orientifold involution.Thus, D9-brane tadpole charges are canceled.Similar things happen for D5-branes with constant magnetic fluxes as well.These conditions play a role of the cancellation of 4D chiral anomalies, but K-theory conditions require extra constraints.Indeed, probe D3 and D7-branes with U Sp(2) ≃ SU (2) gauge group suffer from a global gauge anomaly if the number of 4D fermions charged in the fundamental representation of SU (2) is odd [26].It imposes the following K-theory constraints [27]: Since magnetized D9-branes with negative n 1,2,3 will carry anti D3-and D7-brane charges, it will be possible to construct semi-realistic 3-generation models on the flux background (see, e.g., [28,29]).In these analyses, we have not introduced anti-D3 branes satisfying tadpole cancellation condition, but it would be possible to construct realistic models, taking into account the effect of anti-D3 brane annihilations with flux [30].Note that the N = 1 supersymmetry on the orientifold background will be preserved when the following condition is satisfied [31]: with where A i denote the area of the torus (T 2 ) i .
For concreteness, let us consider the local brane configurations with SM spectra as shown in Table 2 [28], leading to g generation of quarks and leptons Table 2. D-brane configurations leading to left-right symmetric Minimal Supersymmetric Standard Model (MSSM).The magnetic flux g determines the generations of quark and lepton chiral multiplets in the visible sector.
The supersymmetry condition (2.34) is satisfied when Furthermore, some of U (1)s become massive by absorbing axions associated with Ramond-Ramond fields through the Green-Schwarz mechanism.Indeed, the dimensional reduction of the Chern-Simons couplings in the D-brane action induces the corresponding 4D couplings: with To satisfy the tadpole cancellation conditions, we have also supposed the existence of magnetized D9-branes to satisfy the tadpole cancellation conditions.It means that the D3-brane charge induced by the magnetic flux on D9-branes Since there are several possibilities for the choice of magnetized D9-brane sectors,we freely change the value of Q hid D3 to reveal the mutual relation between the generation number g and the flux quanta N flux . 6In Fig. 2, we change the maximum value of Q hid D3 as |Q hid D3 | = 400, 1200, 2000, each which we analyze the distribution of flux vacua at fixed τ with respect to g.It turns out that the number of flux vacua increases when g is smaller.Thus, the small generation number is favored in the string landscape.Furthermore, when we restrict ourselves to three-generation models, that is, g = 3, left-right MSSM-like models are still peaked at the Z 3 fixed point τ = ω as shown in Fig. 3.This phenomenon is similar to the analysis of Sec.2.1, but the percentage of three-generation clustered regions differs from before.One can further study the Yukawa couplings derived in Type IIB magnetized D-brane models [10].In the current brane configuration, the Yukawa couplings of quarks and leptons are rank one, and the flavor structure is trivial due to the fact that the flavor structure is realized from two different tori.Thus, we move on to the other magnetized D-brane model, inducing the non-trivial flavor structure of quarks and leptons.

Distribution of g-generation models with discrete B field
In this section, we add a discrete value of Kalb-Ramond B-field along one of the two-tori [32], in particular, (T 2 ) 3 , corresponding to the twisted torus in the T-dual IIA string theory.Since B-field induces the half-integer flux, the magnetic flux on the third torus is modified as ñ3 a = n 3 a + 1 2 m 3 a .According to it, the tadpole cancellation conditions are given by [33] 7 D3 : (2.40) The cancellations of D5-and D9-brane charges are realized as mentioned below Eq. (2.32).
The other SUSY condition (2.34) and K-theory condition (2.33) are also written in terms of ñ3 a .For concreteness, let us consider the local brane configurations with SM spectra as shown in Table 3, leading to g generation of quarks and leptons I ab = I ca = g. 8The supersymmetry condition (2.34) is satisfied when In this model, there U (1)s in the gauge symmetry Table 3. D-brane configurations leading to Pati-Salam-like model.The magnetic flux g determines the generations of quark and chiral chiral multiplets in the visible sector, where ñ = n + m/2.
For the same reason as the analysis of the previous section, we allow several values of 7 When we consider a different tilted direction, the effective flux is given by m3 a = m 3 a + 1 2 n 3 a as discussed in the T-dual IIA side [34]. 8Similar brane configurations are discussed in T-dual Type IIA string theory, e.g., [35].
to reveal the mutual relation between the generation number g and the flux quanta N flux .In Fig. 4, we change the maximum value of Q hid D3 as |Q hid D3 | = 200, 400, 800, each which we analyze the distribution of flux vacua at fixed τ with respect to g.It turns out that the number of flux vacua also increases when g is smaller, although the behavior is different from the previous analysis.Thus, the string landscape leads to the small generation number.Furthermore, when we restrict ourselves to three-generation model, that is, g = 3, Pati-Salam models are still peaked at the Z 3 fixed point τ = ω in a similar to the analysis of Sec.2.1.In contrast to the previous models in Sec.2.2, the Yukawa couplings of quarks and leptons are rank 3 and the flavor structure is non-trivial due to the fact that the flavor structure is originated from one of tori.We will discuss the relation between flavor symmetries and modular symmetries in the next section.

Eclectic Flavor Symmetry in Type IIB flux vacua
So far, we have studied distribution of moduli fields and remaining modular symmetry in the low-energy effective action.In this section, we discuss a flavor and CP symmetries of degenerate chiral zero-modes on D-branes and its relation to modular symmetry.In Secs.3.2 and 3.3, we show that the metaplectic modular symmetry introduced in Sec.3.1 is useful to describe the matter wavefunctions and Yukawa couplings on T 2 and T 2 /Z 2 with magnetic fluxes in an uniform way.Remarkably, the CP symmetry can be regarded as an outer automorphism of the modular symmetry as discussed in Sec.3.4.These 6D bottom-up models can be embedded in 10D Type IIB magnetized D-brane models with stabilized moduli.In Sec.3.5, we discuss the metaplectic modular flavor symmetries together with traditional flavor and CP symmetries in the framework of eclectic symmetry.

Metaplectic modular symmetry
Since the Yukawa couplings of quarks and leptons are described by a half-integer modular form, they are formulated in the context of metaplectic group M p(2, Z).Following Ref. [36], let us briefly review the notion of M p(2, Z) which is a twofold covering group of SL(2, Z).
Let us rewrite SL(2, Z), its quotient group and metaplectic group by respectively.Note that the complex structure moduli space of the torus τ is governed by Γ due to the fact that τ is invariant under S 2 .By introducing the principal congruence subgroups: with v(γ) = c d being the Kronecker symbol, one can define the finite modular groups: where Γ 2,3,4,5 correspond to S 3 , A 4 , S 4 , A 5 discrete groups, respectively. 9In addition, the finite metaplectic modular groups are given by where the generators satisfy10 and additional relations are required to ensure the finiteness for N > 1, e.g., for Γ4N=8 of order 768 ([768, 1085324] in GAP system [37]), for Γ4N=12 of order 2304, respectively.Under the finite modular groups, modular forms of the modular weight k/2 and level 4N transform as where ρ r (γ) α β denotes an irreducible representation matrix in Γ4N .

T 2 with magnetic fluxes
As discussed in Secs.2.2 and 2.3, the magnetic fluxes generate the semi-realistic MSSMlike models with 3 generations of quarks and leptons.In the following, we address the flavor structure of chiral zero modes with an emphasis on the transformations of these wavefunctions under the modular symmetry.It was known that the magnetic fluxes on extra-dimensional spaces induce the degenerate chiral zero-modes, which are counted by the index theorem.
For concreteness, let us begin with the six-dimensional (6D) Super Yang-Mills theory on T 2 .The Kaluza-Klein reduction of 6D Majorana-Weyl spinor λ is given by with ψ n (z) denotes the n-th excited mode of two-dimensional (2D) Weyl spinors on T 2 .In particular, we focus on zero-mode wavefunctions ψ(z):11 where ψ + and ψ − denote the positive and negative chirality modes on T 2 .The U (1) magnetic flux is given by obtained by the corresponding vector potential with ζ being a Wilson line phase.Note that the boundary conditions of the gauge field as well as the 2D Weyl spinors are respectively chosen as with Here, the normalization of the wavefunctions is fixed as12 The Yukawa couplings of chiral zero-modes are obtained by integrals of three wavefunctions: Remarkably, these wavefunctions show non-trivial transformations under the modular symmetry [10,[38][39][40][41][42][43][44].Indeed, when |M | = even, under S and T transformations of the modular symmetry: the zero-mode wavefunctions respectively transform indicating the wavefunctions with the modular weight 1/2. 13Note that the Wilson line with α, β = 0, 1, ..., 2|M ′ | − 1 and Indeed, the representation matrix ρ( γ) is unitary and satisfies It was known that the boundary conditions of the fermions in Eq. (3.20) and the T transformation are consistent with each other only if M is even.The S transformation is consistent with the boundary conditions.However, the existence of Wilson line modifies the boundary condition as well as the modular transformation [43].Taking into account the modular transformation of the Wilson line ζ in the case of M = odd14 , the wavefunction transforms under the T transformation: with Note that the authors of Ref. [44] proposed that this expression holds for vanishing Wilson lines even in the case of odd units of magnetic flux M .Recall that the exponential factor can be canceled in the Yukawa coupling due to the U (1) gauge invariance as argued in Ref. [44].Thus, T -transformed wavefunction with odd M cannot be expanded in terms of the original wavefunction, but it will be possible to be written in the different coordinate z + 1/2.Since this statement is also true for even units of M , we adopt the T transformation of wavefunction is described by Eq. (3.33) for a general M .The S transformation is still given by Eq. (3.29) with odd units of M .
The modular transformations also act on the 4D fields.When the 4D N = 1 SUSY is preserved, the 4D Lagrangian is written in terms of Kähler potential and superpotential.The Kähler potential and the superpotential of matter fields are derived from the dimensional reduction of 6D Super Yang-Mills theory: where {I ab , I ca , I cb } denote the generation number counted by the index theorem, corresponding to one of the torus in Eq. (2.25).It satisfies I ab + I bc + I ca = 0 to preserve the U (1) gauge symmetry.Then, the modular transformations of matter superfield are given by where explicit forms of ρ(γ) are given in Eqs.(3.29) and (3.34) by replacing M with I ab .In addition, it was known that the holomorphic Yukawa couplings (3.25) are also described by Jacobi theta function [10]: with σ abc = sign(I ab I bc I ca ), where ζa = n a ζ a /m a denotes the redefined Wilson lines and we omit the 6D gauge coupling in the above expression.Since the Yukawa couplings are described by the half-integer modular form, the Yukawa couplings belong to r representation of Γ4N whose transformation is of the form15 : Recalling the condition I ab + I bc + I ca = 0, the Yukawa terms are invariant under the following U (1) symmetry: Φ α,I ab → e iqαI ab Φ α,I ab , (3.40) with q being the U (1) charge of Φ α,I ab .Thus, we redefine the S transformation of matter fields following Ref.[44]: Although we add e 3iπI ab /4 in the S transformation, it is still the unitary representation matrix.Such a redefinition will be convenient to discuss the metaplectic modular symmetry as will be shown later.In this way, the T 2 compactifications with magnetic background fluxes lead to the metaplectic modular flavor symmetries.Before going into details about the relation between the metaplectic modular flavor symmetries, the traditional flavor and CP symmetries, we will discuss the metaplectic modular symmetry on T 2 /Z 2 background.

T 2 /Z 2 with magnetic fluxes
On the T 2 /Z 2 orbifold, the wavefunctions of Z 2 -even and -odd modes are given by the linear combination of these on T 2 as mentioned in Sec.2.2.The explicit forms are given by [47] with . (3.43) The modular transformations are extracted from the matter wavefunctions on T 2 (3.34) and (3.41): for Z 2 -even mode with α, β = 0, 1, ..., I even and for Z 2 -odd mode with α, β = 0, 1, ..., I odd .Here, I even and I odd are defined in Eq. (2.29).In contrast to the analysis of Ref. [41], we added extra phase factors as argued in the previous section.Since Yukawa couplings on T 2 /Z 2 are described by those on T 2 [48]: with where m = 0 and m = 1 respectively correspond to Z 2 -even and -odd modes, they transform under the metaplectic modular symmetry as in the matter fields.
We find that the unitary matrices (3.45) and (3.47) obey the required relations in the metaplectic modular symmetry (3.12).Specifically, for even |M | and odd |M |, modular transformations of both the Z 2 -even and -odd modes are described by Γ2|M| and Γ4|M| , respectively, We checked the additional relations (3.13) and (3.14) for Γ8 and Γ12 , respectively.However, we have not checked these additional relations with Γ4N (N ≥ 4) which will be reported elsewhere.In the next section, we derive such 6D bottom-up models from 10D Type IIB magnetized D-brane models with stabilized moduli.In particular, we discuss the metaplectic modular flavor symmetries together with traditional flavor and CP symmetries in the framework of eclectic symmetry.

Generalized CP
We first discuss the unification of the metaplectic modular symmetry and 4D CP symmetry, which was discussed in T 2 background [41].It was known that the 4D CP and 6D orientation reversing are embedded into the 10D proper Lorentz symmetry [49,50]. 16Since the 6D orientation reversing is realized by z i → −z i for the coordinates of (T 2 ) i , the torus modulus transform under the CP symmetry τ i → −τ i .Note that such a transformation leads to the negative determinant in the transformation of 6D space.In the following, we focus on the CP transformation on T 2 and T 2 /Z 2 .
The multiplication law in the context of metaplectic modular symmetry is defined as When we redefine φ(γ * , τ ) = ±(cτ + d) 1/2 =: ϵJ 1/2 (γ * , τ ) with ϵ = ±1, the above law is written by where the two-cocyle Here, we define and introduce the Hilbert symbol: Since the CP transformation of matter wavefunctions on T 2 is given by corresponding to the basis of canonical CP transformation 18 , it allows us to define the CP transformation in the framework of metaplectic modular symmetry: Thus, Eq. (3.59) is rewritten as Remarkably, the CP transformation does not commute with the metaplectic modular transformations.When we consider the following chain: From the results of Sec.2.3, it turned out that the string landscape leads to a few number of generation of quarks and lepton.In the following analysis, we thus focus on matter wavefunctions on (T 2 ) 1 with vanishing Wilson lines whose explicit forms are given in Eq.
(3.42) with with M = I ab , I bc , I ca .Note that the orbifold projections split the wavefunctions to Z 2 -even and Z 2 -odd modes.For illustrative purposes, we focus on traditional flavor symmetries on three Z 2 -even modes with g = 4.19 • g = 4 The traditional flavor symmetries are described by whose generators {Z ′ , P, C, Z} are of the form [55]: for Z 2 -even mode and for Z 2 -odd mode.Note that such flavor symmetries do not change Yukawa couplings and only act on three generations of quarks and leptons.
In this case, the wavefunctions on T 2 /Z 2 enjoy the modular flavor group Γ 8 and the explicit representations are given in Eq. (3.68).In particular, the flavor generators do not commute with those of modular flavor symmetries G modular = Γ 8 .Indeed, we find that S even C even S −1 even = Z even , S even Z even S −1 even = C even , T even C even T −1 even = C even Z even (Z ′ even ) 2 , T even Z even T −1 even = Z even ,

Conclusions
In this paper, we have examined the vacuum structure of Type IIB flux vacua with SM spectra.The background fluxes play an important role in stabilizing moduli fields and determining the generation number of chiral zero-modes.Since the background fluxes are constrained by the tadpole cancellation conditions, the moduli distribution and the generation number are mutually related with each other.By studying the T 6 /(Z 2 ×Z ′ 2 ) orientifolds with magnetized D-brane models in Secs.2.2 and 2.3, it is found that the string landscape leads to the small generation number of quarks and leptons.Furthermore, the moduli values are peaked at Z 3 fixed point in the complex structure moduli space.It motivates us to study whether such a discrete symmetry is related to the flavor and/or CP symmetries in the low-energy effective action.
To investigate the relation between the modular symmetry of the torus and the flavor symmetries of quarks and leptons, we have focused on the concrete magnetized D-brane model of Sec.2.3.Since the wavefunctions of chiral zero-modes and the corresponding Yukawa couplings are written by Jacobi theta function with the modular weight 1/2, they are described in the framework of metaplectic modular flavor symmetry.Note that the flavor structure of quarks and leptons is originated from one of tori.We found that the modular transformations of both the Z 2 -even and -odd modes are described by Γ2|M| and Γ4|M| for the magnetic flux with even |M | and odd |M |, respectively, Furthermore, the CP symmetry can be regarded as the outer automorphism of the metaplectic modular group.For illustrative purposes, we focus on the M = 4 case, where three Z 2 -even modes transform under a certain traditional flavor symmetry.We found that the traditional flavor, modular flavor and CP symmetries in Type IIB chiral flux vacua are uniformly described in the context of eclectic flavor symmetry: (G flavor ⋊ G modular ) ⋊ G CP as discussed in the heterotic orbifolds [11,12].It would be interesting to explore the realization of eclectic flavor symmetry on other corners of string models.Furthermore, we have stabilized the moduli fields in the framework of flux compactifications.Although the moduli vacuum expectation values are distributed around Z 3 fixed point 20 , a part of eclectic flavor symmetry (G flavor ⋊ G modular ) ⋊ G CP still remains in the low-energy effective action.Since the coefficient of 4D higher-dimensional operators will be described by the product of modular forms with half-integer modular weights, the eclectic flavor symmetry would control the flavor structure of higher-dimensional operators.We leave a pursue of these interesting topics for future work.

Figure 1 .
Figure 1.The numbers of stable flux vacua on the fundamental domain of τ for N max flux = 192 × 10 in the left panel and for N max flux = 192 × 1000 in the right panel, respectively [9].

Figure 2 .Figure 3 .
Figure2.The numbers of models as a function of the generation number g at τ = i and τ = ω, respectively.Note that there exists Z 2 symmetry at τ = i generated by {1, S}.Here, the vertical axis represents the ratio of the number of models to the total number of models.There are three plots in each panel, and each of them corresponds to the maximum value of the D3-brane charge |Q hid D3 | = 400, 1200, 2000.
absorb axions through the Green-Schwarz couplings (2.37).The remaining gauge symmetry is described by SU (4) C × SU (2) L × SU (2) R .Furthermore, the Pati-Salam gauge symmetry can be broken to MSSM gauge group by the splitting of a and c stack of D-branes, but we leave the detailed study of open string moduli for future work.

Figure 4 .Figure 5 .
Figure 4.The numbers of models as a function of the generation number g at τ = i and τ = ω, respectively.Here, the vertical axis represents the ratio of the number of models to the total number of models.There are three plots in each panel, and each of them corresponds to the maximum value of the D3-brane charge |Q hid D3 | = 200, 400, 800.

. 21 )
By solving the Dirac equation for the massless mode with U (1) charge q = 1, we find |M | degenerate zero-mode solutions; ψ + (z) for M > 0 and ψ − (z) for M < 0. Specifically, |M | degenerate zero-mode wavefunctions are written in terms of Jacobi theta function ϑ and the torus area A [10]: τ ) := ℓ∈Z e πi(a+ℓ) 2 τ e 2πi(a+ℓ)(ν+b) .(3.23) ) and the other generators of G flavor commute with G modular .It means that the modular transformation is regarded as an automorphism of the traditional flavor group.Furthermore, we can construct the outer automorphism u CP : G CP → Aut(G flavor ⋊ G modular ).Indeed, the following relations can be verified in the semi-direct product group: modular flavor and CP symmetries are treated in a uniform manner, the traditional flavor, modular flavor and CP symmetries are described by(G flavor ⋊ G modular ) ⋊ G CP ,(3.78)as discussed in heterotic orbifold models.From the analysis of Sec. 2, the moduli fields can be stabilized in flux compactifications, in particular, Z 3 fixed point.It leads to Z 3 modular symmetry generated by {1, ST, (ST ) 2 }.It turned out that such a Z 3 symmetry still enhances the flavor symmetry due to the relation:( S T )C even ( S T ) −1 = C even Z even (Z ′ even ) 2 , ( S T )Z even ( S T ) −1 = Z even .(3.79)Thus, the discrete non-abelian symmetry (G flavor ⋊ Z 3 ) ⋊ G CP remains in the lowenergy action.So far, we have focused on specific magnetized D-brane models with stabilized moduli, but it is quite interesting to explore other flavor models, which left for future work.