Completing the eclectic flavor scheme of the $\boldsymbol{\mathbb Z_2}$ orbifold

We present a detailed analysis of the eclectic flavor structure of the two-dimensional $\mathbb Z_2$ orbifold with its two unconstrained moduli $T$ and $U$ as well as $\mathrm{SL}(2,\mathbb Z)_T\times \mathrm{SL}(2,\mathbb Z)_U$ modular symmetry. This provides a thorough understanding of mirror symmetry as well as the $R$-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the $(T,U)$ modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, CP and $R$-symmetries on the superpotential and Kaehler potential of the theory.


Introduction
In this paper we extend our previous discussion [1] of the T 2 /Z 2 orbifold. T 2 /Z 2 is the only two-dimensional orbifold with two unconstrained moduli T , U that transform under SL(2, Z) T × SL(2, Z) U and under mirror symmetry, which interchanges T and U . Hence, it can serve as a building block for the discussion of six-dimensional orbifolds. 1 In our previous study, ref. [1], we had identified the traditional flavor symmetries and the finite modular symmetries Γ N for the T 2 /Z 2 orbifold. The groups Γ N (for small N ) are isomorphic to groups like S 3 , A 4 , S 4 and A 5 that could be suitable for a description of discrete flavor symmetries in particle physics [7][8][9]. Modular symmetries, however, require more than just a discussion of the finite modular groups Γ N . In addition, we have to include automorphy factors corresponding to the explicit modular weights of matter fields. In the present paper, we discuss the implications of these automorphy factors in the case of the T 2 /Z 2 orbifold. Once they are taken into account, we find an extension of the finite modular flavor symmetry in form of an R-symmetry, which implies further restrictions to the superpotential and Kähler potential of the theory. This is one of the reasons why a modular flavor symmetry has more predictive power than traditional flavor symmetries. In the top-down approach (which we adopt here), this extension of the symmetry reflects the symmetries of the underlying string theory, which restrict the modular weights to well-defined specific values. 2 In the bottom-up approach to modular flavor symmetries, the choice of the modular weights of matter fields is part of model building and can be used to obtain so-called "shaping symmetries" that appear as additional accidental symmetries for specific choices of the modular weights [10,11].
The main results of the paper can be summarized as follows: • We identify the full eclectic flavor symmetry [12] of the T 2 /Z 2 orbifold to be It includes a Z R 4 R-symmetry that originates from the discussion of the automorphy factors and extends the order of the eclectic flavor group from 2304 to 4608. With CP, the order of the eclectic flavor symmetry is further enhanced to a group of order 9216.
• We provide a discussion of the landscape of flavor symmetries in (T, U )-moduli space and identify the local unified flavor groups at specific points and lines in this moduli space. The results are given in figure 3, accompanied by an explicit discussion of the flavor symmetries in the cases of two specific geometrical shapes (the tetrahedron and the squared raviolo) as well as T ↔ U mirror symmetry in section 4.
• We observe a specific relation between mirror symmetry and the allowed values of modular weights of matter fields (discussed explicitly in section 3).
• The additional R-symmetry is closely related to the modular symmetry and leads to further constraints on the allowed values of modular weights of matter fields. Hence, it further restricts the form of superpotential and Kähler potential, as explicitly discussed in section 5.
• We discover the appearance of continuous gauge symmetries for specific configurations in moduli space.
The paper is structured as follows. In section 2, we recall the results of our previous study [1]. Section 3 discusses the automorphy factors and modular weights of matter fields. We identify the additional R-symmetry and the extended eclectic flavor group accordingly. This includes a discussion of the interplay of the modular weights with both, T ↔ U mirror symmetry and the R-symmetry. In section 4 we analyze the unified local flavor groups that appear at specific points, lines and other hyper-surfaces in moduli space. The results including CP are illustrated in figure 3. Section 5 is devoted to the discussion of the superpotential and Kähler potential. We observe the appearance of continuous gauge symmetries for certain configurations of the moduli (naïvely, they might appear as accidental symmetries, but they are consequences of underlying symmetries in string theory). In section 6 we give conclusions and outlook. Technical details are relegated to several appendices that complete the general discussion of ref. [1].

What do we know already?
Technical details of the eclectic flavor symmetries of T 2 /Z K orbifolds (K = 2, 3, 4, 6) have been given in section 2 of ref. [3]. In the cases K > 2, the complex structure modulus U has to be fixed to allow for the orbifold twist. For T 2 /Z 2 , in contrast, we have two unconstrained moduli T and U with the corresponding modular transformations SL(2, Z) T × SL(2, Z) U . For generic values of the moduli, we find the traditional flavor symmetry (we use the Small Group notation from GAP [13]) (D 8 × D 8 )/Z 2 ∼ = [32,49] (2.1) as the result of geometry and string selection rules (see refs. [14,15]) or, equivalently, as a result of the outer automorphisms of the (Narain) space group that describes the orbifold [4,5]. Furthermore, the finite modular symmetry for the T 2 /Z 2 orbifold is shown to be the multiplicative closure of Γ T 2 × Γ U 2 = S T 3 × S U 3 and mirror symmetry (which exchanges T and U ), as discussed in ref. [1]. The full mirror symmetry acting on the matter fields turns out to be ZM 4 (which acts on the moduli as Z 2 , cf. [16]). This leads to the finite modular group [144,115]. If we include a CP-like symmetry acting on the moduli as T → −T and U → −U , the finite modular group enhances to In combination with the generators of the traditional flavor symmetry [32,49], we obtained an eclectic flavor group with 4608 elements (2304 without CP).
Only some of the eclectic flavor symmetries are linearly realized. For generic values of the moduli just the traditional flavor group [32,49] remains unbroken. For specific "geometrical" configurations, this symmetry is enhanced to a larger subgroup of the eclectic flavor group (via the so-called stabilizer subgroups). The generators of the unbroken groups are displayed explicitly in figure 7 of ref. [1]. Relevant values correspond to the moduli U = i (the squared raviolo) and U = exp(πi/3) (the tetrahedron) as well as the line T = U as a consequence of mirror symmetry. At T = U , we find the enhancement of [32,49] to [64,257]. For the tetrahedron, the group [32,49] is enhanced to [96,204] as discussed in section 4.2 of ref. [1], while for the raviolo, we shall see here in section 4 that [32,49] is enhanced to [128,523]. If we include the CP-like transformation, we gain a further enhancement of the number of elements by a factor of two. The largest linearly realized subgroup of the eclectic flavor group (including CP) was found (in ref. [1]) to be [1152,157463] at T = U = exp(πi/3).
So far the results are based on the finite modular groups. A full discussion of modular symmetries should, however, not only include the finite symmetries Γ N (here Γ T 2 × Γ U 2 = S T 3 × S U 3 ), but also the so-called automorphy factors that arise from the non-trivial (fractional) modular weights (n T , n U ) of SL(2, Z) T × SL(2, Z) U . This leads to further restrictions on the action (given by Kähler and superpotential) of the theory with an enhancement of the symmetries. As discussed in refs. [2,3], these automorphy factors lead to discrete phases resulting in R-symmetries. In our previous paper [1], for the sake of clarity and simplicity, we had not included these automorphy factors in our discussion. We shall include them in the following in full detail.

Discrete R-symmetries and mirror symmetry
In this section, we show that a T 2 /Z 2 orbifold sector gives rise to a Z R 4 symmetry that originates from modular transformations, where the automorphy factors of certain modular transformations give rise to the R-charges. As the T 2 /Z 2 orbifold sector is equipped with two moduli, T and U , there exists a modular group for each of them, SL(2, Z) T and SL(2, Z) U , each associated with a modular weight (n T , n U ). Since R-charges can be defined in terms of both modular groups, these modular weights are highly constrained. Furthermore, we give a detailed discussion about the action of mirror symmetry on matter fields and discover a new relation between mirror symmetry and the R-symmetry.

Automorphy factors of modular transformations
Let us consider a general matter field Φ (n T ,n U ) originating from string theory with modular weights n T and n U corresponding to SL(2, Z) T and SL(2, Z) U . Then, under a (non CP-like) modular transformationΣ ∈ Oη(2, 2, Z), the field transforms as Here, j (n T ,n U ) (Σ, T, U ) is the automorphy factor of the modular transformation and ρ r (Σ) is the representation matrix ofΣ that forms a representation r of the finite modular group, as derived in appendix D.
The modular weights of matter fields can be computed in string theory, as reviewed in appendix B, and it turns out that, apart from n T = n U , there is also the possibility in string theory. In order to determine the automorphy factor j (n T ,n U ) (Σ, T, U ), we might use as a first step the analogy to Siegel modular forms based on Sp(4, Z). However, Siegel modular forms are defined for parallel weights n := n T = n U only. In this case, following refs. [16,17], we have as defined in ref. [18]. Then, eq. (3.3) yields Using the dictionary [18] that relates Sp(4, Z) with the modular group Oη(2, 2 + 16, Z) of our string setting, the Sp(4, Z) element M (γ T ,γ U ) is equivalent toΣ (γ T ,γ U ) ∈ Oη(2, 2+16, Z) defined in appendix A.1. We take this as a motivation to define also for the case n T = n U . It turns out that for the other casesΣ =Σ (γ T ,γ U ) one can use the automorphy factor eq. (3.3) for the element M ∈ Sp(4, Z) that corresponds toΣ using again the dictionary of ref. [18]. It is important to note that the resulting automorphy factor will be independent of the specific choice n = n T or n = n U , since n T = n U mod 2. In the following, we will see this explicitly at some examples.

Discrete R-symmetry
In the T 2 /Z 2 orbifold sector, a Z 2 sublattice rotation is given bŷ i.e. in the Narain formulation,Θ (2) = −1 4 is a left-right symmetric 180 • rotation in the T 2 /Z 2 orbifold sector that leaves the orthogonal compact dimensions invariant, see refs. [2,3]. It is an outer automorphism of the full Narain space group of the six-dimensional orbifold and, bulk matter twisted matter As the generalized metric H(T, U ) is invariant under a transformation (A.29) withΘ (2) , this Z 2 sublattice rotation leaves T and U invariant. Hence, the sublattice rotationΘ (2) corresponds to a traditional flavor symmetry. Still, the transformation withΘ (2) originates from a modular transformation,Θ (2) =Ĉ 2 S =K 2 S . So, we expect the appearance of an automorphy factor.
Since R ∈ SL(2, Z) T and R ∈ SL(2, Z) U are identified in Oη(2, 2, Z), we have to ensure that we compute the automorphy factor correctly: we can use either the factor (c T T + d T ) n T = (−1) n T or (c U U + d U ) n U = (−1) n U for the transformation R. Yet, the resulting automorphy factor must coincide in both cases, (−1) n T = (−1) n U . Hence, we see that This relation is satisfied for all (massless) matter from the T 2 /Z 2 orbifold sector, as one can see from table 1. Moreover, eq. (3.9) also holds for all massive strings, as shown in appendix B.
Consequently, having control over the automorphy factor, we can choose R ∈ SL(2, Z) U and the modular weight n U in the following.
The action ofΘ (2) on matter fields Φ (n T ,n U ) with SL(2, Z) U modular weights n U ∈ {0, −1, −1 /2, 1 /2, −3 /2}, as listed in table 1, is given by (3.11) Here, we used that ρ r (Θ (2) ) = ρ r (K S ) 2 = ρ r (Ĉ S ) 2 = 1. For the allowed modular weights n U ∈ {0, −1, −1 /2, 1 /2, −3 /2} the multivalued phase factor gives rise to Z R 4 R-charges R ∈ {0, 2, 3, 1, 1}, respectively. Thus, for the T 2 /Z 2 orbifold sector we find that the R-charge R is given in terms of the modular weight n U (or n T ) as cf. ref. [19]. Note that due to the fractional modular weights n U , (Θ (2) ) 2 gives a non-trivial transformation with charges 2R = 4n U mod 4 that turns out to be equivalent to the point group selection rule of eq. (D.24a). Since the R-symmetry transformation acts trivially on all moduli, it belongs to the traditional flavor symmetry, which gets enhanced to where the Z 2 in the latter quotient identifies the point group selection rule of T 2 /Z 2 contained in both the Z R 4 and the traditional symmetry (D 8 × D 8 ) /Z 2 . In string theory, modular symmetries are anomaly-free (see e.g. [20,21] for details on anomaly cancellation for modular symmetries). Hence, since the Z R 4 R-symmetry arises from modular symmetries, it is anomaly-free.
Due to the Z R 4 R-symmetry, the eclectic flavor group of a T 2 /Z 2 orbifold sector gets extended to which results in a group of order 4608. Including a CP-like transformation, the order of the eclectic flavor group is further enhanced to a group of order 9216.

The action of mirror symmetry on matter fields
In order to analyze the action of mirror symmetryM ∈ Oη(2, 2, Z) on matter fields in our string setup, we have to determine the automorphy factor first. Using the results of section 3.1, we consider the mirror element in Sp(4, Z), which reads Thus, the automorphy factor (3.3) of a mirror transformation is given by (−1) n . Since n T = n U mod 2 as derived in eq. (3.9), one can assign n = n U and the automorphy factor of a mirror transformationM is given as (−1) n U , without loss of generality. Moreover, note that the Z R 4 R-charges given in eq. (3.11) are analogously defined. Thus, the automorphy factor ofM can be removed using the Z R 4 R-symmetry, as we will do in the following. Now, let us assume that under a mirror transformationM we have for a matter field Φ (n T ,n U ) that transforms in the representation r of the finite modular group 115] (where we have absorbed the automorphy factor (−1) n U using Z R 4 ). In the following, we will see that the transformation (3.16) is correct for n T = n U , but must be modified in the case n T = n U . To do so, let us consider the following chain of transformations under the assumption that eq. (3.16) were correct. However, a mirror transformation maps an elementΣ (1 2 ,γ U ) ∈ SL(2, Z) U to an elementΣ (γ U ,1 2 ) ∈ SL(2, Z) T , see eq. (A.25). Thus, eq. (3.17c) must be equal to where the 2 × 2 matrix γ T ∈ SL(2, Z) T inΣ (γ T ,1 2 ) has to be equal to the matrix γ U used in eq. (3.17b). Now, we have to distinguish between two cases: First, in the case of so-called parallel weights (i.e. if n T = n U ) the representation matrices ρ r (Σ (γ U ,1 2 ) ) and ρ r (Σ (1 2 ,γ U ) ) have to be related as follows and eq. (3.17c) coincides with eq. (3.18). In the second case (i.e. if n T = n U ), the (preliminary) chain of transformations given in eq. (3.17) has to be modified Then, we have to impose condition (3.19) and, consequently, eq. (3.20c) coincides with eq. (3.18) usingΣ (γ T ,1 2 ) with γ T equal to γ U .
In other words, for each matter field Φ (n T ,n U ) with n T = n U (satisfying the constraint (3.9)) there must exist a partner field, denoted by Φ (n U ,n T ) , which coincides in all properties with Φ (n T ,n U ) but has interchanged modular weights. Then, a mirror transformation has to act on matter fields (Φ (n T ,n U ) , Φ (n U ,n T ) ) as and eq. (3.19) has to hold. On the other hand, the transformationsK S ,K T ,Ĉ S andĈ T act diagonally on (Φ (n T ,n U ) , Φ (n U ,n T ) ). For example, after an appropriate basis change one can check using the character (see table 1). Finally, in appendix B we show in string theory why a mirror transformation interchanges Φ (n T ,n U ) and Φ (n U ,n T ) if n T = n U , as we have seen in this bottom-up discussion that has lead to eq. (3.21).
At these points, the couplingsŶ where c T , d T , c U , d U are integers that define γ. This relation can be straighforwardly extended to include stabilizer elements that involve the mirror symmetryM and the CP-likeΣ * generators, as they do not induce automorphy factors, see the discussion in section 3.3 and appendix D.3.2. Since the representation ρ 4 3 (γ) of the finite modular group is unitary, its eigenvalues and hence the automorphy factors must be phases, see also ref. [3, section 6]. Note that eq. (4.5) corresponds to the mechanism of flavon alignment in the context of modular flavor symmetries, as also discussed in e.g. refs. [23][24][25].
A consequence of the automorphy factors being phases at ( T , U ) is that the modular transformations from H ( T , U ) act linearly on matter fields. Hence, the stabilizer enhances the traditional flavor symmetry to the multiplicative closure of the traditional flavor group and the stabilizer modular subgroup, i.e. to the so-called unified flavor group (4.6) Explicitly, from eqs. (3.1) and (3.6), the action of a (non-CP-like) stabilizer element γ ∈ H ( T , U ) on a field Φ (n T ,n U ) with modular weights (n T , n U ) is given by where ρ t,( T , U ) is a t-dimensional representation t of the unified flavor group, whereas r is a representation of the finite modular group [144,115]. We stress here the presence of the automorphy factors in the transformation (4.7), which can enhance the order of the unbroken transformations due to the possibility of fractional weights of matter fields.

Unified flavor groups at generic points in moduli space
Even for generic values of the moduli, the traditional flavor group is enhanced for T = U .
In this case, the mirror transformationM is left unbroken. Considering its ZM 4 action on matter fields, as given by eqs. (3.16) or (3.21), we find that the unified flavor group in this case is given by [128,2316].
We can consider also the case that one of the moduli has a generic value while the second modulus is fixed at one of the special points, i or e πi /3 . In figure 1, we display the different unified  flavor groups achieved by incorporating the unbroken modular transformations at those special points in moduli space. Let us consider the results for generic U , as presented in figure 1a. At T = i, we know (cf. ref. [1]) that the stabilizer subgroup is generated byK S , which becomes a

Unified flavor groups of the raviolo
At U = i (cf. ref. [11]), the geometry of the T 2 /Z 2 orbifold adopts the form of a square raviolo, where the corners correspond to the singularities of the orbifold and the edges are perpendicular and have the same length. As just mentioned, in this case the traditional flavor   There are two special points in moduli space for the raviolo, where further enhancements occur if the CP-like modular transformationΣ * is considered. First, at T = i we find the stabilizer H (i,i) = Ĉ S ,M ,Σ * , which enhances the traditional flavor symmetry to [32,49] is the traditional flavor group without Z R modulus fixed at T = i Im T > i is left invariant by Ĉ S ,Σ * . Acting on matter fields along with their corresponding automorphy factors, this yields the unified flavor group [256,25882]. Furthermore, the traditional flavor symmetry is enhanced to [256,6341] along the regions of the locus λ T where T = e iϕ with π /3 < ϕ < π /2, and T = 1 /2 + i Im T with Im T > √ 3 /2. In these regions, the stabilizers are given by Ĉ S ,K SΣ * and Ĉ S ,K TΣ * , respectively.

Unified flavor groups of the tetrahedron
When the complex structure is stabilized at U = e πi /3 , the T 2 /Z 2 orbifold sector has the shape of a tetrahedron, cf. ref. [1, figure 5]. As can be read off from figure 1b, in the tetrahedron with a generic value for T , theĈ TĈS modular transformation leaves the moduli invariant and enhances the traditional flavor group to the unified flavor group [192,1509], the generic flavor symmetry of the tetrahedron. This contains discrete R-symmetries due to the inclusion of the discrete rotation in the compact space generated byĈ TĈS . Along the locus λ T , there are two more enhancements of the traditional flavor group. For T = i Im T > i, not onlyĈ TĈS leaves the moduli invariant but alsoĈ TΣ * . This implies that the flavor symmetry of the tetrahedron [192,1509] is enhanced to the unified flavor group [384,20097]. Besides, for T = e iϕ with π /3 < ϕ < π /2, the stabilizer is Ĉ TĈS ,K SĈSΣ * and leads to the unified flavor group [384,20098]. The same flavor enhancement is obtained if the Kähler modulus sits at T = 1 /2 + i Im T with Im T > √ 3 /2, where the stabilizer is generated byĈ TĈS andK TĈSΣ * .

A 4 flavor symmetry from the tetrahedron
Let us turn back to the case of the tetrahedron, U = e πi /3 , with a generic value for the Kähler modulus. In this case, according to eq. (4.7), the stabilizer modular generatorĈ TĈS acts on matter fields as where due to the automorphy factor (c U U + d U ) n U we get The admissible modular weights of massless matter fields are n U ∈ {0, −1} for bulk matter, and n U ∈ { −3 /2, −1 /2, 1 /2} for twisted matter. Hence, eq. (4.10) describes a Z 6 transformation that we can write as Z 2 × Z 3 , generated by respectively. Using the admissible modular weights, we note that the Z 2 factor in eq. (4.12a) corresponds to the Z (PG) 2 point group selection rule of the T 2 /Z 2 orbifold sector. Moreover, the Z 3 factor eq. (4.12b) acts on the superpotential W as which is a discrete Z R 3 R-symmetry using the definition ω := exp ( 2πi /3). The group generated by the traditional flavor group elements ρ 4 (h 1 ), ρ 4 (h 2 ) from eq. (D. 21) together with the Z (PG) 2 factor from eq. (4.12a) and the Z R 3 factor from eq. (4.12b) turns out to be (4.14) Here, the alternating group A R 4 is a non-Abelian R-symmetry as it arises from ρ 4 (h 1 ), ρ 4 (h 2 ) and the Z R 3 R-symmetry, cf. ref [26] for a general discussion on non-Abelian R-symmetries. The matter fields and the superpotential W build the following representations of A R where we denote the irreducible representations of Z 2 by 1 0 and 1 1 and the ones of A R 4 by 3, 1, 1 and 1 , see appendix E.4.
Combined with the Z R 4 symmetry associated with the sublattice rotationΘ (2) and the Z 2 × Z 2 generators ρ 4 (h 3 ) and ρ 4 (h 4 ) of the traditional flavor group, associated with the space group selection rule of the T 2 /Z 2 orbifold sector, the group A R 4 × Z 2 enhances to the unified flavor group of the tetrahedron, [192,1509], see figure 2b.
Compared to the literature (see e.g. refs. [27][28][29]), we see that in the consistent string approach the naïve A 4 symmetry obtained by the compactification of two extra dimensions on a tetrahedron has to be extended in two ways: First, the Z 3 generator of A R 4 turns out to be an R-symmetry. This can be understood equivalently either as a discrete remnant of the extra-dimensional Lorentz symmetry or as a discrete remnant of an SL(2, Z) U modular symmetry. In addition, A R 4 is enhanced in the full string approach by stringy selection rules to [192,1509] of order 192, which still contrasts with previous results [14].   symmetry arising from embedding the orbifold sector in higher dimensions, ii) the CP-like modular transformationΣ * , and iii) the phases associated with the automorphy factors of modular transformations acting on matter fields Φ (n T ,n U ) . We use as reference axes the straightened lines describing the boundaries λ T and λ U of the T and U moduli spaces, respectively. Mirror symmetryM acts on the moduli and the modular generators as U ↔ T ,Ĉ S ↔K S , andĈ T ↔K T . As a first consequence, the points along the diagonal in figure 3, defined by T = U , are left invariant byM . Furthermore, the points below and above this diagonal are connected by a mirror transformation. It then follows thatM identifies the unified flavor groups in these two sectors of moduli space.

Other CP-enhanced unified flavor groups
Focusing on the lower half of the plane, below the diagonal of figure 3, we see that the sole enhancements that have not been discussed in the preceding subsections are those that lie at the diagonal, and those that are valid in the squared and triangular regions of figure 3. Let us consider two examples. In the lowest part of the diagonal, the stabilizer modular subgroup is H (ix,ix) = M ,Σ * , with x > 1. Considering the associated transformations on matter fields with their corresponding automorphy factors, as given in appendix D.3.2, we find that the traditional flavor group [64,266] is in this case enhanced to [256,56079]. Similarly, in the bottom triangle of the figure, the stabilizer is given just by H (ix,iy) = Σ * , with x, y > 1 and y > x, which yields the unified flavor group [128,2326]. Other cases can be easily determined by using the proper stabilizer subgroups provided in our previous work [1, figure 7].

Effective field theory of the Z 2 orbifold
In this section, we focus on four 4-plets of twisted matter fields that we label by the winding numbers (n 1 , n 2 ) ∈ {(0, 0), (1, 0), (0, 1), (1, 1)} and not by the modular weights n T and n U . Hence, each twisted matter field φ i (n 1 ,n 2 ) is localized at one of the four fixed points of the T 2 /Z 2 orbifold sector, see appendix A and ref.
[1, figure 1]. In other words, we consider four 4-plets We assume that the 4-plets differ in some additional charges, for example with respect to the unbroken gauge group from E 8 × E 8 (or SO (32)). Then, we use the eclectic flavor symmetry to write down the most general Kähler and superpotential to lowest order in these fields.

The Kähler potential
The Hermitian Kähler potential K of a single twisted matter field Φ ( −1 /2, −1 /2) reads to leading order [30] K whereK is an Hermitian bilinear polynomial of the formK = c n 1 n 2 n 1 n 2 φ (n 1 ,n 2 )φ(n 1 ,n 2 ) . In the following, we constrainK by imposing the traditional flavor symmetry step by step. First, the space and point group selection rules (D.24) enforce n 1 = n 1 and n 2 = n 2 , resulting iñ K = c n 1 n 2 φ (n 1 ,n 2 )φ(n 1 ,n 2 ) . Then, invariance under eq. (D.21a) forces all coefficients c n 1 n 2 to be equal (and we normalize them to 1). Hence, Now, we can generalize this easily to four 4-plets of twisted matter fields Φ i ( −1 /2, −1 /2) , for i ∈ {1, 2, 3, 4}. Due to our assumption of additional (gauge) charges that distinguish between φ i (n 1 ,n 2 ) and φ j (n 1 ,n 2 ) for i = j, we obtaiñ Consequently, for the T 2 /Z 2 orbifold sector the traditional flavor symmetry already enforces the Kähler potential to be diagonal in twisted matter fields. Hence, this diagonal structure can not be changed by the full eclectic flavor symmetry: Since additional terms involving modular formsŶ (T, U ) (as suggested by ref. [31]) are singlets of the traditional flavor group, the Kähler potential must remain diagonal, cf. ref. [6]. Yet, additional corrections to the Kähler potential that involve flavons are still possible, cf. ref. [32].

The superpotential
To lowest order in twisted matter fields Φ i ( −1 /2, −1 /2) , the superpotential reads schematically Here, we have imposed the Z R 4 R-symmetry and the fact that the modular weights of matter fields and couplings have to add up to (−1, −1) for the superpotential, see table 1. Thus, Y (0) (T, U ) has to carry modular weights (0, 0), while the modular formŶ (2) (T, U ) has modular weights (2, 2). There exists a unique modular form of weight (2, 2), which we denote bŷ Y (2) 4 3 (T, U ) in the following (see eq. (4.1)). In addition, the superpotential has to be covariant under the full eclectic flavor symmetry, i.e. it has to be invariant simultaneously under the traditional non-R symmetries and the finite modular flavor symmetry but transform with the appropriate phases (automorphy factors) under the R-symmetry (modular symmetry).

Constraints from the traditional flavor symmetry
Let us start with invariance under the traditional flavor symmetry (D 8 × D 8 ) /Z 2 ∼ = [32,49]. First, we consider the product of two twisted matter fields Φ i ( −1 /2, −1 /2) Φ j ( −1 /2, −1 /2) needed for the terms (5.4a) in the superpotential W. The fields Φ i ( −1 /2, −1 /2) transform as irreducible 4-plets of the traditional flavor group [32,49]. Hence, we need to consider the tensor product This tensor product contains one trivial singlet 1 ++++ , which corresponds to the terms for i, j ∈ {1, 2, 3, 4}. The total R-charge is 2 as one can see easily from table 1. As a remark, one can check the invariance of the terms I ij 0 explicitly using the orthogonality of the representation matrices given in eq. (D.21). Since Φ (0,0) is a trivial singlet 1 ++++ of [32,49] with R-charge 0, the terms Φ (0,0) I ij 0 ⊂ W are allowed by both, the traditional flavor symmetry and Z R 4 . Next, we study the product of four twisted matter fields in order to construct the superpotential terms in eq. (5.4b). Since we know from eq. (5.5) that there are 16 invariant combinations I i , i ∈ {1, . . . , 16}. We list them in appendix E.2. Consequently, out of the 4 4 = 256 possible terms from , invariance under the traditional flavor symmetry [32,49] allows only 16.

Constraints from the modular symmetry
As explained in ref.
of the finite modular group [144,115]. Only the two 4 3 representations yield invariant terms when they are combined with the modular formŶ (5.10b) Note that the quartic polynomial Q 1 is antisymmetric when The invariant terms in the superpotential then read Hence, the number of unfixed superpotential parameters in eq. (5.4b) is reduced from 16 in the case of imposing only the traditional flavor symmetry to 2 (i.e. c 1 and c 2 ) when we include the constraints from the full eclectic flavor symmetry. This is in contrast to the leading order Kähler potential, see eq. (5.3), where the finite modular symmetry did not yield additional constraints compared to the traditional flavor symmetry. Finally, the superpotential of eq. (5.4) is thus explicitly given by where Q 1 and Q 2 are the quartic polynomials in the twisted matter fields Φ i ( −1 /2, −1 /2) , given in eq. (5.10).

Gauge symmetry enhancement in moduli space
Let us analyze the "accidental" continuous symmetries of the superpotential eq. (5.12) that appear at special points in (T, U ) moduli space, cf. ref. [3] for the analogous discussion in the case of the T 2 /Z 3 orbifold sector. We assume that the four twisted matter fields Φ i transform identically under the enhanced symmetry. To identify continuous symmetries, we define a general U(4) transformation that leaves the Kähler potential of Φ i  Table 2: Gauge symmetry enhancements by Lie algebra elements t in the case of the T 2 /Z 2 orbifold sector at special points in moduli space, uncovered as "accidental" symmetries of the superpotential due to the alignment of couplings in flavor space. Note that the field basis of twisted matter fields Φ i ( −1 /2, −1 /2),g with well-defined gauge charges differs from the field basis of localized twisted strings Φ i ( −1 /2, −1 /2) using the basis change M g in eq. (5.14).
symmetries for general values of the moduli (T, U ), we do observe subgroups of U(4) being unbroken at special values of the moduli. We discuss such cases in the following.
Note that from the top-down perspective of string theory, the appearance of continuous symmetries is expected. As discussed in appendix C, these "accidental" symmetries are actually gauge symmetries: at special points in moduli space some winding strings become massless, giving rise to the gauge bosons of enhanced gauge symmetries. Consequently, the enhanced symmetries that we uncover in this section are exact symmetries to all orders in the superpotential.
In the following, we briefly discuss the results for three special configurations: i) T = U for generic U , ii) T = U = ω, and iii) T = U = i. In each case, we first evaluate the superpotential eq. (5.12) in the respective vev configuration by analyzing the alignment of the couplingsŶ (2) 4 3 ( T , U ) in flavor space. Then, we identify the unbroken Lie algebra elements t := α a T a from the transformation (5.13). Afterwards, we perform a (unitary) basis change such that the unbroken Lie algebra elements t g := M g t M −1 g are (block-)diagonalized. Finally, we identify the continuous symmetry and the charges (or representations) of the twisted matter fields Φ i ( −1 /2, −1 /2),g . The results are summarized in table 2. At T = U , there appears an enhanced U(1) symmetry. Note that the traditional flavor subgroup Z 4 ⊂ (D 8 × D 8 )/Z 2 generated by h 1 h 3 is a subgroup of this U(1). Hence, one can verify that the traditional flavor symmetry gets enhanced to where the Z 2 quotient identifies (h 1 h 3 ) 2 from the left factor with (h 2 h 4 ) 2 from the right factor.
At T = U = i in moduli space, there is an enhanced U(1) 2 symmetry. In this case, the traditional flavor subgroup generated by the order 4 elements h 1 h 3 and h 2 h 4 is a subgroup of this U(1) 2 symmetry. Hence, the traditional flavor symmetry is enhanced to where the Z 2 quotient identifies (h 1 h 3 ) 2 with (h 2 h 4 ) 2 , as before.

Conclusions and Outlook
We performed a detailed analysis of the modular symmetries of the T 2 /Z 2 orbifold which (among others) might be relevant for the (discrete) flavor symmetries of string compactifications with an elliptic fibration. The T 2 /Z 2 case has two unconstrained moduli with SL(2, Z) T × SL(2, Z) U modular symmetry and allows contact with previous bottom-up constructions that have more than one modulus [16,17,[33][34][35]. In the present paper, we completed the discussion of our earlier work [1] now including the automorphy factors of modular symmetry. This leads to an additional R-symmetry Z R 4 (for the given modular weights of matter fields) that plays the role of a so-called "shaping symmetry" and extends the discrete flavor symmetry. In more detail, the traditional flavor symmetry of ref. [1] is extended from [32,49] via Z R This picture reveals the fact that the top-down discussion of modular flavor symmetry constitutes an extremely restrictive scenario, which is confirmed in other top-down scenarios [36][37][38][39][40]. As in the case of the bottom-up discussion, firstly the role of (otherwise freely chosen) flavons is played by the moduli T and U , and secondly we arrive at a specific finite modular group, being Γ T 2 × Γ U 2 = S T 3 × S U 3 for the T 2 /Z 2 orbifold. In addition, we have to consider the restrictions from the automorphy factors with modular weights fixed from string theory (in contrast to the bottom-up case where these values can be chosen freely). Moreover, in addition to the finite modular symmetry, string theory provides a traditional flavor symmetry, which gives severe restrictions for Kähler-and superpotential of the theory (discussed in section 5). Finally, the representations of the relevant matter fields of the traditional and modular flavor symmetries are determined by the theory. We summarize them (along with the corresponding modular weights) in table 1.
Compared to the earlier discussions [4,5] where one modulus was frozen, the two-modulus case allows a full understanding of mirror symmetry (as discussed in section 3 and appendix B), including the situation of matter fields whose modular weights n T and n U differ from each other, see eq. (3.9). In this case, mirror symmetry requires the presence of matching representations where n T and n U are interchanged.
We observe enhancements of the traditional flavor group at specific locations in moduli space. These unified flavor groups are discussed in section 4 and summarized in figure 3. The largest group is located at T = U = exp( πi /3) and has 2304 elements (including CP).
We also provide a detailed discussion of the tetrahedral T 2 /Z 2 orbifold, which leads to the group [192,1509] as extension of the traditional flavor group. It includes a "geometrical" A 4 as an R-symmetry, where twisted matter fields transform as 3 ⊕ 1 of this A R 4 , as explained in section 4.3.1.
The restrictions on Kähler-and superpotential are discussed in section 5. The traditional flavor symmetry is extremely powerful towards the restrictions on the Kähler potential. As in the T 2 /Z 3 discussed earlier [6], the traditional flavor group restricts the Kähler potential to its trivial diagonal form (5.3): a fact that seems to hold in full generality. In contrast, both symmetries are relevant for the form of the superpotential given in eq. (5.12). The traditional flavor symmetry reduces the 256 terms in eq. (5.4b) down to 16, and the modular flavor symmetry reduces the remaining 16 to 2, see eq. (5.12b).
A further special feature of string theory is the possible appearance of continuous gauge (flavor) symmetries in moduli space. At special points in moduli space, winding modes of the string can become massless and are candidates for the gauge bosons, as discussed in section 5.3 (see table 2) and appendix C (with table 3). These symmetries, of course, reflect themselves in the symmetries of the superpotential. From a bottom-up perspective they might appear as accidental symmetries, but from the top-down point of view they correspond to continuous gauge symmetries of string theory.
Together with our earlier discussion [3] of the T 2 /Z K orbifolds with K = 3, 4, 6, we now have uncovered the basic properties of the flavor symmetries of two-dimensional orbifold compactifications for the case of up to two unconstrained moduli. One might expect that some of these properties will generalize from the case of toroidal orbifolds to more general string compactifications with an elliptic fibration. Moreover, from the string theory point of view, the next step would be the consideration of orbifolds with Wilson lines as additional moduli.
This would require the embedding of SL(2, Z) T × SL(2, Z) U and mirror symmetry in the Siegel modular group Sp(4, Z), as discussed in ref. [18], see also refs. [16,17], where bottom-up model building based on Sp(4, Z) has been initiated.
We focus here on the case without background Wilson lines. In the Narain formulation of the heterotic string [44][45][46], the string coordinate in extra-dimensional space y is split into right-and left-moving string modes y R and y L , respectively. Then, we define and eq. (A.1) is extended to Here, k ∈ {0, . . . , K − 1} for a Narain twist Θ, with θ R , θ L ∈ SO(2), that is of order K, i.e. Θ K = 1 4 . The Narain twist generates the Narain point group P Narain ∼ = Z K and the orbifold action Y → Θ k Y + EN defines the so-called Narain space group S Narain , Each (conjugacy class) g ∈ S Narain defines a closed string and, therefore, we call g the constructing element. We focus on symmetric orbifolds by setting θ R = θ L = θ and choose Therefore, the Narain lattice Γ is an even, integer, self-dual lattice of signature (2,2). In the absence of Wilson lines, the Narain vielbein E can be parameterized in terms of the geometrical vielbein e, its inverse transposed e −T , the geometrical metric G = e T e and the B-field background B, see for example refs. [47,48] (where we changed the convention from B to −B). Then, a twotorus compactification can be parameterized by a Kähler modulus T and a complex structure modulus U , defined as In the last equation of U , we have taken both two-dimensional column vectors e i of the geometrical vielbein e to be complex numbers, e i ∈ C, so that e 2/e 1 is defined. Note that T determines the strength of the B-field and the area of the extra-dimensional two-torus, while U specifies the shape of the two-torus. It is convenient to associate a generalized metric H to the Narain vielbein E and express H in terms of the moduli T and U , (A.8) see for example ref. [3]. Furthermore, we define the Narain twist in the lattice basis aŝ Let us focus in the following on bulk strings, i.e. on strings that close under the identification eq. (A.3) with constructing element (1 4 , EN ) ∈ S Narain . Then, right-and left-moving momenta p R and p L of a string have to be quantized, because the extra dimensions are compact. As the Narain lattice Γ is self-dual, p R and p L must belong to Γ, too. Hence, In order to identify (massless) string states from the bulk, one has to consider the right-and left-moving mass equations where In eq. (A.13a), q = (q 0 , q 1 , q 2 , q 3 ) denotes the bosonized momentum of the right-moving worldsheet fermions. It is called the H-momentum. q has to be an element of one of the following weight lattices of SO(8): either the vector lattice 8 v or the spinor lattice 8 s , see for example ref. [49]. The shortest H-momenta q satisfy q 2 = 1, i.e.
Here, in the first case (8 v ), the underline denotes all permutations and, in the second case (8 s ), the number of plus-signs must be even. The first component q 0 of q defines the fourdimensional chirality. For example, q 0 = 0 yields a scalar. Note that in the four-dimensional effective quantum field theory, we use the convention that the scalar components of left-chiral superfields φ from the bulk are associated with string states having q ∈ { 0, +1, 0, 0 }, such that string states with q ∈ { 0, −1, 0, 0 } give rise to their CPT partners.
Furthermore, we demand that Σ leaves the metric η invariant, In other words, we demand that Σ leaves any Narain scalar product P T ηP for P, P ∈ Γ invariant. The resulting transformationsΣ form a group Oη(2, 2, Z) := Σ Σ ∈ GL(4, Z) withΣ TηΣ =η , the so-called modular group of the Narain lattice Γ. It is easy to see that Oη(2, 2, Z) contains two factors of SL(2, Z), i.e. we can definê As a remark,Σ (γ T ,γ U ) satisfies the property of being a representation, for all γ T , δ T ∈ SL(2, Z) T and γ U , δ U ∈ SL(2, Z) U . The generators S and T of the modular group SL(2, Z) can be represented by the 2 × 2 matrices S = 0 1 −1 0 and T = 1 1 0 1 , (A. 22) respectively. Then, we can definê where one can easily show that mirror symmetryM interchanges the SL(2, Z) factors, Having identified the modular symmetries of a toroidal compactification, the modular symmetries of an orbifold are given by the rotational outer automorphisms of the Narain space group S Narain that preserve the Narain metric η. They can be understood as those modular transformationsΣ ∈ Oη(D, D, Z) (with D = 2 in the present case) that are also from the normalizer of the Narain point group, Note that the Narain twist is not an outer automorphism, but an inner automorphism of S Narain . For example, for the T 2 /Z 2 orbifold we haveΘ = −1 4 andP Narain ∼ = Z 2 . Hence, Oη(2, 2, Z)/Z 2 is the modular group of the T 2 /Z 2 orbifold. However, we consider the twodimensional T 2 /Z 2 orbifold to be contained in a full six-dimensional orbifold. Hence, we assume that the underlying six-dimensional torus is factorized as T 6 = T 2 ⊕ T 2 ⊕ T 2 and that the Narain twist of the (6, 6)-dimensional Narain lattice takes the form Here,Θ (K i ) denotes an order K i Narain twist of the i-th (2, 2)-dimensional Narain sublattice for i ∈ {1, 2, 3}, where K 1 = 2 andΘ (2) = −1 4 . Then, we can define a so-called sublattice rotationΘ (2) ⊕ 1 4 ⊕ 1 4 which is an outer automorphism of the Narain space group of the full six-dimensional orbifold. Consequently, the modular group in the T 2 /Z 2 orbifold sector is Oη(2, 2, Z).

A.2 Transformation of bulk fields under modular symmetries
In this section, we analyze the action of modular transformations from Oη(2, 2, Z) on those fields of the effective four-dimensional theory that originate from the bulk of the extra dimensions. The transformation of twisted matter fields will be discussed later in appendix D.
First, we discuss the moduli (i.e. the Kähler modulus T and the complex structure modulus U , see eq. (A.7)). According to eq. (A.16), a modular transformationΣ ∈ Oη(2, 2, Z) acts as on the Narain vielbein E. Consequently, we can use the generalized metric to compute the transformation of the moduli, This can be used to show thatΣ (γ T ,γ U ) from eq. (A. 19) acts on the moduli as Moreover, using eq. (A.29) we can confirm that the mirror transformationM interchanges the moduli, T ↔ U , while the CP-like transformationΣ * acts as As a remark, we can now understand the conditions (A.10) on a Narain twist as follows: A Narain twistΘ ∈P Narain must be a modular transformation (Θ ∈ Oη(2, 2, Z)) that leaves the moduli invariant (compare eq. (A.29) to eq. (A.10)).
Next, we consider a general (massive) bulk field φ (N ) labeled by its winding and KK numbersN ∈ Z 4 that corresponds to a closed string with boundary condition (A.4) given by the constructing element (1 4 , EN ). Its total mass M 2 (N ; T, U ) is moduli dependent viâ N T H(T, U )N , as shown in eq. (A.14). Then, the corresponding mass terms in the superpotential read schematically Under a (non-CP-like) modular transformationΣ ∈ Oη(2, 2, Z), moduli and bulk fields trans- where we suppress the automorphy factor for φ (N ) . In addition, we haveN =Σ −1N as shown in eq. (A.16) and the factor ±1 of φ (N ) will be derived later in eq. (D.15). Then, due to its moduli-dependence, the total string mass M 2 (N ; T, U ) transforms as where ∆N i := N i −Nī ∈ N 0 is the total number of holomorphic minus anti-holomorphic oscillators with internal index i = 1 orī =1 in the direction of the T 2 /Z 2 orbifold sector, i.e.
and n T and n U coincide if the associated string state carries no oscillator excitations. For example, a massless matter field from the bulk (k = 0) has no oscillators and q ∈ { 0, +1, 0, 0 }, so that n T = n U ∈ {0, −1}, while a massless twisted matter field (k = 1) without oscillators has q 1 sh = 1 2 and n T = n U = −1 /2, see table 1. In addition, there are twisted string states (massless or massive) that are excited by oscillators. According to eq. (B.2), the modular weights are increased/decreased by adding oscillator excitations, add holomorphic oscillator : n T → n T − 1 , n U → n U + 1 (B.4a) add anti-holomorphic oscillator : n T → n T + 1 , In both cases, the resulting string states transform identically under the Z 2 orbifold projection (this is a special property of Z 2 orbifolds and not true for Z K orbifolds with K = 2). Hence, for each matter field Φ (n T ,n U ) with n T = n U , there exists a partner with exactly the same mass and identical quantum numbers except for interchanged weights, i.e. Φ (n T ,n U ) has a partner Φ (n U ,n T ) if n T = n U , cf. table 1.
Mirror symmetry interchanges holomorphic and anti-holomorphic left-moving oscillators. In order to see this, we rewriteM (given in eq. (A.24)) into the left-right coordinate basis (y R , y L ) at T = U in moduli space. This results in Recall that a general transformation Σ := EΣ E −1 acts on the coordinate Y eq.
Hence, the mirror transformation M acts on the complex leftmoving string coordinate z 1 in the direction of the T 2 /Z 2 orbifold sector as Hence, a mirror transformation interchanges holomorphic and anti-holomorphic oscillators resulting in eq. (3.21).

C Gauge symmetry enhancement
It is a well-known feature of string theory that at special points (T, U ) in moduli space, additional gauge symmetries arise whose gauge bosons are associated with massless winding strings. These massless strings become massive by moving in moduli space away from the special points. Hence, the enhanced gauge symmetry gets broken spontaneously by the moduli vevs. In order to identify the enhanced gauge symmetries, we look for additional massless strings from the orbifold bulk that become massless only at certain points in moduli space. We do this in two steps: first, we construct the massless string states on the torus T 2 and then move on to the T 2 /Z 2 orbifold by projecting the massless torus states onto Z 2 -invariant states.
In general, a massless string has to satisfy M 2 R = M 2 L = 0. Then, from eq. (A.13a) together with q 2 = 1 it follows that N R = 0 and p R = 0. Hence, for p R = 0 eqs. where the indices i = 1 andī =1 lie in the two-torus that will be orbifolded by the Z 2 action.
Note that the string states (C.7) correspond to the Cartan generators, while the string states (C.6) correspond to raising operators (with +N ∈ N g (T, U )) and lowering operators (with −N ∈ N g (T, U )) of some non-Abelian, enhanced gauge symmetry. The root lattice of this symmetry group is spanned by the left-moving momenta p L that correspond to the solutionsN ∈ N g (T, U ) using eqs.  where ±p L is given by ±N ∈ N g (T, U ), respectively.
We analyze three special points in moduli space: 4 i) T = U , ii) T = U = i and iii) T = U = ω and summarize the results in table 3. Consequently, the enhanced continuous symmetries identified in section 5.3 are actually gauge symmetries.

D Vertex operators of the Z 2 Narain orbifold
The spectrum of the T 2 /Z 2 orbifold sector includes untwisted strings, associated with constructing elements (1,N ) ∈Ŝ Narain , and twisted strings constructed by elements (Θ,N ) ∈ S Narain . In this appendix, we study how the symmetries of the theory act on these strings by inspecting the transformations of their corresponding vertex operators.

D.1 Untwisted vertex operators
The zero-mode vertex operator corresponding to a bosonic string on a toroidal background with winding and Kaluza-Klein numbersN = (n, m) T ∈ Z 4 is given by [50, eq. (3.41)] where the string coordinate operator Y results from promoting E −1 Y to an operator (see eq. (A.2)). Y satisfies the commutation relations 5 (derived from the action of the sigma model) is the symplectic structure in the Narain basis. The nonzero value of the commutator (D.2) is a result of intrinsic non-commutative effects of closed strings [50]. The zero-mode vertex operators (D.1) in combination with the commutator (D.2) are subject to the so-called Weyl quantization relation According to ref. [51], this relation is instrumental to evaluate the time ordering of operators as required in the computation of scattering amplitudes. The quantization relation (D.3) must hold independently of whether the vertex operators have been affected by modular transformations. As we shall shortly see (cf. eqs. (D.13)), this helps determine the phases required for the modular generators to act consistently on twisted vertex operators [52].
Using eq. (D.1), one finds that the Z 2 orbifold-invariant untwisted vertex operators are given by To determine the transformation of VN 0 under a translationĥ i , we observe that This implies that V (N ) acquires a Z 2 phase, which is identical for V (−N ). Consequently, the orbifold invariant vertex operator (D.4) inherits the same phase. It thus follows that the untwisted vertex operator class VN 0 gets a Z 2 phase too, Under a rotational outer automorphismΣ,N transforms toΣ −1N . Then, we expect that the vertex operator V (N ) transforms according to 1N ) . (D.10) Here, we propose, due to the nontrivial commutation relations (D.2), a phase ϕΣ(N ) that is given by the ansatz The 4 × 4 matrix AΣ (with only half-integral off-diagonal entries) and the vector CΣ ∈ Z 4 will be determined next. Note that, with these conditions, ϕΣ(N ) can only be a Z 2 phase.
By demanding that the Weyl quantization relation (D.3) be preserved byΣ and using the abbreviationμ = 1 2 (η +ω), one arrives at In contrast, CΣ cannot be constrained by the quantization condition. However, the effect of CΣ is equivalent to the one of a translationĥ i in the Narain lattice, given in eq. (D.9). These translations generate the traditional flavor symmetry, which is unbroken independently of the moduli. Therefore, the traditional flavor symmetry allows for a free choice of the vector CΣ.

D.2 Operator product expansions of twisted vertex operators
Even though vertex operators of twisted string states are more involved than untwisted vertex operators, OPEs of two twisted states in two-dimensional orbifolds are known. Let us consider the twisted vertex operators φ n a and φ n b of twisted strings localized at orbifold fixed points given by the winding numbers n a , n b ∈ {(0, 0) T , (0, 1) T , (1, 0) T , (1, 1) T }. Up to a constant overall factor, they satisfy the OPE [53] φ n a φ n b = These expressions together with the transformation properties of untwisted operators, eqs. (D.9) and (D. 16), can lead to the corresponding transformations of the twisted vertex operators φ n , as we now discuss.

D.3 Transformations of twisted vertex operators
With the help of the explicit relations (D. 19) between the OPEs of twisted string states and the untwisted states, and the transformations (D.9) and (D.16) of the latter, we can deduce the action of Out(Ŝ Narain ) onφ n a φ n b . One can then infer the action of those transformations on the single twisted operators, arranged in a twisted multiplet (φ (0,0) , φ (1,0) , φ (0,1) , φ (1,1) ) T . Note that, since no oscillator excitation is present in these twisted string states, this multiplet must correspond to the components of the twisted matter field Φ ( −1 /2, −1 /2) , i.e. with n T = n U = −1 /2. In these terms, the transformations of twisted states can be encoded in transformation matrices ρ r (Σ) or ρ r (h i ), which denote r-dimensional representations of the outer automorphismsΣ and h i . Our goal here is to present those transformation matrices.
Since twisted strings transform in the (real) representation 4 of [32,49], the OPEsφ n a φ n b can be associated with the tensor product Finally, since oscillator excitations are not affected by the transformations associated withĥ i , all other twisted matter fields Φ (n T ,n U ) (see table 1) must transform in the same 4-dimensional representation defined by eq. (D.21).

E.4 A 4 character table
We denote as a, b and c the abstract A 4 generators associated with h 1 , h 2 and (Ĉ TĈS ) 2 , respectively, see section 4.3.1. In these terms, A 4 is defined by the presentation A 4 = a, b, c a 2 = b 2 = c 3 = (ab) 2 = 1, bca = bacb = c , (E. 12) and has four conjugacy classes: (E.13) The character table of A 4 is shown in table 5, where we also present the order and number (size) of the elements in each conjugacy class.