Eclectic flavor scheme from ten-dimensional string theory -- II. Detailed technical analysis

String theory leads to a flavor scheme where modular symmetries play a crucial role. Together with the traditional flavor symmetries they combine to an eclectic flavor group, which we determine via outer automorphisms of the Narain space group. Unbroken flavor symmetries are subgroups of this eclectic group and their size depends on the location in moduli space. This mechanism of local flavor unification allows a different flavor structure for different sectors of the theory (such as quarks and leptons) and also explains the spontaneous breakdown of flavor- and CP-symmetries (via a motion in moduli space). We derive the modular groups, including CP and R-symmetries, for different sub-sectors of six-dimensional string compactifications and determine the general properties of the allowed flavor groups from this top-down perspective. It leads to a very predictive flavor scheme that should be confronted with the variety of existing bottom-up constructions of flavor symmetry in order to clarify which of them could have a consistent top-down completion.


Introduction
The consideration of finite modular flavor groups has been suggested in the pioneering work of Feruglio [1]. A combination of these modular symmetries and traditional flavor symmetries leads to the eclectic flavor scheme [2,3]. Here, we present a study of the eclectic flavor approach towards ten-dimensional string theory with D = 6 compactified spatial dimensions 1 . This generalizes previous work with D = 2 compact dimensions which is shown to fail to capture all the relevant properties of the eclectic picture. A few of the highlights of this discussion have been presented in an earlier paper [12] without explaining the full technical details. These highlights include • a further enhancement of the eclectic group, • a new interpretation of discrete R-symmetries originating from modular transformations of the complex structure modulus, • a relation between the R-charges and the modular weights of matter fields, and • new insights in the nature of CP-symmetry and its spontaneous breakdown.
These results have been illustrated previously [3,12] in an example of a D = 2 sublattice based on the T 2 /Z 3 orbifold. Here we shall now present the complete results in the generic situation and provide the full technical details of the derivation. In addition, we shall illustrate an interesting novel feature: the existence of accidental continuous (gauge) symmetries of the effective theory at special loci in moduli space, which turn out to be further enhancements of the traditional flavor symmetries.
In section 2 we shall discuss basic background material concerning the D = 2 Narain lattice formulation [13][14][15][16] and its outer automorphisms relevant for (discrete) modular symmetries [17,18]. These include the groups SL(2, Z) T and SL(2, Z) U for the Kähler modulus T and the complex structure modulus U , respectively, mirror symmetry as well as a CP-like transformation. Results for the generators are summarized in table 1 for various T 2 /Z K orbifolds. We include a discussion of the modular properties of the superpotential and Kähler potential in the modular invariant field theory, including modular weights and automorphy factors.
Section 3 concentrates on the discussion of those aspects of the modular group SL(2, Z) U of the complex structure modulus which will become relevant for the analysis in D = 6 compact dimensions. In contrast to SL(2, Z) T (which exchanges windings and momenta), SL(2, Z) U can be given a fully geometrical interpretation. In previous discussions of D = 2 orbifolds T 2 /Z K , the subtleties of this analysis have not yet been completely considered, because the modulus U is frozen as a result of the orbifold twist (except for the case T 2 /Z 2 which has been discussed elsewhere [19]). We show here that, even with a frozen value U of the complex structure U , there are nontrivial elements of SL(2, Z) U that remain unbroken. These lead to additional discrete R-symmetries according to the stabilizer subgroups given in formula (81). These symmetries will be relevant for the D = 6 case in the form of sublattice rotations.
In section 4, we consider the T 6 /P orbifold (with point group P ). We start with the discussion in full generality (including the restrictions of N = 1 supersymmetry in four spacetime dimensions) and then illustrate the results for the Z 6 -II orbifold. We shall identify the sublattice rotations of SL(2, Z) U , their action on matter fields and determine which (fractional) modular weights are relevant for the emerging R-symmetries. Section 4.3 includes a discussion of CP-transformations, including the full D = 6 picture (summarized in table 3).
Section 5 presents the complete description for a sublattice T 2 /Z 3 of the Z 6 -II orbifold.
This includes the traditional flavor symmetry ∆ (54), the finite modular symmetry T from SL(2, Z) T , a discrete R-symmetry Z R 9 from SL(2, Z) U as well as a CP-like transformation (which is separately discussed in subsections 5.6 and 5.7). The maximal eclectic group is shown to have 3, 888 elements, extending Ω(2) ∼ = [1944,3448] (see table 5) by the CP-like transformation. We discuss restrictions of this symmetry on the form of the superpotential and Kähler potential of the N = 1 supersymmetric field theory.
Only part of the eclectic flavor symmetry is realized linearly. For generic values of the moduli only the traditional flavor symmetry is unbroken. This symmetry will be enhanced at special points (or sub-loci) of the moduli space. In section 6 we present this phenomenon of local flavor unification for the T 2 /Z 3 orbifold and identify these enhanced groups (see table 6 and figure 3) as well as the specific forms of the superpotential. Moving away from these sub-loci in moduli space will correspond to a spontaneous breakdown of the enhanced flavor groups (in particular also for the violation of CP-symmetry). A novel observation concerns the emergence of (accidental) continuous gauge symmetries at special loci in moduli space, here discussed in section 6.4. Section 7 will present conclusions and an outlook for future applications of our results. Note to the reader: As our discussion is partially very technical we include a short specific summary at the end of each section 2-6 to make the results of the discussion as transparent as possible.

Modular symmetries in string theory
In general, the modular group SL(2, Z) can be defined as the group generated by two abstract generators, S and T, that satisfy the defining relations It turns out convenient to represent these generators by the matrices S := 0 1 −1 0 and T := 1 1 0 1 .
In this context, the elements γ ∈ SL(2, Z) can be expressed by 2 × 2 matrices with integer entries and determinant one, i.e.

Narain lattice of torus compactifications
In string theory, target-space modular symmetries can arise from the two-dimensional worldsheet CFT. To see this, let us first consider string theory compactified on a D-dimensional torus T D , which is most naturally described in the Narain formulation [13][14][15]. There, one considers right-and left-moving string coordinates y R and y L , which are related to the ordinary internal coordinates y and their dual coordinatesỹ via y := α 2 (y L + y R ) andỹ := α 2 (y L − y R ) .
Then, (y R , y L ) are compactified on a 2D-dimensional auxiliary torus. This torus is defined by the so-called Narain lattice Γ that is spanned by the so-called Narain vielbein E. In more detail, the Narain torus boundary condition in this formalism reads y R y L ∼ y R y L + λ , where λ ∈ Γ := EN N = n m ∈ Z 2D , (6) and n, m ∈ Z D are known as winding and Kaluza-Klein (KK) numbers, respectively. Due to worldsheet modular invariance of the one-loop string partition function, the Narain lattice Γ spanned by E has to be an even, integer, self-dual lattice with metric of signature (D, D). This translates into the following condition on the Narain vielbein E: denotes the metric of signature (D, D) and 1 D is the (D × D)-dimensional identity matrix. Then, the Narain scalar product of two Narain vectors λ i = EN i ∈ Γ withN i = (n i , m i ) ∈ Z 2D for i ∈ {1, 2} reads λ T 1 η λ 2 =N T 1ηN 2 = n T 1 m 2 + m T 1 n 2 ∈ Z .
This confirms that the lattice Γ is integer and even, as the scalar product is integer and, if N 1 =N 2 , even. Also, as expected from eq. (7), we note thatη is the Narain metric in the lattice basis.
In the absence of (discrete) Wilson lines [20], one can choose the (2D × 2D)-dimensional Narain vielbein E fulfilling eq. (7) as see refs. [16][17][18]. Here, α is the Regge slope that enters the Narain vielbein E, such that E is dimensionless. Then, E is parameterized by the (D × D)-dimensional geometrical vielbein e, yielding the metric G := e T e of the geometrical torus T D , and the (D × D)-dimensional anti-symmetric B-field B.
Using the explicit parameterization (9), we can translate the Narain boundary condition (6) to the coordinates y and their duals. We obtain Note that the boundary condition y ∼ y +e n for the geometrical coordinates y of closed strings motivates the name of n ∈ Z D as winding numbers of the geometrical torus T D spanned by e.
The Narain lattice Γ is beneficial as it incorporates the winding numbers n ∈ Z D and the KK numbers m ∈ Z D on equal footing. Hence, it allows for a natural formulation of T -duality in string theory, where KK and winding numbers get interchanged.

Outer automorphisms of the Narain lattice
In the Narain lattice basis, the Narain metricη can be used to define the group of outer automorphisms 2 of the Narain lattice Γ that preserve the Narain metricη as such that each Narain lattice vector λ = EN ∈ Γ is transformed by an outer automorphism Σ ∈ Oη(D, D, Z) according to Here, we useΣ −1 instead ofΣ for later convenience, see section 2.5. Then,Σ −1N ∈ Z 2D and the Narain scalar product λ T η λ from eq. (8) is invariant under the transformation (12). Note 2 Here, Oη(D, D, Z) gives the discrete "rotational" outer automorphisms of the Narain lattice. Furthermore, there are continuous translations that act by conjugation on Narain lattice vectors and, therefore, correspond to the trivial automorphism of Γ.
that the transformation of the Narain lattice vector eq. (12) can be interpreted as induced by the transformation of the Narain vielbein under outer automorphisms given by The group of outer automorphisms of the Narain lattice Oη(D, D, Z) with these properties contains the modular group, as we discuss next.
For concreteness, let us focus on a geometrical two-torus T 2 with D = 2. One can easily verify that Oη(2, 2, Z) contains two copies of SL(2, Z) as follows: First, we can define two Oη(2, 2, Z) elementŝ which satisfy the defining relations (1) of SL(2, Z), We denote this modular group by SL(2, Z) T . Moreover, we identify the Oη(2, 2, Z) elementŝ which also satisfy the SL(2, Z) defining relations, Hence,Ĉ S andĈ T give rise to another factor of the modular group, denoted by SL(2, Z) U . Even though all elements of SL(2, Z) T commute with those of SL(2, Z) U , these two factors are not fully independent since they share a common element, given by The group Oη(2, 2, Z) of outer automorphisms of the Narain lattice Γ contains two additional generators, which we can choose for example aŝ They give rise to Z 2 × Z 2 . Motivated by the relationŝ we callΣ * a CP-like transformation. Furthermore, we callM a mirror transformation as it interchanges SL(2, Z) T and SL(2, Z) U , i.e.

Moduli and modular symmetries of T 2
The modular subgroups SL(2, Z) T and SL(2, Z) U of Oη(2, 2, Z) are associated with the transformations of the moduli of a T 2 torus, as we now discuss.
The Narain vielbein E of the geometrical two-torus T 2 with background B-field allows for certain deformations while satisfying the conditions eq. (7) required for E to span a Narain lattice. These deformations correspond to two moduli fields: Comparing these definitions with eq. (9), we realize that the Narain vielbein can be expressed in terms of the T 2 moduli, E = E(T, U ), up to some unphysical transformations, see for example ref. [16]. Hence, the action ofΣ ∈ Oη(2, 2, Z) on the Narain vielbein given by eq. (13) is equivalent to a transformation of the T and U moduli. To determine explicitly the transformation properties of the moduli, it is convenient to define the generalized metric (valid for an arbitrary dimension D) For a two-torus T 2 , the generalized metric in terms of the torus moduli reads Due to the transformation of the vielbein eq. (13), the action ofΣ ∈ Oη(2, 2, Z) on the generalized metric is given by From this equation and the expression (24), given a transformationΣ, one can readily obtain the forms of the transformed moduli T and U . For example, considering asΣ the SL(2, Z) T modular generatorsK S andK T in eq. (14), we observe that the associated moduli transformations are which explains the index chosen for the modular group SL(2, Z) T . From this observation, as shown e.g. in ref. [18], we know that under a modular transformation γ T ∈ SL(2, Z) T the moduli T and U transform as in the 2 × 2 matrix notation used in eq. (3).
Analogously, letting the generalized metric (24) transform under the SL(2, Z) U generators defined in eq. (16), one finds that the T and U moduli transform according to Hence, a general modular element γ U ∈ SL(2, Z) U acts on the moduli as as we will also re-derive later in section 3.1 using an alternative approach. Moreover, repeating the previous steps, the mirror transformationM interchanges T and U , while the CP-like transformationΣ * acts as A brief remark is in order: The modular transformation (K S ) 2 , which equals (Ĉ S ) 2 , acts trivially on both moduli. Hence, the modular groups restricted to their action only on the moduli is PSL(2, Z) T and PSL(2, Z) U , where for example γ T ∈ SL(2, Z) T and −γ T ∈ SL(2, Z) T are identified in PSL(2, Z) T . However, in full string theory (K S ) 2 acts nontrivially, for example, on massive winding strings.

Orbifold compactifications in the Narain formulation
Defining Y := (y R , y L ) T for the Narain coordinates, a T D /Z K orbifold in the Narain formulation is obtained by extending the boundary conditions of closed strings eq. (6) to Moreover, the so-called Narain twist Θ shall not interchange right-and left-movers. Thus, it is given by Hence, Θ generates a Z K rotation that is modded out from the Narain torus in the orbifold.
The Narain twist has to be a (rotational) symmetry of the Narain lattice of the torus, i.e. Θ Γ = Γ. Thus, Θ has to be an outer automorphism of the Narain lattice Γ. This translates into the following condition on the so-called Narain twist in the Narain lattice basiŝ Furthermore, from eq. (33) we find that the Narain metric (7) and the generalized metric (23) are orbifold invariant,Θ TηΘ =η andΘ T HΘ = H .
In summary, the Narain twist in the Narain lattice basisΘ := E −1 Θ E has to be a Narainmetric preserving outer automorphism of the Narain lattice, which, in addition, leaves the generalized metric (and therefore all torus moduli) invariant, i.e.
where T a and U b denote all Kähler and complex structure moduli of a T D /Z K orbifold, respectively. This invariance condition of the moduli results in a stabilization of some of them: One says that some moduli are frozen geometrically in order to satisfy eq. (36).

Modular symmetries of orbifold compactifications
We shall restrict ourselves here to symmetric orbifolds T D /Z K , for which right-and leftmovers are equally affected by the orbifold twist, which is granted if we additionally impose in eq. (33) the condition θ := θ R = θ L ∈ O(D). Then, the orbifold twist θ generates the so-called The group structure and the symmetries of the orbifold can be realized by expressing the boundary conditions eq. (32) of the Narain orbifold in terms of the so-called Narain space group S Narain as The Narain space group is equipped with the product defined by An arbitrary element g ∈ S Narain can be expressed in the Narain lattice basis by conjugating with h := (E, 0) aŝ where we have used the Narain twist in the Narain lattice basis defined in eq. (34). In our notation, the Narain space group and its elements are indicated by a hat in the Narain lattice basis, i.e.ĝ ∈Ŝ Narain .
To identify the modular symmetries of the four-dimensional effective theory after orbifolding amounts to determining the outer automorphisms of the Narain space groupŜ Narain . An outer automorphism of the Narain space group is given by a transformation such thatĥ −1ĝĥ ∈Ŝ Narain for allĝ = (Θ k ,N ) ∈Ŝ Narain is satisfied. Explicitly, the condition forĥ to be an outer automorphism ofŜ Narain reads see refs. [19,21] for an algorithm to classify the outer automorphisms of a (Narain) space group. IfŜ Narain is generated by pure rotations (Θ, 0) and pure translations (1 2D ,N ) withN ∈ Z 2D , the condition (41) is equivalent to demanding that for all k = 0, . . . , K −1 andN ∈ Z 2D . Then, also the group of outer automorphism is generated by pure rotations (Σ, 0) and pure translations (1 2D ,T ), such thatΣ =Θ k andT / ∈ Z 2D . The translational outer automorphisms contribute to the so-called traditional flavor symmetry of the theory, as we will review in section 5.2 in the case of a T 2 /Z 3 orbifold sector. Let us focus now on the rotational elements (Σ, 0). Note that choosing k = 0 in eq. (42) delivers precisely the outer automorphisms of the Narain lattice, studied in section 2.2. In particular, we find the conditionΣ −1N ∈ Z 2D , which is satisfied only ifΣ ∈ Oη(D, D, Z), cf. eqs. (11) and (12).
However, for an orbifold compactification, the outer automorphismΣ is further constrained by the first condition in eq. (42) for k = 0.
Let us focus now on D = 2. As we have seen in section 2.2, in this case the outer automorphisms of T 2 build the group of modular symmetries of a torus compactification, generated byĈ S ,Ĉ T ,K S ,K T ,M andΣ * , given in eqs. (14), (16) and (19). Thus, the outer automorphismsΣ of the orbifold must be a modular subgroup thereof, depending on the orbifold geometry T 2 /Z K , with K ∈ {2, 3, 4, 6}. Table 1 presents the Narain twistsΘ and a choice of generators of the modular symmetries arising from the outer automorphisms of all symmetric T 2 /Z K orbifolds. As we shall discuss explicitly in section 3.2, even though the Narain twistΘ is an inner automorphism of the orbifold T 2 /Z K , it becomes an outer automorphism if the T 2 /Z K orbifold is only a subsector of a six-dimensional factorized orbifold.

Modular invariant field theory from strings
In string models, matter fields are associated with the excitation modes of strings. In a twodimensional compactification of heterotic strings on a T 2 /Z K orbifold, matter fields arise from strings that close on the orbifold. They can be arranged in three categories: (i) Untwisted strings, which are trivially closed strings, even before compactification; (ii) Winding strings, which close only after winding along the directions e 1 and/or e 2 of the T 2 torus. At a generic point in moduli space, winding strings have masses at or beyond the string scale. Thus, they do not lead to matter fields that appear in the low-energy effective field theory; and (iii) Twisted strings, which close only due to the action of the twist.
This classification can be stated in terms of elements of the orbifold space group, which set the boundary conditions for the strings to close. This relation is revealed by regarding eq. (37) as the boundary conditions of closed strings in the orbifold, according to corresponds to the (bosonic) worldsheet string field in terms of the worldsheet coordinates τ and σ. Hence, we note that untwisted strings are associated with the trivial space group element (1 4 , 0), winding strings are connected with translational space group elements (1 4 , EN ), and twisted strings are related to more general space group elements (Θ k , EN ), The properties and dynamics of string matter fields are inextricably linked to the attributes of the compact space on which these strings live. In particular, if two of the extra dimensions are associated with a T 2 /Z K orbifold sector, matter fields inherit target space modular symmetries, whose generators are listed in table 1 for each twist order K. Matter fields of heterotic orbifold compactifications are endowed with a modular weight for each SL(2, Z) modular symmetry, and we denote the set of all modular weights by n. Let us discuss the example of two modular symmetries SL(2, Z) T and SL(2, Z) U of the Kähler modulus T and the complex structure modulus U , where we have n = (n T , n U ) ∈ Q 2 (see e.g. refs. [22,23]). We denote untwisted and twisted matter fields of the orbifold theory as Φ n given that matter fields can be distinguished in general by their modular weights n. In the case of twisted matter fields, they build multiplets Φ n = (φ 1 , φ 2 , . . .) T , where the fields φ i are associated with different closed strings, whose boundary conditions are produced by inequivalent space group elements with the same power k of Θ (i.e. by space group elements from the same twisted sector but different conjugacy classes). Consequently, in the absence of discrete Wilson lines [20] in the T 2 /Z K orbifold sector, the twisted fields φ i in the multiplet Φ n share identical gauge quantum numbers and modular weights.
Ignoring CP-like and mirror modular transformations that shall be addressed in the following section, a general matter field Φ n transforms under general modular transformations γ T ∈ SL(2, Z) T and γ U ∈ SL(2, Z) U , as defined by eqs. (27) and (29), according to where (c T T + d T ) n T and (c U U + d U ) n U are the so-called automorphy factors with modular weights n = (n T , n U ) ∈ Q 2 . The matrices ρ s (γ T ) and ρ s (γ U ) build a (reducible or irreducible) representation s of some finite modular group (for example Γ N T × Γ N U , where in terms of the 2 × 2 matrices S, T given in eq. (2), the pair of transformation matrices ρ s (γ T = S) and ρ s (γ T = T) satisfy the defining relations eq. (4a) for some integer N T and analogously for γ U ).
As we can infer from the modular symmetry generators listed in table 1 and shall be discussed in detail in section 3, in all T 2 /Z K orbifold compactifications but those with K = 2, the complex structure U is not dynamic as it is fixed at a value U . We will see that this fact together with the transformation (44b) leads to new insights about the effective nature of these symmetry transformations: they give rise to the well-known R-symmetries in heterotic orbifold compactifications. If the complex structure modulus U is fixed, the strength of a superpotential coupling is given by a modular formŶ s Y (T ) of the Kähler modulus T . Then, under a general modular transformation γ T ∈ SL(2, Z) T such a modular form transforms aŝ Here, s Y denotes an s Y -dimensional representation of the finite modular group (e.g. Γ N T or Γ N T ) and (c T T + d T ) n Y is the automorphy factor of γ T with modular weight n Y . If the finite modular group is Γ N , n Y must be even. However, for the double cover groups Γ N that appear in string orbifold compactifications also odd n Y are allowed [24].
String models based on orbifold compactifications yield typically an N = 1 effective field theory. Here, we are mainly interested in its Kähler potential K and superpotential W. The superpotential is a holomorphic function of moduli and matter fields Φ n . Under γ T ∈ SL(2, Z) T and γ U ∈ SL(2, Z) U modular transformations, the superpotential becomes i.e. W behaves similar to a matter field with modular weights n U = n T = −1 and which is invariant under the finite modular group of the theory. Consequently, each allowed superpotential coupling of matter fields Φ n has to satisfy the conditions n Y + matter fields Φn for γ T ∈ SL(2, Z) T with n = (n T , n U ), ρ s is the representation s associated with Φ n , and the couplingŶ (n Y ) s Y of weight n Y transforms in the representation s Y . An analogous result holds On the other hand, the general Φ-independent contribution to the Kähler potential is given by [25] K This universal Kähler contribution transforms under γ T ∈ SL(2, Z) T and γ U ∈ SL(2, Z) U as can be removed by performing a Kähler transformation after each modular transformation, rendering the Kähler potential modular invariant. Therefore, any additional Φ-dependent contribution to K must be invariant under modular transformations, cf. ref. [26].
On the other hand, it turns out that the superpotential is also modular invariant since e.g. the modular transformation eq. (46a) followed by the Kähler transformation applied for achieving invariance of K yields

Summary
The modular symmetry groups SL(2, Z) and PSL(2, Z) as well as their respective finite modular groups Γ N and Γ N are natural to string compactifications. For example, as explained in section 2.3 strings on an internal two-torus yield two moduli: a complex structure modulus U as well as a Kähler modulus T . This is in contrast to bottom-up models of flavor that typically consider only one modulus. Using the Narain description of a torus compactification introduced in section 2.1, one can find the modular transformations acting on both moduli by computing the outer automorphisms of the Narain lattice associated with the compact space (see section 2.2). These modular transformations build in general a large group that includes SL(2, Z) T and SL(2, Z) U for the standard modular transformations of T and U , as well as two additional special transformations: a Z 2 mirror duality that exchanges SL(2, Z) T and SL(2, Z) U , and a Z 2 CP-like transformation. From this result, one can explore how this changes for all admissible T 2 /Z K toroidal orbifolds, whose Narain formulation is introduced in section 2.4. As displayed in table 1, we find that only T 2 /Z 2 orbifolds preserve all the modular symmetries of the torus, while for K ∈ {3, 4, 6} the modular groups are subgroups of the group of modular symmetries of the torus. Interestingly, under modular transformations string matter fields Φ n and the couplings among them transform as representations of finite modular groups Γ N . In these terms, as detailed in section 2.6, we review the modular properties of the superpotential and Kähler potential that yield a modular invariant effective field theory.
3 SL(2, Z) U of the complex structure SL(2, Z) T of the Kähler modulus T is of stringy nature as it relates, for example, compactifications on compact spaces with small and large volumes. In contrast, in this section we will show explicitly using the Narain formulation of D = 2 extra dimensions that the SL(2, Z) U factor of the complex structure modulus U allows for a pure geometrical interpretation: On the one hand, SL(2, Z) U modular transformations are defined in terms of a special class of outer automorphisms of the Narain lattice. On the other hand, SL(2, Z) U only affects the geometrical vielbein e. Hence, we will see in a second step that rotational Z N symmetries of the two-torus are described by those SL(2, Z) U modular transformations that leave the complex structure modulus U invariant. Moreover, the automorphy factors of SL(2, Z) U modular transformations turn out to be related to discrete Z N charges. Furthermore, they promote the two-dimensional rotational Z N symmetry to a discrete R-symmetry.

Geometrical interpretation of SL(2, Z) U of the complex structure
In the Narain lattice basis, the generatorsĈ S andĈ T of SL(2, Z) U (and also the CP-like transformationΣ * ) as defined in section 2.2 yield elements of the form whereσ ∈ GL(2, Z). Note that, in contrast to SL(2, Z) T , modular transformations of this kind do not interchange winding and KK numbers. Next, we can change the basis to right-and left-moving coordinates (in analogy to eq. (39)) Hence, Σ acts on right-and left-moving string coordinates as see section 2.5. Consequently, the coordinates y and their dual coordinatesỹ, as defined in eq. (5), transform as where σ := eσ e −1 . It is easy to see that andσ ∈ GL(2, Z). Now let us discuss the case det(σ) = +1, i.e. whenσ ∈ SL(2, Z) corresponds to a modular transformation without CP. Then,σ ∈ SL(2, Z) is compatible with the B-field in the two-torus T 2 , i.e.σ T Bσ = B, and we obtain Furthermore, eq. (54) yields in the case det(σ) = +1 the simple transformations These transformations can be absorbed completely into a redefinition of the geometrical torus vielbein e. Explicitly, the torus boundary condition is mapped under y → σ y to Hence, an SL(2, Z) U transformationΣ can be performed alternatively by a transformation of only the vielbein e, which then induces a change of the torus metric G = e T e, i.e.
while the B-field is invariant. Note that the boundary conditioñ of the dual coordinatesỹ (as given in eq. (10)) transforms analogously underỹ → σ −Tỹ from eq. (57). Sinceσ ∈ SL(2, Z), the two-dimensional lattice spanned by e is mapped to itself underσ −1 in eq. (60): in other words,σ ∈ SL(2, Z) is an outer automorphism of the twodimensional lattice spanned by the geometrical vielbein e. For example, under modular S and T transformations of the complex structure modulus U , the geometrical lattice transforms as using eq. (60) withσ = − forĈ S andσ = γ forĈ T as given in eq. (16). Moreover, using eq. (62) the complex structure modulus U defined in eq. (22b) transforms as Note that these transformations (63) can also be obtained directly following ref. [9] if one takes the torus lattice vectors e 1 and e 2 to be complex numbers, i.e. e 1 , e 2 ∈ C, and rewrites the complex structure modulus as U = e 2 /e 1 . Then, the transformations (62) of e 1 and e 2 imply eqs. (63).
In addition, we take the transformation eq. (60) of the metric G underΣ given in eq. (51) in order to obtain the general transformation property of the complex structure modulus U , i.e. (see also ref. [27]) To summarize and to be specific, on the one hand we have identified a special class of outer automorphismsΣ of the Narain lattice Γ for D = 2, given bŷ On the other hand, we have shown that an outer automorphism of this class acts geometrically on the vielbein e that defines the two-torus as Hence, it gives rise to a modular transformation of the complex structure modulus while T is invariant. In other words, there exist outer automorphisms of the Narain lattice Γ for D = 2 that are specified by their geometrical actionσ ∈ SL(2, Z) on the torus vielbein e and translate to modular transformations γ U ∈ SL(2, Z) U of the complex structure modulus U , where the dictionary between these two reads explicitly 3.2 Geometrical rotations from SL(2, Z) U Based on the discussion from the last section, we now analyze a special class of outer automorphisms of the Narain lattice that corresponds to geometrical rotations in the extra-dimensional space, see also ref. [9] for a related discussion. We focus on two extra dimensions (D = 2), but the generalization to six extra dimensions is straightforward, since a rotation in six dimensions can be decomposed into three rotations in three orthogonal two-dimensional planes. Thus, the results of this section will be important for both: i) to define an orbifold twist and ii) to perform a two-dimensional "sublattice rotation" of a six-dimensional orbifold.
We begin with a discrete rotation θ (K) ∈ Z K ⊂ SO(2) that acts as while it leaves all orthogonal coordinates inert. The geometrical rotation angle In order to be a rotational symmetry of the geometrical torus, θ (K) has to map the lattice spanned by the torus vielbein e to itself. Hence, we have to impose the condition This constrains the order K to the allowed orders of the Z K wallpaper groups, being K ∈ {2, 3, 4, 6}, see e.g. ref. [28]. Now, we translate this discrete geometrical rotation into the Narain formulation of string theory. To do so, we split y into right-and left-moving string coordinates using eq. (5) and define the action on these coordinates as and (2) are of order K. Using eq. (69) we obtain such that the rotation has to be left-right symmetric Thus, the left-right-symmetric rotation Θ (K) defined in eq. (72) together with eq. (74) corresponds to the transformation Σ from eq. (56) with σ = θ (K) ∈ SO(2), i.e. σ −T = σ. Consequently, we can follow the discussion around eq. (56) and write the rotation in terms of an outer automorphism of the Narain lattice aŝ Due to this block-structure together withσ (K) := e −1 θ (K) e ∈ SL(2, Z), the Narain twistΘ (K) can be decomposed in terms of the generatorsĈ S andĈ T , given in eq. (16). Then, using eq. (68) we can translate the action of the outer automorphismΘ (K) to a corresponding modular transformation from SL(2, Z) U as where the modular transformation γ (K) is of order K, i.e. (γ (K) ) K = 1 2 .

Stabilizing the complex structure modulus by geometrical rotations
Now, we can use the transformation property of the torus metric G = e T e given in eq. (60) for a rotational symmetry eq. (71) in order to find Hence, the metric G and, therefore, the complex structure modulus U (as defined in eq. (22b)) must be invariant under a Narain twistΘ (K) . In other words, the vev of the complex structure modulus U is a fixed point of those modular transformations γ (K) ∈ SL(2, Z) U that correspond to rotational symmetries of the geometrical torus, where the vacuum expectation value U of the complex structure modulus parameterizes the torus vielbein e, except for the overall size of its two-torus.
In our case of two-dimensional rotational symmetries of order K, the Kähler modulus T is invariant but the complex structure modulus U is frozen geometrically to, for example, The detailed results for Z K rotational symmetries of the Narain lattice in D = 2 for K ∈ i.e. U is a fixed point of the elements of H U . Hence, the vev U of the stabilized complex structure modulus spontaneously breaks SL(2, Z) U to an unbroken symmetry where the Z 2 factors are generated by S 2 ∈ SL(2, Z) U and Z 3 is generated by S T ∈ SL(2, Z) U . These unbroken modular transformations will be of importance for the symmetries after orbifolding.
ii) As we will analyze next, stabilizing the complex structure modulus U promotes the automorphy factor (c U U + d U ) to a phase. Table 2: Geometrical Z K symmetriesΘ (K) ∈ Oη(2, 2, Z) of the Narain lattice, translated to modular transformations γ (K) ∈ SL(2, Z) U that leave the vev U of the associated complex structure modulus invariant. As a consequence,Θ (K) acts left-right symmetrically and rotates the geometrical coordinate y by an angle v (K) . At the same time, the superpotential picks up a phase exp (2πi v (K) ) such that the geometrical rotationΘ (K) generates an R-symmetry, as shown in section 3.4 .

R-symmetries from SL(2, Z) U
In addition, we can compute the automorphy factor (c U U + d U ) −1 of the superpotential W, see eq. (46), for a rotational Z K symmetry that is described by a modular transformation γ (K) ∈ SL(2, Z) U . It has to be a phase of order K since (γ (K) ) K = 1 2 . Indeed, we obtain where c U and d U are given by γ (K) ∈ SL(2, Z) U and v (K) = 1 /K turns out to coincide with the geometrical rotation angle corresponding to γ (K) as defined in section 3.2 and listed in table 2. This yields W Since the modular transformation γ (K) ∈ SL(2, Z) U leaves the (stabilized) complex structure modulus U invariant (cf. eq. (78)), the Kähler potential eq. (48) is invariant under γ (K) . Hence, the phase in eq. (83) has to be compensated by a transformation of the Grassmann number ϑ of N = 1 superspace. Thus, the rotational modular transformation γ (K) ∈ SL(2, Z) U of the complex structure modulus is an R-transformation. In more detail,

Summary
In summary, a two-dimensional rotational Z K symmetry of the extra dimensional space is an outer automorphism of the Narain lattice that leaves the torus metric G and the B-field invariant. Because of its geometric nature, it corresponds to a modular transformation γ (K) from SL(2, Z) U of the complex structure modulus U that leaves U evaluated at its vev and T invariant, as might have been expected from geometrical intuition. Moreover, a geometric rotation in two dimensions acts as an R-transformation, where the R-charge of the superpotential originates from the automorphy factor (c U U + d U ) −1 of the modular transformation

R-symmetries and CP in six-dimensional orbifolds
It is known that toroidal orbifold compactifications of string theory lead to discrete Rsymmetries of the four-dimensional effective theory. In the traditional approach, R-symmetries are explained to originate from rotational isometries of the six-dimensional orbifold geometry [29][30][31][32][33]. Here, we give a novel interpretation for the origin of R-symmetries as a special class of outer automorphisms of the Narain space group: R-symmetries correspond to elements from the modular group SL(2, Z) U that leave the vev of the (stabilized) complex structure modulus U invariant.

Six-dimensional orbifolds T 6 /P
In order to specify a geometrical point group P ∼ = Z K of a six-dimensional orbifold, we define a six-dimensional orbifold twist θ ∈ SO(6) by combining three two-dimensional Z K i rotations θ (K i ) ∈ SO(2) given in eq. (69) for i ∈ {1, 2, 3}. In detail, the orbifold twist is chosen as for k i ∈ {0, 1, . . . , K i − 1} and y ∈ R 6 . One can choose complex coordinates z i := y 2i−1 + i y 2i for i ∈ {1, 2, 3}. Then, each two-dimensional rotation θ (K i ) acts only in the i-th complex plane z i . The orbifold twist θ generates a Z K rotation group, whose order K is given by the least common multiple of K 1 , K 2 and K 3 . By combining the three rotation angles k We have extended v by an additional null entry for its action in string light-cone coordinates, including those of the uncompactified space. Using the results from table 2, the orbifold twist vector v corresponds to a modular transformation This transformation leaves the three complex structure moduli U i invariant, where i ∈ {1, 2, 3} corresponds to the three complex planes, i.e.
and U i is given in eq. (79) for the cases K i ∈ {2, 3, 4, 6}. Hence, a complex structure modulus In addition, using the three automorphy factors arising from the modular transformations  87), if the associated orbifold twist vector v preserves N = 1 supersymmetry, i.e. if On the other hand, the Kähler potential is invariant under the modular transformation γ given in eq. (87): For K i ∈ {3, 4, 6}, the complex structure modulus U i is fixed to the values U i provided in eq. (79). Hence, the Kähler potential of the theory does not exhibit any field-dependence on U i . For K i = 2, the Kähler potential does depend on the (unstabilized) modulus U i explicitly, but is left invariant by the rotational modular transformation γ because neither the complex structure modulus itself nor the Φ-dependent terms of the form |Φ| 2 are altered under these transformations.
Factorized orbifolds. In the following we concentrate on factorized T 6 /Z K orbifold geometries, where the six-torus factorizes as T 6 = T 2 × T 2 × T 2 and each two-dimensional Z K i rotation θ (K i ) is a symmetry of the respective two-torus T 2 . In this case, the orbifold twistΘ in the Narain lattice basis readŝ where the Z K i Narain twistsΘ (K i ) ∈ Oη(2, 2, Z) of order K i ∈ {2, 3, 4, 6} are given in table 2.

Sublattice rotations from SL(2, Z) U of the complex structure modulus
For simplicity, we have chosen the six-torus T 6 that underlies the T 6 /Z K orbifold geometry to be factorized as T 2 × T 2 × T 2 . Then, the six-dimensional orbifold has three independent Z K i rotational isometries, whose orders K i ∈ {2, 3, 4, 6} for i ∈ {1, 2, 3} are given in general by the components of the Z K orbifold twist vector defined in eq. (86): there is one rotational isometry per two-torus T 2 . These Z K i rotations are also called sublattice rotations.
In detail, as the T 6 /Z K orbifold geometry (with Z K orbifold twist vector given in eq. (86)) is assumed to be factorized, it contains three two-dimensional subsectors, which we call T 2 /Z K i orbifold sectors for i ∈ {1, 2, 3}. Then, a sublattice rotation θ (K i ) acts on the geometrical coordinate y of the T 2 /Z K i orbifold sector as while it leaves the orthogonal coordinates invariant. Following section 3.2, these three geometrical transformations for i ∈ {1, 2, 3} correspond to three outer automorphismsR i of the full six-dimensional Narain space group, given bŷ whereR i ∈ Oη(2, 2, Z) × Oη(2, 2, Z) × Oη(2, 2, Z) ⊂ Oη(6, 6, Z) and for i ∈ {1, 2, 3}. Each of them is associated with a modular transformation corresponding to the complex structure modulus U i , which leaves all moduli invariant. Note that the productR 1R2R3 from eq. (98) equals the Narain twistΘ given in eq. (93). Hence, R 1R2R3 is an inner automorphism of the Narain space group.
Next, we analyze the action of the modular transformation γ (K i ) ∈ SL(2, Z) U i on matter fields Φ n , where n collectively denotes the set of all modular weights, containing the modular weights n U i of SL(2, Z) U i for i ∈ {1, 2, 3}, see e.g. ref. [22,23] for the definition of the (fractional) modular weights for various orbifold geometries. In this case, the modular transformation (44) of matter fields reads Φ n such that the superpotential picks up a phase Here, we used that the automorphy factor evaluated at U i becomes a modulus-independent phase, see eq. (82). In order to ensure that (γ (K i ) ) K i = 1 2 , the matrix ρ s (γ (K i ) ) of the modular transformation has to be of order K i . Furthermore, note that ρ s (γ (K i ) ) can be diagonal or nondiagonal: On the one hand, it is diagonal for example for K i ∈ {2, 3} since the corresponding Oη(6, 6, Z) sublattice rotation maps each string (i.e. each conjugacy class of the constructing element) to itself, possibly multiplied by a phase. On the other hand, ρ s (γ (K i ) ) is non-diagonal for example for a K i = 4 sublattice rotation in a T 6 /Z 4 orbifold geometry, see ref. [32].
Eq. (101) defines the discrete R-charges of matter fields Φ n for modular transformations γ (K i ) ∈ SL(2, Z) U i that correspond to geometrical sublattice rotations. On the other hand, it is known [29][30][31][32][33] that a matter field Φ n transforms under a geometrical sublattice rotation Using K 2 i as denominator, the R-charges R i of superfields Φ n become integer, as defined in ref. [31] for K i ∈ {2, 3, 6}. Consequently, R-charges R i and modular weights n U i are related. For example, if the representation matrix ρ s (γ (K i ) ) from eq. (101) is diagonal, we find where α i gives the order K i phase of the i-th entry on the diagonal of ρ s (γ (K i ) ), i.e. ρ s (γ (K i ) ) = diag(exp( 2πiα i (φ 1 ) /K i ), . . . , exp( 2πiα i (φs) /K i )).
do not depend on the complex structure modulus U i if K i ∈ {3, 4, 6}. In contrast, for K i = 2 they do depend on U i but are invariant under the "rotational" modular transformation γ (2) . Hence, we can set n Y = 0 and ρ s Y (γ (2) ) = 1. Then, we obtain from eq. (47) the invariance conditions matter fields Φn n U i = − 1 and matter fields Φn The later condition can be rewritten (for diagonalized representation matrices ρ s (γ (K i ) )) as matter fields Φn Consequently, we see that using eqs. (105) we have to impose the constraint matter fields Φn on the R-charges R i for each superpotential coupling of matter fields Φ n to be allowed, cf. refs. [29][30][31][32][33]. Hence, the order K i sublattice rotation yields an R-symmetry that is in gen- where the nontrivial right-hand side of eq. (107) follows from eq. (102). As we have shown, this R-symmetry originates from unbroken modular transformations SL(2, Z) U i (that leave the complex structure modulus U i invariant), even in the case when U i is frozen geometrically by the orbifolding.
Finally, let us note that for special points in moduli space (e.g. for special values of a complex structure modulus or for vanishing discrete Wilson lines) there can be additional Rsymmetries originating from the modular group SL(2, Z) U of a complex structure modulus and from SL(2, Z) T of the Kähler modulus. A detailed example of sublattice rotations in a T 2 /Z 3 orbifold sector will be discussed later in section 5.4.

CP for six-dimensional orbifolds
A CP-like transformation has to map a field to a CP-conjugate partner with inverse (discrete and gauge) charges. For heterotic orbifolds, a string state with constructing elementĝ ∈Ŝ Narain finds its CP-partner in a string state with constructing elementĝ −1 ∈Ŝ Narain . Hence, a CP-like transformation is an outer automorphism of the Narain space groupŜ Narain , such that for all constructing elementsĝ ∈Ŝ Narain . This means in particular that the Narain twist is mapped to its inverse under a CP-like transformation. In detail, in the Narain lattice basis we have to impose the conditionŝ As in section 4.1, we now assume our six-dimensional orbifold geometry T 6 /Z K under consideration to be factorized, i.e. T 6 = T 2 × T 2 × T 2 . In more detail, we take the Z K Narain twistΘ to be built out of three two-dimensional orbifold sectorŝ Θ :=Θ (K 1 ) ⊕Θ (K 2 ) ⊕Θ (K 3 ) ∈ Oη(2, 2, Z) × Oη(2, 2, Z) × Oη(2, 2, Z) ⊂ Oη(6, 6, Z) , (110) see eq. (91). Then, eq. (109) is solved bŷ whereĈ This means that for each complex plane i ∈ {1, 2, 3} (or each T 2 /Z K i orbifold sector) on which a Narain twistΘ (K i ) of order K i ∈ {2, 3, 4, 6} acts, we look for a corresponding CPlike transformation that we callĈP (K i ) . In addition to eq. (111), a two-dimensional CP-like transformation has to act on the Kähler modulus T i of the T 2 /Z K i orbifold sector as and in addition, if K i = 2, on the complex structure modulus as U i → −Ū i , see ref. [17,36,37]. The results are given in table 3. Hence, the six-dimensional CP-like transformation given by eq. (112) acts simultaneously on all six dimensions of the compactified space, see also ref. [38].
At first sight, there seems to be a special case if K i = 2, i.e. if the six-dimensional orbifold geometry contains a T 2 /Z 2 orbifold sector. Let us discuss an example, where the first complex plane contains a T 2 /Z 2 orbifold sector (K 1 = 2), i.e.
and K i = 2 for i ∈ {2, 3}. Then, we can define two independent transformationŝ withĈP (2) =Σ * given in table 3. Both transformations,ĈP andĈP , satisfy eq. (109) since (Θ (2) ) −1 =Θ (2) forΘ (2) := −1 4 . However, onlyĈP is a consistent transformation but not CP :ĈP acts nontrivially on the Kähler modulus T 1 and the complex structure modulus U 1 associated with the first complex plane of the six-dimensional orbifold, but leaves the moduli associated with the four orthogonal compact dimensions invariant. Nonetheless, the superpotential W has to be a holomorphic function of all fields (matter fields and moduli) and after CP-conjugation it needs to become anti-holomorphic. Therefore, assuming that there are some superpotential couplings that depend on moduli associated with all six compact dimensions, CP has to act nontrivially in all three complex planes of the six-dimensional orbifold simultaneously. In this case, CP is broken spontaneously if any modulus departs from one of its CP-conserving points in moduli space. order orbifold twistΘ (K i ) ∈ Oη(2, 2, Z)ĈP (K i ) ∈ Oη(2, 2, Z)

Summary
Many properties of factorized six-dimensional orbifolds can be understood in terms of three two-dimensional T 2 /Z K i orbifold sectors, i ∈ {1, 2, 3}, as we show in section 4.1. Nevertheless, these six-dimensional orbifolds display a richer structure compared to two-dimensional ones. In particular, we observe two important new features: i) Inner automorphisms of a Narain space group of a two-dimensional orbifold sector can become outer automorphisms of a Narain space group of a suitable six-dimensional orbifold. This applies especially to two-dimensional sublattice rotations of six-dimensional orbifolds, which are given by (unbroken) SL(2, Z) U i modular transformations. Interestingly, this uncovers an unexpected relation, eq. (104), between the R-charge R i of a matter field associated with a sublattice rotation in the i-th two-torus and its SL(2, Z) U i modular weight.
ii) CP-like transformations in six-dimensional factorized orbifolds must act simultaneously in all two-dimensional orbifold sectors, in order to not spoil the holomorphicity of the superpotential. This is based on the assumption that at least one superpotential coupling depends on the moduli from all three two-dimensional orbifold sectors. We list in table 3 the CP-like transformations for all T 2 /Z K i orbifold sectors (K i ∈ {2, 3, 4, 6}).

R-symmetries and CP in a T 2 /Z 3 orbifold sector
The four-dimensional effective field theory (obtained from a heterotic compactification on an orbifold geometry that contains a T 2 /Z 3 orbifold sector without discrete Wilson lines) is equipped with various types of symmetries: • a ∆(54) traditional flavor symmetry, • an SL(2, Z) T modular symmetry that acts as a T finite modular group on matter fields and couplings, • a Z R 9 discrete R-symmetry, • a CP-like transformation, see also refs. [29][30][31][32][33][39][40][41][42][43]. These discrete symmetries combine to the eclectic flavor symmetry Ω(2) ∼ = [1944,3448]. Furthermore, as we have seen in the previous sections, all of them have a common origin from string theory: they can be defined as outer automorphisms of the Narain space group [12,17,18]. This novel approach has revealed several interconnections between discrete symmetries of the various types, with the ultimate aim to connect bottom-up flavor model building with a consistent top-down approach from strings.
In this section, we first recall the main results from the analysis of the effective field theory obtained from a heterotic string compactification on a T 6 /Z K orbifold geometry that contains a T 2 /Z 3 orbifold sector, for example in the first complex plane. The moduli in this plane are denoted for simplicity by T and U (without subindex). In addition, we focus on three copies (labeled by an index i ∈ {1, 2, 3}) of twisted matter fields (X i , Y i , Z i ) without string oscillator excitations. These matter fields are localized in the extra dimensions at the three fixed points of the T 2 /Z 3 orbifold sector. We call them collectively Φ i where n T = −2 /3 is the modular weight with respect to SL(2, Z) T . In this section, we will obtain new results concerning the origin and nature of discrete R-symmetries and CP, related to the modular symmetries of the theory.

Modular symmetry in a T 2 /Z 3 orbifold sector
One defines a so-called finite modular group Γ N using the relations (1) combined with the additional constraint which renders the group finite. For the T 2 /Z 3 orbifold sector, we are interested in the case N = 3 that corresponds to the finite modular group Then, a modular form of T with modular weight n Y ∈ N is defined as a (vector of) holomorphic function(s) of the Kähler modulus T that transforms under general modular transformations Here, s Y denotes an s-dimensional representation of T and (c T T + d T ) n Y is the automorphy factor of γ T with modular weight n Y .
It turns out that there exists only one (two-component) modular form of weight n Y = 1 Using eq. (118) it can be checked explicitly that this modular form transforms in a twodimensional (unitary) representation 2 of T , where Note that we use the T conventions from ref. [44] (with p = i).

Traditional flavor symmetry in a T 2 /Z 3 orbifold sector
It is well-known that a T 2 /Z 3 orbifold sector yields a ∆(54) traditional flavor symmetry, where twisted matter fields build triplet-representations [39,45]. New insight was gained using the Narain space group and its outer automorphisms [17,18]: In the absence of discrete Wilson lines, there exist two translational outer automorphisms, A and B, of the T 2 /Z 3 Narain space group, see the discussion around eq. (42) in section 2.5. Similar to eq. (10), one can see that A and B induce the following geometrical translations as illustrated in figure 1a. Note that, in addition to eq. (124), there is a nontrivial action onỹ. These Narain-translations leave the Kähler modulus inert. Hence, they belong to the traditional flavor symmetry. Acting on twisted matter fields Φ i −2 /3 , they are represented as In addition, there is a rotational outer automorphism C := (−1 4 , 0) of the T 2 /Z 3 Narain space group, which corresponds to a geometrical rotation by 180 • in the T 2 /Z 3 orbifold sector.
Hence, C generates an R-symmetry. As we have seen in eq. (81), C corresponds to a modular transformation S 2 ∈ SL(2, Z) U from the stabilizer subgroup H U =ω . For the twisted matter fields, C is represented as By comparing with eq. (122), we note that ρ(C) = ρ(K S ) 2 , which is indeed a consequence of the more fundamental identity C =K 2 S , see section 2.2. of ref. [3] for more details. This additional element enhances ∆(27) to a non-Abelian R-symmetry [49] ∆ (54), where the twisted matter fields Φ i −2 /3 transform in the representation 3 2 . Note that the non-diagonal structure of ρ(C) is an example of a non-diagonal representation matrix ρ s (γ (K i ) ) in eq. (101). Furthermore, one can understand geometrically the fact that Y i and Z i get interchanged under the action of C, while X i is mapped to itself, as illustrated in figure 1b.

Superpotential in a T 2 /Z 3 orbifold sector
Taking only twisted matter fields from one twisted sector into account, the modular group SL(2, Z) T , its associated finite modular group T and the traditional flavor group ∆(54) con- , 2, 3}, such that there is only one free parameter, denoted by c (1) . The precise result reads [3] As a remark, assuming that Φ 1 −2 /3 = (X 1 , Y 1 , Z 1 ) T develops a non-trivial vev, this superpotential results in a (symmetric) mass matrix This observation can be useful in semi-realistic models e.g. to determine the mass textures of quarks and leptons, considering that Φ 1 −2 /3 corresponds to three Higgs fields coupled to the three generations of quark and lepton superfields.

Z R
9 R-symmetry in a T 2 /Z 3 orbifold sector We consider a Narain space group S Narain of a T 6 /Z K orbifold, which contains a T 2 /Z 3 orbifold sector (for example in the first complex plane). Then, there exists an outer automorphism of the Narain space group S Narain , wherê see eqs. (69) with y = e i and (71). In other words, the two-dimensional sublattice spanned by e 1 and e 2 , illustrated for example in figure 1, is left invariant by a 120 • rotation.
As a remark, let us assume that the order three discrete Wilson line in the T 2 /Z 3 orbifold sector is chosen to be non-zero. Then, this 120 • rotational symmetry remains unbroken. In contrast, the Z 2 outer automorphism C := (−1 4 , 0), discussed in section 5.2, is broken in this case.
As we have seen in section 3.2, the outer automorphism eq. (132) translates to a modular transformation which leaves (by definition of the stabilizer subgroup H U =ω ) the complex structure modulus U of the T 2 /Z 3 orbifold sector invariant, Finally, the superpotential transforms under the modular transformation γ (3) ∈ SL(2, Z) U as using that the complex structure modulus of the T 2 /Z 3 orbifold sector is frozen at U = ω.
This sublattice rotationR 1 acts on matter fields as where the R-charges R 1 are normalized to be integer for superfields, see table 4. Hence, the Z 3 sublattice rotationR 1 leads to a Z R 9 R-symmetry of the theory, where the superpotential Table 4: Discrete Z R 9 R-charges R 1 for a T 2 /Z 3 orbifold sector with trivial Wilson line of: matter fields Φ n T , the Grassmann number ϑ, the superpotential W, and the Kähler potential K in conventions, similar to those of ref. [31]. The Z R 9 R-symmetry originates from a Z 3 sublattice rotation of the T 2 /Z 3 orbifold sector in the normalization where all R-charges of bosons (but not of fermions) are integer.
Note that the Z (PG) 3 point group symmetry eq. (128a) is a subgroup of Z R 9 (using that ϑ → −ϑ and fermions → −fermions follows from four-dimensional Lorentz invariance). eq. (130) transforms according to eq. (137) with a Z R 9 R-charge 3. Note that fermions have halfinteger R-charges. Hence, this symmetry acts as a Z R 18 symmetry of the fermionic sector [31].
In more detail, one can show that the Grassmann number ϑ of N = 1 superspace has a Z R 9 R-charge R(ϑ) = 3 /2. Furthermore, using dϑ ϑ = 1 we know that the R-charge of dϑ is − 3 /2, such that the Lagrangian L ⊃ d 2 ϑ W is invariant.

5.5
The full eclectic flavor group without CP in a T 2 /Z 3 orbifold sector Hence, the full eclectic flavor group of the T 2 /Z 3 orbifold sector without CP is given by where Ω(1) ∼ = ∆(54) ∪ T , see ref. [50] for the nomenclature. The various groups and their origins in string theory are summarized in table 5. Note that Ω(2) contains the ∆(54) traditional flavor group and all of its enhancements (without CP) to the various unified flavor groups at special points in T moduli space, as we shall discuss later in section 6. T Ω(2) Table 5: Table from ref. [12]: eclectic flavor group Ω(2) for orbifolds T 6 /P that contain a T 2 /Z 3 orbifold sector. In this case, SL(2, Z) U of the stabilized complex structure modulus U = exp ( 2πi /3) is broken, resulting in a remnant Z R 9 R-symmetry. Including Z R 9 enhances the traditional flavor group ∆(54) to ∆ (54, 2, 1) ∼ = [162, 44] and, thereby, the eclectic group to Ω(2) ∼ = [1944,3448]. Note that Ω(1) ⊂ Ω(2).

CP in a T 2 /Z 3 orbifold sector
For heterotic orbifold compactifications, the modular group SL(2, Z) T of the Kähler modulus T originates from outer automorphisms of the Narain space group, as we reviewed in section 2.
For a T 2 /Z 3 orbifold sector, it can be enhanced naturally by an additional outer automorphism that acts CP-like [17,18], i.e.
This gives rise to a reflection at the line perpendicular to e 2 , as illustrated in figure 2. Finally, note that CP has to act in all extra dimensions simultaneously, following our discussion in section 4.3.

CP as a modular symmetry
On the level of 2 × 2 matrices, the CP-like transformation withK * can be represented by Since det(K * ) = −1, it enhances the modular group SL(2, Z) T to GL(2, Z) T . Then, the modular transformations in eq. (27) of the Kähler modulus T are complemented by see also ref. [37] for a bottom-up derivation.
Moreover, the transformation of matter fields eq. (44) can be generalized to CP-like transformations (with det(γ) = −1) as where the fieldΦ n T on the right-hand side is evaluated at the parity-transformed point in spacetime. In addition, as derived in ref. [18] for our concrete case of a T 2 /Z 3 orbifold sector, twisted matter fields transform under the CP-like generatorK * as where we used that the automorphy factor (cT + d) n T in eq. (146) equals 1 for K * defined in eq. (144).
Consequently, one can show that the finite modular group T ∼ = SL(2, 3) gets enhanced by CP to GL (2,3). Simultaneously, the eclectic flavor group Ω(2) is enhanced by CP to a group of order 3,888 (while the Ω(1) subgroup of Ω(2) is enhanced to [1296, 2891] by CP).
Modular forms are also affected by the inclusion of CP-like modular transformations. Under a CP-like transformation T → −T , we find for the Dedekind η-function η(r T ) → η(−rT ) = (η(r T )) * if r ∈ R. Hence, the weight 1 modular forms of T defined in eq. (119) transform asŶ cf. ref. [52]. We have chosen here a basis of T where all Clebsch-Gordan coefficients in T tensor products are real (see refs. [44,53] with p = i, p 1 = 1, and p 2 = −1). Therefore, all irreducible T modular forms of modular weight n Y ≥ 1 are complex conjugated byK * , analogously to eqs. (148), i.e.
Finally, we discuss the transformation of the superpotential W and the Kähler potential K under general CP-like transformations for a CP invariant theory. Again, we take γ ∈ GL(2, Z) T with det(γ) = −1. Then, if the superpotential and the Kähler potential transform as the theory is CP invariant, using the transformed matter fields Φ n T given in eq. (146) and a Kähler transformation to remove the holomorphic function f (T ) = ln(c T + d), see section 2.6. Let us remark that our convention for K * given in eq. (144) is chosen such that the automorphy factors in eqs. (146), (150) and (151) have a standard form.
Applied to our explicit trilinear superpotential W eq. (130), we note that W satisfies the transformation property of a CP invariant theory. For example, underK * provided that the coefficient c (1) is real as can be chosen in our superpotential by appropriate field redefinitions. However, at a generic point in moduli space, the CP-like transformation K * does not leave the vacuum T invariant, i.e. T K * −→ − T = T , which can indicate spontaneous CP breaking. Moreover, even though the Clebsch-Gordan coefficients in T tensor products are real, this is not true for ∆(54) tensor products: tensor products of ∆(54) triplets can also yield ∆(54) doublets with complex Clebsch-Gordan coefficients [44], indicating that CP can be broken [53]. However, in order to see this effect in our superpotential one has to consider higher order couplings that, from a stringy point of view, originate from the exchange of heavy ∆(54) doublets, cf. ref. [54].

Summary
In this section, we illustrate our previous findings on R-symmetries and CP in the T 2 /Z 3 orbifold sector. It is known that this case is endowed with a T finite modular symmetry and a ∆(54) traditional flavor symmetry. However, when embedded into a six-dimensional orbifold, sublattice rotations (from SL(2, Z) U ) yield a Z R 9 R-symmetry, which acts linearly on matter fields, enhancing the traditional flavor symmetry to ∆ (54, 2, 1) ∼ = [162,44]. This leads to the largest eclectic symmetry without CP, being Ω(2) ∼ = ∆ (54, 2, 1) ∪ T ∼ = [1944,3448]. On the other hand, the modular group in T 2 /Z 3 orbifolds is naturally enhanced by a CP-like generator K * that promotes SL(2, Z) T to GL(2, Z) T . We confirm that a general modular invariant theory is also invariant under CP-like transformations from GL(2, Z) T (cf. eq. (151)), provided that matter fields and modular forms transform according to eqs. (146) and (150), respectively. Note that these conditions are expected to be valid also in general bottom-up models involving modular symmetries with CP. Including CP, the full eclectic symmetry of a T 2 /Z 3 orbifold sector embedded in a higher-dimensional orbifold is Ω(2) ∪ Z CP

Local flavor unification
In this section, we discuss first in general the mechanism of local flavor unification [18] for T 2 /Z K i orbifold sectors, taking into account the effect of their embedding in six-dimensional orbifolds. We will then focus on the T 2 /Z 3 orbifold sector as our working example, where we additionally study the implications of this scheme on the superpotential.
Consider a T 2 /Z K i orbifold sector, whose complex structure and Kähler moduli shall be denoted here as U and T (instead of U i and T i ), in order to simplify the notation as in section 5. Let us focus on the case K i = 2, for which the U modulus is geometrically fixed at a value U according to table 2. (Some of the differences appearing in the T 2 /Z 2 orbifold sector, which has been studied in ref. [19], shall be briefly mentioned below.) We focus first on modular transformations that are not CP-like and assume that the corresponding Kähler modulus T has been stabilized [55,56] at a self-dual point in moduli space. In other words, we suppose that T is stabilized at a vacuum expectation value (vev) T , at which there exists a nontrivial finite [57] subgroup H T ⊂ SL(2, Z) T that leaves T invariant. Just as H U ⊂ SL(2, Z) U defined in eq. (80), H T is also a modular stabilizer subgroup, defined as Compared to traditional flavon models, eq. (154) corresponds to the mechanism of flavon vev alignment, see also refs. [52,58,59]. In this scenario, since the vev T of the Kähler modulus breaks some modular symmetry but leaves the traditional flavor symmetry unbroken, an additional flavon will be needed for its breaking.
ii) On the other hand, matter fields Φ n with SL(2, Z) T modular weight n T ∈ n (where n denotes the set of all modular weights of the matter field Φ n ) transform in general nontrivially under γ T ∈ H T , according to see eq. (44). Since the moduli are invariant under a modular transformation γ T ∈ H T , we realize that γ T is linearly realized. Thus, the transformation matrix ρ t, T (γ T ) enhances the traditional flavor symmetry (now, also containing the discrete Z R (K i ) 2 Rsymmetry that originates from SL(2, Z) U as discussed in section 4.2) to the so-called unified flavor symmetry [18] at the point T in moduli space. Moreover, the transformation matrix ρ t, T (γ T ) builds a t-dimensional representation t of the unified flavor group. It is important to stress the presence of the extra phase in ρ t, T (γ T ) that originates from the automorphy factor (c T T + d T ) n T in eq. (156).
iii) In addition, the superpotential transforms under γ T ∈ H T , according to eq. (46), as where (c T T + d T ) −1 is a complex phase, see eq. (155), and U is the fixed value of the complex structure U of the T 2 /Z K i orbifold sector, see table 2. By definition of H T , the vev T is invariant under a transformation γ T ∈ H T . Thus, also the Kähler potential is invariant under γ T ∈ H T . Since the Lagrangian L ⊃ d 2 ϑ W must be invariant under these transformations as well, the Grassmann number ϑ must transform with a phase In summary, taking into account in the T 2 /Z K i orbifold sector the traditional flavor symmetry G fl , the discrete Z R (K i ) 2 R-symmetry from SL(2, Z) U , and the stabilizer subgroup H T which may also include CP-like transformations, the unified flavor symmetry is given by the multiplicative closure of these groups. Thus, the unified flavor symmetry can be expressed as where also the transformations from Z R (K i ) 2 ∪H T are linearly realized in spite of their modular origin.
Let us now spend a few words on the case of the T 2 /Z 2 orbifold sector. As detailed in ref. [19], in this case both T and U moduli are not fixed. However, one can choose special In the following, we illustrate the scheme of local flavor unification by assuming that the Kähler modulus is stabilized at various different points in moduli space of the T 2 /Z 3 orbifold sector. This moduli space is illustrated in figure 3. We explore first the resulting unified flavor symmetries at the boundaries of the moduli space, depicted by the curves in the figure, where we realize that a unique purely CP enhancement of the flavor symmetry occurs. Then, we focus on two self-dual points T = i and T = ω, displayed in the figure 3 as a (blue) square and a (black) bullet, respectively. Since we suppose the T 2 /Z 3 orbifold sector to be embedded in a six-dimensional orbifold, our starting point is the traditional flavor symmetry ∆ (54, 2, 1) ∼ = [162, 44] ∼ = ∆(54) ∪ Z R 9 , cf. table 5. Depending on the value of T in moduli space, different unbroken modular transformations contribute to enhance this traditional flavor group to various unified flavor symmetries contained in the full eclectic flavor group, Ω(2)∪Z CP 2 , which is of order 3,888. A brief summary of these findings is shown in table 6. Furthermore, we study the details of the twisted matter fields Φ i −2 /3 = (X i , Y i , Z i ) T and their superpotential eq. (130) compatible with the unbroken symmetries at the different points in moduli space. at Besides, we find the unbroken symmetryK 2 S that leaves the whole moduli space invariant and satisfiesK 2 S = C, becoming a symmetry generator already included in the traditional flavor group. Therefore, we find that the stabilizer subgroup is H T = Z 2 × Z CP 2 . Note that the stabilizer elements that can enhance the traditional flavor group include the CP-like generator K * in all cases. Hence, there is no flavor enhancement without CP-like transformations at a generic point on the boundary of the fundamental domain in T moduli space. That is, ignoring the possibility of CP, the traditional flavor symmetry ∆ (54, 2, 1) ∼ = ∆(54)∪Z R 9 is not enhanced at the boundaries of the fundamental domain.
In order to figure out the flavor enhancements that the CP-like stabilizer elements in eq. (160) induce, we must consider i) the matrix representations of these stabilizer elements when acting on the twisted matter fields Φ−2 /3 (and their CP-conjugateΦ−2 /3 ), and ii) the automorphy factors associated with these transformations. The representations are obtained as products of ρ(K S ) and ρ(K T ) given in eq. (122), as well as ρ(K * ) defined in eq. (147). Further, by using the 2 × 2 matrices that represent the GL(2, Z) T generatorsK S ,K T (see eq. (2)) and K * (see eq. (144)), we find that the automorphy factors are trivial for Re T = 0, −1 /2, i.e.
As the vev T = i of the modulus is invariant under H i , the coupling strengthsŶ 1 ( T ) andŶ 2 ( T ) are also invariant. Using the general modular transformations eq. (118), this implies particularly that theK S transformation on the coupling strengths leads tô Consequently, the doublet of couplingsŶ which provides a nontrivial relation betweenŶ 1 (i) andŶ 2 (i). As a side remark, because of the vev alignment eq. (166), the T modular formsŶ   in ref. [3] get aligned at T = i aŝ Y (4) Furthermore, both modular forms,Ŷ 1 (i) andŶ 2 (i), are real at T = i (as one can also infer directly from eq. (148)) and we find We can apply the vev alignment eq. (166) also to the trilinear superpotential eq. (130) of twisted matter Under a modularK S transformation with its representation given by eq. (122), twisted matter fields transform, according to eq. (156), as where the multivalued factor (−i) −2 /3 can be fixed to −ω 2 = exp( πi /3) by the Z 3 point group symmetry, and the representation t = 3 2 following the notation of eq. (156) corresponds to the triplet that the matter fields build in the resulting unified flavor symmetry at T = i. It then follows thatK S acts on the superpotential evaluated at T = i as an R-symmetry, as one can also see directly from eq. (46a). Moreover, due to ρ 3 2 ,i (K S ), the modularK S transformation at T = i generates a Z 12 ∼ = Z 4 × Z 3 traditional flavor symmetry when acting on matter fields, cf. eq. (170), where Z 4 × Z 3 can be generated by respectively. It is easy to see that the Z 4 factor generated by (ρ 3 2 ,i (K S )) 3 is an R-symmetry, while the Z 3 factor generated by (ρ 3 2 ,i (K S )) 4 is equivalent to the point group symmetry (128a) of the orbifold sector. 4 Using that (ρ 3 2 ,i (K S )) 6 = ρ 3 2 (C), without including CP-like transformations the traditional flavor symmetry is enhanced from ∆ (54, 2, 1) to Ξ(2, 2) ∼ = [324, 111] at T = i .
As a side remark, the relation ρ 3 2 ,i (K 4 S ) = (ρ 3 2 ,i (K S )) 4 = ω 2 1 in eq. (172) is a consistent representation ofK 4 S = 1 (see eq. (1)) because the latter indicates thatK 4 S must act trivially on the fields up to the action of the traditional flavor symmetry.
So far, we have not yet considered the Z CP 2 CP-like transformationK * at T = i. Using the CP-like transformation (161a) of matter fields, one can explicitly show that using the superpotential eq. (169) with c (1)Ŷ 2 (i) ∈ R and the discussion on CP from section 5.7.
Consequently, if we stabilize the Kähler modulus T at T = i, the trilinear superpotential eq. (169) respects CP. Including CP, the unified flavor group at T = i is given by where Ξ(2, 2) ∼ = ∆ (54, 2, 1) ∪ Z 4 is the maximal flavor subgroup of G uf, T =i which does not act CP-like.
6.3 Flavor symmetry at T = ω in moduli space of T 2 /Z 3 Let us now assume that the Kähler modulus is stabilized at T = ω in moduli space. There, as illustrated in figure 3, the unbroken GL(2, Z) T symmetries can be generated by the Z CP 2 CP-like transformationK * KT , and the Z 3 modular transformationK SKT , In addition, we have the Z 2 transformationK 2 S = C that is already included in the traditional flavor group. The resulting stabilizer subgroup reads H ω ∼ = Z 3 × Z 2 × Z CP 2 . At this point in moduli space, the coupling strengths governed by the modular formŝ Y 1 ( T ) andŶ 2 ( T ) with modular weight n Y = 1 get aligned aŝ 4 The transformationK 3 S at T = i acts as a Z4 on bosons and on W. However, due to eq. (158), it is actually an Z8 R-symmetry when acting on fermions. Since discrete R-symmetries are relevant for phenomenology [60,61], it may be interesting to explore the consequences of this symmetry.
i.e. the order increases by a factor of three (see ref. [50] for the nomenclature).
It follows that the Z 9 ⊂ H(3, 2, 1) transformation eq. (180) acts on the superpotential eq. (179) as This is expected due to the automorphy factor (− T − 1) −1 = ω of the superpotential W for γ T =K SKT evaluated at the point T = ω in moduli space. Thus, the Z 9 symmetry enhancement yields a discrete R-symmetry.
symmetries of the superpotential at T = ω, we have shown that the ∆(54) flavor symmetry is enhanced to (U(1) × U(1)) S R 3 , which is broken spontaneously elsewhere in moduli space.

Conclusions and outlook
In this paper we have presented a new step of the eclectic approach towards ten-dimensional string theory. This discussion goes beyond previous studies in D = 2 that concentrated on the modular transformations SL(2, Z) T of a single Kähler modulus T . We now include a detailed analysis of SL(2, Z) U of the complex structure modulus U . In the orbifolds T 2 /Z K with K = 3, 4, 6 the U -modulus is fixed because of the orbifold twist. We show that even in the case of frozen U we encounter remnant symmetries of SL(2, Z) U in form of R-symmetries that extend the eclectic group. When embedded in D = 6 compact dimensions these symmetries are relevant as sublattice rotations. We illustrate the full eclectic picture in the case of T 2 /Z 3 , where we obtain G ecl = ∆(54) ∪ T ∪ Z R 9 ∪ Z CP 2 , a discrete symmetry group of order 3, 888. In a next step towards D = 6 we shall have to consider more than one unconstrained modulus beyond T . The simplest case with two unconstrained moduli (T and U ) is the T 2 /Z 2 orbifold that has already been discussed in ref. [19]. There, it was shown that the remnants of the full modular group Oη(2, 2, Z) (that contains SL(2, Z) T × SL(2, Z) U ) lead to the finite modular group ((S 3 ×S 3 ) Z 4 )×Z CP 2 , where mirror symmetry acts as a Z 4 on the level of matter fields but only as a Z 2 permutation of T and U , and Z CP 2 is a CP-like transformation [19]. Together with the traditional flavor group (D 8 × D 8 )/Z 2 this leads to an eclectic group with 4, 608 elements. The full technical details of the analysis of ref. [19] will be presented in ref. [62]. This includes a discussion of modular forms with more than one unconstrained modulus, such as e.g. Siegel modular forms. Incidentally, in a recent work of Ding, Feruglio and Liu [63] such a situation has been discussed from the bottom-up perspective, including the aforementioned finite modular group (S 3 × S 3 ) Z 2 (on the level of the moduli and without CP). It will be interesting to see how our top-down approach from string theory can make contact with the bottom-up discussion of ref. [63].
After having control of the T 2 /Z 2 case, all of the relevant building blocks for the D = 6 case are at our disposal. A discussion of the generic case with D = 6 will be too difficult to be analyzed in detail. Simpler cases can be found in models of elliptic fibrations of D = 6 compact space. Sublattice rotations of the various D = 2 sectors would be relevant for the eclectic group. Finally, the next step would require the selection of specific string models that could successfully describe the flavor structure of quarks and leptons to make connections to the bottom-up constructions (see e.g. [64] and references therein).