MSSM-like from $SU_{5}\times D_{4}$ Models

Using finite discrete group characters and symmetry breaking by hyperflux as well as constraints on top- quark family, we study minimal low energy effective theory following from SU$_{5}\times D_{4}$ models embedded in F-theory with non abelian flux. Matter curves spectrum of the models is obtained from SU$_{5}\times S_{5}$ theory with monodromy $S_{5}$ by performing two breakings; first from symmetric group $S_{5}$ to $S_{4}$ subsymmetry; and next to dihedral $D_{4}$ subgroup. As a consequence, and depending on the ways of decomposing triplets of $S_{4}$, we end with three types of $D_{4}$- models. Explicit constructions of these theories are given and a MSSM- like spectrum is derived.


Introduction
Recently, there has been an increasing interest in building SU 5 × Γ GUT models, with discrete symmetries Γ, embedded in Calabi-Yau compactification of F-theory down to 4d space time [1]- [11]; and in looking for low energy minimal prototypes with broken monodromies [12]- [19]. This class of supersymmetric GUTs with discrete groups lead to quasi-realistic field spectrum having quark and lepton mass matrices with properties fitting with MSSM requirements. In the geometric engineering of these F-GUTs, splitting spectral cover method together with Galois theory tools are used to generate appropriate matter curves spectrum [20]- [25]; and a geometric Z 2 parity has been also introduced to suppress unwanted effects such as exotic couplings and undesired proton decay operators [26,27,28,29]. In this paper, we develop another manner to deal with monodromy of F-GUT that is different from the one proposed first in [18], and further explored in [27,30,31], where matter curves of the same orbit of monodromy are identified. In our approach, we use the non abelian flux conjecture of [15,16] to think of monodromy group of F-theory SU 5 models as a non abelian flavor symmetry Γ. Non trivial irreducible representations of the non abelian discrete group Γ are used to host the three generations of fundamental matter; a feature that opens a window to build semi-realistic models with matter curves distinguished from each other in accord with mass hierarchy and mixing neutrino physics [32,33,34]. In this work, we study the family of supersymmetric SU 5 × Γ p × U (1) 5−p models in the framework of F-theory GUT; with non abelian monodromies Γ p contained in the permutation group S 5 [30]- [42]; and analyse the realisation of low energy constraints under which one can generate an effective field spectrum that resembles to MSSM. A list of main constraints leading to a good low energy spectrum are described in section 5; it requires amongst others a tree-level Yukawa coupling for top-quark family. To realise this condition with non abelian Γ p , we consider the case where Γ p is given by the order 8 dihedral group D 4 ; this particular non abelian discrete symmetry has representations which allow more flexibility in accommodating matter generations. Recall that the non abelian alternating A 4 group has no irreducible doublet as shown on the character relation 12 = 3 2 + 1 2 + 1 2 + 1 2 ; and the irreducible representation of non abelian S 4 and S 3 , which can be respectively read from 24 = 3 2 + 3 2 + 2 2 + 1 2 + 1 2 and 6 = 2 2 + 1 2 + 1 2 , have a doublet and two singlets. The non abelian dihedral group D 4 however has representations R i with dimensions, that can be read from 8 = 2 2 + 1 2 + 1 2 + 1 2 + 1 2 , seemingly more attractable phenomenologically; it has 5 irreducible R i 's; four singlets, indexed by their basis characters as 1 ++ , 1 +− , 1 −+ , 1 −− ; and an irreducible doublet 2 00 ; offering therefore several pictures to accommodate the three generations of matter of the electroweak theory; in particular more freedom in accommodating top quark family. To deal with the engineering of SU 5 ×D 4 -models, we develop a new method based on finite discrete group characters χ R i ; avoiding as a consequence the complexity of Galois theory approach. The latter is useful to study F-theory models with the dihedral D 4 and the alternating A 4 subgroups of S 4 as they are not directly reached by the standard splitting spectral cover method; they are obtained in Galois theory by putting constraints on the discriminant of underlying spectral covers; and introducing other monodromy invariant of the covers such a resolvent [14,15,29]. To derive the D 4 -matter curves spectrum in SU 5 × D 4 -models, we think of it in terms of a two steps descent from S 5 -theory; a first descent down to S 4 ; and a second one to D 4 by turning on appropriate flux that will be explicitly described in this work; see also appendix C. By studying all scenarios of breaking the triplets S 4 -theory in terms of irreducible D 4 -representations, we end with three kinds of D 4 -models; one having a field spectrum involving all D 4 -representations including doublet 2 00 (model I ); the second theory (model II ) has no doublet 2 00 nor the singlet 1 −− ; and the third model has no 2 00 ; but does have 1 −− . We have studied the curves spectrum of the three D 4 -models; and we have found that only model III allows a tree level 3-couplings and exhibits phenomenologically interesting features. The presentation is as follows: In section 2, we study the SU 5 × S 5 model; and describes the picture of the two steps breaking S 5 → S 4 → S 3 by using standard methods. In section 3, we introduce our method; and we revisit the construction of the S 4 -and S 3models from the view of discrete group characters. In section 4, we use character group method to build three SU 5 × D 4 × U ⊥ 1 models. In section 5, we solve basic conditions for deriving MSSM-like spectrum from SU 5 × D 4 × U ⊥ 1 models. In section 6, we conclude and make discussions. Last section is devoted to three appendices: In appendix A, we give relations regarding group characters. In appendix B, we report details on other results obtained in this study; and in appendix C we exhibit the link between non abelian monodromies and flavor symmetry.

Spectral Covers in SU 5 × Γ models
In F-GUT models with SU 5 gauge symmetry, matter curves carry quantum numbers in SU 5 × SU ⊥ 5 bi-representations following from the breaking of E 8 as given below 248 → (24, 1 ⊥ ) ⊕ (1, 24 ⊥ ) ⊕ (10, 5 ⊥ ) ⊕ 10,5 ⊥ ⊕ (5, 10 ⊥ ) ⊕ 5, 10 ⊥ (2.1) In this SU 5 theory, the perpendicular SU ⊥ 5 is restricted to its Cartan-Weyl subsymmetry U ⊥ 1 4 , see appendix C for some explicit details; and the matter content of the model is labeled by five weights t i like with traceless condition t 1 + t 2 + t 3 + t 4 + t 5 = 0 (2. 3) The components of the five 10-plets 10 t i and those of the ten 5-plets5 t i +t j are related to each other by monodromy symmetries Γ; offering a framework of approaching GUT -models with discrete symmetries originating from geometric properties of the elliptic Calabi-Yau fourfold CY 4 which, naively, can be thought of as given by the 4-dim complex space CY 4 ∼ E × B 3 (2.4) In this fibration, the complex 3-dim base B 3 contains the complex GUT surface S GU T wrapped by 7-brane; and the complex elliptic curve E fiber is as follows where the homology classes Matter curves of SU 5 ×U (1) 5−k ×Γ k models live on GUT surface S GU T with monodromy symmetries Γ k contained in S 5 , the Weyl group of SU ⊥ 5 ; see eq(9.8) of appendix C. In the case of Γ 5 = S 5 ; these curves organise into reducible multiplets 1 of S 5 with the following characteristic properties matters curves weights S 5 repres homology classes holomorphic sections where the t i ' s as above; T ij = t i + t j with i < j; and S ij = t i − t j with i = j. These t i 's, T ij 's; and S ij 's are respectively interpreted as the simple zeros of the spectral covers C 5 = 0 describing ten-plets, C 10 = 0 describing five-pelts and C 20 = 0 for flavon singlets 1 An equivalent spectrum can be also given by using irreducible representations of S 5 and their characters; to fix ideas see the analogous S 4 -and S 3 -models studied in section 3. [45]-[50] The homology classes of the complex curves in (2.7) are nicely obtained by defining the spectral covers in terms of the usual holomorphic sections; for the 5-sheeted covering of S GU T , we have with b 1 = 0 due to traceless condition; and homology classes of the complex holomorphic sections b k as follows holomorphic sections homology classes with canonical homology class η given by with c 1 and −t as in eqs (2.6). From these relations, the homology class [10 but as approached in [14,15] in dealing with local models. For simplicity, we use a short way to introduce this parity by requiring, up to an overall phase, invariance of C 5 = 0, C 10 = 0, C 20 = 0 under the following transformations along the spectral fiber; see [14,15,16] for explicit details, Under this phase change, the spectral covers eqns transform like Focussing on 10-plets, and equating above C ′ 5 with the one deduced from construction of [16] If we put ζ = 0, we get (b ′ 0 , b ′ 5 ) = (+b 0 , −b 5 ); while by taking ζ = π, we have (b ′ 0 , b ′ 5 ) = (−b 0 , +b 5 ); below we set ζ = π. To get the parity of the holomorphic sections d k and g k of eqs (2.8), we use their relationships with the b k coefficients. By help of the relations [27,30,31] in agreement with the homology class properties η ′ = 3η and η ′′ = 9η.

Models with broken S 5
To engineer matter curves with monodromy Γ k ⊂ S 5 ; we generally use spectral cover splitting method combined with constraints inspired from Galois theory [14,15,16,26,27]. In this study, we develop a new method without need of the involved tools of Galois group theory; our approach uses characters χ R (g) of discrete group representations; and relies directly the roots of the spectral covers. To illustrate the method; but also for later use, we first study the two interesting cases by using the standard method: The case Γ 4 = D 4 requires more tools; it will be studied later after revisiting S 4 -and S 3models from the view of characters of their representations.

S 4 -model in standard approach
To engineer the breaking of S 5 down to S 4 , we proceed as follows: First, we use S 5invariance to rewrite the holomorphic polynomial C 5 like and similarly for C 10 and C 20 . To break S 5 down to S 4 , we impose a condition fixing one of the weight [51]; for example This requirement breaks S 5 down to one of the five possible S 4 subgroups living inside S 5 ; and leads to the following features: (a) the traceless condition (2.3) of the orthogonal SU ⊥ 5 is solved as t 5 = − (t 1 + t 2 + t 3 + t 4 ); it is manifestly S 4 -invariant. To deal with this t 5 weight, we shall think about the breaking of S 5 down to S 4 in terms of the descent of the symmetry SU 5 × U (1) 5−k × Γ k from k=5 to k=4 as follows [58,59] (b) the spectral covers C 5 and C 10 split as the product of two factors: (α) the spectral cover C 5 factorises like C 4 × C 1 with together with the transformations following from (2.14-2.15). Notice that the above factorisations put conditions on the field K where live the holomorphic sections; a feature 2 The holomorphic sections A l and a m eqs(2.21) are directly derived by expanding the factorised forms of the spectral covers C 4 and C 1 ; we will not give these details here; for example the relevant A 4 and a 1 are given by that is also predicted by Galois theory [28,29]. As a naive illustration, we use the comparison with arithmetics in the set of integers Z; an integer number like 6 can be factorised in Z as 6 = 2 × 3; while a prime integer like 5 has no factorisation. By using C ′ 5 = C ′ 4 × C ′ 1 and equating e i(ζ−φ) (C 4 × C 1 ) with e iξ C 4 × e iψ C 1 ; it follows that ζ − φ = ξ + ψ; and from which we learn that A 4 and a 1 sections transform differently; and then Z 2 (b 4 ) = Z 2 (A 4 ) × Z 2 (a 1 ) . (β) the C 10 splits in turns likeC 6 ×C 4 with as well asC 6 = e 2iξC 6 andC 4 = e 2iψC 4 withξ +ψ =ζ − φ. Under the above splitting, the spectrum (2.7) decomposes in terms of reducible S 4 multiplets as follows and where κ i andκ k refer to Z 2 parities; for instance The last column of eq(2.26) refers to the hyperflux of the U(1) Y gauge field strength; it breaks SU 5 gauge symmetry down to standard model gauge invariance; and also pierces the matter curves of the model as shown on table.

S 3 -model in standard approach
The breaking of S 5 down to S 3 may be obtained from above S 4 model by further breaking S 4 down to S 3 ; this corresponds to SU 5 × U (1) 5−5 × S 5 → SU 5 × U (1) 5−3 × S 3 . This can be realised by fixing one of the four t i roots; say t 4 ; so that the breaking pattern is given by Setting U (1) 2 = U ⊥ 1 ×U ⊥ 1 , the previous S 4 spectrum decomposes into reducible S 3 multiplets as follows, given by relations of form as in (2.27). An extra column for Z 2 -parity can be also added as in (2.26) with the property Observe also that here we have two new homology class cycles χ and χ ′ with The non zero P is responsible for the second splitting; this is because the breaking of S 5 down to S 3 has been undertaken into two stages: first S 5 → S 4 ; and second S 4 → S 3 . In what follows we extend this idea to the breaking pattern of S 5 down to D 4 .

Revisiting S 4 and S -models
In this section, we develop tools towards the study of the breaking of S 5 monodromy down to its D 4 sub-symmetry. To our knowledge these tools, have not been used before; even for S n permutation groups; so we begin by revisiting the S 4 -and S 3 -models from the view of characters of their irreducible representations; and turn in next section to develop the D 4 theory.
In the canonical t i -weight basis, the matter spectrum of S 4 -model is given by (2.26); there matter curves are organised into reducible multiplets of S 4 × U ⊥ 1 . Below, we give another manner to approach the spectrum of S 4 -model. By help of the standard relation 24 = 1 2 + 1 2 + 2 2 + 3 2 + 3 2 showing that S 4 has 5 irreducible representations R i and 5 conjugacy classes C i [39,40,41,42]; and by using properties of the irreducible R i representations of S 4 given in appendix; eq(2.26) may be expressed in terms of the R i 's and their χ (a,b,c) R characters as follows Notice that S 4 has three generators denoted here by (a, b, c) and chosen as given by 2-, 3-and 4-cycles; they obey amongst others the cyclic properties a 2 = b 3 = c 4 = I id ; these three generators are non commuting permutation operators making extraction of full information from them a difficult task; but part of these information is given their χ (a,b,c) R 's; these characters are real numbers as collected in following table [39,40,41,42], Notice also that the 4-and 6-representations of S 4 , which have been used in the canonical formulation of section 2, are decomposed in (3.1) as direct sums of irreducible components as follows: 4 (2,1,0) = 1 (1,1,1) ⊕ 3 (1,0,−1) 6 (0,0,0) = 3 (1,0,−1) ⊕ 3 ′ (−1,0,1) Notice moreover that the previous t i -weights are now replaced by new quantities x i given by some linear combinations of the t i 's fixed by representation theory of S 4 . One of these weights; say x 4 , is given by the usual completely S 4 -symmetric term transforming in the trivial representation of S 4 ; the three other x i are given by some orthogonal linear combinations of the four t i 's that we express as follows These three weights transform as an irreducible triplet of S 4 ; but seen that we have two kinds of 3-dim representations in S 4 namely 3 and 3 ′ , the explicit expressions of (3.5) depend in which of the two representations the x i 's are sitting; details are reported in appendix where one also finds the relationships t µ = U µρ x ρ and t µ ± t ν = (U µρ ± U νρ ) x ρ . Notice finally that the explicit expressions of X µν weights in (3.1) are not needed in our approach; their role will be played by the characters of the representations.

SU
The spectrum of GUT-curves of the SU 5 × S 3 × U ⊥ 1 2 model follows from the spectrum of the SU 5 × S 5 theory by using splitting spectral method. By working in the canonical basis for t i -weights, this spectrum, expressed in terms of reducible multiplets, is given by (2.30). Here, we revisit the SU 5 × S 3 × U ⊥ 1 2 curves spectrum by using irreducible representations of S 3 and their characters. We start by recalling that S 3 has three irreducible representations as shown of the usual character relation 6 = 1 2 + 1 2′ + 2 2 linking the order of S 3 to the squared dimensions of its irreducible representations; these irreducible representations are nicely described in terms of Young diagrams [42] 1 : The group S 3 is a non abelian discrete group; it has two non commuting generators (a, b) satisfying a 2 = b 3 = 1 with characters as follows The spectrum of matter curves in the S 3 -model is obtained here by starting from the S 4 spectrum (t 1 , t 2 , t 3 ) (2.30); and then breaking S 4 monodromy to S 3 × S 1 . We find where the integers P and N are as in eq(2.33).

SU 5 × D models
First notice that the engineering of the SU 5 × D 4 × U ⊥ 1 theory has been recently studied in [16] by using Galois theory; but here we use a method based on characters of the irreducible representations of D 4 ; and finds at the end that there are in fact three kinds of SU 5 × D 4 × U ⊥ 1 models; they are explicitly constructed in this section. To that purpose, we first review useful aspects on characters of the dihedral group; then we turn to construct the three D 4 × U ⊥ 1 models.

Characters in D 4 models
The dihedral D 4 is an order 8 subgroup of S 4 with no 3-cycles; there are three kinds of such subgroups inside S 4 ; an example of D 4 subgroup is the one having the following elements with non commuting generators a = (24) and b = (1234) satisfying a 2 = b 4 = I and aba = b 3 . The two other D ′ 4 and D ′′ 4 have similar contents; but with other transpositions and 4-cycles. In terms of (a, b) generators, the eight elements (4.1) of the dihedral D 4 reads as they form 5 conjugacy classes as follows The dihedral group D 4 has also 5 irreducible representations R i ; this can be directly learnt on the character formula 8 = 1 2 1 + 1 2 2 + 1 2 3 + 1 2 4 + 2 2 , linking the order of D 4 with the sum of d 2 i , the squares of the dimensions d i of the irreducible R i representations of D 4 . So, the order 8 dihedral group has four irreducible representations with 1-dim; and a fifth irreducible D 4 -representation with 2-dim [42]. The character table of D 4 representations is given by from which we learn the following characters of the (a, b) generators For other features see [41]. With these tools at hand, we turn to engineer the SU 5 × D 4 × U ⊥ 1 models with dihedral monodromy symmetry.

Three D 4 -models
As in the case of S 3 monodromy, the breaking of S 4 down to D 4 is induced by non zero flux piercing the curves of the SU 5 × S 4 × U ⊥ 1 model. Using properties from the character table of D 4 , we distinguish three kinds of models depending on the way the S 4irreducible triplets have been pierced; there are three possibilities and are as described in what follows: In this model, the various irreducible triplets of S 4 ; in particular those involved in: (i) the five 10-plets namely 5 = 1 ⊕ 3 ⊕ 1 t 5 , and (ii) the ten 5-plets which includes the four 10-plets charged under U ⊥ 1 namely 4 t 5 = 1 t 5 ⊕ 3 t 5 , and the six uncharged 10-plets given by 6 = 3 ⊕ 3 ′ , are decomposed as sums of two singlets 1 p,q + 1 p ′ ,q ′ and a doublet 2 0,0 . The character properties of the D 4 -representations indicate that the decompositions of the triplets should be as By substituting these relations back into the restricted spectrum resulting from (3.1), we end with the following SU 5 × D 4 × U ⊥ 1 spectrum • five 10-plets where χ (a,b) R stands for the character of the generators in the R representation; ϕ = χ + χ ′ , and the integers N and P as in eqs (2.33). Notice that the multiplets 10 y 4 and 10 t 5 transform in the same trivial D 4 -representation; but having different t 5 -charges; the 10 y 3 transforms also as a singlet; but with character (1, −1); it is a good candidate for accommodating the top-quark family.
• ten 5-plets where we have set ϕ = χ + χ ′ . From this table, we learn that among the ten 5-plets, two sit in the 1 +,− representation with character (1, −1); but with differen t 5 charges; one in 1 −,+ with character (−1, 1) with no t 5 charge; and a fourth in the trivial representation of D 4 with a unit t 5 charge.
This is a completely reducible model; under restriction to dihedral subsymmetry, the 3 and 3 ′ triplets of S 4 are decomposed as follows by substituting these decompositions back into the spectrum of SU 5 × S 4 × U ⊥ 1 -theory given by (3.1), we obtain the curves spectrum of the second SU 5 × D 4 × U ⊥ 1 -model: • five 10-plets The spectrum of the 10-plets in the D 4 -model II can be also deduced from (4.7) by splitting the 2 0,0 doublet as Here we have two matter multiplets namely 10 +,+ and 10 t 5 +,+ ; they transform in the same trivial D 4 -representation with character (1, 1); but having different t 5charges. We also have two 10 +,− multiplets transforming in 1 +,− with character (1, −1); but with different fluxes; and one multiplet 10 −,+ with character (−1, 1); it will be interpreted in appendix B as the one accommodating the top-quark family.

Third case
This D 4 -model differs from the previous one by the characters of the singlets; since in this case the S 4 -triplets 3| S 4 and 3 ′ | S 4 are decomposed in terms of irreducible representations of D 4 like Substituting these relationships back into (3.1), we get the curve spectrum of the third model namely: • five 10-plets Here we have three 10 p,q matter multiplets in the trivial D 4 -representation with character (p, q) = (1, 1); one of them namely 10 t 5 +,+ having a t 5 charge and the two others not. A fourth curve 10 +,− in 1 +,− without t 5 charge nor a flux; and a fifth 10 −,− in 1 −,− with no t 5 but carrying a flux.

MSSM like spectrum
First, we describe the breaking of the SU 5 × D 4 × U ⊥ 1 theory down to supersymmetric standard model; then we study the derivation of the spectrum of MSSM like model with D 4 monodromy; and where the heaviest top-quark family is singled out.

Breaking gauge symmetry
Gauge symmetry is broken by U(1) Y hyperflux; by assuming doublet-triplet splitting produced by N units of U(1) Y , but still preserving D 4 × U ⊥ 1 , the 10-plets and 5-plets get decomposed into irreducible representations of standard model symmetry. The 5-plets of the SU 5 × D 4 × U ⊥ 1 models with multiplicity M 5 split as [60,61] leading to a difference between number of triplets and doublets in the low energy MSSM effective theory. These two relations are important since for N = 0 the correlation is some how relaxed; by choosing M multiplicities, we can also have the desired matter curve properties for accommodating fermion families; in particular the chirality property n (1,2) +1/2 = n (1,2) −1/2 which is induced by hyperflux. Furthermore, due to the flux, we also have different numbers of down quarks d c L and lepton doublets L. For the 10-plets of the GUT-model with multiplicity M 10 , we have the following decompositions [27,62,63] 3) The first relation with M 10 = 0 generates up-quark chirality since the number n In what follows, we study the derivation of an effective matter curve spectrum that resembles to the field content of MSSM. In addition to three families and as well as total hyperflux conservation f luxes we demand the following: • only a tree-level Yukawa coupling is allowed; and is given by the top-quark family, • the heaviest third generation is the least family affected by hyperflux, • MSSM matter generations are in D 4 × U ⊥ 1 representations, • no dimension 4 and 5 proton decay operators are allowed, • no µ-term at a tree level, • two Higgs doublets H u and H d as required by MSSM.

Building the spectrum
Seen that there are three possible SU 5 × D 4 × U ⊥ 1 models, we focus on the first model with curve spectrum given by eqs(4.7-4.8); and consider first the 10-plets; then turn after to 5-plets. Results regarding the two other models II and III are reported in appendix B.
10 = 0). Notice that the top-quark generation can a priori be taken in any one of the three D 4 -singlets; that is either 10 3 or 10 4 ; or 10 5 ; the basic difference between these D 4 -singlets is given by t 5 charge and hyperflux. But the choice of the 10 3 -multiplet looks be the natural one as it is unaffected by hyperflux, a desired property for MSSM and beyond; and has no t 5 charge Monodromy invariance of (5.9) under D 4 × U ⊥ 1 requires 5 Hu in the trivial representation with no t 5 charge; i.e: 5 Hu ∼ 1 +,+ . However, an inspection of the characters of the U ⊥ 1 chargeless 5-plets revels that there is no (5 +,+ ) t 5 =0 in the spectrum of the the D 4 × U ⊥ 1models I and II constructed above. To bypass this constraint, we realise the role of the Higgs 5 Hu by allowing VEVs to come from flavons as well; in other words by thinking of 5 Hu as follows where ϑ p ′ ,q ′ stands for a flavon in the representation 1 p ′ ,q ′ .
(ii) second it gives an important tool to distinguish between matter and Higgs in the 5plets sector as manifestly exhibited by the tri-coupling 10 +,− ⊗5 M ⊗5 H d . This interaction requires matter5 M 3 and Higgs5 H d to be in different D 4 -singlets 1 p,q and 1 p ′ ,q ′ with pp ′ = 1 and qq ′ = −1; see discussion given later on. By choosing the hyperflux units as N = P = 1; and using (5.3) we obtain the matter content curves D 4 U ⊥ 1 flux matter content Z 2 parity 10 i 10 3 10 4 10 5 Notice that by following [16] using Galois theory, the 10-plets have been attributed Z 2 parity charges as reported by the last column of above table. In our formulation these parities correspond to s i → −s i and κ 1 and κ 4 = κ 41 κ 42 κ 43 as in eq(2.28); by help of (2.14) and (2.22) we obtain in agreement with (2.17).
We also learn that the 5-plet (5 M +,− ) 0 is the least multiplet affected by hyperflux; and because of our assumptions, it is the candidate for matter5 M 3 ; the partner of 10 3 in the underlying SO 10 GUT-model. With this choice, the down-type quarks tri-coupling for the third family namely 10 3 ⊗5 M 3 ⊗5 H d ; and which we rewrite like This coupling requires the matter5 M 3 and the down-Higgs5 H d multiplets to belong to different D 4 singlets seen that 10 3 is in 1 +,− representation. However, the candidates Non diagonal 4-order coupling superpotentials with one (10 +,− ) 0 are as follows 3 Below, we discuss some properties of these couplings.

More on couplings in D 4 model I
First, we study the quark sector; and turn after to the case of leptons. 3 a complete classification requires also use Z 2 partity; see [16]. with right hand sides capturing same monodromy representations as left hand sides; that is Q, u c same D 4 × U ⊥ 1 representations as 10 M ; and so on. In what follows, we study each of these terms separately by taking into account ϑ p,q flavon contributions up to order four couplings; some of these flavons are interpreted as right neutrinos; they will be discussed at proper time. By restricting to VEVs ϑ −,+ = ρ 0 and H u = v u ; this non renormalisable coupling leads to the top quark mass term m t Q 3 u c 3 with m t equal to α 3 v u ρ 0 . Such a term should be thought of as a particular contribution to a general up-quark mass terms u c i M ij u j with 3×3 mass matrix as follows

Quark sector
where the ( * )'s refer to contributions coming from other terms including non diagonal couplings; one of them is it involves a 10-plet doublet (10 0,0 ) 0 ≡ (10 i ) 0 and a flavon doublet (ϑ 0,0 ) ′ 0 ≡ (ϑ i ) ′ 0 with VEVs (ρ 1 , ρ 2 ); the latter (ϑ 0,0 ) ′ 0 will be combined the 10 i -plet doublet like (10 0,0 ) 0 ⊗(ϑ 0,0 ) ′ 0 to make a scalar. Indeed, the tensor product can be reduced as direct sum over irreducible representations of D 4 having amongst others the D 4 -component with (−, −) charge character. This negative charge is needed to compensate the (−, −) charge coming from (10 +,− ) 0 ⊗ (5 Hu −,+ ) 0 . Restricting to quarks, this reduction corresponds to (10 Putting back into (5.24), and thinking of S −,− in terms of the linear combination α 2 (Q 1 ρ 2 − Q 2 ρ 1 ) of quarks, we obtain α 2 v u (Q 1 ρ 2 − Q 2 ρ 1 )u c 3 ; which can be put into the form u c i M ij u j with mass matrix as One can continue to fill this mass matrix by using the VEV's of other flavons; however to do that, one needs to rule out couplings with those flavons describing right neutrinos ν c i . Extending ideas from [16], the 3 generations of the right handed neutrinos ν c i in with the following features among the set of 15 flavons of the model However, though monodromy invariant, this couplings cannot generate the mass term mQ 1,2 u c 1,2 since the matter curve (10 0,0 ) 0 don't contain the quark u c 1,2 ; so the mass matrix (5.27) for the up-type quarks is it is a rank one matrix; it gives mass to the third generation (top-quark); while the two first generations are massless.
masses for lighter families The rank one property of above mass matrix (5.31) is a known feature in GUT models building including F-Theory constructions; see for instance [36,43,44,64]. To generate masses for the up-quarks in the first two generations, different approaches have been used in literature: (i) approach based on flux corrections using non perturbative effects [20] or non commutative geometry [21]; and (ii) method using δW deformations of the GUT superpotential W by higher order chiral operators [14,43,44,64,65,66]. Following the second way of doing, masses to the two lighter families are generated by higher dimensional operators corrections that are invariant under D 4 symmetry and Z 2 parity. This invariance requirement leads to involve 6-and 7-dimensional chiral operators which contribute to the up-quark mass matrix as follows and Notice that the adjunction of (ϑ +,+ ) t 5 chiral superfield is required by invariance under Z 2 parity. Using this deformation, a higher rank up-quark mass matrix is obtained as usual by giving VEVs to flavons as in in (5.29) and (ϑ +,− ) −t 5 = ϕ. By calculating the product of the operators in eqs(5.33-5.34) using D 4 fusion rules, we obtain The operator contributes in the up-quark mass matrix (5.31) as a correction to the matrix elements m 1,1 and m 1,2 ; it has the same role as the higher operator (5.35); so we will not take it into account in the quark mass matrix. Expanding the remaining operators by help of the D 4 rules, we have Summing up all contributions, we end with the following up-quark matrix • Down-type Yukawa Following the same procedure as in up-Higgs type coupling, we can build invariant operators for the down-type Yukawa leading to the mass matrix y 1,1 σ 1 y 1,2 σ 1 y 1,3 σ 1 y 1,1 σ 2 y 1,2 σ 2 y 1,3 σ 2 y 3,1 ω y 3,2 ω y 3,3 ω    (5.40)

Lepton sector
First we consider the charged leptons; and then turn to neutrinos.
• Charged leptons Charged leptons masses are determined by the same operators used in the case of the down quark sector 10 M ⊗ 5 M ⊗ 5 H d ; using spectrum eqs (5.11,5.16), the appropriate operators which provide mass to charged leptons are giving the lepton mass term m ij e c i L j with mass matrix • Neutrinos Right handed neutrinos are as in eq(5.28), they have negative R-parity. Dirac neutrino term is embedded in the coupling ν c i ⊗ 5 M ⊗ 5 Hu where the right neutrino ν c i is an SU 5 singlet; it allows a total neutrino mass matrix using see-saw I mechanism [18]. The invariant operators that give the Dirac neutrino in SU 5 × D 4 × U ⊥ 1 model are Using the D 4 algebra rules and flavon VEV's, these couplings lead to and then to a Dirac neutrino mass matrix as The Majorana neutrino term is given by Mν c i ⊗ν c j ; by using eqs (5.11,5.16), the Majorana neutrino couplings in SU 5 × D 4 × U ⊥ 1 model are as follows we can also add the singlet (ϑ −,+ ) 0 as a correction of the last two operators. The operators in above (5.46) lead to and ends with a Majorana neutrino mass matrix like The general neutrino mass matrix is calculated using see-saw I mechanism; it reads as and leads to the following effective neutrino mass matrix and and where we have set To obtain neutrino mixing compatible with experiments we need a particular parametrization and some approximations on M ν . To that purpose, recall that there are three approaches to mixing using: (i) the well know Tribimaximal (TBM) mixing matrix, (ii) Bimaximal (BM) and (iii) Democratic (DC); all of the TBM, BM and DC mixing matrices predict a zero value for the angle θ 13 . However recent results reported by MINOS [24], Double Chooz [25],T2K [54], Daya Bay [55], and RENO [56] collaborations reaveled a non-zero θ 13 ; such non-zero θ 13 has been recently subject of great interest; in particular by perturbation of the TBM mixing matrix [57].

Conclusion and discussions
In this paper, we have developed a method based on characters of discrete group representations to study SU 5 × D 4 × U ⊥ 1 -GUT models with dihedral monodromy symmetry. After having revisited the construction of SU 5 × S 4 × U ⊥ 1 and SU 5 × S 3 × U ⊥ 1 2 models from the character representation view, we have derived three SU 5 × D 4 × U ⊥ 1 models (referred here to as I, II and III) with curves spectrum respectively given by eqs(4.7-4.8), (4.11-4.12) and (4.14-4.15). These models follow from the three different ways of decomposing the irreducible S 4 -triplets in terms of irreducible representations of D 4 ; see eqs (4.6,4.10,4.13); such richness may be interpreted as due to the fact that D 4 has four kinds of singlets with generator group characters given by the (p, q) pairs with p, q = ±1. Then we have focussed on the curve spectrum (4.7-4.8) of the first SU 5 × D 4 × U ⊥ 1 model; and studied the derivation of a MSSM-like spectrum by using particular multiplicity values and turning on adequate fluxes. We have found that with the choice of: (i) topquark family 10 3 as (10 +− ) 0 , transforming into a D 4 -singlet with χ (a,b) character equal to (1, −1); and (ii) a 5 Hu up-Higgs as (5 −,+ ) 0 , transforming into a different D 4 -singlet with character equal to (−1, 1); there is no tri-Yukawa couplings of the form (10 +,− ) 0 ⊗ (10 +,− ) 0 ⊗ 5 Hu ++ as far as D 4 × U ⊥ 1 invariance is required; this makes SU 5 × D 4 × U ⊥ 1 model with two quark generations accommodated into a D 4 -doublet non interesting phenomenologically. Monodromy invariant couplings require implementation of flavons ϑ p,q by thinking of 5 Hu ∼ (5 −,+ ) 0 ⊗ (ϑ −,+ ) 0 leading therefore to a superpotential of order 4. The same property appears with the down-Higgs couplings where By analysing the conditions that a D 4 × U ⊥ 1 -spectrum has to fulfill in order to have a tri-Yukawa coupling for top-quark family 10 3 , we end with the constraint that the character of 5 Hu up-Higgs should be equal to (1, 1) as clearly seen on 10 +,− ⊗ 10 +,− ⊗ 5 Hu . This constraint is valid even if 10 3 was chosen like 10 +,+ . By inspecting the spectrum of the three studied SU 5 × D 4 × U ⊥ 1 models; it results that the spectrum of the third model given by eqs(4.14-4.15) which allow tri-Yukawa coupling; for details on contents and couplings of models II and III; see appendix B.

Appendix A: Characters in S 4 -models
In this appendix, we give details on some useful properties of Γ-models studied in this paper; in particular on the representations of S 4 and their characters.

Irreducible representations of S 4
First, recall that S 4 has five irreducible representations; as shown on the character formula 24 = 1 2 + 1 ′2 + 2 2 + 3 2 + 3 ′2 ; these are the 1-dim representations including the trivial 1 and the sign ǫ = 1 ′ ; a 2-dim representation 2; and the 3-dim representations 3 and 3 ′ , obeying some "duality relation". This duality may be stated in different manners; but, in simple words, it may be put in parallel with polar and axial vectors of 3-dim euclidian space. In the language of Young diagrams; these five irreducible representations are given by 1 : , 2 : , 3 : (7.1) and 3 ′ : This diagrammatic description is very helpful in dealing with S 4 representation theory [40,41,42]; it teaches us a set of useful information; in particular helpful data on the three following: i) Expressions of (3.5) In the representation 3 of the permutation group S 4 , the three x i -weights in (3.5) read in terms of the t i 's as is the completely symmetric term. The normalisation coefficient 1 2 is fixed by requiring the transformation x i = U ij t j as follows For the the representation 3 ′ , we have The entries of these triplets are cyclically rotated by the (234) permutation.
ii) S 4 -triplets as 3-cycle (234) The {|t i } and {|x i } weight bases are related by the orthogonal 5×5 matrix with U as in (7.4); and then From these transformations, we learn t i = U ki x k ; and then t i ± t j = (U ki ± U kj ) x k which can be also expressed t i ± t j = V ±kl ij X ± kl . Similar relations can be written down for {|x ′ i }.
In our approach the character of these generators have been used in the engineering of GUT models with S 4 monodromy; they are as follows In the SU 5 × S 4 theory considered in paper, the various curves of the spectrum of the GUT-model belong to S 4 -multiplets which can be decomposed into irreducible representation of S 4 . In doing so, one ends with curves indexed by the characters of the generators of S 4 as follows    The allowed Yukawa couplings that are invariant under D 4 × U ⊥ 1 are: • Down-type quark Yukawa couplings The Yukawa couplings down-type are: • Up-type quark Yukawa couplings The allowed Yukawa couplings that are invariant under D 4 × U ⊥ 1 and preserving parity symmetry are: (10 +,− ) 0 ⊗ (10 +,− ) 0 ⊗ (5 Hu +,+ ) 0 (8.10) for third generation; and (10 −,− ) 0 ⊗ (10 −,− ) 0 ⊗ (5 Hu +,+ ) 0 (10 +,+ ) 0 ⊗ (10 +,+ ) 0 ⊗ (5 Hu +,+ ) 0 (10 +, • Down-type quark Yukawa couplings The Yukawa coupling down-type are: for third generation; and For the neutrino sectors in both models II and III, the couplings are embedded in the Dirac and Majorana operators as for model I; their mass matrix depend on the choice of the localization of right neutrino in the singlet curves ϑ ±,± .

Appendix C: Monodromy and flavor symmetry
We begin by recalling that in F-theory GUTs, quantum numbers of particle fields and their gauge invariant interactions descend from an affine E 8 singularity in the internal Calabi-Yau Geometry: CY 4 ∼ E → B 3 . The observed gauge bosons, the 4D matter generations and the Yukawa couplings of standard model arise from symmetry breaking of the underlying E 8 gauge symmetry of compactification of F-theory to 4D space time.
In this appendix, we use known results on F-theory GUTs to exhibit the link between non abelian monodromy and flavor symmetry which relates the three flavor generations of SM. First, we briefly describe how abelian monodromy like Z p appear in F-GUT models; then we study the extension to non abelian discrete symmetries such the dihedral D 4 we have considered in present study.

Abelian monodromy
One of the interesting field realisations of the F-theory approach to GUT is given by the remarkable SU 5 × SU ⊥ 5 model with basic features encoded in the internal geometry; in particular the two following useful ones: (i) the SU 5 × SU ⊥ 5 invariance follows from a particular breaking way of E 8 ; and (ii) the full spectrum of the field representations of the model is as in eq(2.1). From the internal CY4 geometry view, SU 5 and SU ⊥ 5 have interpretation in terms of singularities; the SU 5 lives on the so called GUT surface S GU T ; it appears in terms of the singular locus of the following Tate form of the elliptic fibration y 2 = x 3 + b 5 xy + b 4 x 2 z + b 3 yz 2 + b 2 xz 3 + b 0 z 5 ; it is the gauge symmetry visible in 4D space time of the GUT model. Quite similarly, the SU ⊥ 5 may be also imagined to have an analogous geometric representation in the internal geometry; but with different physical interpretation; it lives as well on a complex surface S ′ ; another divisor of the base B 3 of the complex four dimensional elliptic CY4 fibration. Obviously these two divisors are different, but intersect. Here, we want to focus on aspects of the representations of SU ⊥ 5 appearing in eq(2.1) and too particulary on the associated matter curves Σ t i , Σ t i +t j , Σ t i −t j ; which are nicely described in the spectral cover method using an extra spectral parameter s. If thinking of the hidden SU ⊥ 5 in terms of a broken symmetry by an abelian flux or Higgsing down to its Cartan subgroup, the resulting symmetry of the GUT model becomes U (1) 4 × SU 5 with 4 The extra U (1)'s in the breaking U (1) 4 × SU 5 put constraints on the superpotential couplings of the effective low energy model; the simultaneous existence of U (1) 4 is phenomenologically undesirable since it does not allow a tree-level Yukawa coupling for the top quark. This ambiguity is overcome by imposing abelian monodromies among the U (1)'s allowing the emergence of a rank one fermion mass matrix structure; see eqs(9.4-9.5) given below. Following the presentation of section 2 of this paper, the spectral covers describing the above invariance are given by polynomials with an affine variable s as in eq(2.8); see also (2.9,2.12,2.13). To fix the ideas, we consider monodromy properties of 10-plets Σ t i encoded in the spectral cover equation The location of the seven branes on GUT surface associated to this SU 5 representation is given by b 5 = 0. Using the method of [18,27,30,31], the possible abelian monodromies are Z 2 , Z 3 , Z 4 , Z 2 × Z 3 and Z 2 × Z 2 ; they lead to factorizations of the C 5 spectral cover as and to the respective identification of the weights {t 1 , t 2 }, {t 1 , t 2 , t 3 }, {t 1 , t 2 , t 3 , t 4 }, {t 1 , t 2 } ∪ {t 3 , t 4 , t 5 } and {t 1 , t 2 } ∪ {t 3 , t 4 }.
The algebraic equations for the matter curves Σ t i , Σ t i +t j , Σ t i −t j in terms of the t i weights associated with the SU ⊥ 5 fundamental representation are respectively given by t i = 0; (t i + t j ) i<j = 0 and ± (t i − t j ) i<j = 0; they are denoted like 10 t i ,5 t i +t j and 1 ±(t i −t j ) ; see eq(2.2). As a first step to approach non abelian monodromies we are interested in here, it is helpful to notice the two useful following things: (a) the homology 2-cycles in the CY4 underlying SU 5 × U (1) 4 invariance has monodromies captured by a finite discrete group that can be used as a constraint in the modeling. (b) from the view of phenomenology, these monodromies must be at least Z 2 in order to have top-quark Yukawa coupling at tree level as noticed before. Notice moreover that under this Z 2 , matter multiplets of the SU 5 model split into two Z 2 sectors 5 : even and odd; for example the two tenplets {10 t 1 , 10 t 2 } are interchanged under t 1 ↔ t 2 ; the corresponding eigenstates are given by 10 t ± with eigenvalues ±1. By requiring the identification t 1 ↔ t 2 , naively realised by setting t 1 = t 2 = t, matter couplings in the model get restricted; therefore the off diagonal tree level Yukawa coupling 10 t 1 .10 t 2 .5 −t 1 −t 2 (9.4) which is invariant under SU 5 × U (1) 4 , becomes after t 1 ↔ t 2 identification a diagonal top-quark interaction invariant under Z 2 monodromy. The resulting Yukawa coupling reads as follows [27,30,31] 10 t .10 t .5 −2t (9.5) the other diagonal coupling 10 0 .10 0 .5 −2t is forbidden by the U(1) symmetry; see footnote 5. Notice that for bottom-quark the typical Yukawa coupling 10 t .5 t i +t j .5 t k +t l is allowed by Z 2 while 10 0 .5 t i +t j .5 t k +t l is forbidden. In this monodromy invariant theory, the symmetry of the model is given by SU 5 × U (1) 3 × Z 2 ; it may be interpreted as the invariance that remains after taking the coset with respect to Z 2 ; that is by a factorisation of type G = H × Z 2 with H = G/Z 2 . Indeed, starting from SU 5 × U (1) 4 and performing the two following operations: (i) use the traceless property of the fundamental representation of SU ⊥ 5 to think of (9.1) like U (1) t i /J (9.6) with J = {t i | t 1 + t 2 + t 3 + t 4 + t 5 = 0} ≃ U (1) diag ; this property is a rephrasing of the usual U (5) factorisation; i.e SU (5) = U (5) U (1) . (ii) substitute the product U (1) t 1 × U (1) t 2 by the reduced abelian group U (1) t × Z 2 where monodromy group has been explicitly exhibited. In this way of doing, one disposes of a discrete group that may be promoted to a symmetry of the fields spectrum. To that purpose, we need two more steps: first explore all allowed discrete monodromy groups; and second study how to link these groups to flavor symmetry. For the extension of above Z 2 , a similar method can be used to build other prototypes; in particular models with abelian discrete symmetries like SU 5 × U (1) 5−k × Z k with k = 3, 4, 5; or more generally as where 1 < p + q ≤ 5 and Z 1 ≡ I id , Z 0 ≡ I id . Notice that the discrete groups in eq(9.7) are natural extensions of those of the theories with SU 5 × U (1) 5−k × Z k symmetry; and that the condition p + q ≤ 5 on allowed abelian monodromies is intimately related with the Weyl symmetry W SU ⊥ 5 of SU ⊥ 5 . Therefore, we end with the conclusion that the Z p × Z q abelian discrete groups in above relation are in fact particular subgroups of the non abelian symmetric group W SU ⊥ 5 ≃ S 5 .

Non abelian monodromy and flavor symmetry
To begin notice that the appearance of abelian discrete symmetry in the SU 5 based GUT models with invariance (9.7) is remarkable and suggestive. It is remarkable because these finite discrete symmetries have a geometric interpretation in the internal CY4; and constitutes then a prediction of F-theory GUT. It is suggestive since such kind of discrete groups, especially their non abelian generalisation, are highly desirable in phenomenology; particularly in playing the role of a flavor symmetry. In this regards, it is interesting to recall that it is quite well established that neutrino flavors are mixed; and this property requires non abelian discrete group symmetries like the alternating A 4 group which has been subject to intensive research during last decade [32,33,34,52,53]. Following the conjecture of [15,16], non-abelian discrete symmetries may be reached in F-theory GUT by assuming the existence of a non abelian flux breaking the SU ⊥ 5 down to a non abelian group Γ ⊂ W SU ⊥ 5 . In this view, one may roughly think about the Z p × Z q group of (9.7) as special symmetries of a family of SU 5 based GUT models with invariance given by SU 5 × U (1) 5−k × Γ k (9.8) where now Γ k is a subgroup of S 5 that can be a non abelian discrete group. In this way of doing, one then distinguishes several SU 5 GUT models with non abelian discrete symmetries classified by the number of surviving U (1)'s. In presence of no U (1) symmetry, we have prototypes like SU 5 × S 5 and SU 5 × A 5 ; while for a theory with one U (1), we have symmetries as follows where the alternating A 4 and dihedral D 4 are the usual subgroups of S 4 itself contained in S 5 . In the case with two U (1)'s, monodromy gets reduced like SU 5 × U (1) 2 × S 3 . Moreover, by using non abelian discrete monodromy groups Γ k , one ends with an important feature; these discrete groups have, in addition to trivial representations, higher dimensional representations that are candidates to host more than one matter generation. Under transformations of Γ k ; the generations get in general mixed. Therefore the non abelian Γ k 's in particular those having 3-and/or 2-dimensional irreducible representations may be naturally interpreted in terms of flavor symmetry.
In the end of this section, we would like to add a comment on the splitting spectral cover construction regarding non abelian discrete monodromy groups like A 4 and D 4 . In the models (9.9), the spectral cover for the fundamental C 5 is factorised like C 5 = C 4 × C 1 and similarly for C 10 and C 20 respectively associated with the antisymmetric and the adjoint of SU ⊥ 5 . In the C 4 × C 1 splitting, we have C 4 = a 5 s 4 + a 4 s 3 + a 3 s 2 + a 2 s + a 1 C 1 = a 7 s + a 6 (9.10) where the a i 's are complex holomorphic sections. For the generic case where the coefficients a i are free, the splitted spectral cover C 4 × C 1 has an S 4 monodromy. To have splitted spectral covers with monodromies given by the subgroups A 4 and D 4 , one needs to put constraints on the a i 's; these conditions have been studied in [14,16]; they are non linear relations given by Galois theory. Indeed, starting from SU 5 × SU ⊥ 5 model and borrowing tools from [16], the breaking of SU 5 × SU ⊥ 5 down to SU 5 × D 4 × U (1) model considered in this paper may be imagined in steps as follows: first breaking SU ⊥ 5 to subgroup SU ⊥ 4 × U (1) by an abelian flux; then breaking the SU ⊥ 4 part to the discrete group S 4 by a non-abelian flux as conjectured in [15,16]; deformations of this flux lead to subgroups of S 4 . To obtain the constraints describing the D 4 splitted spectral cover descending from C 4 × C 1 , we use Galois theory; they are given by a set of two constraints on the holomorphic sections of C 4 × C 1 ; and are obtained as follows: (i) the first constraint comes from the discriminant ∆ C 4 of the spectral cover C 4 which should not be a perfect square; that is ∆ C 4 = δ 2 . The explicit expression of the discriminant of C 4 has been computed in literature; so we have 108a 0 (λa 2 6 + 4a 1 a 7 )(κ 2 a 2 7 + a 0 (λa 2 6 + 4a 1 a 7 )) 2 = δ 2 (9.11) where dependence into a 6 and a 7 is due to solving the traceless condition b 1 = 0 in C 5 = C 4 × C 1 . (ii) the second constraint is given by a condition on the cubic resolvent which should be like R C 4 (s)| s=0 = 0. The expression of R C 4 (s) is known; it leads to a 2 2 a 7 = a 1 a 0 a 2 6 + 4a 3 a 7 (9.12) where a 0 is a parameter introduced by the solving the traceless condition b 1 = 0; for explicit details see [16].