Flavon alignments from orbifolding: $SU(5)\times SU(3)$ model with $\mathbb{T}^6/\Delta(54)$

We systematically develop the formalism necessary for ensuring that boundary conditions of flavon fields in extra dimensions are consistent with heterotic string theory. Having developed a set of consistency conditions on the boundary conditions, we explore a series of examples of orbifolds in various dimensions to see which ones can satisfy them. In addition we impose the further phenomenological requirements of having non-trivial flavon vacuum alignments and also of having quarks and leptons located appropriately in extra dimensions. The minimal successful case seems to be a 10d theory with a $SU(3)_{fl}$ gauged flavour symmetry, where the six-dimensional torus is compactified on a $\mathbb{T}^6/\Delta(54)$ orbifold. We construct a realistic $SU(5)$ grand unified theory along these lines, leading to tribimaximal-reactor lepton mixing, which we show to be consistent with current neutrino data.


Introduction
The Standard Model (SM), though remarkably successful, gives no understanding of either the origin of the three generations of quarks and leptons or their curious pattern of masses and mixings. In particular the observed very light neutrino masses and approximate tribimaximal lepton mixing requires new physics beyond the SM. To address some of these questions, it has been suggested that the three generations of quarks and leptons may be unified into a triplet of an SU(3) fl gauged flavour symmetry. The three generations are then analogous to the three colours of quarks in QCD. However, unlike QCD, the SU(3) fl gauged flavour symmetry must be broken spontaneously in such a way as to result in the observed quark and lepton masses and mixings [1][2][3][4][5].
In order to spontaneously break the SU(3) fl gauged flavour symmetry, one introduces additional Higgs-like scalars, usually referred to as flavons. Such flavons must have certain vacuum alignments in SU(3) fl flavour space in order to account for the observed quark and lepton masses and mixings. Unfortunately, the sectors introduced in order to achieve such vacuum alignments are typically rather ad hoc. However, there is a top-down method for achieving flavon vacuum alignments coming from string theory formulated in extra dimensions. For example, E 8 × E 8 heterotic string theory in 10d can accommodate a larger gauge symmetry than the SM which in principle could also include a flavon sector whose vacuum alignments may be understood from a more robust theoretical point of view.
A recent example of the above approach to flavons in extra dimensions was discussed in ref. [28]. There it was suggested that a SU(3) fl gauged flavour symmetry in 6d, when compactified on a torus with the orbifold T 2 /(Z 2 × Z 2 ) and supplemented by a Z 6 × Z 3 discrete symmetry, together with orbifold boundary conditions, may generate all the desired SU(3) fl breaking vacuum expectation values (VEVs). The analysis considered the (0, 1, −1) and (1, 3, −1) vacuum alignments (CSD3) in SU(3) fl flavour space of the Littlest Seesaw model [29,30] suitable for atmospheric and solar neutrino mixing. Although this idea of using orbifold boundary conditions to dial the desired vacuum alignments of flavons is very attractive, it is non-trivial to ensure that such boundary conditions are consistent with the constraints arising from string theory. These constraints were ignored in [28], loosening the connection of that model with string theory.
In this paper we systematically develop the formalism necessary for ensuring that boundary conditions of flavon fields in extra dimensions are consistent with heterotic string theory. Having developed a set of consistency conditions on the boundary conditions, we explore a series of examples which satisfy them plus the further phenomenological requirements of yielding non-trivial flavon vacuum alignments and of having the SM fermions located on orbifold fixed points, so that their massless modes may include Orbifold Flavon alignment GUT breaking SM matter localization Section T 2 /Z 2 3 T 6 /(Z 2 × Z 2 ) 4 T 6 /S 4 5 T 6 /∆(54) 6 Table 1: Orbifolds studied in this paper. We demand three necessary conditions to build a realistic and predictive model: a non-trivial flavon alignment, the possibility of orbifold GUT breaking and appropriate localizations for SM matter. Only T 6 /∆(54) fulfils all of them.
complete multiplets under the gauged flavour symmetry. It turns out to be non-trivial to satisfy all of these conditions (theoretical and phenomenological) together. For instance, the simple T 2 /Z 2 orbifold, while allowing SM fermion matter localisation on fixed points, does not permit non-trivial flavon vacuum alignments, consistently with the formal requirements of the boundary conditions. This motivates us to go to 10d models.
However, the simple orbifold T 6 /(Z 2 × Z 2 ) fares no better than the previous case. We find that the boundary conditions must exhibit some non-Abelian structure so that we can have non-trivial VEV alignments. Hence, we are led to consider 10d non-Abelian orbifolds [31][32][33], where the torus is modded out, for example, by S 4 or ∆ (54). The latter case is an example where we can locate the SM fermions on fixed points in extra dimensions. Since the ∆(54) orbifold is not so well studied in the literature, we develop this case in some detail, and eventually show that we can choose the extra dimensions in a way that we can build a complete, consistent, predictive and realistic model which is in principle compatible with string theory. The resulting model is capable of achieving the flavon vacuum alignments consistent with tri-bimaximal lepton mixing [34], which may be corrected to yield tribimaximal-reactor lepton mixing [35], which we show to be consistent with the latest neutrino data. The layout of the paper is summarised in table 1 which not only summarises the foregoing situation but also gives the organisation of the main body of this paper in terms of the section numbers 3, 4, 5, and 6 shown. These sections are bracketed by the extra dimensional heterotic string friendly formalism in section 2 and a realistic model based on the ∆(54) orbifold in section 7. Section 8 concludes the paper.

Constraints on the gauge embedding
Let us consider a field theory with D extra dimensions z compactified on an orbifold O = R D /S with space group S, see appendix A for details. The geometric orbifold action z → θ z + λ of a space group element h = (θ, λ) ∈ S (where θ ∈ P and the so-called point group P is a finite subgroup of SO(D)) is embedded into an action on a general field Φ(x, z) of the theory as where for each h ∈ S there is an element R h of the symmetry group G of the theory. In general, the symmetry group G contains the internal part of the higher-dimensional Lorentz symmetry. For example, if Φ(x, z) is a four-dimensional scalar equipped with an internal vector-index we get Φ(x, z) → θ Φ(x, θ −1 z) for h = (θ, 0). In addition, G can contain a higher-dimensional gauged flavour symmetry such that R h is given by a constant gauge transformation, see for example ref. [12]. We call R h the (gauge) embedding of h and denote the associated discrete group by R S := {R h | h ∈ S}. One can apply two transformations g and h on a field Φ(x, z), either combined or one transformation after the other, i.e.
In both cases one has to obtain the same result. Hence, the consistency condition follows. Mathematically, this condition says that R has to be a group homomorphism from the space group S into the (gauge) group G. It is easy to obtain some immediate consequences of eq. (3), e.g for g, h ∈ S, and for a space group element (θ, λ) ∈ S of order N , i.e. (θ, λ) N = (1, 0), i, j ∈ {1, . . . , D}, and we have defined the embedding of a pure translation as R e i = R (1,e i ) , see also section 2 of ref. [36]. Hence, the choices for the embedding R are strongly constrained as we will also see in more detail in the examples of the next sections.

Standard embedding
There exists a simple solution to the group homomorphism condition (3): the so-called standard embedding. In this case, (ignoring for a moment the higher-dimensional Lorentz symmetry for simplicity) one chooses a gauge group G such that the point group P is a discrete subgroup of it. Furthermore, for supersymmetric orbifolds in D = 6 dimensions we have P ⊂ SU(3) such that there exists a globally defined constant spinor on the orbifold O [22,23]. Hence, in complex coordinates each point group element θ ∈ P is given by a 3 × 3 unitary matrix (with det(θ) = 1) and the choice trivially satisfies the group homomorphism condition (3) for G = SU(3) fl . This SU(3) gauge symmetry can naturally be identified with a flavour symmetry, hence the subscript "fl". In other words, the geometrical element h = (θ, λ) that acts on the (complex) extradimensional coordinates z ∈ C 3 as z → θ z + λ is accompanied by an identical gauge Note that eq. (6) implies for instance that R e i = 1 for i ∈ {1, . . . , 6}, i.e. there are no Wilson lines along the torus-directions [12] in the case of standard embedding, and the discrete gauge embedding group is isomorphic to the point group, R S ∼ = P . As a final remark, in addition to the condition (3) on the (gauge) embedding, string theory orbifolds are constrained by world-sheet modular invariance of the one-loop string partition function [23]. However, modular invariance is automatically satisfied in the case of standard embedding. Thus, the standard embedding R S ∼ = P in our field theory discussion fulfils all necessary conditions for a full string completion.

Orbifold boundary conditions
Next, we discuss the various origins of fields Φ(x, z) on an orbifold O and the conditions to make these fields well-defined within the orbifold construction. This discussion is crucial in order to understand the orbifold-alignment of flavon VEVs in flavour space.
For each space group element g ∈ S that has a non-trivial fixed point set one can define a field Φ g (x, z) that is localized on F g , i.e.
This localized field Φ g (x, z) transforms in some representation of G. For example, for a flavour symmetry G = SU(3) fl we will mostly assume that Φ g (x, z) is either a singlet or a triplet of SU(3) fl .
In the language of string theory, a field Φ g (x, z) with g = 1 corresponds to a so-called twisted string localized at the fixed point set F g of the so-called constructing element g ∈ S. In contrast, a field Φ g (x, z) with g = 1 has a trivial fixed point set F g = R D and, hence, lives in the full bulk O of the extra dimensions.
Importantly, a field Φ g (x, z) from eq. (8) is in general not yet invariant under the orbifold action eq. (1), i.e. it transforms as for h ∈ S. Let us analyze eq. (9) in more detail. While the field Φ g (x, z) on the lefthand side is localized at z ∈ F g , the field Φ g (x, h −1 z) on the right-hand side is localized at h −1 z ∈ F g which is equivalent to z ∈ F h g h −1 . However, the only field localized on the fixed have to be related, By definition, the space group element h g h −1 belongs to the conjugacy class [g]. Thus, fields from the same conjugacy class have to be identical up to some proportionality factors, i.e. up to a symmetry transformation from G.
A special case appears if h g h −1 = g for h = g, i.e. in the case when g and h commute. Hence, we define the set of commuting elements (the so-called centralizer C g ) of the constructing element g as Now, we have to distinguish between two cases, depending on whether g and h commute or not: The case h ∈ C g is not of great importance to us and is therefore relegated to appendix A.1. On the other hand, if h ∈ C g we get Φ h g h −1 (x, z) = Φ g (x, z). Then, eqs. (9) and (10) yield a non-trivial boundary condition for h ∈ C g and h = g . (12) Since z and h −1 z are identified on the orbifold O, this boundary condition ensures that the field Φ g evaluated at identified points has a unique value up to a symmetry transformation with R h ∈ G. The orbifold boundary condition using the constructing element h = g ∈ C g itself is special, since Hence, Φ g (x, g −1 z) = Φ g (x, z) for all z ∈ R D and the transformation (9) reads while the relation (10) becomes trivial for h = g. In more detail, for h = g we get Φ g (x, h −1 z) = Φ g (x, z) on the left-hand side and Φ h g h −1 (x, z) = Φ g (x, z) on the righthand side of eq. (10). From the string construction, we know that a string with constructing element g ∈ S survives the orbifold projection at least under the action of g ∈ C g , see for example appendix A.5 in ref. [37]. Hence, eq. (14) imposes the boundary condition In summary, in order to build an orbifold-invariant field Φ g (x, z) that is localized on the fixed point set F g we have to impose a non-trivial boundary condition (12) for each space group element h ∈ C g that commutes with the constructing element g ∈ S of the localized field Φ g (x, z).
Let us briefly discuss the trivial example with constructing element g = 1 ∈ S. In this case, the fixed point set is given by F 1 = R D and the field Φ 1 (x, z) lives in the full bulk O of the extra dimensions. Furthermore, the centralizer C 1 equals the full space group, i.e. C 1 = S, and we have to impose boundary conditions (12) for all elements h ∈ S, i.e. for all generators of S.

VEV alignment from orbifold boundary conditions
Our main focus is to interpret G as a gauged flavour group in extra dimensions and the field Φ g (x, z) as a flavon. Therefore, we separate the higher-dimensional Lorentz symmetry from G and consider G as a pure gauge symmetry. Then, orbifold boundary conditions (12) break the flavour symmetry G and simultaneously align the vacuum expectation values of the flavons, as we discuss next.
Consider a field Φ g (x, z) localized at z ∈ F g with constructing element g ∈ S. We take an element h = (θ h , λ h ) ∈ C g , where the order of θ h is denoted by N h , i.e. (θ h ) N h = 1. After choosing h ∈ {0, . . . , N h − 1}, the field Φ g (x, z) has to satisfy the boundary condition (12), The h -dependent phase originates from diagonalizing the higher-dimensional Lorentz transformation, which is possible for all h ∈ C g simultaneously if the centralizer C g is Abelian.
We denote the four-dimensional zero mode of the field Φ g (x, z) by Φ g (x). Then, the orbifold boundary condition (15) evaluated at the vacuum expectation value of the zero mode Φ g reads for all elements h ∈ C g from the centralizer of the constructing element g. This condition can align the VEV of a localized field to a specific direction in flavour space. In other words, the VEV of a flavon located in F g must be an eigenvector of the matrices R h with an h -dependent phase as eigenvalue. However, the magnitude of the VEV cannot be constrained by orbifold boundary conditions. For example, take an SU(3) fl flavour group and a triplet flavon Φ g with constructing element g ∈ S and assume that the centralizer contains an element h ∈ C g with h 3 = 1 such that (R h ) 3 = 1. By choosing a special embedding R h ∈ SU(3) fl and h = 0, the flavon VEV Φ g is aligned according to However, one can always choose a basis in flavour space such that a given embedding matrix R h becomes diagonal and the VEV aligns, for example, into the first component of Φ g . To avoid this rather trivial situation, one has to ensure that the discrete embedding group R S ⊂ G is non-Abelian such that one cannot diagonalize all elements simultaneously. This is the key observation towards successful flavon alignment.
3 Flavour from a T 2 /Z 2 orbifold Let us begin with an easy example with D = 2 extra dimensions, parametrized in complex coordinates by z ∈ C. We choose a general two-torus T 2 spanned by two vectors e 1 and e 2 . Then, z is compactified on a T 2 /Z 2 orbifold, i.e. with point group P ∼ = Z 2 , where the Z 2 orbifold action ϑ is generated by z → −z. This orbifold has four inequivalent fixed points F g , corresponding to the constructing elements g, g ∈ {(ϑ, 0) , (ϑ, e 1 ) , (ϑ, e 2 ) , (ϑ, e 1 + e 2 )} , respectively, see figure 1.
To define a field theory on this Z 2 orbifold, we have to choose a (gauge) embedding R ϑ , R e 1 , and R e 2 , for each generator of the Z 2 space group S, i.e. for each (ϑ, 0), (1, e 1 ), and (1, e 2 ) ∈ S. We have to ensure that this embedding satisfies the following conditions, obtained from eq. (3), where i ∈ {1, 2}, see also eqs. (4) and (5).

Example of a trivial VEV alignment
A trivial solution to eq. (20) is given by the choice of a gauged flavour symmetry G = U(2) and   In this case, the embedding group R S ∼ = Z 2 is Abelian and, consequently, VEVs can only be aligned trivially in flavour space. In detail, take a doublet flavon Φ g (x, z) from the bulk (i.e. with trivial constructing element g = (1, 0) ∈ S and centralizer C g = S). Then, its VEV can only be aligned according to originating from the boundary conditions where the extra ± sign in the ϑ-boundary condition is motivated from the transformation properties of Φ g (x, z) under higher-dimensional Lorentz symmetry, see section 2.4.
In order to avoid this trivial VEV alignment eq. (22) in the case of this Z 2 orbifold, we have to choose non-trivial Wilson lines R e i = 1 for some i ∈ {1, 2}. However, choosing (20) would also result in an Abelian embedding group 5 and, consequently, would yield a trivial VEV alignment.

Example of a non-trivial VEV alignment
Let us give a non-trivial example of matrices R ϑ , R e 1 and R e 2 for a gauged flavour Since (ϑ) 2 = 1 the gauge embedding R has to satisfy (R ϑ ) 2 = 1. We might choose 5 This choice corresponds to the Abelianization R S ∼ = S/[S, S] of the space group S, see e.g. [38][39][40].
Furthermore, we have to satisfy condition (20) we are left with the freedom to choose R e i for i ∈ {1, 2} subject to the previous condition. One can check that a solution is given by using the matrices where (SU ) 2 = 1. These matrices are chosen since they are part of a specific representation of the discrete group S 4 , which is known for generating predictive flavour structures [28,41]. Importantly, for this choice the discrete gauge embedding group R S is non-Abelian and, furthermore, the matrices R e i have infinite order, i.e. for all N i ∈ N we have Hence, R S is not a finite group, see ref. [42] and, furthermore, ref. [14] for a discussion on rank reduction in the case when the discrete gauge embedding group R S is non-Abelian. Now, consider a triplet flavon Φ g (x, z) with constructing element g = (1, 0) ∈ S.
It is localized in the bulk O of the extra-dimensional space. Then, the corresponding centralizer C g can be generated by The flavon is subject to boundary conditions (16) which result in the following non-trivial conditions on the VEV (choosing h = 0 for h = (ϑ, 0)) The solution is given by a fixed VEV alignment where the magnitude of the VEV cannot be determined by orbifold boundary conditions.
We can try to locate another flavonΦ g in the bulk with a different Z 2 phase, explicitly to obtain a different flavon alignment. However, it turns out that Φ g = 0. In other words,Φ g is projected out by the orbifold action in this case. We have achieved the flavon alignment (30), which is necessary for the CSD3 setup [29]. This is highly predictive for the lepton sector and usually complicated to obtain through a vacuum alignment superpotential [21,29,[43][44][45][46]. However, it is not enough by itself. We have shown that there are no other alignments we can obtain through boundary conditions in this setting. Consequently, after a brief discussion on GUT breaking and the localization of SM matter in the following, we will go to higher-dimensional orbifolds with larger point groups to align various flavons simultaneously.

GUT breaking
We assume that the extra-dimensional theory before orbifolding contains a GUT gauge symmetry in addition to the gauged flavour symmetry G, where we will choose G = SU(3) fl or U(3) fl as our prime examples. Then, the full gauge symmetry in extra dimensions is given by Both, the GUT group and the flavour group, have to be broken by orbifold boundary conditions, determined by the gauge embedding R. In this specific T 2 /Z 2 orbifold the GUT-breaking boundary conditions can be chosen to correspond to any of the translations R e i , while the flavor symmetry can be broken by the gauge embedding R ϑ of the orbifold twist ϑ.
If the symmetry is SU(5) GUT the GUT-breaking boundary condition can be chosen as (5) does not act on the flavour symmetry G, it commutes with all flavour breaking conditions and, hence, is consistent with eqs. (20) (using for example R e 1 = P SU(5) and (P SU(5) ) 2 = 1).
In the case of an SO(10) GUT symmetry, SO(10) can be broken by two independent Z 2 boundary conditions [13] where σ i are the Pauli matrices. Each condition separately breaks SO(10) GUT according to while together they break SO(10 (1), see e.g. [18]. The two boundary conditions P GG and P PS commute with each other, as well as with any flavour-breaking condition. Thus, we can choose each one to be one of the R e i consistently.

SM fermion localization
The SM matter fields, as any other field in this T 2 /Z 2 orbifold, must be located somewhere in extra dimensions: i) either on a fixed point set F g for g = 1, being points in compactified dimensions, see eq. (18), or ii) in the two-dimensional bulk O. In principle, localized fields feel boundary conditions (16) with respect to their centralizers C g . Consequently, some zero modes of SM matter fields can be projected out by the orbifolddepending on the respective centralizers C g . First, consider a localized field Φ g with constructing element g from eq. (19). In this case, the centralizer C g = {1, g} is trivial as discussed in section 2.3. Consequently, a localized field Φ g in the T 2 /Z 2 orbifold is not subject to orbifold boundary conditions and the four-dimensional zero mode Φ g (x) is not projected out. Therefore, SM matter fermions from localized fields appear in complete GUT multiplets. This is the fieldtheoretical analogue to string-theoretical local GUTs with complete SM generations [47][48][49].
In contrast, a bulk field is subject to non-trivial orbifold boundary conditions, especially to those that induce GUT breaking. Let us discuss the consequences of SM matter in the bulk for SU(5) GUT and SO(10) GUT in the following: In the SU(5) GUT , a complete SM generation is given by the representations5 + 10 + 1 of SU(5) GUT . What happens if we assume that these matter fields live in the bulk of the orbifold? In this case, the singlet (of the right-handed neutrino) is not affected by the GUT-breaking boundary condition P SU (5) . For the other SU(5) GUT representations each matter field can be a positive or negative eigenstate of P SU (5) , which determines the respective zero modes. For example, let us denote the5 of SU(5) as F = (d c , ). Then, eq. (16) yields while the 10 of SU(5) GUT (being an anti-symmetric 5 × 5 matrix T ) is subject to the boundary conditions T → P SU(5) T P SU(5) = ±T . In total, we get To have the full SM matter content after compactification, we have to have each eigenstate. Consequently, the number of SM matter bulk fields before compactification must be duplicated.
In the SO(10) GUT , a complete SM generation fits into the 16 of SO(10) GUT . Depending on the eigenstate of each boundary condition P GG and P PS we obtain the zero modes Hence, if the SM matter is supposed to originate from the extra-dimensional bulk we need four copies of 16-plets in the orbifold bulk for a single generation of SM quarks and leptons. Note that split GUT multiplets can be beneficial to explain the different masses for charged leptons and down quarks. Beside GUT-breaking boundary conditions, SM matter fields will in general be subject to the boundary conditions of the flavour group SU(3) fl . Obviously, SU(3) fl singlets are not affected by the flavour-breaking boundary conditions. However, in order to get non-trivial predictions from the SU(3) fl flavour group, we assume that some SM matter fields transform as SU(3) fl triplets corresponding to the number of three generations. Then, in order to keep the structure in the fermion mass matrices dictated by the flavour symmetry, these flavour-triplets must be kept as triplets after compactification. Therefore, the SU(3) fl matter triplets must be localized at zero-dimensional fixed points in the extra dimensions of the orbifold with trivial centralizers such that they are not subject to flavour-breaking boundary conditions. Then, the point group P ∼ = Z 2 × Z 2 is generated by where z = (z 1 , z 2 , z 3 ) ∈ C 3 denotes the complex coordinates on the three two-tori and ϑ and ω are given by Since ϑ, ω ∈ SU(3), four-dimensional N = 1 supersymmetry can be preserved in this orbifold. This orbifold has 16 + 16 + 16 = 48 inequivalent fixed point sets F g corresponding to the constructing elements g being where n 3 , n 4 , n 5 , Note that each fixed point set F g is two-dimensional, e.g.
see figure 2 for a three-dimensional illustration.
To define a field theory on this T 6 /Z 2 × Z 2 orbifold, we have to choose a (gauge) embedding R ϑ , R ω , and R e i for each generator of the space group S, i.e. for each (ϑ, 0), (ω, 0), and (1, e i ) for i ∈ {1, . . . , 6}. We have to ensure that this embedding satisfies the following conditions, obtained from eq. (3), and see also eqs. (4) and (5). Then, using the homomorphism property (3) a general element of the space group g ∈ S has a (gauge) embedding R g given by Figure 2: Illustration of the six-dimensional T 6 /Z 2 ×Z 2 orbifold with an orthonormal six-torus, projected onto (Re(z 1 ), Re(z 2 ), Re(z 3 )). The orbifold has 16 + 16 + 16 two-dimensional fixed tori corresponding to the ϑ, ω and ϑω twisted sectors, respectively. They become 4 + 4 + 4 fixed lines in this projection, c.f. ref. [51].
Compared to the T 2 /Z 2 orbifold in section 3 we have more R-matrices that could potentially allow us to fix more flavon alignments. However, the conditions (43) and (44) are quite restrictive. It turns out that if we were to build a flavon-setup as in section 3.2 the stringent conditions would not allow new useful alignments. We end up with the same alignment capabilities as the smaller orbifold T 2 /Z 2 . Furthermore, the T 6 /Z 2 ×Z 2 orbifold only has two-dimensional fixed tori and no zero-dimensional fixed points. Consequently, the centralizers C g are non-trivial and induce in general projection conditions on localized SM matter fields, compare to section 3.4.

Flavour from a T 6 /S 4 orbifold
In the previous section, we studied boundary conditions in the T 6 /Z 2 × Z 2 orbifold which do not allow for predictive flavour alignments. Hence, we enlarge the orbifolding symmetry. We know that the alignment CSD3 can be obtained with the flavour group S 4 [43], and noticing that Z 2 × Z 2 ⊂ S 4 , the orbifold T 6 /S 4 seems a fair choice.
and define a corresponding orthonormal six-torus T 6 (ignoring the possibility to change the overall radius of T 6 ). In complex coordinates z = (z 1 , z 2 , z 3 ) ∈ C 3 the basis vectors see appendix C.2 in ref. [31]. One can check that ϑ and ω generate the permutation group S 4 , i.e.
After fixing the geometry, we have to choose the gauge embeddings R ϑ , R ω , and R e i corresponding to the twists (ϑ, 0) and (ω, 0) and the translations (1, e i ), respectively. The twists (ϑ, 0) and (ω, 0) generate the permutation group S 4 . Since R must be a group homomorphism, see eq. (3), R ϑ and R ω can be chosen to generate also S 4 or a subgroup thereof (for example, ignoring world-sheet modular invariance from string theory one could also choose R ϑ = R ω = 1).
we have to find matrices R ϑ , R ω , R e 1 and R e 2 such that where the matrix R ϑ 2 ω ϑ 2 = (R ϑ ) 2 R ω (R ϑ ) 2 is of order 2. If these conditions are satisfied we can define R e i for i = 3, 4, 5, 6 using eqs. (123) in appendix B. This choice will satisfy all conditions, i.e. those from the presentation of S 4 in eq. (48) and, additionally, those from eqs. (121) and (122). flavon localization centralizer generators of centralizer Table 3: Localizations of the three flavons φ 1 , φ 2 and φ 3 in the various sectors of the T 6 /S 4 orbifold and their centralizers, which indicates which boundary condition the respective flavon is feeling.

Alignment of flavon VEVs
We assume to have an SU (3) as where α = exp( 2πi 6 ). This choice satisfies all gauge embedding conditions for an S 4 orbifold.
Next, we choose to have three flavons φ 1 , φ 2 , and φ 3 (each being a triplet of the SU(3) fl flavour group) and localize them in different sectors of the T 6 /S 4 orbifold as listed in table 3. Let us begin with specifying the flavon φ 1 in great detail so that our discussion for φ 2 and φ 3 can be shorter later on. We want the flavon φ 1 to be subject to the boundary condition R ϑ R ω . Hence, we localize it in the ϑ ω twisted sector (e.g. on the fixed torus z f ∈ C 3 given by the solutions of ϑ ω z f = z f ). In order to identify all boundary conditions that act on φ 1 we have to compute the centralizer of ϑ ω, i.e. we have to identify all elements of S 4 that commute with ϑ ω. It turns out that the centralizer of ϑ ω is generated by ϑ ω and ϑ 2 ω 2 ϑ , and corresponds to Z 2 ×Z 2 . Consequently, the flavon φ 1 will feel the boundary conditions R g of all elements g of the centralizer (up to some phases as introduced in section 2.4 that can be chosen freely). Hence, φ 1 is subject to where the ± signs in both conditions can be chosen independently. If this does not work out, there are ways to make the centralizer smaller. But then S 4 orbifold becomes more complicated. For example, one can use various different six-tori that can not be written as T 2 × T 2 × T 2 , c.f. ref. [31].
After we have chosen the localization of each flavon, the flavon VEVs must comply with the respective boundary conditions. We assume that the flavons obtain a nonvanishing VEV through some other mechanism. However, the alignment of the flavon VEV in flavour space is fixed to a specific direction through the boundary conditions. In more detail, the flavon φ 3 is chosen to be localized in the ϑ sector. Its VEV must comply with the boundary condition R ϑ = T . It has the freedom of having any of the three phases α 2n for n = 0, 1, 2, and we choose it to be α 4 so that which aligns the VEV completely. This VEV must be invariant (up to a phase) under the full centralizer of ϑ, which is generated by ϑ itself, so it is consistent. The flavon φ 2 is chosen to be localized in the ωϑ 2 ω 2 sector, so that its VEV must be invariant under R ω R 2 ϑ R 2 ω = U , up to a sign, which we choose to be negative. This enforces the VEV to be which aligns the VEV completely. The VEV must also be invariant (up to a sign) with the corresponding centralizer, which in this case is generated by ω ϑ 2 ω 2 and ϑ 2 ω 2 ϑ. Hence, the VEV eq. (57) must also be invariant under the boundary condition using R 2 ϑ R 2 ω R ϑ = S, up to a sign. We choose the sign to be negative (the positive sign would force the VEV to vanish) so that with arbitrary a, b. This alignment is compatible with the previous condition when a = 0. This fixes the VEV φ 2 completely and consistently through boundary conditions. The flavon φ 1 obtains a VEV that, due to the choice of the localization in the sector ϑω, must be invariant under the boundary conditions R ϑ R ω = SU up to a sign. We choose the positive sign so that This VEV is aligned in the general CSDn direction which is defined with a = 1, b = n. It must also comply with the boundary conditions of the centralizer, up to a sign, which is generated by ϑ ω and ϑ 2 ω 2 ϑ. This VEV must also be invariant under the boundary condition R 2 ϑ R 2 ω R ϑ = S, up to a sign. We choose the sign to be positive (the negative sign would force the VEV to vanish) so that which is consistent with the previous condition fixing a = b = 1. This is the CSD1 alignment which is widely used in the tribimaximal (TBM) alignment [52].
We conclude that the flavon VEV alignments can be fixed completely and consistently to the TBM alignment and we can arrange for a situation with three flavons φ i such that These flavons are enough to fit all masses predictibly, specially in the lepton sector.

Roto-translations
Another realization of the S 4 orbifold is based on a space group with roto-translations . This way we can obtain the general CSDn complete alignment. However, we did not find a mechanism to fix n = 2, 3 so that we have a highly predictive fermion mass setup as in ref. [21,44,46].

GUT breaking
Up to now, we have only chosen specific embedding matrices R ϑ and R ω in eq. (53) to align flavons in the CSDn or TBM alignment. We have not fixed any R e i in the process. From eqs. 51, we see that there are only two free matrices (R e 1 and R e 2 ) to choose and they must comply with their specific conditions. Hence, one option is to choose R e 1 and R e 2 to break the GUT but not flavour.
In this case, the matrices R e 1 and R e 2 would commute with R ϑ and R ω . The only remaining constraints from eqs. 51 state that R e 1 and R e 2 must commute with each other and be of order two. Hence, they generate Z 2 × Z 2 . Consequently, we are allowed to apply the GUT breaking mechanism as discussed in section 3.3.

SM fermion localization
We have shown that this S 4 orbifold is enough to align three different flavons in the TBM or the CSDn setups and break GUTs.
However, we want some of the SM fermions (e.g. the charged leptons ) to form a triplet of SU(3) fl flavour. Thus, these fermions must be located where they are not affected by the flavour breaking conditions associated to R ϑ or R ω . In this S 4 orbifold the only places to locate fields are the bulk and invariant tori, see table 2, which all are affected by the conditions R ϑ or R ω . Hence, some components of SU(3) fl flavour triplets for SM matter are necessarily projected out in this S 4 orbifold. This would destroy any predictability coming from the flavon alignments (61). Consequently, we move on to another orbifold that allows for suitable fermion localizations.

Flavour from a T /∆(54) orbifold
We want to enlarge the orbifolding symmetry once again, to allow a place to locate the fermions consistently but keep the alignments we have achieved. We can note that S 4 ∆(24) so that we can continue in the ∆(6n 2 ) discrete group series by choosing the next one, ∆ (54).
To define a field theory on this ∆(54) orbifold, we have to choose a (gauge) embedding We have to ensure that this embedding satisfies the following conditions, obtained from eq. (3), such that R ϑ , R ω and R ρ generate ∆(54) or a subgroup thereof. Furthermore, where i, j ∈ {1, . . . , 6}, see also eqs. (4) and (5). Explicitly, eqs. (71b), (71c) and (71d) read and where we use R ω 2 ρ instead of R ρ in order to keep the conditions (74) simple. One possibility to solve eqs. (72), (73) and (74) is given by assuming that for all point group elements θ ∈ ∆(54). Importantly, one can show that in this case the (gauge) embeddings of the translations have to be trivial, i.e.
Then, we are left with R ϑ , R ω and R ρ that have to satisfy eq. (70). In the following, we will choose standard embedding R ϑ = ϑ, R ω = ω and R ρ = ρ with gauged flavour symmetry G = SU(3) fl , c.f. section 2.2.

VEV alignment
Consider a (flavon) field Φ g (x, z) localized at z ∈ F g with constructing element g ∈ S.
We denote the order of g by N g , i.e. g (Ng) = 1. Then, the field Φ g (x, z) has to satisfy the boundary conditions where h ∈ C g has to be taken from the centralizer of g. We choose standard embedding eq. (6), where the gauge embedding R g is identical to the geometrical action, i.e. R g = θ for g = (θ, λ) ∈ S, and the flavon Φ g (x, z) is a triplet of SU(3) fl . Note that the additional phases in eq. (77) (which dependent on k and h , respectively) can originate from additional U(1) charges or from higher-dimensional Lorentz symmetry. These boundary conditions (77) result in the following conditions on the VEV of the zero mode Φ g (x), For each sector g from table 4 we find some k and h such that the VEV Φ g is non-trivial. For example, consider the sector g = (ϑ, 0) with N g = 2. Then, eq. (78a) has two non-trivial solutions for v, w ∈ C. Next, we have to ensure that these VEV alignments are invariant under transformations h ∈ C g from the centralizer. In this case, the centralizer C g is generated by h = (ρϑρ, 0) which is of order N h = 6. One can verify that eq. (78b) has the same non-trivial solutions as before provided that h takes some special values, i.e.
We repeat this analysis for the other sectors of the ∆(54) orbifold, listed in table 4, and find the following invariant VEV directions. The ϑ-sector allows for two different boundary conditions that yield two flavon VEV alignments For the ω-sector we obtain three different VEV alignments while the ρ-sector yields The matrices ω and ρ are very similar only involving different phases, and we can only obtain one different matrix built from them Multiplying ω to ρ or ϑ would only cyclicly rotate the entries of the VEVS. The only other possibility would be to study the matrix We conclude that we can completely fix three flavons to have the TBM alignment choosing them to be eigenvectors of while adding other matrices can introduce powers of α 2 in any entry while keeping the same alignment.

GUT breaking
As we stated before, in principle we could choose the gauge embedding R e i of the translations to break the GUT, for example, to break SU (5). However, in this ∆(54) orbifold with standard embedding the consistency conditions force them to be unity, see (76).
The simplest choice to avoid this situation in this orbifold would be to enlarge the Z 2 generator which is consistent with all conditions and breaks SU(5). Our model outlined in section 7 is based on this GUT breaking.

SM fermion localization
Quarks and leptons that transform as triplets under the SU(3) fl flavour symmetry should not feel any flavour breaking boundary conditions. Otherwise, some of them would be projected out by the orbifold and, hence, we would lose the predictivity from the flavon VEV alignments. We can see from table 4 that this specific ∆(54) orbifold has specific locations with zero-dimensional fixed points. Any field localized at such a point in extra dimensions is already a 4d field and, hence, is not be subject to any boundary condition. Consequently, we localize our SU(3) fl lepton triplet at such a fixed point. We start with N = 1 SUSY in 10 dimensions and with a SU(5) GUT × SU(3) fl gauge symmetry. In addition, we impose a U(1) shaping symmetry that allows the required Yukawa sector. Then, we define the ∆(54) orbifold boundary conditions as where α = exp( 2πi 6 ) and P SU(5) = diag(1, 1, −1, −1, −1). Since the embedding R acts as standard embedding on SU(3) fl , one can check easily that these matrices fulfil all the necessary conditions of the T 6 /∆(54) orbifold.
As also discussed in section 6.2 these boundary conditions break SU(5) GUT ×SU(3) fl → SU(3) C × SU(2) L × U(1) Y with only the MSSM superfields and some pure flavons left after compactification.
The list of chiral superfield is given in table 5. There, the ± superscript indicates that there are two copies of each ten-plet T i , i = 1, 2, 3, i.e. one for each parity under the boundary condition P SU (5) .
We assume a standard Kähler potential with canonical normalized fields (and without Field Representation Localization extra dim. Zero mode SU(3) fl SU(5) GUT U(1) large corrections [54]). Then, the Yukawa sector after compactification reads where We have defined The U(1) shaping symmetry allows only these terms and there are no higher order contributions.
Note that the terms coming from H 5 T − i T + j and H5F φ k T ± i ξ satisfy the basic string selection rule for allowed interactions: the point group selection rule demands that the point group elements of the respective constructing elements multiply to the identity element. For example, H5 and ξ originate from the bulk with constructing element 1 , F is localized in the ϑρ sector, the fields φ k for k = τ, µ, e live in the ω 2 ρ sector and, finally, the fields T ± i for i = 1, 2, 3 are localized in the ϑ sector. Then, using the conjugacy classes of ∆(54) given in eq. (67). We assume all dimensionless couplings to be O(1) complex numbers, so that all the hierarchies are due to the flavon VEVs which is an assumption. With these assumptions we may approximateξ +ξ ≈ξ. We write the up quark mass matrix, coming from the first line of eq. 89 6 The next lines of eq. 89 give masses to down quarks and charged leptons. The down quark matrix is while the charged lepton mass matrix is Since e c comes from T + and q comes from T − the Yukawa terms have different and independent couplings y ± ij for each one. This way the charged lepton mass matrix is completely independent of the down quark mass matrix.
The final two lines in eq. 89 give the Dirac neutrino mass matrix and the right handed neutrino Majorana mass matrix From the assumed VEV hierarchies from eq. 93, theξ terms in the first column are expected to be one order of magnitude smaller than the ones withoutξ, so we may safely ignore them, leading to In the limit that the small entry denoted by is ignored, the Dirac mass matrix is of the CSD form and leads to tribimaximal neutrino mixing [52]. The presence of has the effect of switching on the reactor angle θ 13 , without modifying very much the solar and atmospheric angles from their tribimaximal values [34]. This corresponds to so called tribimaximal-reactor lepton mixing [35]. The RHN are very heavy so that the left handed neutrinos become very light after the Seesaw mechanism has been implemented, where y ν 12 = y ν eṽ e /y ν aṽ a . Looking at all the mass matrices, as noted above, we may see that the VEVξ only appears in the neutrino mass matrix. Knowing that the charged lepton correction to the PMNS are negligible, if we sent thisξ → 0, we would have the tribimaximal (TBM) setup for the PMNS. Therefore the sole role of theξ is to deviate from the TBM [35,55,56].

Numerical fit in the neutrino sector
In this section we perform a fit to the PMNS observables and the neutrino masses assuming a diagonal charged-lepton mass matrix (the off-diagonal elements in eq. 96 are negligible due to the appearance ofξ). The complex parameters in the up-type and down-type quark mass matrices in eqs. 94 and 95 have enough freedom to fit all the quark masses and the observed CKM mixing angles. Therefore, in this section we focus only on the neutrino sector.
The effective neutrino mass matrix in eq. 99 can be rewritten in terms of input parameters as where we have defined the input parameters We implement a numerical fit using a χ 2 test function where we sum over the 6 observables given by P obs n = {θ 12 , θ 13 , θ 23 , δ l , ∆m 2 21 , ∆m 2 31 } with statistical errors σ n . The predictions of the model for these observables are given by P n (x), where x = {µ a ,ξ 12 , ηξ, µ s , η} refers to the different input parameters. We are doing the numerical fit in terms of the effective neutrino mass matrix in eq. (100) and we ignore any renormalisation group running corrections.
We use the recent global fit values of neutrino data from NuFit4.1 [57]. Most of the observables follow an almost Gaussian distribution and we take a conservative approach using the smaller of the given uncertainties in our computations except for θ l 23 and δ l . The best fit from NuFit4.1 is for normal mass ordering with inverted ordering being disfavoured with a ∆χ 2 = 6.2(10.4) without (with) the Super-Kamiokande atmospheric neutrino data analysis.
The model predictions are shown in table 6. The neutrino mass matrix in eq.100 predicts near maximal atmospheric mixing angle θ l 23 = 45.66 • and solar mixing angle θ 12 = 35 • as expected since we only have a small correction from tribimaximal mixing which allows a correct non-zero reactor angle θ l 13 = 8.615 • . The CP violation prediction is δ l ≈ 225 • . All the model predictions for the PMNS observables and the neutrino masssquared differences are within the 3σ region from the latest neutrino oscillation data and reproduce a χ 2 = 4.8 value. Furthermore, since we only have 2RH neutrinos, m 1 = 0 and there is only one physical Majorana phase α 23 [58]. The bound on the effective Majorana mass m ββ is taken from [59] while the prediction is also given in  Table 6: Latest values of PMNS observables and neutrino masses given by NuFit4.1 [57] together with the model predictions with χ 2 ≈ 4.8. The neutrino masses m i as well as the Majorana phase α 23 are pure predictions of our model. The bound on m i is taken from [60]. The bound on m ββ is taken from [59]. There is only one physical Majorana phase since m 1 = 0. Table 7 shows the input parameter values. There are 3 real parameters {µ a ,ξ 12 , µ s } plus two additional phases {ηξ , η}, a total of 4 input parameters to fit 6 data points. Naively, we can measure the goodness of the fit computing the reduced χ 2 , i.e. the χ 2 per degree of freedom χ 2 ν = χ 2 /ν. The number of degrees of freedom is given by ν = n − n i , where n = 6 is the number of measured observables, while n i = 4 is the number of input parameters. A good fit is expected to have χ 2 ν ∼ 1. We have 2 degrees of freedom and the best fit has a reduced χ 2 ν 2.4. We view this as a good fit and it also remarks the predictivity of the model, not only fitting to all available quark and lepton data but also fixing the neutrino masses and Majorana phases.
Using the definition of the input parameters in eq. 101 and their values in table 7 for the best fit point, we can give a naive estimation of the value ofξ. If we assume the dimensionless parameters to be O(1) and the VEVsṽ a ∼ṽ s , then we find µ s /µ a ≈ξ ≈ 0.1, which justifies the assumption of an approximate diagonal charged-lepton matrix in eq. 96

Conclusions
The flavour puzzle, in particular the large mixing observed in the lepton sector, provides a strong motivation for going beyond the Standard Model. The literature is replete with flavour models involving some family symmetry spontaneously broken by flavon fields with certain vacuum alignments motivated by phenomenological considerations, but highly non-trivial to achieve without resort to extra symmetries and driving fields.
In order to overcome this obstacle, one promising approach is to attempt to formulate such theories in extra dimensions, where the desired vacuum alignments may emerge from orbifold boundary conditions. We have systematically developed the formalism necessary for ensuring that boundary conditions of flavon fields in extra dimensions are consistent with heterotic string theory. Having developed a set of consistency conditions on the boundary conditions, we have then explored a series of examples of orbifolds in various dimensions to see which ones can satisfy them. In addition we have imposed the further phenomenological requirement of having non-trivial flavon vacuum alignments. We have also demanded that quarks and leptons be located appropriately in extra dimensions so that their massless modes may include complete multiplets under the gauged flavour symmetry.
It turns out that it is highly non-trivial to satisfy all of these conditions (theoretical and phenomenological) together. For instance, the simple T 2 /Z 2 orbifold, while allowing SM fermion matter localisation on fixed points, does not permit non-trivial flavon vacuum alignments, consistently with the formal requirements of the boundary conditions. This motivates us to go to 10d models. However, the simple orbifold T 6 /(Z 2 × Z 2 ) fares no better than the previous case, since it too can only provide one non-trivial alignment. We find that the boundary conditions must exhibit some non-Abelian structure so that we can have non-trivial VEV alignments.
Following the above logic, we were led to consider 10d non-Abelian orbifolds, where the torus is modded out by a non-Abelian group. We have studied the orbifold T 6 /S 4 which can fix flavons into the highly predictive CSDn structure. This orbifold however, does not have 4d branes where the SM matter could be localized. This motivated us to consider the orbifold T 6 /∆(54), as an example where we can locate the SM fermions on fixed points in extra dimensions. Since the ∆(54) orbifold is not so well studied in the literature, we have developed this case in some detail, and eventually shown that we can choose the extra dimensions in such a way that we can build a realistic model.
The minimal successful flavour theory seems to be a 10d theory with a SU(3) fl gauged flavour symmetry, where the six extra dimensions are compactified on a T 6 /∆(54) orb-ifold. The SU(3) fl flavour symmetry is broken by flavon VEVs which are completely aligned by the boundary conditions of the orbifold. The vacuum alignment of the flavons is of the tribimaximal form, but the theory can allow for some small corrections leading to tribimaximal-reactor lepton mixing, which we have shown to be consistent with current neutrino data. We have constructed a fully realistic SU(5) × SU(3) fl grand unified theory along these lines, which is complete, predictive and in principle consistent with heterotic string theory.

A Definition of an orbifold by its space group
In order to define a D-dimensional toroidal orbifold O geometrically as a quotient space of D-dimensional space, i.e.
one has to specify a space group S first. A general element g of a space group S consists of a rotation θ ∈ SO(D) (also called twist) and a translation λ ∈ R D , i.e. g = (θ, λ) ∈ S . (104) By definition, g acts on the internal coordinates z ∈ R D as z g −→ g z = (θ, λ) z = θ z + λ .
Furthermore, the inverse element g −1 ∈ S of g = (θ, λ) ∈ S is given by and the neutral element is Hence, one can see that S is a discrete group, actually a discrete subgroup of the extradimensional Euclidean group. Practically, one defines a space group by a (finite) list of generators, which are pure translations and rotations. In this work, we focus on the case of up to three rotational generators 7 , i.e.
The rotations ϑ, ω (and ρ) generate the so-called point group P , where we are dealing with the cases P = Z 2 × Z 2 in section 4, P = S 4 in section 5 and P = ∆(54) in section 6.
Having defined a space group, the orbifold O given in eq. (103) is defined by identifying those points z (1) and z (2) in R D that are mapped to each other by some element of the space group, i.e. z (1) ∼ z (2) ⇔ there is g ∈ S such that z (1) = g z (2) .
This equivalence relation can be used to define a fundamental domain of the orbifold. A given element g ∈ S of the space group can have a set of fixed points F g , defined by F g := {z ∈ R D | g z = z} .
(112) For a given space group element g = (θ, λ) ∈ S (with appropriate translation λ) the dimension of F g depends on the eigenvalues of the rotation matrix θ ∈ P : Each eigenvalue +1 corresponds to an invariant direction in F g . Our main concern is the case of supersymmetric orbifolds in D = 6, where we find fixed point sets of dimensions six (i.e. the bulk O for g = 1), two (i.e. fixed tori) and zero (i.e. fixed points). By acting with h ∈ S onto the fixed point equation g z = z (i.e. z ∈ F g ), one obtains h g h −1 (h z) = (h z) .
Hence, h z ∈ F h g h −1 . However, due to eq. (111) points z and h z are identified on the orbifold O. Thus, the corresponding fixed point sets are identified on the orbifold O as well, for all h ∈ S. Consequently, the inequivalent fixed point sets correspond to the conjugacy classes [g] of S, where If the point group P is Abelian, each elementg of a conjugacy class [g] has the same point group element θ ∈ P , i.e. g = (θ, λ) ⇔ h g h −1 = (θ, λ ) for all h ∈ S and some λ ∈ Γ .

A.1 Orbifold-invariant fields
In this appendix we complete the discussion from section 2.3 in the case g h = h g. In this case, we can choose the proportionality in eq. (10) to be trivial, i.e.
where a possible phase has been absorbed in a redefinition of Φ h g h −1 . Consequently, all fields Φ h g h −1 from the same conjugacy class [g] = {h g h −1 | h ∈ S} are identified and eqs. (9) and (117) yield Then, we can construct an orbifold-invariant field, denoted by Φ [g] (x, z), on the covering space R D of O. To do so, we have to build the following linear combination
Now, we repeat these steps for eqs. (122d) and (122f) and obtain Again, using R ϑ R ω R ϑ = (R ω ) −1 and R e 2 R ω = R ω R e 2 we see that condition (127b) is trivial.