The eclectic flavor symmetry of the ℤ2 orbifold

Modular symmetries naturally combine with traditional flavor symmetries and CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{CP} $$\end{document}, giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional ℤ2 orbifold, which is equipped with two modular symmetries SL(2, ℤ)T and SL(2, ℤ)U associated with two moduli: the Kähler modulus T and the complex structure modulus U. The resulting finite modular group is ((S3× S3) ⋊ ℤ4) × ℤ2 including mirror symmetry (that exchanges T and U) and a generalized CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{CP} $$\end{document}-transformation. Together with the traditional flavor symmetry (D8× D8)/ℤ2, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and T=U=expπi3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\langle T\right\rangle =\left\langle U\right\rangle =\exp \left(\frac{\pi \mathrm{i}}{3}\right) $$\end{document}. This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.


Introduction
In the present paper we further explore the eclectic flavor picture in the framework of orbifold compactifications of string theory. The eclectic flavor symmetry is the maximal discrete symmetry that can arise from the nontrivial combination of traditional flavor symmetry and modular flavor symmetry [1]. In this approach, the benefits of finite modular symmetries, uncovered by ref. [2] and further developed for example in refs. [3][4][5][6], are merged with the appeal of traditional flavor symmetries (see e.g. [7]), possibly improving [8] their predictability [9]. Up to now the detailed analysis has concentrated on two-dimensional orbifolded tori where the complex structure modulus U is fixed geometrically to allow for the specific orbifold twist of the torus [10,11]. Aspects of the embedding of the two-tori into six-dimensional compactified space have been discussed in ref. [12] and have shown to be especially relevant for the discussion of R-symmetries. To capture the full eclectic picture one has to consider also those orbifolded tori, where the complex structure modulus is not fixed. This will certainly lead to a richer eclectic structure, as modular JHEP02(2021)018 transformations now act nontrivially on both, the Kähler modulus T and the complex structure modulus U . As a first step in this analysis, we consider the T 2 /Z 2 orbifold that allows the full action of SL(2, Z) T and SL(2, Z) U . This captures all the qualitative aspects of the eclectic flavor picture and can be used as building block for the general discussion in the six-dimensional case.
Flavor symmetries of the T 2 /Z 2 orbifold strongly depend on the values of the moduli. First, we discuss in section 2 the case for generic values of both T and U that gives rise to the traditional flavor symmetry (which leaves the moduli invariant). This leads to the traditional flavor group (D 8 × D 8 )/Z 2 ∼ = [32,49], see ref. [13]. The numbers [32,49] correspond to a unique identifier, assigned by the computer program GAP [14], where the first number (32) gives the order of the group. The finite modular group of the T 2 /Z 2 orbifold is derived in section 3 and turns out to be [144,115], the multiplicative closure of mirror symmetry and the S 3 × S 3 finite modular groups arising from SL(2, Z) T and SL(2, Z) U . If we further include a CP-like modular transformation [3,10,11,15], this group is enhanced to [288,880]. Thus, by combining the traditional flavor group and the finite modular group we are led to an eclectic flavor group with maximally 2304 elements (without CP) and 4608 elements (including CP).
Typically, only a subgroup of the eclectic flavor group is linearly realized and its size depends on the values of the moduli. This leads to an enhancement of the traditional flavor group (D 8 × D 8 )/Z 2 at specific points and hypersurfaces in moduli space: the so-called mechanism of local flavor unification, which is discussed in section 4. There are two specific configurations of the T 2 /Z 2 orbifold that deserve special attention: the raviolo (at U = i, see figure 1) and the tetrahedron (at U = exp πi 3 , see figure 5). There, we observe a further enhancement of the unified flavor symmetry. The largest linearly realized group is found at T = U = exp πi 3 and turns out to be [1152,157463], which includes mirror symmetry and CP. The landscape of unified flavor symmetries (with CP) is illustrated in figure 7. Even for an orbifold as simple as T 2 /Z 2 we find amazingly large flavor groups. In section 5 we shall summarize our results and give an outlook on future research tasks. These should include a full implementation of the automorphy factors (in the spirit of refs. [12,16]) and a road-map towards an embedding in the six-dimensional case. Some technical results are relegated to three appendices.

Flavor from outer automorphisms of the Z Z Z space group
In order to specify the two-dimensional T 2 /Z 2 orbifold, we first define the geometrical space group S. The space group consists of elements g = (θ k , e n) ∈ S, with k ∈ {0, 1}, that act on the coordinates y ∈ R 2 of the extra dimensions as y g → g y := θ k y + e n , (2.1) where the Z 2 twist θ is given by θ = −1 2 and n = (n 1 , n 2 ) T ∈ Z 2 are called winding numbers. One can easily see from eq. (2.1) that two space group elements multiply as θ k , e n θ k , e n = θ k+k , e n + θ k e n . (2.2) JHEP02(2021)018 (1,0) φ (1,1) φ (0, 1) φ (0,0) e 1 e 2 ⇔ Figure 1. The T 2 /Z 2 orbifold. Left: the two-torus T 2 is defined by a two-dimensional lattice, spanned by the vectors e 1 and e 2 . The Z 2 orbifold twist θ = −1 2 maps the fundamental domain of T 2 to the fundamental domain of the orbifold, depicted in yellow. The orbifold action y → θ k y + e n has four inequivalent fixed points, indicated by blue bullets. Localized at these fixed points, there are four (left-chiral) twisted strings (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ). The boundaries of the yellow fundamental domain are identified according to the arrows. Right: after identifying the boundaries of the fundamental domain, the T 2 /Z 2 orbifold has a pillow-like shape with the four fixed points at the corners of the pillow (or raviolo).
The 2 × 2 matrix e (called the geometrical vielbein) consists of two columns e 1 and e 2 . These vectors have to be linearly independent, so that they span a two-dimensional lattice that defines a two-torus T 2 . Then, the T 2 /Z 2 orbifold O is defined as a quotient space where y ∼ y if there exists g ∈ S such that y = g y , (2.3) i.e. points y, y ∈ R 2 in extra dimensions are identified if they are related by the orbifold action with some space group element g ∈ S. This yields a reduced fundamental domain of the orbifold, see figure 1. Closed strings on O are defined by boundary conditions for the string world-sheet degrees of freedom [17][18][19], with world-sheet coordinates τ and σ. Concentrating on the world-sheet bosons y(τ, σ) that describe two extra spatial dimensions y, a boundary condition for a closed string is given by where g = (θ k , e n) ∈ S with k ∈ {0, 1} is the so-called constructing element of the string. In fact, inequivalent strings correspond not only to constructing elements g ∈ S but to their conjugacy classes [g] := {f −1 g f | f ∈ S}, since y(τ, σ) and f y(τ, σ) are identified on the orbifold for all f ∈ S. If k = 0 in eq. (2.4), the string is called untwisted and lives in the bulk of the orbifold. In this case, it can still wind around the two-torus depending on its winding numbers n ∈ Z 2 . If k = 1, the string is called a twisted string. Then, its center of mass is given by the fixed point y g of g ∈ S. In more detail, y g denotes the solution of the fixed point equation g y g = θ y g + e n = y g . For g = (θ, e n) ∈ S, it reads JHEP02(2021)018 y g = 1 2 e n. Furthermore, the internal momentum of a twisted string vanishes such that a twisted string stays localized at its fixed point y g .
Moreover, the matter field φ (n 1 ,n 2 ) is localized at the fixed point y g = 1 2 (n 1 e 1 + n 2 e 2 ) in O, as illustrated in figure 1.
Discrete flavor symmetries of the effective four-dimensional field theory from strings on orbifolds find their origin in the outer automorphisms of the so-called Narain space group [10,11]. Since the Narain construction of strings on orbifolds is rather technical, we refer here only to a short discussion in appendix A and to the literature [20][21][22]. Still, one can gain some insights by considering the outer automorphisms of the geometrical space group S (instead of the Narain space group). An outer automorphism of S is given by a transformation h = (σ, e t) ∈ S, such that see e.g. ref. [11] for a similar discussion in the case of a T 2 /Z 3 orbifold. For the T 2 /Z 2 orbifold, the outer automorphisms of the geometrical Z 2 space group that leave the moduli unaltered are generated by two translations, In the absence of nontrivial discrete Wilson lines [19], they give rise to symmetries of the full string construction, see appendix A for the corresponding Narain construction. Then, one is interested in the action ρ 4 (h) of a (geometrical) transformation h ∈ S on the four twisted matter fields On the level of constructing elements, one already realizes that, for example, the translation h 1 acts as = (θ, (n 1 − 1) e 1 + n 2 e 2 ) . (2.10b)
In addition, there are string selection rules restricting the ability of strings on orbifolds to join and split [23]. The associated symmetry can be determined in two ways: i) as the Abelianization of the space group S [24], or ii) as additional outer automorphisms of the Narain space group. Combined with the geometrical transformations h 1 and h 2 , which exchange orbifold fixed points pairwise, these string selection rules yield a non-Abelian flavor symmetry as follows: in the T 2 /Z 2 orbifold, the string selection rules give rise to a Z 2 × Z 2 × Z 2 symmetry, under which twisted matter fields transform as Here, h 3 and h 4 are connected to the Z 2 × Z 2 space group selection rule for twisted strings, while h 5 is associated with the Z 2 point group selection rule. Then, as first shown in ref. [13], the full traditional flavor group (without R-symmetry) of the T 2 /Z 2 orbifold, at a generic point in moduli space, is generated by the transformations (2.11) and (2.12), resulting in Here, D 8 ∼ = [8,3] denotes the dihedral group of order 8 (sometimes also called D 4 using a different naming convention). Furthermore, the first D 8 factor in eq. (2.13) is generated by h 1 and h 3 . This D 8 is associated with the e 1 direction of the orbifold, cf. ref. [25]. The second D 8 in eq. (2.13) is generated by h 2 and h 4 and is associated with e 2 . Note that JHEP02(2021)018  the transformation h 5 in eq. (2.12c), linked to the point group selection rule, is not an independent generator of the traditional flavor group eq. (2.13) as it can be written as (2.14) This identity gives rise to the Z 2 quotient in eq. (2.13). Moreover, the four twisted matter fields (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ) T build a four-dimensional, irreducible, and faithful representation of the traditional flavor group eq. (2.13). Even though we have discussed the origin of the traditional flavor symmetry (D 8 × D 8 )/Z 2 based on the outer automorphisms of the geometrical Z 2 space group, the results can be confirmed using the full Narain approach, see appendices A and B. Even more, the full Narain space group and its outer automorphisms reveal a common origin of all discrete symmetries for strings on orbifolds, giving rise to the eclectic flavor symmetry that consists of traditional flavor, modular, CP and R-symmetries. In the next section, we will analyze in detail the modular symmetries and their finite modular groups that arise in the T 2 /Z 2 orbifold.

Flavor from modular symmetries
In general, the deformations of a two-dimensional torus used to compactify a string theory can be parameterized by a complex structure modulus U and a Kähler modulus T . The complex structure modulus U can be interpreted geometrically as the shape of the torus, while the Kähler modulus T gives the overall size of the torus and the value of the anti-symmetric B-field background. Moreover, a toroidal compactification exhibits several JHEP02(2021)018 symmetries that act nontrivially on these moduli (see e.g. ref. [11]): These transformations are actually not defined on the level of the moduli but on the level of outer automorphisms of the Narain space group, see appendix A.2. Then,Ĉ S and C T generate a modular group SL(2, Z) U associated with the complex structure modulus U , whileK S andK T are the generators of another factor SL(2, Z) T associated with the Kähler modulus T . Note that the two factors of SL(2, Z) share a common element: C := (Ĉ S ) 2 = (K S ) 2 . Even though C acts trivially on both moduli, it can in principle still act nontrivially on matter fields, see refs. [8,11] (and also ref. [26]). In addition, there are two special transformations The former is the origin of the so-called mirror symmetry that interchanges SL(2, Z) U and SL(2, Z) T , while the latter induces a CP-like transformation, see refs. [10,15] and [3]. In principle, performing an orbifold of a torus can stabilize some moduli geometrically. Consequently, some of the symmetry transformations generated by eqs. (3.1) and (3.2) can be broken by the orbifolding. For example, in the case of a T 2 /Z 3 orbifold sector the U modulus needs to be stabilized, e.g. at U = exp 2πi 3 , and the unbroken modular symmetry after Z 3 orbifolding is generated byK S ,K T , the R-symmetryĈ SĈT and the CP-like transformationK * :=Ĉ SĈTĈSΣ * (i.e. by those modular transformations that leave U = exp 2πi 3 invariant). In contrast, the T 2 /Z 2 orbifold with θ = −1 2 is equipped with both moduli: the complex structure modulus U and the Kähler modulus T . Each of them remains associated with its own unbroken modular group, SL(2, Z) U and SL(2, Z) T for U and T , respectively. Moreover, the transformations (3.2) remain symmetries after the torus has been modded out by the Z 2 orbifold action θ = −1 2 . Hence, the T 2 /Z 2 orbifold gives a simple example of a string setup with multiple modular symmetries (see e.g. ref. [27]), with the extension by mirror symmetry eq. (3.2a) that interchanges both moduli.
As shown in refs. [12,16], the modular group SL(2, Z) U of the complex structure modulus acts geometrically on the compact dimensions. In particular, the SL(2, Z) U generatorŝ C S andĈ T act on the T 2 basis vectors e i according to
Next, we are interested in the action of SL(2, Z) U on the four twisted matter fields without taking the automorphy factors (c U U + d U ) n U of γ U with modular weight n U into account. Then, one can use the geometrical SL(2, Z) U transformations (3.3) as illustrated in figure 3 in order to obtain the matrix representations of the modular S and T transformations, given byĈ S andĈ T , for the four twisted matter fields φ (n 1 ,n 2 ) . The results read One can check that these representation matrices ρ 4 (Ĉ S ) and ρ 4 (Ĉ T ) generate a so-called finite modular group S 3 ∼ = [6, 1]: even though the symmetry is SL(2, Z) U , twisted matter fields transform under SL(2, Z) U in unitary representations of S 3 . We denote this factor by S U 3 as it is associated with the complex structure modulus U . Also the mirror symmetryM has a non-trivial action on twisted matter fields. It turns out that it can be represented by the matrix see appendix A.4. Interestingly, one can verify easily that Hence, ρ 4 (M ) is of order 4. Moreover, eq. (3.7) shows that, althoughM 2 acts trivially on the moduli, it acts nontrivially on twisted matter fields: in fact, (ρ 4 (M )) 2 acts like the traditional flavor transformation ρ 4 (h 5 ) = −1 4 associated with the point group selection rule, see eq. (2.12c). Consequently, the finite modular group can not be disentangled from the traditional flavor group completely: the elementM 2 belongs to both groups. The situation is similar to the T 2 /Z 3 orbifold, where the modular S transformation squared equals a traditional flavor transformation: (Ĉ S ) 2 = (K S ) 2 = C from ∆(54), see ref. [11].

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Then, we can translate the finite modular group S U 3 of the complex structure modulus U to the one of the Kähler modulus T . Using eq. (A.13) from appendix A.2, we obtain see ref. [29]. We denote the resulting finite modular group associated with the Kähler modulus T by S T 3 . The final transformation from the list of generators given in eqs. (3.1) and (3.2) is the CP-like transformationΣ * . It acts on twisted matter fields as where we suppress the spacetime dependencies. Hence, if one considers CP-like transformations, it is beneficial to extend the 4 × 4 representation matrices to 8 × 8 matrices acting on the eight-dimensional vector (Φ,Φ) T of twisted matter fields Φ := (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ) T and their CP-partners.
In summary, the matrices ρ 4 (K S ), ρ 4 (K T ), ρ 4 (Ĉ S ) and ρ 4 (Ĉ T ) generate the finite modular group of order 6 × 6 = 36. Combined with the Z 4 mirror element ρ 4 (M ), we obtain the finite modular group without CP which is of order 36 × 4 = 144. We observe, as a side remark, that this finite modular group is related to the group of outer automorphisms of the traditional flavor group, where the Z 2 on the right-hand side is generated byM 2 . Moreover, by including CP we get [288, 880] ∼ = [144, 115] × Z 2 , which is the maximal finite modular group of the T 2 /Z 2 orbifold. Next, we combine the finite modular group with the traditional flavor group (D 8 × D 8 )/Z 2 and construct the eclectic flavor group G eclectic . The traditional flavor group is a normal subgroup of G eclectic , as expected from the general framework of eclectic flavor JHEP02(2021)018 groups [1]. However, since (D 8 × D 8 )/Z 2 and the finite modular group share the common element (ρ 4 (M )) 2 = ρ 4 (h 5 ), the eclectic flavor group is not a semi-direct product of these two factors. In the case without CP, the finite modular group is (S T 3 ×S U 3 ) ZM 4 and G eclectic turns out to be of order 2304. This order can be understood easily since (144×32)/2 = 2304 using the fact that (ρ 4 (M )) 2 belongs to both factors. As a side remark, note that all finite groups of order 2304 have been classified in ref. [30]. In appendix C we examine the representation ρ 4 of twisted matter fields with respect to G eclectic and its various subgroups. In addition, if we include CP, the eclectic flavor group gets enhanced further to a group of order 4608.

Local flavor unification
As discussed in section 2, for generic values of the moduli, the traditional flavor symmetry of the [32,49], cf. eq. (2.13), associated with the discrete symmetries of the theory that do not affect the moduli.
On the other hand, as explained in section 3, omitting the CP-like transformationΣ * in a first step, the finite modular group of the

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This group can be constructed by the order 4 generator ρ 4 (M ) associated with the mirror transformationM , see eq. (3.6), and the S T 3 × S U 3 finite modular transformations arising from SL(2, Z) T and SL(2, Z) U . These modular symmetries are, in general, independent of the traditional flavor symmetry because traditional flavor transformations act trivially on the moduli T and U . In contrast, modular transformations act by definition nontrivially on these moduli. We assume now that the moduli are fixed at some vacuum expectation values (vevs) ( T , U ), see e.g. ref. [31]. Then, we have to distinguish between two cases: first, there are modular transformations that do not leave the moduli vevs invariant. They are broken spontaneously. Second, some modular transformations may leave the moduli vevs invariant. They build the so-called stabilizer subgroup  However, since string matter fields carry (fractional) modular weights (n T , n U ) of SL(2, Z) T and SL(2, Z) U , the matrix representation ρ(γ) of a modular transformation γ at the point ( T , U ) in moduli space is accompanied by so-called automorphy factors of the form As discussed in ref. [12], these automorphy factors evaluated at the vevs ( T , U ) are discrete phases. This fact can i) change the order of the flavor symmetry associated with γ, and/or ii) reveal that γ acts as a discrete R-symmetry, which in general promotes the unified flavor group at ( T , U ) to a non-Abelian discrete R-symmetry of N = 1 supersymmetry [32]. However, for the sake of clarity and simplicity, we ignore the automorphy factors in the following. Hence, hereafter we shall provide the actual groups of the unified flavor symmetries only in cases where the automorphy factors JHEP02(2021)018 do not affect the results, but shall give in all cases the generators of the corresponding stabilizer subgroups. The subtleties about the consequences of the automorphy factors as well as their relevance in six-dimensional orbifolds shall be discussed in detail later in ref. [33].
A first example of a unified flavor symmetry arises at T = U with generic vev. This is a two-dimensional hypersurface in four-dimensional (real) moduli space. At these points, the mirror symmetry transformationM , acting on the moduli as TM ←→ U , generates a Z 2 stabilizer subgroup. Considering the representation ρ 4 (M ) ofM for twisted matter fields, eq. (3.6), one finds that the traditional flavor group Consequently, the associated unified flavor symmetries, resulting from combining these stabilizer subgroups with the universal traditional flavor group, as prescribed by eq. (4.2), are isomorphic too. This implies that mirror symmetryM halves the fundamental domain in the full moduli space and we must not explore both isomorphic cases eq. (4.5) independently. Thus, we shall explore only those symmetry enhanced points and hypersurfaces in moduli space that are associated with a geometric interpretation of U . Then, for each case of U , we identify the symmetry enhanced points in the T -moduli space. Hence, we can restrict ourselves to four special cases: iii) the raviolo with U = i, and iv) T 2 /Z 2 orbifolds with CP-enhancement.
In the following, we will discuss these four cases in detail, restricting to points of the fundamental domain in moduli space.

The generic T T T 2 /Z Z Z 2 orbifold
Let us consider first the case of a generic vev U of the complex structure modulus U . In this case, CP is broken by the generic vev of U . Furthermore, as shown in figure 4a, there are only two inequivalent special values of the Kähler modulus T associated with a nontrivial stabilizer modular subgroup: T = i and T = e πi 3 . At these points, we see that the Kähler modulus is invariant under the transformations and U is not affected, cf. eq. (3.1). The corresponding stabilizer subgroups are at T = i :

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Note that (K S ) 2 is a trivial element of all stabilizer subgroups because the stabilizer subgroup is defined up to point group transformations (see the discussion around eq. (4.1)) and (K S ) 2 = (Ĉ S ) 2 = −1 4 is equivalent to the Z 2 point group generatorΘ = −1 4 . It is easy to confirm the mirror duals of eqs. (4.7), as shown in figure 4b for generic T .
To exemplify the enhancement of the traditional flavor symmetry to unified flavor groups, let us focus on the hypersurface T = e πi 3 for generic U . We use the ρ 4 representation ofK T andK S given in eqs. (3.8), where (ρ 4 (K S )) 2 = 1 4 , together with those of the generators of the traditional flavor group, eqs. (2.11) and eqs. (2.12). In this way, we find that the nontrivial stabilizer subgroup enhances [96,204] at U = generic, T = e for generic T . After this first case, we can now proceed to study the more complex cases of the T 2 /Z 2 orbifold adopting the shapes of a tetrahedron and a raviolo, where the CP-like modular transformationΣ * plays an important role.

The tetrahedron with U = e πi 3
Setting U = e πi 3 leads to the T 2 /Z 2 illustrated in figure 5. On the left, we see the corresponding two-dimensional torus lattice spanned by e 1 and e 2 of equal length and enclosing an angle of π 3 . Modding out a Z 2 symmetry of the two-torus reduces the fundamental domain to the yellow region, whose boundaries are identified as the arrows indicate. These identifications allow for the fundamental domain of the T 2 /Z 2 orbifold to adopt the shape of the tetrahedron displayed on the right of the figure. The corners of the tetrahedron correspond to the four fixed points of the orbifold, where the twisted matter fields (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ) T are localized.
The generic flavor symmetry group of the tetrahedron can be found by considering the stabilizer subgroup at U = e Since the stabilizer generatorĈ TĈS is now a symmetry everywhere in Kähler moduli space, it is displayed in the fundamental domain (yellow area) of figure 6a. If we now consider the CP-like transformationΣ * , for particular values of T , the flavor group of the tetrahedron is enhanced further. These enhancements occur at the points T along the curve λ T of figure 6a, which is the boundary of the fundamental domain in T moduli space. The orientation of the curve λ T is indicated by bold arrows. It will be used later in section 4.4. For example, at the points of the curve λ T with Re T = 1 2 and Im T > √ 3 2 we find the CP-like transformationK TĈSΣ * . It acts on the moduli as , which is known as SW 4 and can be associated with all pure rotations of a four-dimensional cube [34]. In contrast to ref. [13], we realize the SW 4 symmetry only if both moduli T and U take special values. The same enhancement results for T = e iϕ with π 3 < ϕ < π 2 , where, as indicated in figure 6a, the stabilizer subgroup H e iϕ ,e πi 3 is generated byĈ TĈS andK SĈSΣ * .

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Another interesting example is the maximally symmetric point T = U = e πi 3 . There, in addition toĈ TĈS , also the mirror transformationM and the CP-like transformation T build the stabilizer subgroup  to the largest enhancement of the traditional flavor group in the T 2 /Z 2 orbifold, including mirror symmetry and CP. We observe that even in the simplest case of a T 2 /Z 2 orbifold, the flavor symmetry can be very large.
Other nontrivial stabilizer subgroup generators are displayed along the curve λ T of figure 6a.

The raviolo with U = i
Fixing the complex structure modulus to U = i amounts to setting |e 1 | = |e 2 | with an angle of π 2 between them. This corresponds to a toroidal lattice whose basis vectors are orthogonal and have equal length. In this case, the T 2 /Z 2 orbifold takes the shape of a raviolo, similar to the one depicted in figure 1, but whose edges are perpendicular and all have the same length.
The stabilizer subgroup for generic T and U = i is dual to the one given in eq. (4.7a): it is generated byĈ S . Note that at U = i the transformationĈ S gives rise to a π 2 rotation in the compact dimensions. Hence,Ĉ S is a discrete remnant of the higher-dimensional Lorentz symmetry. Thus, it generates an R-symmetry. In figure 6b, this universal generator of the stabilizer subgroups is displayed in the fundamental domain in the T moduli space (yellow area). Similarly to the previous case, taking into account the CP-like transformation Σ * , further enhancements arise from the elements of the stabilizer subgroups shown in figure 6b at the points T along the curve λ T .

T T T 2 /Z Z Z 2 orbifolds with CP-enhancement
In figure 7, we display the landscape of all nontrivial stabilizers that contain CP-like generators. To do so, we use the straightened curves λ T and λ U as the axes in figure 7. To be specific, the curve λ T is defined as the boundary of the fundamental domain of the S T 3 modular symmetry in figure 6, where the orientation is illustrated by the bold arrows. Its mirror dual λ U is defined analogously in the U -modulus plane. Hence, the horizontal λ T axis must be interpreted as follows: • between the leftmost point and λ T = i, λ T corresponds to ∞ > Im T ≥ 1 and Re T = 0, • for λ T between i and e πi 3 , the curve λ T describes T = e iϕ with π 2 > ϕ > π 3 , and • between λ T = e πi 3 and the rightmost point, the curve λ T is associated with  figure 7, i.e. the middle left panel (equivalently defined by T = ix with x > 1 and U = e iϕ for π 2 > ϕ > π 3 ). In this case, the stabilizer subgroup is generated by the CP-like transformationĈ SΣ * . In summary, we obtain all nontrivial stabilizers that involve CP-enhancement and illustrate them in figure 7.

Conclusions and outlook
We have seen that modular transformations of the T 2 /Z 2 orbifold lead to an exceedingly rich eclectic structure. From the traditional flavor symmetry [32,49], we obtain an eclectic flavor group with as many as 4608 elements. We also observe large flavor groups that are linearly realized at specific regions in T -and U -moduli space, the largest being [1152, 157463] at T = U = exp πi 3 . For reasons of clarity and simplicity, the discussion in the present paper has concentrated on group theoretical studies of finite modular groups and we have not yet included a full implementation of the automorphy factors of SL(2, Z) T and SL(2, Z) U . Future work [33] will include this along the lines explained in refs. [12,16] for the T 2 /Z 3 orbifold. As discussed in ref. [12], the automorphy JHEP02(2021)018 for T ∈ λ T and U ∈ λ U . The axes λ T and λ U correspond to the curves on the boundaries of the two-dimensional fundamental domains of T and U , see e.g. figure 6 for λ T . The diagonal depicts the hypersurface where U = T on the curves λ T and λ U . The stabilizers above and below the diagonal are related by mirror symmetryM .
factors lead to discrete phases. These give rise to discrete R-symmetries that complete the full eclectic structure. Together with the results presented in this paper, this will be a crucial step towards a discussion of orbifolds with six compact extra dimensions. The general case in D = 6 will be too difficult to be analyzed in detail. Using our D = 2 building blocks we can simplify the discussion and consider elliptic fibrations of the D = 6 case with various D = 2 sublattice rotations. In a further step we would then have to consider specific string models that could successfully describe the flavor structure of quarks and leptons and make contact with available botton-up constructions [7]. The predictions of the model will crucially depend on the value of the moduli that potentially break flavor and CP in a desirable way. Thus, the ultimate step in realistic model building will be a discussion of moduli stabilization. We think it is premature to discuss this mechanism at the moment, as the process of moduli stabilization will crucially depend on the specific models under JHEP02(2021)018 consideration. So we rather first would like to construct models that have a chance for a realistic description of flavor for some values of the moduli and relegate the discussion of moduli stabilization to future work.

A Narain lattice
In this appendix, we give a brief discussion on the Narain space group of our T 2 /Z 2 orbifold, its outer automorphisms and their actions on the twisted matter fields that are localized at the fixed points of the T 2 /Z 2 orbifold.

A.1 Orbifolds defined by the Narain space group
The string y(τ, σ) can be split into right-and left-moving degrees of freedom y R and y L , respectively, where ỹ y = 1 √ 2 andỹ denotes the so-called dual string coordinate. Then, one defines Narain coordinates Y and compactifies them on a Narain torus, i.e.
Here, the integer vector N = (n 1 , n 2 , m 1 , m 2 ) T ∈ Z 4 gives the winding numbers (n 1 , n 2 ) and the Kaluza-Klein numbers (m 1 , m 2 ). In addition, E is a 4 × 4 vielbein matrix, the so-called Narain vielbein (given for example in refs. [10,11], changing B to −B). To ensure two-dimensional worldsheet modular invariance, the Narain vielbein has to span an even, integer, self-dual lattice with metric η = diag(−1, −1, 1, 1) of signature (2, 2): the so-called Narain lattice Γ. In a second step, one can mod out the Narain lattice by a Z K rotational symmetry, generated by the Narain twist Hence, one defines an orbifold in the Narain formulation of string theory as where g = (Θ k , E N ) is an element of the so-called Narain space group S Narain . The orbifold is called symmetric if the Narain twist Θ acts identically on right-and left-movers, i.e. θ := θ R = θ L .

A.2 Outer automorphisms of the Z Z Z 2 Narain space group
Discrete symmetries of the four-dimensional effective theory obtained from orbifold compactifications are associated with the outer automorphisms of the Narain space group S Narain . In analogy to the definition in eq. (2.7) for the geometrical space group, an outer automorphism of S Narain is defined as a mapping from S Narain to itself. In more detail, a transformation h = Σ, E T ∈ S Narain is an outer automorphism of S Narain if for all g = Θ k , E N ∈ S Narain . In addition, Σ has to preserve the Narain metric Next, we translate the Narain space group and its outer automorphisms into the lattice basis, in which we denote all quantities in general by a hat. For example, eq. (A.5) reads in the lattice basisĝĥ −→ĥ −1ĝĥ ! ∈Ŝ Narain , (A.7) whereĝ Here, we have definedΘ := E −1 Θ E andΣ := E −1 Σ E. Furthermore, due to eq. (A.6) we have to imposeΣ ∈ Oη(2, 2, Z), where the group Oη(2, 2, Z) of "rotational" outer automorphisms of the Narain lattice Γ is defined as As discussed in detail in ref. [11], the group Oη(2, 2, Z) contains the generatorŝ Note that, compared to refs. [8,11] we redefinedK S ,K T andĈ S .K S andK T generate SL(2, Z) T of the Kähler modulus T , whileĈ S andĈ T generate SL(2, Z) U of the complex structure modulus U . Moreover, the group Oη(2, 2, Z) contains two additional generators, given byΣ * :=

A.3 Transformation of Z Z Z 2 twisted strings
From eq. (A.16) we find that Narain translationsĥ i = 1 4 , T i ∈Ŝ Narain with T i given in eq. (A.17) act on the constructing elements Θ , N ∈Ŝ Narain of Z 2 twisted matter fields φ (n 1 ,n 2 ) as

A.4 Details on mirror symmetry
Let us consider the constructing elementsĝ = Θ , N ∈Ŝ Narain of Z 2 twisted strings with N = (n 1 , n 2 , m 1 , m 2 ) T and n 1 , n 2 , m 1 , m 2 ∈ {0, 1}. These 16 elements are associated with four twisted matter fields φ (n 1 ,n 2 ) as follows: where we usedM −1ΘM =Θ for the Z 2 Narain twistΘ = −1 4 , andM is given in eq. (A.12). Hence, we obtain Using the correspondence between constructing elements of the Narain space group and the twisted matter fields stated in eq. (A.19), we get where T i are defined up to integers such that −T i ∼ T i . Let us embed these equations into their action on twisted matter fields This fixes all unknowns α ij except for α 11 , and we obtain a representation of mirror transformation that is of order two by setting α 11 = 1 √ 2 . However, in order to disentangle the two finite modular groups (associated with the Kähler modulus and the complex structure modulus) we have decided to redefinẽ ρ 4 (M ) according to as stated in eq. (3.6) in section 3. This redefinition is possible, since each transformation h i associated with ρ 4 (h i ) in eq. (A.27) belongs to the traditional flavor symmetry that does not affect the moduli and, hence, is valid everywhere in moduli space. Further, the redefinition eq. (A.27) does not alter the physics of the theory because ρ 4 (h i ) is a symmetry transformation. Additional details will be given in ref. [33].

B How to classify the outer automorphisms of a space group
The group of outer automorphisms Out(Ŝ Narain ) of a Narain space groupŜ Narain is the key to uncover all discrete symmetries resulting from the orbifold compactification encoded inŜ Narain . Interestingly, there exists a general algebraic construction of Out(S) for any kind of space group S due to Lutowski, see refs. [35,36]. In this appendix, we briefly demonstrate its application to the Narain space group of the T 2 /Z 2 orbifold in order to confirm the results presented in appendix A.2. The Z 2 Narain space group of our T 2 /Z 2 orbifold is defined aŝ in the lattice basis. Then, Lutowski's algorithm states that Out(Ŝ Narain ) is given by two factor groups, denoted by Ξ and H 1 (P , Z 4 ). These groups combine semi-directly in the same way as the lattice and the point group combine to defineŜ Narain , i.e.

JHEP02(2021)018
The constituents of Out(Ŝ Narain ) are the so-called stabilizer Ξ of the space group and the first cohomology group H 1 (P , Z 4 ), whereP is the Narain point group in the lattice basis (P = Θ = {±1 4 } ∼ = Z 2 in our case). Therefore, one can write an outer automorphism aŝ h = Σ , T ∈Ŝ Narain with a rotational partΣ and a translational part T that acts as for all Θ k , N ∈Ŝ Narain . By applying Lutowski's algorithm, one observes that H 1 (P , Z 4 ) accounts for pure translations. In detail, one finds see eq. (A.17). The four generators of this group T i correspond exactly to the transformationsĥ i introduced earlier as the geometrical translations for i = 1, 2 and the space group selection rule for i = 3, 4. Acting on the twisted strings of the Z 2 orbifold, these outer automorphisms give rise to the traditional flavor symmetry. The other factor of Out(Ŝ Narain ) in eq. (B.2) is the stabilizer Ξ, which is governed by the explicit form of the space group. In general, an element of Ξ might admit both, a rotational partΣ and a translational part s(Σ) such that a general element is given by a so-called roto-translation Σ , s(Σ) . Since the Z 2 point groupP = Θ considered here acts as a pure rotation on the Narain lattice, we can set s(Σ) = 0 and the algorithm finds the stabilizer group to be GL(4, Z)/ Θ . (B.5) However, we have to impose the physical condition (related to level-matching) that the rotationΣ has to preserve the Narain metric in the lattice basisη, given in eq. (A.10). This means thatΣ TηΣ =η has to hold. As defined in eq. (A.9), these rotations form the subgroup Oη(2, 2, Z) of GL(4, Z). Therefore, the stabilizer group Ξ of the Z 2 Narain space group is given by Ξ ∼ = Oη(2, 2, Z)/ Θ .

C Irreducible representations of twisted matter fields
The four twisted matter fields (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ) T are localized at the four fixed points of the T 2 /Z 2 orbifold. They transform under both, the traditional flavor symmetry JHEP02(2021)018 [32,49], where they transform as an irreducible 4, and also under the modular symmetries discussed in section 3. There, it was found that twisted matter fields do not transform faithfully under SL(2, Z) T and SL(2, Z) U but in a four-dimensional unitary representation ρ 4 of the respective finite modular groups S T 3 and S U 3 . Moreover, this representation turns out to be reducible. In detail, by studying the characters of the representation matrices one can show that the four-dimensional representation decomposes i) into irreducible representations of each S 3 factor as and ii) into irreducible representations of S T 3 × S U 3 as 4 = (2, 1) ⊕ (1, 2) . (C.2) Moreover, including mirror symmetry, S T 3 × S U 3 is enhanced to [144, 115] (cf. eq. (3.11)), where the four twisted matter fields build an irreducible representation 4. These decompositions can be made explicit by the following orthogonal basis change B: In this basis, the matrix representations, eqs. where we indicate the 2 × 2 block structure by horizontal and vertical lines, cf. ref. [37]. This proves the (2, 1) ⊕ (1, 2) block structure with respect to S T 3 × S U 3 . Furthermore, in this basis it is easy to see that the mirror transformationM interchanges S T 3 and S U 3 , i.e.  Table 1. Irreducible representations of twisted matter fields (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ) T with respect to the various flavor symmetries (in the absence of string oscillator excitations).
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