Gauge Origin of Discrete Flavor Symmetries in Heterotic Orbifolds

We show that non-Abelian discrete symmetries in orbifold string models have a gauge origin. This can be understood when looking at the vicinity of a symmetry enhanced point in moduli space. At such an enhanced point, orbifold fixed points are characterized by an enhanced gauge symmetry. This gauge symmetry can be broken to a discrete subgroup by a nontrivial vacuum expectation value of the K\"ahler modulus $T$. Using this mechanism it is shown that the $\Delta(54)$ non-Abelian discrete symmetry group originates from a $SU(3)$ gauge symmetry, whereas the $D_4$ symmetry group is obtained from a $SU(2)$ gauge symmetry.


Introduction
It is important to understand the flavor structure of the standard model of particle physics. Quark and lepton masses are hierarchical. Two of the mixing angles in the lepton sector are large, while the mixing angles in the quark sector are suppressed, except for the Cabibbo angle. Non-Abelian discrete flavor symmetries may be useful to understand this flavor structure. Indeed, many works have considered field-theoretical model building with various non-Abelian discrete flavor symmetries (see [1][2][3] for reviews).
Understanding the origin of non-Abelian flavor symmetries is an important issue we have to address. It is known that several phenomenologically interesting non-Abelian discrete symmetries can be derived from string models. 1 In intersecting and magnetized D-brane models, the non-Abelian discrete symmetries D 4 , (27) and (54) can be realized [5][6][7][8]. Also, their gauge origins have been studied [6]. In heterotic orbifold compactifications [9][10][11][12][13][14][15][16][17] (also see a review [18]), non-Abelian discrete symmetries appear due to geometrical properties of orbifold fixed points and certain properties of closed string interactions [19]. First, there are permutation symmetries of orbifold fixed points. Then, there are string selection rules which determine interactions between orbifold sectors. The combination of these two kinds of discrete sym-metries leads to a non-Abelian discrete symmetry. In particular, it is known that the D 4 group emerges from the one-dimensional orbifold S 1 /Z 2 , and that the (54) group is obtained from the two-dimensional orbifold T 2 /Z 3 . The phenomenological applications of the string-derived non-Abelian discrete symmetries are analyzed e.g. in [20].
In this paper we point out that these non-Abelian discrete flavor symmetries originate from a gauge symmetry. To see this, we consider a heterotic orbifold model compactified on some sixdimensional orbifold. The gauge symmetry G gauge of this orbifold model is, if we do not turn on any Wilson lines, a subgroup of E 8 × E 8 which survives the orbifold projection. In addition, from the argument in [19], we can derive a non-Abelian discrete symmetry G discrete . Then, the effective action of this model can be derived from G gauge × G discrete symmetry invariance. 2 However, this situation slightly changes if we set the model to be at a symmetry enhanced point in moduli space. At that special point, the gauge symmetry of the model is enlarged to G gauge × G enhanced , where G enhanced is a gauge symmetry group. The maximal rank of the enhanced gauge symmetry G enhanced is six, because we compactify six internal dimensions. At this specific point in moduli space, orbifold fixed points are characterized by gauge charges of G enhanced , and the spectrum is extended by additional massless fields charged under G enhanced . Furthermore, the Kähler moduli fields T in the untwisted sector obtain G enhanced -charges and a non-zero vacuum expectation value (VEV) of T corresponds to a movement away from the enhanced point. This argument suggests the possibility that the non-Abelian discrete symmetry G discrete is enlarged to a continuous gauge symmetry G enhanced at the symmetry enhanced point. In other words, it suggests a gauge origin of the non-Abelian discrete symmetry. Moreover, the group G enhanced originates from a larger non-Abelian gauge symmetry that exists before the orbifolding. We will show this explicitly in the following.

Gauge origin of non-Abelian discrete symmetry
In this section we demonstrate the gauge origin of non-Abelian discrete symmetries in heterotic orbifold models. We concentrate on the phenomenologically interesting non-Abelian discrete symmetries D 4 and (54) which are known to arise from orbifold models.

D 4 non-Abelian discrete symmetry
First, we study a possible gauge origin of the D 4 non-Abelian discrete symmetry. This symmetry is associated with the onedimensional S 1 /Z 2 orbifold. Here, we consider the heterotic string on a S 1 /Z 2 orbifold, but it is straightforward to extend our argument to T 2 /Z 2 or T 6 /(Z 2 × Z 2 ). The coordinate corresponding to the one dimension of S 1 is denoted by X . It suffices to discuss only the left-movers in order to develop our argument. Let us start with the discussion on S 1 without the Z 2 orbifold. There is always a U (1) symmetry associated with the current H = i∂ X . At a specific point in the moduli space, i.e. at a certain radius of S 1 , two other massless vector bosons appear and the gauge symmetry is enhanced from U (1) to SU (2). Their currents are written as where α = √ 2 is a simple root of the SU(2) group. These currents, H and E ± , satisfy the su(2) Kac-Moody algebra. Now, let us study the Z 2 orbifolding X → −X. The current H = i∂ X is not invariant under this reflection and the corresponding U (1) symmetry is broken. However, the linear combination E + + E − is Z 2 -invariant and the corresponding U (1) symmetry remains on S 1 /Z 2 . Thus, the SU(2) group breaks down to U (1) by orbifolding. Note that the rank is not reduced by this kind of orbifolding. It is convenient to use the following basis, The introduction of the boson field X is justified because H and E ± satisfy the same operator product expansions (OPEs) as the original currents H and E ± . The invariant current H corresponds to the U (1) gauge boson. The E ± transform as under the Z 2 reflection and correspond to untwisted matter fields U 1 and U 2 with U (1) charges ±α. In addition, there are other untwisted matter fields U which have vanishing U (1) charge, but are charged under an unbroken subgroup of E 8 × E 8 .
From (4), it turns out that the Z 2 reflection is represented by a shift action in the X coordinate, where w = 1/ √ 2 is the fundamental weight of SU (2). That is, the Z 2 -twisted orbifold on X is equivalent to a Z 2 -shifted orbifold on Table 1 Field contents of U (1) Z 2 model from Z 2 orbifold. U (1) charges are shown. Charges under the Z 4 unbroken subgroup of the U (1) group are also shown.

Sector
Field X with the shift vector s = w/2 (see e.g., [21]). In the twist representation, there are two fixed points on the Z 2 orbifold, to each of which corresponds a twisted state. Note that the one-dimensional bosonic string X with the Z 2 -twisted boundary condition has a contribution of h = 1/16 to the conformal dimension. In the shift representation, the two twisted states can be understood as follows. Before the shifting, X also represents a coordinate on S 1 at the enhanced point, so the left-mover momenta p L lie on the momentum lattice Then, the left-mover momenta in the Z 2 -shifted sector lie on the original momentum lattice shifted by the shift vector s = w/2, i.e.
Thus, the shifted vacuum is degenerate and the ground states have momenta p L = ±α/4. These states correspond to charged matter fields M 1 and M 2 . Note that p 2 L /2 = 1/16, which is exactly the same as the conformal dimension h = 1/16 of the twisted vacuum in the twist representation. Indeed, the twisted states can be related to the shifted states by a change of basis [21]. Notice that the twisted states have no definite U (1) charge, but the shifted states do. Table 1 shows corresponding matter fields and their U (1) charges.
From Table 1, we find that there is an additional Z 2 symmetry of the matter contents at the lowest mass level (in a complete model, these can correspond to massless states): Transforming the U (1)-charges q as q → −q, (8) while at the same time permuting the fields as U 1 ↔ U 2 and M 1 ↔ M 2 maps the spectrum onto itself. The action on the U i and M i fields is described by the 2 × 2 matrix 0 1 1 0 . (9) This Z 2 symmetry does not commute with the U (1) gauge symmetry and it turns out that one obtains a symmetry of semi-direct product structure, U (1) Z 2 .
In the twist representation, this model contains the Kähler modulus field T , which corresponds to the current H and is charged under the U (1) group. In the shift representation, the field T is described by the fields U i as Now we consider the situation where our orbifold moves away from the enhanced point by taking a specific VEV of the Kähler modulus field T which corresponds to the VEV direction Note that this VEV relation maintains the Z 2 discrete symmetry (9). Moreover, since the fields U 1 and U 2 are charged under the U (1) gauge symmetry and due to the presence of the M i fields, the VEV breaks U (1) down to a discrete subgroup Z 4 . The Z 4 charge is 1/4 for M 1 and −1/4 for M 2 as listed in Table 1. Written as a 2 × 2 matrix, the Z 4 action is described by The matrices (9) and (12)  reproduces the known result for a general radius of S 1 [19]. The pattern of symmetry breaking we have shown here is summarized as follows: The other VEV directions of U 1 and U 2 break U (1) Z 2 to Z 4 .
However, while the VEV direction defined by Eq. (11) is D-flat, the other cases do not correspond to D-flat directions and the resulting symmetries have no geometrical interpretation.

(54) non-Abelian discrete symmetry
Next, we consider the two-dimensional T 2 /Z 3 orbifold case which is associated with the (54) non-Abelian discrete symmetry. Here, we study the heterotic string on a T 2 /Z 3 orbifold.
However, our argument straightforwardly extends to orbifolds such as T 6 /Z 3 . The coordinates on T 2 are denoted by X 1 and X 2 . We start with the discussion of the two-dimensional torus, T 2 , without orbifolding. There is always a U (1) 2 symmetry corresponding to the two currents, H 1 = i∂ X 1 with E n 1 ,n 2 = e i i=1,2 (n 1 α i Now, let us study the Z 3 orbifolding, where Z = X 1 + i X 2 and ω = e 2π i/3 . The currents H i and their linear combinations are not Z 3 -invariant and the corresponding gauge symmetries are broken. On the other hand, two independent linear combinations of E n 1 ,n 2 are Z 3 -invariant and correspond to a U (1) 2 symmetry that remains on the T 2 /Z 3 orbifold. Thus, the SU(3) gauge group is broken to U (1) 2 by the orbifolding. It is convenient to use the following basis, where The E n 1 ,n 2 correspond to states with charges (n 1 α 1 1 + n 2 α 1 2 , n 1 α 2 1 + n 2 α 2 2 ) under the unbroken U (1) 2 . They transform under the Z 3 twist action as follows: Thus, the first three E n 1 ,n 2 correspond to untwisted matter fields with charges −α 1 , −α 2 and α 1 + α 2 under the unbroken U (1) 2 .
We denote them as U 1 , U 2 and U 3 The Z 3 twist action on X i can then be realized as a shift action on X i as In the twist representation there are three fixed points on the  Then, the momenta p L in the k-shifted sector lie on the momentum lattice shifted by the Z 3 shift vector s = α 1 /3, Table 2 Field contents of U (1) 2 S 3 model from Z 3 orbifold. U (1) 2 charges are shown.
Charges under the Z 2 3 unbroken subgroup of the U (1) 2 group are also shown.

Sector Field
For k = 1, there are three ground states with p L ∈ {α 1 /3, α 2 /3, −(α 1 + α 2 )/3}. They correspond to (would-be-massless) matter fields which we denote by M 1 , M 2 and M 3 , respectively. These matter fields are shown in Table 2. The states for k = −1 correspond to CPT-conjugates. As expected, the shifted ground states have conformal dimension h = p 2 L /2 = 1/9, which coincides with the twisted ground states. Indeed, the shifted states are related to the twisted states by a change of basis [21]. The shifted states have definite U (1) 2 charges.
From Table 2, it turns out that the matter contents at the lowest mass level possess a S 3 permutation symmetry (in a complete model, these can correspond to massless states). Let S 3 be generated by a and b, with a 3 = b 2 = (ab) 2 = 1. Then, for a point (q 1 , q 2 ) on the two-dimensional U (1) 2 charge plane, a and b shall act as a: b: The action of a is equivalent to the replacement α 1 → α 2 → −(α 1 + α 2 ) → α 1 . Then, the spectrum is left invariant if at the same time we transform the fields The action of a on the F i is described by the 3 × 3 ma- The action of b corresponds to α 1 ↔ α 1 and α 2 ↔ −(α 1 + α 2 ), so simultaneously transforming F 1 ↔ F 1 and F 2 ↔ F 3 results in a symmetry of the spectrum. This transformation corresponds to the matrix The S 3 symmetry just shown does not commute with U (1) 2 . Rather, S 3 and U (1) 2 combine to semi-direct product U (1) 2 S 3 .
Next we shall consider the situation where our orbifold moves away from the enhanced point by taking a certain VEV of the Kähler modulus field T , which corresponds to H ω . The Kähler modulus can be described by the U i fields as The deformation is realized by the following VEV direction, Note that this VEV relation preserves the S 3 discrete symmetry generated by (36) and (37). However, the U (1) 2 gauge symmetry breaks down to a discrete Z 2 3 subgroup due to the presence of the M i fields. The two Z 3 charges (z 1 , z 2 ) are determined by U (1) 2 charges (u 1 , u 2 ) as 3 charges are listed in Table 2. The Z 3 actions are described by The matrices (36), (37), (40) and (41) are nothing but the generators of (54) (Z 3 × Z 3 ) S 3 in the 3 1(1) representation [22]. Thus, the fields (M 1 , M 2 , M 3 ) transform as the 3 1(1) under (54), and the field U is the (54) trivial singlet 1. This reproduces the known properties of ordinary Z 3 orbifold models at a general point in moduli space [19]. Summarizing, the origin of the (54) discrete symmetry in orbifold models can be explained as follows: There are other VEV directions that one might consider. For Finally, when all VEVs are different, i.e. U 1 = U 2 = U 3 = U 1 the symmetry is broken to Z 3 × Z 3 . However, while the VEV direction defined by (39) is D-flat, the other directions are not D-flat and do not allow for a geometrical interpretation.

Conclusion
We showed that non-Abelian discrete symmetries in heterotic orbifold models originate from a non-Abelian continuous gauge symmetry. The non-Abelian continuous gauge symmetry arises from torus-compactified extra dimensions at a special enhanced point in moduli space. In the two-dimensional orbifold case, by acting with Z 3 on the torus-compactified SU(3) model, the non-Abelian gauge group SU(3) is broken to a U (1) 2 subgroup. We observed that the matter contents of the orbifold model possess a S 3 symmetry which is understood to act on the two-dimensional U (1) 2 charge plane. The resulting orbifold model then has a symmetry of semi-direct product structure, U (1) 2 S 3 . In the untwisted sector, the orbifold model contains a Kähler modulus field which is charged under the unbroken Abelian gauge group. By assigning a VEV to the charged Kähler modulus field, the orbifold moves away from the enhanced point and the U (1) 2 gauge symmetry breaks to a discrete Z 2 3 subgroup. Thus, effectively the non-Abelian discrete symmetry (54) ( The other VEV directions of the untwisted scalar fields break the In the onedimensional Z 2 orbifold case, we showed that the non-Abelian gauge symmetry SU(2) is the origin of the discrete symmetry D 4 Z 4 Z 2 . The other VEV directions of the untwisted scalar fields break the symmetry to Z 4 .
The resulting non-Abelian discrete flavor symmetries are exactly those that have been obtained from heterotic string theory on symmetric orbifolds at a general point in moduli space [19]. In [19], the geometrical symmetries of orbifolds were used to derive these discrete flavor symmetries. However, in this paper, we have not used these geometrical symmetries on the surface, although obviously the gauge symmetries and geometrical symmetries are tightly related with each other. At any rate, our results also indicate a procedure to derive non-Abelian discrete symmetries for models where there is no clear geometrical picture to begin with, such as in asymmetric orbifold models [23][24][25][26] or Gepner models [27].
We give a comment on anomalies. Anomalies of non-Abelian discrete symmetries are an important issue to consider (see e.g. [28]). We start with a non-Abelian (continuous) gauge symmetry and break it by orbifolding and by moduli VEVs to a non-Abelian discrete symmetry. The original non-Abelian (continuous) gauge symmetry is anomaly-free and if it were broken by the Higgs mechanism, the remaining symmetry would also be anomaly-free. That is because only pairs vector-like under the unbroken symmetry gain mass terms. But this does not hold true for orbifold breaking, as it is possible to project out chiral matter fields. Thus, in our approach the anomalies of the resulting non-Abelian discrete symmetries are a priori nontrivial. However, in our mechanism we obtain semi-direct product structures such as U (1) 2 S 3 . Since the corresponding U (1) 2 is broken by the Higgs mechanism, the remnant Z 2 3 symmetry is expected to be anomaly-free if the original U (1) 2 is anomaly-free (the semi-direct product structure automatically ensures cancellation of U (1)-gravity-gravity anomalies, but other anomalies have to be checked). Thus, the only discrete anomalies that remain to be considered are those involving S 3 . We also comment on applications of our mechanism to phenomenological model building. In our construction the non-Abelian gauge group is broken by the orbifold action. This situation could be realized in the framework of field-theoretical higherdimensional gauge theory with orbifold boundary conditions. Furthermore, our mechanism indicates that U (1) m S n or U (1) m Z n gauge theory can be regarded as a UV completion of non-Abelian discrete symmetries. 3 Thus, it may be possible to embed other phenomenologically interesting non-Abelian discrete symmetries into such a gauge theory and investigate their phenomenological properties.