$A_4$ Family Symmetry from SU(5) SUSY GUTs in 6d

We propose a model in which $A_4$ Family Symmetry arises dynamically from a six dimensional orbifold SU(5) Supersymmetric Grand Unified Theory. The SU(5) is broken to the Standard Model gauge group by a particular orbifold compactification leading to $A_4$ Family Symmetry, low energy Supersymmetry and Higgs doublet-triplet splitting. The resulting four dimensional effective superpotential leads to a realistic description of quark and lepton masses and mixing angles including tri-bimaximal neutrino mixing and an inter-family mass hierarchy provided by a Froggatt-Nielsen mechanism. This model is the first which combines the idea of orbifold GUTs with $A_4$ family symmetry resulting from the orbifolding.


Introduction
It is well known that the solar and atmospheric data are consistent with so-called tribimaximal (TB) mixing [1], The ansatz of TB lepton mixing matrix is interesting due to its symmetry properties which seem to call for a possibly discrete non-Abelian Family Symmetry in nature [2]. There has been a considerable amount of theoretical work in which the observed TB neutrino flavour symmetry may be related to some Family Symmetry [3,4,5,6,7,8,9,10,11,12]. These models may be classified according to the way that TB mixing is achieved, namely either directly or indirectly [13]. The direct models are based on A 4 or S 4 , or a larger group that contains these groups as a subgroup, and in these models some of the generators of the Family Symmetry survive to form at least part of the neutrino flavour symmetry. In the indirect models, typically based on ∆(3n 2 ) or ∆(6n 2 ), none of the generators of the Family Symmetry appear in the neutrino flavour symmetry [13]. The most ambitious models combine Family Symmetry with grand unified theories (GUTs). The minimal Family Symmetry which contains triplet representations and can lead to TB mixing via the direct model approach is A 4 . The minimal simple GUT group is SU (5). A direct model has been proposed which combines A 4 Family Symmetry with SU(5) Supersymmetric (SUSY) GUTs [14]. This model was formulated in five dimensions (5d), in part to address the doublet-triplet splitting problem of GUTs, and in part to allow a viable description of the charged fermion mass hierarchies, by placing the lightest two tenplets T 1 , T 2 in the bulk, while the pentaplets F and T 3 are on the brane. An additional U(1) Family Symmetry is also assumed in order to yield hierarchies between different families via the Froggatt-Nielsen mechanism [16].
In the approach in [14] the A 4 is simply assumed to exist in the 5d theory. However it has been shown how an A 4 Family Symmetry could have a dynamical origin as a result of the compactification of a 6d theory down to 4d [17]. Similar considerations have been applied to other discrete family symmetries [18], and the connection to string theory of these and other orbifold compactifications has been discussed in [19]. According to [17], the A 4 appears as a symmetry of the orbifold fixed points on which 4d branes, which accommodate the matter fields, reside, while the flavons which break A 4 are in the bulk. The formulation of a theory in 6d is also closer in spirit to string theories which are formulated in 10d where such theories are often compactified in terms of three complex compact dimensions. The 6d theory here will involve one complex compact dimension z.
The purpose of this paper is to formulate a realistic direct model in which an A 4 Family Symmetry arises dynamically from an SU(5) SUSY GUT in 6d. The A 4 Family Symmetry emerges as a result of the compactification of the extra complex compact dimension z, assuming a particular orbifolding. SO (10) in 6d has been considered in [20], with the extra dimensions compactified on a rectangular torus. In order to realize an A 4 Family Symmetry upon compactification, we shall generalise the formalism of 6d GUTs in [20] to the case of compactification on a twisted torus. Then, starting from an SU(5) SUSY GUT in 6d, we shall show how the A 4 Family Symmetry can result from the symmetry of the orbifold fixed points after compactification, assuming a particular twist angle θ = 60 • and a particular orbifold T 2 /(Z 2 × Z SM 2 ). Unlike the model in [14], the resulting model has all three tenplets T i , as well as the pentaplet F , located on the 3-branes at the fixed points. However, as in [14], we shall assume an additional U(1) Froggatt-Nielsen Family Symmetry to account for inter-family mass hierarchies. We emphasise that this model is the first which combines the idea of orbifold GUTs with A 4 family symmetry resulting from the orbifolding.
The layout of the remainder of the paper is as follows. In Section 2 we generalize the formulation of 6d GUTs (usually compactified on a rectangular torus) to the general case of compactification on a twisted torus with a general twist angle θ. Then we show how compactification of the SU(5) SUSY GUT in 6d on an orbifold T 2 /(Z 2 × Z SM 2 ) leads to an effective 4d theory with N = 1 SUSY preserved but the SU(5) GUT broken to the Standard Model (SM) gauge group. We also show how Higgs doublet-triplet splitting emerges if the Higgs fields are in the bulk. In Section 3 we present the SU(5) SUSY GUT model in 6d in which the A 4 Family Symmetry emerges after the above compactification. We specify the superfield content and symmetries of the model and provide a dictionary for the realization of the 4d effective superpotential in terms of the 6d A 4 invariants. From the effective 4d superpotential we show how a successful pattern of quark and lepton masses and mixing, including tri-bimaximal neutrino mixing, can emerge. In Section 4 we comment on the vacuum alignment and subleading corrections expected in the model. Section 5 concludes the paper. In order to make the paper self-contained we include an Appendix on the A 4 group and it's representations. We also include another Appendix which summarizes how A 4 family symmetry can arise from the orbifold discussed in this paper.
2 SU (5) GUTs in six dimensions on a twisted torus 2.1 The gauge sector of SUSY SU (5) in 6d We are considering a N = 1 supersymmetric Yang-Mills theory in 6 dimensions, the Lagrangian reads, -6 x 5 x 6 2πR 1 2πR 2 θ Figure 1: The twisted torus, R 1 and R 2 are the radii and θ is the twist angle (later we shall specify θ = π/3 and R 1 = 2R 2 ).
where V M = t a V a M and Λ = t a Λ a , here t a are the generators of SU (5).

Compactification on a twisted torus
We compactify the two extra dimensions on a twisted torus T 2 so that the theory lives on M = R 4 × T 2 . The torus is defined by: We can expand the SU(5) gauge multiplet fields Φ = (V M , Λ) using the mode expansion: where R 1 and R 2 are the two radii of the torus and θ is the angle of twist as shown in Fig.1. The vector field is Hermitian so the coefficients satisfy the relation V Note that we are only including terms below O(1/R) so there are only bilinear terms in the 4d Lagrangian. We make a convenient choice of variables for the 4d scalars: where M(m, n) = 1 The 4d Lagrangian for the gauge and scalar fields is then given by: whereṼ (m,n) µν . The gaugino part of the Lagrangian integrates to This is the kinetic term for a dirac fermion λ D = (λ 1 , λ 2 ) with a mass M(m, n). In total there is the vector V and λ D forming a massive N = 1 vector multiplet in 4d. However when we look at the massless sector of the theory we have unwanted N = 2 symmetry which can be removed by orbifolding, as we now discuss.

Compactification on the orbifold T 2 /Z 2
Instead of compactifying on the torus we can compactify on the orbifold T 2 /Z 2 where we assign parities under the reflection (x 5 , x 6 ) → (−x 5 , −x 6 ) to the vectors and scalars: where we chose P = I, so for the Fourier modes we find: This eliminates the scalar zero modes, also the number of massive modes is halved. Because the derivatives ∂ 5,6 are odd under the reflection the two Weyl fermions must have opposite parities: (V µ , λ 1 ) and (V 5,6 , λ 2 ) form vector and chiral multiplets respectively, only the vector multiplets have zero modes. The orbifolding has thus broken the extended N = 2 SUSY in 4d down to N = 1.

Gauge symmetry breaking using the orbifold
The zero modes obtained from the compactification on T 2 /Z 2 form a N = 1 SUSY SU(5) theory in 4d. The breaking of the SU(5) gauge group down to that of the Standard Model can be achieved by another orbifolding. We make a coordinate shift to a new set of coordinates: and introduce a second parity Z SM 2 on these new coordinates By using a single parity P SM , we shall require that: Gauge boson fields of the standard model thus have positive parity and fields belonging to SU(5)/G SM have negative parity. The orbifold is now T 2 /(Z 2 × Z SM 2 ).
Explicitly the expansion for the fields with any combination of parities is: Only fields with both parities positive have zero modes.

Higgs and doublet-triplet splitting
So far we have just considered the gauge sector of SUSY SU (5). Adding the MSSM Higgs to the 6d SUSY theory is straightforward. In the SU(5) GUT theory these are contained in the 5-plet and 5-plet of Higgs fields. These are two complex scalars H and H ′ , and a fermion h = (h, h ′ ). The chiralities are γ 5 h = h, γ 5 h ′ = −h ′ in 4d with an overall positive 6d chrality Γ 7 h = h. The Lagrangian reads: (26) Again we integrate over the compact dimensions to get, For the first orbifolding parity we choose For the gauge breaking orbifold we choose: It is easy to see with the form of P SM that the last three entries gain a minus sign which makes them heavy whereas the first two entries are left unchanged leaving them light, resulting in a light doublet and a heavy coloured triplet.

A 4 Family Symmetry from 6d SU (5) SUSY GUTs
The model will involve an A 4 family symmetry which is not assumed to exist in the 6d theory, but which originates after the compactification down to 4d. The way this happens is quite similar to the discussion in [17] based on the orbifold T 2 /(Z 2 ) but differs somewhat due to the different orbifold considered here, namely T 2 /(Z 2 × Z SM 2 ). This is discussed in Appendix B, where we also briefly summarize all the results required in order to formulate our model, as necessary in order to make this paper self-contained. Using the formalism of the previous section and Appendix B, we now present the model.
The basic set-up of the model is depicted in Fig. 2 and the essential features may be summarized as follows. The model assumes a 6d gauge N = 1 SUSY SU(5) Yang-Mills theory compactified down to 4d Minkowski space with two extra dimensions compactified on a twisted torus with a twist angle of θ = 60 • and R 1 = 2R 2 . Upon compactification, without orbifolding, the 6d supersymmetry would become extended to N = 2 SUSY in 4d. However the N = 2 SUSY is reduced to N = 1 SUSY by use of a particular orbifolding and a further orbifolding is used to break the gauge symmetry to the SM, as discussed in Section 2. Due to the tetrahedral pattern of fixed points on the torus, the compactified extra dimensions have some additional symmetry left over from the 6d Poincaré spacetime symmetry, which is identified as a Family Symmetry corresponding to the A 4 symmetry group of the tetrahedron. The particular gauge breaking orbifolding also leads to the 5-plets of higgs splitting into a light doublet and heavy coloured triplet. It should be noted that the four fixed points of the tetrahedral orbifold are inequivalent in that they have different gauge groups associated with them. The A 4 symmetry is a to an effective 4d theory with the N = 1 SUSY SM gauge group together with A 4 Family Symmetry after compactification. The gauge symmetry at the four fixed points is explicitly labelled. Matter fields are localised at the fixed points as discussed in Appendix B and in [17].
symmetry of the standard model gauge bosons only and not the full SU(5) gauge group. The gauge bosons belonging to SU(5)/G SM have negative parity under the second gauge breaking orbifolding so these fields do not transform as trivial singlets under the A 4 as the standard model gauge bosons do. The model is therefore The model is further specified by matter fields located on the 3-branes in various configurations, at the fixed points shown in Fig. 2. These matter fields are 4d fields with components at the 4 fixed points as described in [17]. Matter fields carry an extra U(1) family dependent charge which is in turn broken by two A 4 singlet Froggatt-Nielsen flavons θ, θ ′ which live on the fixed points. Realistic charged fermion masses and mixings are produced using these Froggatt-Nielsen flavons θ, θ ′ together with the bulk flavon ϕ T which breaks A 4 but preserves the T generator. Tri-bimaximal mixing of the neutrinos is achieved using further bulk flavons ϕ S which breaks A 4 but preserves the S generator, and the singlet bulk flavon ξ. A full list of the particle content of the model is given in Table 1. The superpotential of the theory is a sum of a bulk term depending on bulk fields, plus terms localised at the four fixed points. The 4D superpotential is produced from the 6D theory by integrating over the extra dimensions and assuming a constant background value for the bulk supermultiplets ϕ S (z), ϕ T (z) and ξ S (z) as in ref [17].

Superfield Content
After compactification, an effective 4d superpotential may be written down, using the dictionary for the realisation of the 4d terms in terms of the local 6d A 4 invariants given in Table 2. Using this dictionary, we decompose the effective 4d superpotential into several parts: Superfield  The term w d is concerned with vacuum alignment whose effect will be discussed later.
The first three terms give rise to the fermion masses after A 4 , U(1) and electroweak symmetry breaking and they are: Table 2: A dictionary for the realisation of the 4d terms in the superpotential in terms of the local 6d A 4 invariants. The 4d terms are obtained by integrating out the extra dimensions and assuming a constant background value for the bulk multiplets, as discussed in Appendix B where the notation is defined. The delta function, δ i = δ(z − z i ) where z i are the fixed points, restricts the couplings to the fixed points.
The dimensionless coefficients of each term in the superpotential have been omitted and they aren't predicted by the flavour symmetry, though they are all expected to be of the same order. It should be noted that the up mass matrix m u is not symmetric since the lagranian is invariant under the standard model and not SU (5). The powers of the cut-off Λ are determined by the dimensionality of the various fields, recalling that brane fields have mass dimension 1 and bulk fields have mass dimension 2 in 6d.
The neutrinos have both Dirac and Majorana masses: whereξ is a linear combination of two independent ξ type fields which has a vanishing VEV and therefore doesn't contribute to the neutrino masses.
Using the alignment mechanism in [14], the scalar components of the supermultiplets will be assumed to obtain VEVs according to the following scheme: where i = u, d, e allowing for different messenger masses [5]. Since the brane fields live in 4 dimensions the messengers will also be 4 dimensional particles so that the mechanism in [5], allowing different messenger masses, can be applied in this scenario. Also recall that the dimensions of the torus are now fixed In the remainder of this paper we shall give results in terms of R 1 ,R 2 and sin θ. It should be noted that they are however fixed to the values in Eqn. (41). Note that the flavon VEVs v T , v S and u are defined to be dimensionless since the bulk fields have mass dimension of 2.

Higgs vevs
The Higgs multiplets live in the bulk this gives the required doublet-triplet splitting. The value of the Higgs VEVs at the fixed points is what will enter in the Yukawa couplings, so the values of we are interested in will be averages over the fixed points z i : where v u and v d have mass dimension 1. The electroweak scale will be determined by: Because we are using an extra dimensional setup a suppression factor s will enter into our mass matrices since a bulk field and it's zero mode are given by: which results in the suppression factor: R 1 ,R 2 and sin θ are given by Eqn. (41). The size of s is discussed below in sec. 3.3.2.

Quark and Lepton Mass Matrices
We can now calculate the fermion mass matrices from the effective 4d superpotential, using the flavon and Higgs VEVs and expansion parameters above, (using a left-right convention throughout): where we have achieved different values for t u , t d and t e via different messenger masses Λ u , Λ d and Λ e and the dots represent contributions from subleading operators as discussed in sec. 4.

Down sector
For the down quark mass matrix, m d , we can choose t d ∼ ǫ and t ′ d ∼ ǫ 2/3 to give: For example, assuming a value ǫ ≈ 0.15 allows the order unity coefficients to be tuned to O(ǫ) to give acceptable down-type quark mass ratios. The 11 element of the mass matrix is of order ǫ 3 , which needs to be tuned to order ǫ 4 using the dimensionless coefficients we have omitted to write in the superpotential. The dots again represent subleading operators as discussed in sec. 4.

Up sector
The up quark matrix is given by: Again we have left out the O(1) coefficients for each term, which forǭ ≈ 0.22, may be tuned to give acceptable up-type quark mass ratios. The CKM mixing angles will arise predominantly from the down-mixing angles, but with possibly significant corrections from the up-mixing angles, depending on the unspecified operators represented by dots. In general there will be corrections to all the Yukawa matrices as discussed later. Since the top mass is given by the size of s, we would expect a value around s ∼ 0.5.

Charged lepton mass matrix
The mass matrix for the charged lepton sector is of the form: with t e ∼ ǫ and t ′ e ∼ ǫ 2/3 . The dots again represent subleading operators as discussed in sec. 4.

Neutrino sector
The neutrino sector after the fields develop VEVs and the gauge singlets N become heavy the see-saw mechanism takes place as discussed in detail in [3]. After the see-saw mechanism the effective mass matrix for the light neutrinos is given by: The neutrino mass matrix is diagonalised by the transformation which is of the TB form in Eq. (1). However, although we have TB neutrino mixing in this model we do not have exact TB lepton mixing due to fact that the charged lepton mass matrix is not diagonal in this basis. Thus there will be charged lepton mixing corrections to TB mixing resulting in mixing sum rules as discussed in [4,21].

Vacuum alignment and subleading corrections
Field The resulting A 4 model is of the direct kind discussed in [13] in which the vacuum alignment is achieved via F-terms resulting in the A 4 generator S being preserved in the neutrino sector. The vacuum alignment is achieved by the superpotential w d introduced in [14], where we have absorbed the mass dimension into the coefficients g i , f i .
involving additional gauge singlets, the driving fields ϕ T 0 , ϕ S 0 and ξ 0 in Table 3. The above form of the driving superpotential w d and the vanishing of the F-terms, yields the vacuum alignment anticipated in the previous section. For more details see [14]. Note that the FN flavons θ, θ ′ require no special vacuum alignment and their VEVs may be generated dynamically by a radiative symmetry breaking mechanism. The ratio of VEVs of θ, θ ′ will depend on the details of all the Yukawa couplings involving these flavons from which the desired VEVs can emerge. In general we do not address the question of the correlation of flavon VEVs in this paper.

Subleading corrections
Subleading corrections in the mass matrices arise from shifts in the VEVs of the flavons, and the shifted VEVs including such corrections are of the general form: as discussed in [14], [15]. ϕ T obtains a correction proportional to the VEV of ϕ S , where ϕ S obtains a correction in an arbitrary direction. The VEV ofξ, which was zero at leading order, obtains a small correction. The shift in the VEV of ξ has been absorbed into a redefinition of u since at this stage u is a free parameter.

Corrections to m up
The leading order terms in the up sector are of the form θ m θ ′n H u q i u j . Terms are gauge and A 4 singlets, to create higher order terms we need to introduce flavon fields. The most straighforward way to do this is to introduce two flavon fields ϕ T ϕ T , since ϕ T is an A 4 triplet we need two fields in order to construct a singlet. Such terms will lead to entries in the mass matrix suppressed by a factor of v 2 T . Because of the Z 3 symmetry the flavon fields ϕ S , ξ,ξ must enter at the three flavon level so entries will be supressed by a factor of v 2 S u, v 3 S and u 3 relative to the leading order term.

Corrections to m down and m charged lepton
In the down mass matrix subleading corrections fill in the entries indicated by dots. Entries in the matrix are generated by terms of the form θ m θ ′n H ′ d ((d c ϕ T )q i + (lϕ T )e c i ), higher order terms can come from replacing ϕ T with a product of flavon fields or including the effect of the corrections to the VEV of ϕ T . We can replace ϕ T with ϕ T ϕ T , this is compatible with the Z 3 charges and results in corrections with the same form as m down but with an extra overall supression of v T . If we include the corrections to the VEV of ϕ T then we fill in the entries indicated by dots in eqn. (49), the corrections are of the form: The corrections to the charged lepton mass matrix are, up to O(1) cooefficients, the transpose of the above matrix: Following ref. [14], δv/v ∼ O(ǫ 2 ) leading to negligible corrections to the leading order m d , m e mass matrices.

Corrections to m ν
The Dirac mass term (H u (Nl)) can be modified with an insertion of the ϕ T flavon, producing corrections suppressed by sv T . The leading Dirac mass correction is the term H u (ϕ T Nl). This leads to a correction to the Dirac mass matrix suppressed by a factor of sv T relative to the leading order (LO) term.
The Majorana mass term can receive corrections from a number of higher order terms since the (NN) term can be a 1, 1 ′ , 1 ′′ or 3. The higher order terms all consist of insertions of 2 flavon fields where the leading order terms have only one insertion e.g. the term (NN) ′ (ϕ T ϕ S ) ′′ obeys the Z 3 symmetry, is an A 4 singlet and results in a higher order correction to the terms (x a ξ +x aξ )(NN) + x b (ϕ S NN). If we call the correction to the Majorana mass matrix δm RR then for this example the correction is given below, Such corrections have a relative supression of sv T,S to the leading order term. After the see-saw mechanism this leads to an effective mass matrix with every entry suppressed by a factor of sv T,S . This leads to corrections to the neutrino tri-bimaximal mixing angles of order sv T,S .
The magnitude of v T depends on the ratio of the top and bottom quark Yukawa couplings, but may be roughly between v T ∼ O(ǫ 2 ) − O(ǫ) leading to significant corrections to tri-bimaximal mixing. The flavon shifts δv S also give corrections to the leading order term (x b (ϕ S NN)), however if v T ∼ O(ǫ 2 ) these corrections are of O(ǫ 2 ) and they enter at the same order of magnitude as the corrections from higher order corrections. If however v T ∼ O(ǫ) then the correction enters at the order of ǫ.The effect of the VEV of ξ , which was zero at leading order, and obtains a small correction, leads to a small shift in the overall scale of the right-handed neutrino masses. And, as already remarked, the shift in the VEV of ξ has been absorbed into a redefinition of u, which we are free to do since u is a free parameter.

Conclusion
We have proposed a model in which an A 4 Family Symmetry arises dynamically from an N = 1 SU(5) SUSY GUT in 6d. The A 4 Family Symmetry emerges as a result of the compactification of the extra complex compact dimension z, assuming a particular twist angle θ = 60 • and a particular orbifold T 2 /(Z 2 × Z SM 2 ) which breaks the N = 1 SU(5) SUSY GUT in 6d down to the effective 4d N = 1 SUSY SM gauge group. In this model the A 4 Family Symmetry emerges after compactification as a residual symmetry of the full 6d spacetime symmetry of 6d translations and proper Lorentz transformations. It should be noted that had improper Lorentz transformations been included then the residual symmetry would have been S 4 and not A 4 . The model also involves other symmetries, in particular we assume a Froggatt-Nielsen U(1) Family Symmetry and other Z N symmetries in order to achieve a realistic model.
We emphasize that the SU(5) GUT symmetry is broken by the compactification, while the A 4 Family Symmetry is only realized after the compactification. The matter fields are located at the fixed points on 3-branes, while the Higgs fields live in the 6d bulk. The Froggatt-Nielsen flavons are all located at the fixed point 3-branes while the A 4 flavons all live in the bulk. We have adopted an A 4 classification scheme of quarks and leptons compatible with the SU(5) symmetry. We have also used a Froggatt-Nielsen mechanism for the inter-family mass hierarchies. By placing the 5 and 5 of Higgs in the 6d bulk we have avoided the doublet-triplet splitting problem by making the coloured triplets heavy. The model naturally has TB mixing at the first approximation and reproduces the correct mass hierarchies for quarks and charged leptons and the CKM mixing pattern. The presence of SU(5) GUTs means that the charged lepton mixing angles are non-zero resulting in predictions such as a lepton mixing sum rule of the kind discussed in [4,21].
In conclusion, this paper represents the first realistic 6d orbifold SU(5) SUSY GUT model in the literature which leads to an A 4 Family Symmetry after compactification. We emphasize that the motivation for building such higher dimensional models is purely bottom-up, namely to make contact with high energy theories and to solve the conceptual problems with GUT theories such as Higgs doublet-triplet splitting and the origin of Family Symmetry in a higher dimensional setting. The hope is that 6d models such as the one presented here, based on one extra complex dimension z, may provide a useful stepping-stone towards a 10d fully unified string theory (including gravity, albeit perhaps decoupled in some limit) in which GUT breaking and the emergence of Family Symmetry can both be naturally explained as the result of the compactification of three extra complex dimensions.

Acknowledgements
We acknowledge partial support from the following grant: STFC Rolling Grant ST/G000557/1.

A The group A 4 and it's representations
The A 4 group is the group of even permutations of 4 objects. There are 4!/2=12 elements. This group can be seen as the symmetry group of the tetrahedron, the odd permutations can be seen as the exchange of two vertices which can't be obtained with a rigid solid). Let a generic permutation be denoted by (1,2,3,4) → (n 1 , n 2 , n 3 , n 4 ) = (n 1 n 2 n 3 n 4 ). A 4 can be generated by the two basic permutations S and T where S = (4321) and T = (2314). We can check that This is called the presentation of the group.

A.1 Equivalence classes
There are 4 equivalence classes ( h and k belong to the same equivalence class if there is a member of the group g such that ghg −1 = k): For a finite group the squared dimensions for each inequivalent representation sum to N, the number of transformations in the group (N=12 for A 4 ). There are 4 inequivalent representations of A 4 three singlets 1,1 ′ ,1 ′′ and a triplet 3. The three singlets representations are: Class The triplet representation in the basis where S is diagonal is constructed from:

A.2 Characters
The characters of a group χ R g of each element g are defined as the trace of the matrix that maps the element in a representation R. Equivalent representations have the same characters and the characters have the same value for all the elements in an equivalence class. Characters satisfy g χ R g χ S g * = Nδ RS . Also the character for an element h in a direct product of representations is a product of characters χ R⊗S h = χ R h χ S h and is also equal to the sum of characters in each representation that appears in the decomposition of R ⊗ S.
From the character table 4 we can see that there are no more inequivalent irreducible representations than 1, 1 ′ , 1 ′′ and 3. We can also see the multiplication rules: If we have two triplets 3 a ∼ (a 1 , a 2 , a 3 ) and 3 b ∼ (b 1 , b 2 , b 3 ) we can obtain the irreducible representations from their product:

A.3 Another representation
Previously we used the representation where the matrix S is diagonal. In this paper we shall construct the model in a different basis in which we arrange T to be diagonal through a unitary transformation: In this basis the product composition rules are different: As discussed in Appendix B, this is done by applying a matrix v = Uu which block diagonalises the generators of A 4 . This formula allows us to write triplets and singlets of our 6d theory in terms of brane fields at the four fixed points.

B A 4 Family Symmetry from 6d compactification
In this Appendix we adapt the calculation in [17] to the orbifold T 2 /(Z 2 × Z SM 2 ).

B.1 The
The Orbifold we are using is based on the twisted torus with the twist angle θ = 60 • . We set R 1 = 2R 2 , as shown in figure 3, then under the orbifolding Z SM 2 the fundamental domain is reduced to a rhombus. We then perfom another orbifolding Z 2 which folds the rhombus into a tetrahedron giving rise to the A 4 symmetry, as described in Appendix B, we later exploit as a family symmetry. The fixed points are inequivalent but the symmetry is a symmetry of the Standard Model Gauge bosons only i.e.
where z S , z T are the coordinate transformations that generate the S and T generators of the A 4 group. This makes explicit that the A 4 symmetry is a symmetry of the standard model but not SU(5).

B.2 The orbifold with θ = π/3
We are working with a quantum field theory in 6 dimensions with the 2 extra dimensions compactified onto an orbifold T 2 /(Z 2 × Z SM 2 ). The extra dimensions are complexified such that z = x 5 + ix 6 are the coordinates on the extra space. The torus T 2 is defined by identifying the points (as in Eq. (5)) We have set the length 2πR 2 to unity for clarity. If we first perform the gauge breaking orbifolding Z SM 2 by making a coordinate shift as described in sect. 2.4 (x ′ 5 , x ′ 6 ) = (x 5 + πR 1 , x 6 ) = (x 5 + 1, and introduce a parity Z SM 2 on these new coordinates we are left with a fundamental domain in the shape of rhombus. The second orbifolding is defined,as in sect. 2.3, by the parity Z 2 identifying: leaving the orbifold to be represented by the triangular region shown in Fig. 3. The orbifold has 4 fixed points which are unchanged under the symmetries of the orbifold Eqns. (74),(70),(71). The orbifold can be described as a regular tetrahedron with the fixed points as the vertices. The 6d spacetime symmetry is broken by the orbifolding, previously the symmetry consisted of 6d translations and proper Lorentz tranformations 1 . We are now left with a 4d space-time symmetry and a discrete symmetry of rotations and translations due to the special geometry of the orbifold. We can generate this group with the transformations: These two generators are even permutations of the four fixed points: S : (z 1 , z 2 , z 3 , z 4 ) → (z 4 , z 3 , z 2 , z 1 ) (77) T : (z 1 , z 2 , z 3 , z 4 ) → (z 2 , z 3 , z 1 , z 4 ).
The above two transformations generate the group A 4 which is the symmetry of the tetrahedron (see Appendix A for an introduction to A 4 ). This can be verified by showing that S and T obey the characteristic relations, the presentation, of the generators of A 4 , We can easily represent S and T by 4 × 4 matrices describing their action on the fixed points of the orbifold:

B.3 Parametrising multiplets
If we are to place fields at the fixed points of the orbifold then we will need to parametrise a 4 dimensional representation in terms of singlet and triplet representations as in [17]. We now briefly summarise the results of [17] to build the dictionary in Table 2 from a 6d orbifolded theory to an effective 4d one. If we consider a multiplet u = (u 1 , u 2 , u 3 , u 4 ) T transforming as: S : u → Su T : u → T u, We can decompose the reducible quadruplet into a triplet and invariant singlet irreducible representations: As noted in [17] this parametrisation is not unique, Brane singlets are given by a vector of the form a singlet = (a c /2, a c /2, a c /2, a c /2) T , i.e. brane fields having the same value at each fixed point. Brane Triplets a = (a 1 , a 2 , a 3 )are in one of three representations R 0,±1 given by depending on which singlet the triplets are forming in the superpotential. Bulk singlets depend on the extra coordinates and transform as Sξ(z) = ξ(z+1/2) and T ξ(z) = ξ(ωz).
We require these decompositions because we will want to construct non-invariant singlets from products of triplets and if we were to restrict ourselves to the first parametrisation we would be unable to do so.

B.4 Bulk and Brane Fields
Following [17] we now look at the coupling of a bulk multiplet: B(z) = (B 1 (z), B 2 (z), B 3 (z), transforming as a triplet of A 4 and the brane triplet a = (a 1 , a 2 , a 3 , a 4 ) transforming as R 0 , as in Eqn (82). The transformations of B are: We can write a bilinear in a and B given by: where α iK is a four by three matrix of constant coefficients, and δ i = δ(z − z i ) where z i are the fixed points. We want J to be invariant under A 4 then we choose: Since a is in the R 0 representation after integration and if the triplet B(z) aquires a constant VEV B(z) = (B 1 , B 2 , B 3 ) then J becomes: We can do the same for a bilinear J ′ given by: which transforms as a 1 ′ with the matrix α ′ iK given by: