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Complete structure of Z_n Yukawa couplings

We give the complete twisted Yukawa couplings for all the Z_n orbifold constructions in the most general case, i.e. when orbifold deformations are considered. This includes a certain number of tasks. Namely, determination of the allowed couplings, calculation of the explicit dependence of the Yukawa couplings values on the moduli expectation values (i.e. the parameters determining the size and shape of the compactified space), etc. The final expressions are completely explicit, which allows a counting of the DIFFERENT Yukawa couplings for each orbifold (with and without deformations). This knowledge is crucial to determine the phenomenological viability of the different schemes, since it is directly related to the fermion mass hierarchy. Other facts concerning the phenomenological profile of Z_n orbifolds are also discussed, e.g. the existence of non--diagonal entries in the fermion mass matrices, which is related to a non--trivial structure of the Kobayashi--Maskawa matrix. Finally some theoretical results are given, e.g. the no--participation of (1,2) moduli in twisted Yukawa couplings. Likewise, (1,1) moduli associated with fixed tori which are involved in the Yukawa coupling, do not affect the value of the coupling.


Introduction and brief review
In the last few years an enormous effort has been made in order to establish the connection between string theories [1] (especially E 8 × E 8 heterotic string [2]) and low energy physics. Different schemes for constructing classical string vacua have arisen during this time. Using these schemes it has been possible to build up four-dimensional strings that resemble the Standard Model in many aspects, e.g. SU(3) × SU(2) × U(1) Y gauge group and three generations of particles with the correct representations [3][4][5][6]. In spite of these achievements there remain many pending questions. In particular there is a large number of classical vacuum states, which reduces the predictive power of the theory. At present there are no dynamical criteria to prefer a particular vacuum, so the best we can do is to study their phenomenological characteristics in order to select the viable vacua. In this respect, orbifold compactifications [7] have proved to be very interesting four-dimensional string constructions since they can pass succesfully a certain number of low energy tests [8]. However, not all the experimental constraints have been used in order to probe the phenomenological potential of orbifolds. The best example of this is the observed structure of fermion masses and mixing angles [9].
Concerning the last point, a crucial ingredient in order to relate theory and observation is the complete knowledge of the theoretical Yukawa couplings. This knowledge includes a certain number of aspects: i) Physical states that enter the couplings.
iii) Numerical values of the Yukawa couplings and dependence of these values on the physical parameters that define the string vacuum (e.g. the size of the compactified space). iv) Number of different Yukawa couplings and phenomenological viability of the scheme (i.e. fitting of the observed pattern of fermion masses and mixing angles by the theoretical Yukawa couplings).
In this sense only for prime Abelian orbifolds, i.e. Z 3 and Z 7 are the Yukawa couplings completely known [10,9,11]. For the other orbifolds points i) and ii) have recently been studied in ref. [12]. General expressions of Z n Yukawa couplings have been determined in ref. [13]. However, although very useful, they are not explicit enough to lucidate points iii) and iv), specially when deformations of the compactified space are considered.
Undoubtedly, a better knowledge of the Yukawa couplings is of utmost importance to select or discard explicit string constructions with a highly non-trivial test. This is the main purpose of this paper, i.e. to answer points i), ii), iii), iv) for all the Z n orbifolds. Besides the phenomenological motivation, there are strong theoretical reasons to completely determine the Yukawa couplings. In particular it is the only way to know the moduli dependence of the matter Lagrangian and, in consequence, the superpotential. This allows the examination of the properties of the action under target-space modular transformations (e.g. R → 1/R ) [14]. It is also necessary in order to discuss supersymmetry breaking dynamics [15] and cosmological implications (note that moduli play the role of Brans-Dicke fields in four dimensions) of these theories.
Let us review briefly Z n orbifold constructions. A Z n orbifold is constructed by dividing R 6 by a six-dimensional lattice Λ modded by some Z n symmetry, called the point group P. The space group S is defined as S = Λ × P , i.e. S = {(γ, u); γ ∈ P, u ∈ Λ}. The requirement of having N = 1 supersymmetry in four dimensions and the absence of tachyons restrict the number of possible point groups [7]. The complete list is given in the first two columns of Table 1, where the so-called twist θ (i.e. the generator of P) is represented in an orthogonal complex basis of the six-dimensional space. Λ must be chosen so that θ acts crystallographically on it. If the realization of θ on the lattice coincides with the Coxeter element of a rank-six Lie algebra root lattice, the orbifold is of the Coxeter type. A list of Coxeter orbifolds, taken from ref. [16], is given in the third column of Table 1. Some additional examples of Coxeter orbifolds can be found in ref. [12]. The lattice of the Z 8 -II orbifold, SO(5) × SO (8), corresponds in fact to a generalized Coxeter orbifold where the Coxeter element has been multiplied by an outer automorphism. Non-Coxeter orbifolds can also be constructed. An example of a non-Coxeter orbifold (the Z 4 one with [SO(4)] 3 lattice) is studied in Section 3. The total number of possible lattices associated with each Z n orbifold can be found in ref. [17]. As will become clear in the text, some properties of the Yukawa couplings for a particular Z n orbifold depend on the lattice chosen, while others do not.
It is important to point out that in a string orbifold construction the lattice Λ can get deformations compatible with the point group [9,11]. These degrees of freedom correspond to the untwisted moduli surviving compactification. Deformations play an important role on the value of the Yukawa couplings.
We are interested in the couplings between twisted fields (the untwisted sector has already been studied [10] and is physically less interesting [9]). As we will see, these couplings present a very rich range, which is extremely attractive as the geometrical origin of the observed variety of fermion masses [10,18,9]. A twisted string satisfies x(σ = 2π) = gx(σ = 0) as the boundary condition, where g is an element of the space group whose point group component is non-trivial. Owing to the boundary condition a twisted string is attached to a fixed point of g. Physical twisted fields are associated with conjugation classes of the space group rather than with particular elements [7]. For example {hgh −1 , h ∈ S} is the conjugation class of g. For prime orbifolds conjugation classes are in one-to-one correspondence with the fixed points of P . For non-prime orbifolds the situation is a bit more involved since two different fixed points under θ n may be connected by θ m , m < n. Then both of them correspond to the same conjugation class.
The paper is organized as follows. In Section 2 we expound the various steps necessary to obtain the final spectrum of Yukawa couplings for each orbifold, taking the Z 6 -I as a guide example. These steps include: determination of the geometrical structure of the orbifold and deformation parameters, physical states, calculation of explicit Yukawa couplings, and counting of different couplings. Section 3 is devoted to a comparative study of the [SO(4)] 3 and [SU(4)] 2 Z 4 orbifolds. This shows which properties of the couplings depend on the lattice chosen and which do not. Furthermore, the [SO(4)] 3 case provides an example of a non-Coxeter orbifold. Besides this, the Z 4 orbifold allows one to see the physical meaning of a (1,2) modulus (absent in the Z 6 -I case) and its effect in the Yukawa coupling values. The complete results for the rest of Z n orbifolds are given in Appendix 1 and summarized in Table 1.

The method
Several steps are necessary in order to obtain the final spectrum of Yukawa couplings for each orbifold. We explain these steps in the present section, taking the Z 6 -I orbifold as an illustrative example. The reason for this choice is that prime orbifolds (Z 3 and Z 7 ) have already been studied in depth in references [10,9,11]. A complete exposition of the method followed here can be found in ref. [11]. It is, however, convenient to discuss the present example in some detail, since non-prime orbifolds exhibit certain features which are absent in the prime ones.

Geometrical structure
The twist θ of the Z 6 -I orbifold has the form (see Table 1) in the complex orthogonal basis {(ẽ 1 ,ẽ 2 ), (ẽ 3 ,ẽ 4 ), (ẽ 5 ,ẽ 6 ) }. Very often it is more suitable to work in the lattice basis {e 1 , ..., e 6 }, which in this case is simply a set of simple roots of G 2 2 × SU(3). Then θ acts as the Coxeter element Note that we have labelled as e 5 , e 6 the simple roots of SU(3). As mentioned above Λ can get deformations compatible with the point group. These degrees of freedom correspond to the Hermitian part of the five untwisted (1,1) moduli surviving compactification being the left (right) vacuum, α L is an oscillator operator and i (j) is a holomorphic (antiholomorphic) index. Note that under a deformation the actuation of θ on the lattice basis, eq. (2), remains the same. Then P invariance imposes the following relations: where α ij = cos θ ij and e i e j = |e i ||e j | cos θ ij . Therefore we can take the 5 deformation degrees of freedom R i are global scales of the three sublattices (G 2 , G 2 , SU(3)), and for α 13 = α 14 = 0 we recover the rigid G 2 × G 2 × SU(3) lattice. The connection between the lattice basis (e 1 , ..., e 6 ) and the orthogonal basis (ẽ 1 , ...,ẽ 6 ) in which θ takes the form (1) is given by where ϕ 1 1 , ϕ 3 1 , ϕ 1 2 , ϕ 3 2 , φ are arbitrary angles that are irrelevant for our results and Let us consider now the fixed points under the action of the point group. f n is a fixed point under θ n if it satisfies f n = θ n f n + u, u ∈ Λ. As Z 6 -I is a non-prime orbifold, a point fixed by θ n (n = 1) may be not fixed by θ m (m = n). Consequently, the fixed points under θ, θ 2 and θ 3 must be considered separately (θ 4 , θ 5 are simply the antitwists of θ 2 and θ). It is easy to check from (2) that there are three different fixed points under θ. Working in the lattice basis their coordinates (up to lattice translations) are Similarly under θ 2 there are 27 fixed points. 12 of them are connected to the others by a θ rotation Consequently, there are 15 conjugation classes under θ 2 . Finally, under θ 3 , there are 16 fixed tori that are the direct product of 16 fixed points in the sublattice (e 1 , ..., e 4 ) by the 2-torus defined by the sublattice (e 5 , e 6 ). (Notice that θ 3 is trivial in the SU(3) root lattice.) 15 of these fixed points are connected between themselves by θ rotations (10) with The direct product under the (e 5 , e 6 ) torus has been understood. Consequently, there are 6 conjugation classes under θ 3 . A similar analysis for other orbifolds can be found in Appendix 1.

Physical states
The next step is to determine which are the physical states. These must be invariant under a total Z 6 transformation which, besides the twist θ in the 6-dimensional space, includes a Z 6 gauge transformation, usually represented by a shift V I (the so-called embedding) on Λ E 8 ×E 8 and a shift v t on Λ SO (10) . Accordingly one has to construct for each θ k sector linear combinations of states, associated with θ k fixed points, that are eigenstates of θ [19,12]. If f k is a fixed point of θ k such that l is the smallest number giving θ l f k = f k + u, u ∈ Λ, then the eigenstates of θ have the form with eigenvalue e −iγ (obviously, if k = 1, then l = 1 and eq. (12) is trivial). Under a Z 6 transformation the complete state gets a phase [19,12] where p t is the NSR part momentum put on the SO(8) weight lattice and P I is the transverse 8-dim. momentum (E 8 × E 8 root momentum) fulfilling the right-mover and left-mover massless conditions respectively. Then ∆(k, e iγ ) = 1 for physical states.

Allowed Yukawa couplings
Let us now analyse the allowed Yukawa couplings between physical states. A twisted string associated with a fixed point f and a rotation θ j is closed due to the action of g = (θ j , (I − θ j )f ), so the corresponding conjugation class is given by After some algebra, the general expression of the conjugation class of g is The set of translations (1 − θ j ) { (θ k f + Λ), k = 0, ..., j − 1} is called the coset associated with θ j and f (note that the cosets associated with f and θ k f are obviously the same). For a trilinear coupling of twisted fields T 1 T 2 T 3 to be allowed, the product of the respective conjugation classes should contain the identity. This implies, in particular, that the product of the three point group elements θ j 1 θ j 2 θ j 3 should be 1 (this is the so-called point group selection rule). For the Z 6 -I orbifold this implies that only θθ 2 θ 3 , θ 2 θ 2 θ 2 and θθθ 4 couplings have to be considered. A straightforward application of the H-momentum conservation [10,20,21] shows that the θθθ 4 couplings are also forbidden. Furthermore for the θθ 2 θ 3 couplings we must require which leads to the so-called space group selection rule for the coupling θθ 2 θ 3 in the Z 6 -I orbifold It should be noticed that if the space group selection rule (18) is satisfied for three fixed points f 1 , f 2 , f 3 , then it is also satisfied for θ k 1 f 1 , θ k 2 f 2 , θ k 3 f 3 , and, consequently, for all the physical combinations (12). For the case at hand, i.e. the Z 6 -I orbifold, one can consider 270 kinds of couplings f of the θθ 2 θ 3 type, from which only 90 are allowed: those for which the e 5 , e 6 components (i.e. the SU(3) sublattice projection) of f 1 − f 2 are vanishing. So, if we write the fixed points Let us finally note that the product of the θ eigenvalues of the physical combinations of fixed points involved in the coupling should be one, otherwise the coupling is vanishing. For example, in the Z 6 -I orbifold the following θθ 2 θ 3 coupling is forbidden on these grounds, since, due to θ invariance, it is equal to This result can also be obtained for the standard embedding case from gauge invariance since the state considered in the θ 3 sector corresponds to a (27), while the others are (27)'s. In consequence all the couplings to be considered in the Z 6 -I involve symmetric combinations of fixed points (i.e. θ eigenvalue = 1) exclusively. We have performed the previous analysis for all the Z n orbifolds. In all cases only couplings between symmetric combinations of fixed points survive. We do not really know what is the fundamental principle behind this rule (if any), but it has important consequences. For instance, in ref. [19] it was suggested that the phases of the non-zero θ eigenvalue states (see eq. (12)) could be the geometrical origin of the phases of the Kobayashi-Maskawa matrix. Clearly, the present rule excludes this possibility.

Calculation of Yukawa couplings
We are interested in couplings of the type ψψφ (i.e. fermion-fermion-boson). A trilinear string scattering amplitude is given by the correlator < V 1 (z 1 )V 2 (z 2 )V 3 (z 3 ) > of the vertex operators creating the corresponding states. Complete expressions for the vertex operators of the fields under consideration can be found in refs. [10,20]. As has been pointed out [10] the non-vanishing Yukawa couplings are essentially given by the bosonic twist correlator < σ 1 (z 1 )σ 2 (z 2 )σ 3 (z 3 ) >, where σ i represents a twist field creating the appropriate twisted ground state. According to subsection 2.2 σ fields for physical states are, in general, linear combinations of σ fields associated with specific rotations and fixed points, say σ θ j ,f . For example, for a physical state in the θ j sector whose twist part is given by k=0,...,l−1 e −ikγ |θ k f > (the meaning of γ and l is given in eq. (12)) the corresponding twist field is simply k=0,...,l−1 e −ikγ σ θ j ,θ k f . According to the result of the previous subsection only symmetric combinations (γ = 2π) are relevant for trilinear couplings, so the correlator < σ 1 (z 1 )σ 2 (z 2 )σ 3 (z 3 ) > associated with a θ j 1 θ j 2 θ j 3 will take the form where the square root is a normalization factor. The correlation functions on the righthand side are evaluated following standard lines [10]. They are defined by Owing to the Gaussian character of the action S where X = X 1 + iX 2 and a sum over the three complex coordinates is understood, the scattering amplitude can be separated into a classical and a quantum part [10] The quantum contribution represents a global factor for all the couplings with the same θ j 1 θ j 2 θ j 3 pattern in a given orbifold; so the physical information mostly resides in the classical contribution. Eventually, a total normalization factor, which depends on the size of the compactified space, has to be determined with the help of the four-point correlation function. The final task lies in writing the couplings in terms of the physically significant parameters, i.e. those that parametrize the size and shape of the orbifold. Let us consider, for the sake of definiteness, a θθ 2 θ 3 coupling in our guide example, the Z 6 -I orbifold. The classical contribution, see eq. (29), to a < σ θ σ θ 2 σ θ −3 > correlator has been determined in references [11,13], so we escape here the details of the calculation. The result is that the contribution of the classical (instantonic) solutions to the classical action, eq. (28), for a three-point correlation on the sphere, is where i = 1, 2, 3 denotes the corresponding complex coordinate (and thus the projection over the associated z-plane), the fields X i are twisted by exp(2πk i /N) (k 1 = 1, k 2 = 1, k 3 = 2 and N = 6 for the Z 6 -I orbifold) and (f 2 −f 3 +Λ) ⊥ selects only (f 2 −f 3 +Λ) shifts that are orthogonal to the invariant plane (this means that we can choose (f 2 − f 3 ) i=3 = 0 and Λ = Λ G 2 ×G 2 ). Several comments are in order here. First it is clear that the i = 3 plane (i.e. the invariant plane) does not contribute to the classical action, and the coupling for i = 3 behaves much as an untwisted one. In fact in the invariant plane the three strings must be attached to the same fixed point, i.e.
. These facts are general for all the couplings, in any orbifold, when fixed tori are involved. Second, the v-coset in (30) does not depend on f 1 since in the calculation of X cl (z), z 1 has been sent to infinity by using SL(2, C) invariance. We call this the 2-3 picture (for more details see ref. [11]). Equation (30) can be expressed in the 1-2 and 1-3 pictures as well where We can check by using the space group selection rule (18) that there is a one-to-one correspondence between S cl(1−2) , S cl(1−3) , S cl(2−3) . The 2-3 picture, eq. (30) is the most convenient one since Λ 12 , Λ 13 are subsets of the original lattice Λ. Furthermore S cl(1−2) and S cl(1−3) depend on the three fixed points considered f 1 , f 2 , f 3 ; while S cl(2−3) depends only on f 2 , f 3 . 2 We can now write the complete form of the correlator using eqs. (29,30) with V ⊥ the volume of the G 2 × G 2 unit cell. General expressions for the couplings similar to eq. (34) can be found in ref. [13] for all the Z n orbifolds. We have performed the calculation in all the cases, checking that the results of the mentioned reference are correct.
Expression (34) is not explicit enough for most purposes. For example it does not allow examination of the transformation properties of the Yukawa couplings under target-space modular transformations (e.g. R → 1/R). From a phenomenological point of view eq. (34) does not exhibit the dependence of the value of the coupling on physical quantities, i.e. those that parametrize the size and shape of the compactified space. In fact eq. (34) is not even good enough to calculate the final value of the coupling numerically, especially when deformations are considered. The key point in order to do this is to write (f 23 + u) i in terms of (e 1 , ..., e 6 ), i.e. the lattice basis. This can be done with the help of the results of subsection 2.1, see eq. (5). Then the correlator (34) appears as an explicit function of the deformation parameters of the compactified space. Substituting the resulting expression in (26) we obtain the final Yukawa coupling, which can be writen in a quite compact way where f 23 represents the first four components of (f 2 − f 3 ) (i.e. those corresponding to the G 2 × G 2 sublattice basis (e 1 , ..., e 4 ) ), and with where the deformation parameters R 2 i , α ij have been defined in eqs. (3,4). It is worthwhile to have a look at eq. (36) to realize which are the physical quantities on which the value of the coupling depends. First C θθ 2 θ 3 depends on the relative positions in the lattice of the relevant fixed points to which the physical fields are attached. This information is condensed in f 23 . Second C θθ 2 θ 3 depends on the size and shape of the compactified space, which is reflected in the orbifold compactification parameters (R 2 i , α 13 , α 14 ) appearing in Ω and (implicitely) in V ⊥ . Note that both pieces of information appear in a completely distinguishable way from each other in eq. (36). Notice also that the deformation parameter R 2 5 does not appear in Ω. This is due to the fact that R 5 parametrizes the size of the i = 3 sublattice, i.e. the fixed torus, and we have learnt that for i = 3 the coupling is equivalent to an untwisted one. This is a general fact for all the orbifold couplings in which fixed tori are involved (e.g. it does not occur for the θ 2 θ 2 θ 2 coupling of the Z 6 -I orbifold, see below). We say that R 5 is not an effective deformation parameter for the θθ 2 θ 3 couplings. The number of effective deformation parameters (4 in this case) is physically relevant since it is strongly related to the number of different Yukawa couplings and their corresponding sizes.
For the other twisted coupling θ 2 θ 2 θ 2 in the Z 6 -I orbifold, the expression of the coupling can be calculated in the same way as in the θθ 2 θ 3 case, and is given by where F = 1 for l 1 = 1 or l 2 = 1 or l 3 = 1 and F = 1 √ 2 for l 1 = l 2 = l 3 = 2. l i is the number of elements in the conjugation class associated with f i , see eq. (12). f 23 represents the components of (f 2 − f 3 ) in the lattice basis (e 1 , ..., e 6 ). The global normalization factor and the Ω matrix are given by Clearly the number of effective parameters is 5.
We have performed a similar analysis for all the trilinear twisted couplings in all the Z n orbifolds, giving the number of effective deformation parameters in each case. The results are expounded in Appendix 1.

Accidental symmetries and the number of different couplings
We are now ready to count the number of different couplings that appear in each orbifold. From the physical point of view this is one of the most relevant questions about a string construction, since it is directly related to the possibility of reproducing the observed pattern of fermion masses and mixing angles. Unfortunately, this task is probably the most tedious part of the work presented here. Again, we expound in some detail the analysis for the two possible couplings in the Z 6 -I orbifold. Let us begin with the twisted coupling θθ 2 θ 3 .The corresponding results for other orbifolds can be found in Appendix 1.
The first point is that the Ω matrix appearing in the Jacobi theta function of the coupling, eqs. (36-38), is universal for all the θθ 2 θ 3 couplings. This means that the differences between the Yukawa couplings come exclusively from the sum in (f 23 + u) in the classical part of the correlation. Hence, two couplings will have the same value if there exists an integer unimodular transformation U (i.e. U ∈ GL(4, Z), | U |= ±1) such that Then, if the previous equations are true for some U, there is a one-to-one correspondence between the terms of the series defining ϑ To check the latter, note that when the sum (37) is expressed in the complex orthogonal basis, the exponent takes a diagonal form as can be seen from (34). Then the terms multiplying the coefficients a i are unchanged under θ n twists, since these correspond to make rotations in each i-plane. This argument is always valid because the factorization (44) is a consequence of the fact that the classical contributions can be computed in each i-plane separately. In addition to the group generated by {−I, θ} , there can be "accidental U-symmetries" leaving Ω unchanged in eq. (43). Some of these symmetries con be spontaneously broken when deformations are taken into account. After inspection it turns out that, for the case at hand, these accidental symmetries are generated by (when expressed in the complex orthogonal basis) plus products of these matrices by {−I, θ}. U 1 , U 2 , U 3 are broken when deformations are considered. Now, two Yukawa couplings C, C ′ are equal (in the non-deformed case) if f 23 and f 23 ′ are connected as in (43) by one of the U-matrices mentioned above. The analysis has to be performed for the 90 θθ 2 θ 3 allowed couplings, see subsection 2.3. The result is that for the rigid The meaning of l i and its influence in the couplings are given in eqs. (12),(26). With deformations the symmetry of the Ω matrix is smaller, as explained above, and it turns out to be 30 different couplings The absolute and relative size of these 30 couplings obviously depend on the value of the deformation parameters, as reflected in eqs. (36-38).
Performing an analysis similar to the θθ 2 θ 3 case, we find out the number of inequivalent shifts for the θ 2 θ 2 θ 2 coupling. For the non-deformed case there are 8 different couplings, namely (48) For deformations the number is increased to 12 different couplings given by the following shifts (49) We have performed a similar analysis for all the Z n orbifolds, the results are in Appendix 1. In all cases we have checked by computer that the number of different Yukawa couplings is correct.

A comparative study of the [SO(4)] and [SU (4)] 2 Z 4 orbifolds
Although most of the aspects of orbifold Yukawa couplings have been adequately illustrated in the previous section by the Z 6 -I example, there are still some interesting features that can be exhibited in the framework of a Z 4 orbifold. In particular we will see the physical meaning of a (1,2) modulus (absent in the Z 6 -I case) and its effect in the Yukawa coupling values. Furthermore, the comparison of the Yukawa couplings of a Z 4 orbifold, formulated in an [SO(4)] 3 lattice, with those of a Z 4 orbifold, formulated in an [SU(4)] 2 lattice, will show us which properties of the couplings depend on the chosen lattice and which do not. Moreover, the [SO(4)] 3 case provides an example of a non-Coxeter orbifold. The twist of a Z 4 orbifold in an orthogonal complex basis has the form (see Table 1) Again, the lattice Λ can get deformations compatible with the twist θ. These degrees of freedom correspond to the Hermitian part of the five (1,1) moduli surviving compactification, N 11 , N 22 , N 33 , N 12 , N 21 with N ij = |i > R ⊗α −1 jL |0 > L , and the (1,2) modulus (Notice that no N ij moduli appeared in the Z 6 -I case.) Untwisted moduli can be easily expressed in terms of g mn , b mn ( m, n = 1,...,6), i.e. the internal metric and torsion respectively. It is easy to check, however, that N 33 contains only g mn degrees of freedom, more precisely (g 55 − g 66 ) and g 56 . Therefore both Re(N 33 ) and Im(N 33 ) correspond to deformation parameters. In order to see what these parameters are, let us choose first an [SO(4)] 3 root lattice, with basis (e 1 , ..., e 6 ), as a lattice on which the twist θ, see eq. (50), acts crystallographically as Then, as in subsection 2.1, P invariance impose the following relations where α ij = cos θ ij and e i e j = |e i ||e j | cos θ ij . Therefore we can take the seven deformation degrees of freedom as Now it is easy to see that the two deformation parameters coming from N 33 correspond to a variation of the relative size of |e 5 | and |e 6 | and to the θ 56 angle; thus allowing for a rhomboid-like lattice from the original third SO(4) sublattice. It is remarkable however that, as will be seen shortly, the deformation parameters coming from N 33 are not involved in the Yukawa couplings.
Let us briefly summarize the main results of the [SO(4)] 3 Z 4 orbifold. They have been obtained performing an analysis similar to that followed in the previous section for the Z 6 -I one. There are 16 fixed points under θ in this orbifold, given by where g 1 = (0, 0) , g 1 = (0, 1 2 ). Under θ each fixed point is associated with a conjugation class in a one-to-one correspondence. Under θ 2 there are 16 fixed tori that are the product of 16 fixed points in the sublattice (e 1 , e 2 , e 3 , e 4 ) by the 2-torus defined by the sublattice (e 5 , e 6 ) (with or without deformations). Six of these 16 fixed points are connected to the others through θ rotations. The fixed points are (in the first two SO(4)'s) with g 1 . And we can see that 2 , g 2 , g 2 .
Note that in the two first SO(4)'s g 2 and g 2 are fixed points under θ while θg 2 . Consequently there are 10 θ 2 conjugation classes and, as was explained in subsection 2.3, only symmetric combinations of fixed points (for the conjugation classes with more than one fixed point) take part in the Yukawa couplings. For this orbifold all the twisted couplings are of the θθθ 2 type and the selection rule reads where f 3 is the θ 2 fixed point. Denoting the fixed points by see eqs. (54-55), the selection rule is simply The number of allowed couplings is 160. It is clear now that the third SO(4) lattice always enters in the couplings as the fixed torus associated with the θ 2 field. Then the coupling in this invariant plane is of the untwisted type and, consequently, the deformation parameters for the third SO(4) sublattice (i.e. R 5 , R 6 , α 56 ; see eq. (53)) do not affect the value of the coupling. Two of these parameters are precisely those coming from N 33 . Remarkably enough we have checked that this is a general property for all the orbifolds: (1,2) moduli are not involved in the expressions of the Yukawa couplings. It looks as though there is a selection rule (unknown to us) forbidding this kind of dependences.
For the case where f 3 is also a θ fixed point, the value of the coupling in the 2-3 picture is where V ⊥ is the volume of the unit cell of the first two SO(4)×SO(4) sublattice orthogonal to the invariant plane. If f 3 is not fixed by θ, see eq. (60), the result is exactly the same but multiplying C θθθ 2 by √ 2. Clearly the number of effective deformation parameters is 4. The number of different Yukawa couplings, from the 160 allowed ones, is 6 (without deformations), corresponding to 2 ⊗ g 1 , g 1 ⊗ g 1 , 2 .
The 7 deformation parameters coming from (N 11 , N 22 , N 33 , N 12 , N 21 , N 33 ) are where ( Notice that when deformations are included, (ẽ 1 ,ẽ 2 ,ẽ 4 ,ẽ 5 ) remain orthogonal to (ẽ 3 ,ẽ 6 ). Actually, (ẽ 1 ,ẽ 2 ,ẽ 3 ,ẽ 4 ,ẽ 5 ,ẽ 6 ) generate an [SO(4)] 3 sublattice of the [SU(4)] 2 lattice but they are not a basis of the whole lattice. Anyway theẽ i will be of help below. The number of fixed points is the same in both cases. For the case at hand, [SU(4)] 2 , there are 16 fixed points under θ which can be expressed as with g 1 = ( 3 4 , 1 2 , 1 4 ). Under θ 2 there is a fixed torus generated by (e 1 + e 3 ) and (e 4 + e 6 ). Then we can form 16 fixed tori as products of this fixed torus by the following 16 θ 2 fixed points (six of them connected to the others by θ rotations) 2 ⊗ g 2 , g 2 ⊗ g 2 , g where g Note that g 2 and g 2 are fixed under θ but g 2 and g 2 are connected by a θ rotation, θg 2 . As in the [SO(4)] 3 case there are 10 conjugation classes. The space group selection rule also has the same form Denoting the fixed points by eq. (69) can be expressed as (Note that in spite of eqs. (69-71) being formally identical with (56-58) the meaning of the vectors implicitely involved is quite different.) The number of allowed couplings is again the same, 160. The Yukawa coupling, if f 3 is fixed by θ, is given by As usual, the arrows denote components in the lattice basis (e 1 , ...e 6 ). The subscript ⊥ means that only v shifts orthogonal to the invariant plane (defined by (e 1 + e 3 ) and (e 4 + e 6 )) have to be considered. If f 3 is not fixed by θ the previous expression has to be multiplied by a √ 2 factor.N andM are given bȳ (73) where V ⊥ is the volume of the sublattice orthogonal to the invariant plane (see below). By addition of lattice vectors we can always choose f 2 and f 3 in (72) such that f 2 − f 3 is orthogonal to the invariant plane. Then f 2 − f 3 can be expressed in the "basis" (66) as We can check that x i = 0, 1 2 (up to lattice vectors) for all the choices of f 2 , f 3 . However it is amusing to see that many of the possibilities are in fact equivalent. Consider, for definiteness, the case f 2 − f 3 = 0, i.e. x i = 0 in (74). Now we can add to f 2 − f 3 any shift contained in the invariant plane, Demanding v = f 2 − f 3 + Λ to be orthogonal to the invariant plane we find constraints for α and β. A shift i a i e i is orthogonal to the invariant plane if it satisfies the condition a 1 − a 2 + a 3 = 0 and a 4 − a 5 + a 6 = 0 (with or without deformations). Then v = α(e 1 + e 3 ) + β(e 4 + e 6 ) + is orthogonal if (α, β) = (0, 0), (0, up to lattice vectors. Then v can be expressed as v = (n 1 + α)ẽ 1 + (n 3 + α)ẽ 2 + (n 4 + β)ẽ 4 + (n 6 + β)ẽ 5 .
Therefore we have to sum up four possibilities for (x 1 , x 2 , x 4 , x 5 ), namely This is characteristic of the lattices that cannot be decomposed as the direct product of an invariant sublattice times an orthogonal sublattice. In particular it did not happen in the [SO(4)] 3 lattice. In order to write the coupling we have to add to each case in (79) lattice vectors orthogonal to the invariant plane, i.e. of the form as is reflected in (78). Now we can express the coupling (72), which contained a 6 × 6M matrix, as a sum of four ϑ functions defined in the four-dimensional lattice (ẽ 1 ,ẽ 2 ,ẽ 4 ,ẽ 5 ) where ṽ and f 23 are the components in (ẽ 1 ,ẽ 2 ,ẽ 4 ,ẽ 5 ) of v and (f 2 − f 3 ) respectively. f 23 runs over the possibilities displayed in (79), and Ω ′ is given by Taking into account that the coupling gets a factor √ 2 if f 3 is not fixed by θ, this gives 8 different Yukawa couplings when deformations are considered, and 6 without deformations (the two first possibilities in (83) are equal). This differs from the [SO(4)] 3 case, where there were 10 and 6 respectively. Note that the matrix Ω ′ , eq. (82), appearing in the coupling is formally identical to that of [SO(4)] 3 , eq. (60). However, as we have seen, the structure of possible shifts is very different. In any case the number of effective deformation parameters is the same for both cases.

Conclusions
We have calculated the complete twisted Yukawa couplings for all the Z n orbifold constructions in the most general case, i.e. when deformations of the compactified space are considered. This includes a certain number of tasks. Namely, determination of the allowed couplings, calculation of the explicit dependence of the Yukawa couplings values on the moduli expectation values (i.e. the parameters determining the size and shape of the compactified space), etc. Some progress in this direction has recently been made but without arriving at such explicit expressions as those given in this paper. This is an essential ingredient in order to relate theory and observation. In particular it allows a counting of the different Yukawa couplings for each orbifold (with and without deformations), which is crucial to determine the phenomenological viability of the different schemes, since it is directly related to the fermion mass hierarchy. In this sense some orbifolds (e.g. Z 3 , Z 4 , Z 6 -I, Z 8 -I, Z 12 -I) have much better phenomenological prospects than others (e.g. Z 7 , Z 6 -II, Z 8 -II, Z 12 -II). The results for the whole set of Coxeter orbifolds are summarized in Table 1. Other facts concerning the phenomenological profile of Z n orbifolds are also discussed, e.g. the existence of non-diagonal entries in the fermion mass matrices, which is related to a non-trivial structure of the Kobayashi-Maskawa matrix. In this sense non-prime orbifolds are favoured over prime ones which do not have off-diagonal entries in the mass matrices at this fundamental level.
The results of this paper give the precise form in which moduli fields are coupled to twisted matter. This is essential in order to study in detail other important issues. Namely, the supersymmetry breaking mechanism by gaugino condensation (in which the moduli develop an additional non-perturbative superpotential), and cosmological implications (note that the moduli are also coupled to gravity in a Jordan-Brans-Dicke-like way). The level of explicitness given in the paper is also necessary for more theoretical matters (e.g. the study of the transformation properties of the Yukawa couplings under target-space modular transformations like R → 1/R ). Concerning the last aspect we have found some appealing results, such as the fact that (1,2) moduli never appear in the expressions of the Yukawa couplings. Likewise, (1,1) moduli associated with fixed tori which are involved in the Yukawa coupling, do not affect the value of the coupling. It is worth noticing that the above mentioned moduli are precisely the only ones which contribute to the string loop corrections to gauge coupling constants [22].

APPENDIX 1
We follow a notation as compact as possible. The precise meaning of all the concepts appearing here is explained in detail in the text for the Z 6 -I and Z 4 examples.
Deformation parameters Relations Degrees of freedom (9) R i = |e i |, α i,j , α i,j+1 , i, j = 1, 3, 5 i < j Lattice basis (e i ) in terms of orthogonal basis (ẽ i ) Not necessary in this case.

Number of conjugation classes: 8
Coupling θθ 2 θ 3 Selection rule 0, 1, ..., 5 , j 1 , j 3 = 0, 1 , i 2 = 0, 1, ..., 8 , i 3 = 0, 1, ..., 7 , α, β, γ, δ ∈ R the selection rule reads Number of allowed couplings: 48 Expression of the coupling where l i is the number of elements in the f i conjugation class and (f 3 −f 2 +Λ) ⊥ denotes elements orthogonal to the two invariant planes with n i 1 , n i 2 ∈ Z. The coupling takes the final form with with V ⊥ the volume of the unit cell generated by { 1 2 e 1 +e 2 +e 3 + 1 2 e 4 , e 1 +e 2 −e 4 −e 5 } Number of effective parameters: 1 Number of different couplings without deformations: 4 Number of different couplings with deformations: 4 Coupling θθθ 4 Selection rule the selection rule reads Number of allowed couplings: 72 Expression of the coupling where (f 3 − f 2 + Λ) ⊥ denotes that the coset elements must be orthogonal to the (e 1 + e 3 + e 5 , e 6 ) plane with n i 1 , n i 2 , n i 3 , n i 4 ∈ Z. Finally the coupling takes the form , V ⊥ is the unit cell volume of the sublattice orthogonal to the invariant plane Lattice basis (e i ) in terms of orthogonal basis (ẽ i ) Coupling θθ 2 θ 4 Selection rule the selection rule reads Number of allowed couplings: 49 Expression of the coupling Number of effective parameters: 3 Number of different couplings without deformations: 2 corresponding to the following f 23 shifts 1 , f 1 , f Lattice basis (e i ) in terms of orthogonal basis (ẽ i ) with α = 2π 8 and 1/2 ϕ 1 , ϕ 2 , ϕ 3 are free parameters.
Fixed points of θ (4) 2 = 1 2 (1, 1), g where (f 3 −f 2 +Λ) ⊥ denotes that only coset elements belonging to SO(9) lattice are considered; where V ⊥ is the volume of the SO(9) lattice Number of effective parameters: 2 Number of different couplings without deformations: 8 corresponding to the following f 23 shifts Number of different couplings with deformations: 9 corresponding to the following f 23 shifts Coupling θθ 2 θ 5 Selection rule the selection rule reads Number of allowed couplings: 40 Expression of the coupling with l 2 the number of elements in the f 2 conjugation class Number of effective parameters: 3 Number of different couplings without deformations: 8 corresponding to the following f 23 shifts 2 ⊗ĝ 2 ⊗ĝ (2) 2 , 2 ⊗ĝ Number of different couplings with deformations: 9 corresponding to the following f 23 shifts 2 ⊗ĝ