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COMPLETE STRUCTURE OF Zn YUKAWA COUPLINGS

arXiv:hep-th/9110060v1 21 Oct 1991CERN–TH.6194/91IEM–FT–38/91US–FT–7/91COMPLETE STRUCTURE OF Zn YUKAWA COUPLINGSJ.A. CASAS∗, F. GOMEZ† and C. MU˜NOZ∗∗∗Instituto de Estructura de la Materia (CSIC),Serrano 119, 28006–Madrid, Spain† Departamento de F´ısica de Part´ıculas,Universidad de Santiago, 15706–Santiago, Spain∗∗CERN, CH–1211 Geneva 23, SwitzerlandAbstractWe give the complete twisted Yukawa couplings for all the Zn orbifold constructions inthe most general case, i.e.

when orbifold deformations are considered. This includes acertain number of tasks.

Namely, determination of the allowed couplings, calculation ofthe 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.

Thefinal expressions are completely explicit, which allows a counting of the different Yukawacouplings for each orbifold (with and without deformations). This knowledge is crucial todetermine the phenomenological viability of the different schemes, since it is directlyrelated to the fermion mass hierarchy.Other facts concerning the phenomenologicalprofile of Zn orbifolds are also discussed, e.g.

the existence of non–diagonal entries inthe 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 fixedtori which are involved in the Yukawa coupling, do not affect the value of the coupling.CERN–TH.6194/91IEM–FT–38/91US–FT–7/91July 1991

1Introduction and brief reviewIn the last few years an enormous effort has been made in order to establish the connec-tion between string theories [1] (especially E8 × E8 heterotic string [2]) and low energyphysics. Different schemes for constructing classical string vacua have arisen during thistime.

Using these schemes it has been possible to build up four–dimensional strings thatresemble the Standard Model in many aspects, e.g. SU(3) × SU(2) × U(1)Y gauge groupand three generations of particles with the correct representations [3–6].

In spite of theseachievements there remain many pending questions. In particular there is a large numberof classical vacuum states, which reduces the predictive power of the theory.

At presentthere are no dynamical criteria to prefer a particular vacuum, so the best we can do is tostudy their phenomenological characteristics in order to select the viable vacua. In thisrespect, orbifold compactifications [7] have proved to be very interesting four–dimensionalstring 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 thephenomenological potential of orbifolds. The best example of this is the observed structureof fermion masses and mixing angles [9].Concerning the last point, a crucial ingredient in order to relate theory and obser-vation is the complete knowledge of the theoretical Yukawa couplings.

This knowledgeincludes a certain number of aspects:i) Physical states that enter the couplings.ii) Allowed couplings.iii) Numerical values of the Yukawa couplings and dependence of these values on thephysical parameters that define the string vacuum (e.g. the size of the compactifiedspace).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 thetheoretical Yukawa couplings).In this sense only for prime Abelian orbifolds, i.e. Z3 and Z7 are the Yukawa cou-plings completely known [10, 9, 11].

For the other orbifolds points i) and ii) have recentlybeen studied in ref.[12]. General expressions of Zn Yukawa couplings have been deter-mined in ref.[13].

However, although very useful, they are not explicit enough to lucidatepoints iii) and iv), specially when deformations of the compactified space are considered.1

Undoubtedly, a better knowledge of the Yukawa couplings is of utmost importance toselect or discard explicit string constructions with a highly non–trivial test. This is themain purpose of this paper, i.e.

to answer points i), ii), iii), iv) for all the Zn orbifolds.Besides the phenomenological motivation, there are strong theoretical reasons to com-pletely determine the Yukawa couplings. In particular it is the only way to know themoduli dependence of the matter Lagrangian and, in consequence, the superpotential.This allows the examination of the properties of the action under target-space modulartransformations (e.g.

R →1/R ) [14]. It is also necessary in order to discuss supersym-metry breaking dynamics [15] and cosmological implications (note that moduli play therole of Brans-Dicke fields in four dimensions) of these theories.Let us review briefly Zn orbifold constructions.

A Zn orbifold is constructed bydividing R6 by a six–dimensional lattice Λ modded by some Zn symmetry, called the pointgroup 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 absenceof tachyons restrict the number of possible point groups [7].

The complete list is givenin the first two columns of Table 1, where the so–called twist θ (i.e. the generator ofP) is represented in an orthogonal complex basis of the six–dimensional space.

Λ mustbe chosen so that θ acts crystallographically on it. If the realization of θ on the latticecoincides with the Coxeter element of a rank–six Lie algebra root lattice, the orbifoldis of the Coxeter type.

A list of Coxeter orbifolds, taken from ref. [16], is given in thethird column of Table 1.

Some additional examples of Coxeter orbifolds can be foundin ref. [12].

The lattice of the Z8–II orbifold, SO(5) × SO(8), corresponds in fact to ageneralized Coxeter orbifold where the Coxeter element has been multiplied by an outerautomorphism. Non–Coxeter orbifolds can also be constructed.

An example of a non–Coxeter orbifold (the Z4 one with [SO(4)]3 lattice) is studied in Section 3. The totalnumber of possible lattices associated with each Zn orbifold can be found in ref.

[17]. Aswill become clear in the text, some properties of the Yukawa couplings for a particularZn orbifold depend on the lattice chosen, while others do not.It is important to point out that in a string orbifold construction the lattice Λcan get deformations compatible with the point group [9, 11].

These degrees of freedomcorrespond to the untwisted moduli surviving compactification. Deformations play animportant role on the value of the Yukawa couplings.We are interested in the couplings between twisted fields (the untwisted sector hasalready been studied [10] and is physically less interesting [9]).

As we will see, thesecouplings present a very rich range, which is extremely attractive as the geometricalorigin of the observed variety of fermion masses [10, 18, 9]. A twisted string satisfies2

x(σ = 2π) = gx(σ = 0) as the boundary condition, where g is an element of the spacegroup whose point group component is non–trivial. Owing to the boundary conditiona twisted string is attached to a fixed point of g. Physical twisted fields are associatedwith conjugation classes of the space group rather than with particular elements [7].

Forexample {hgh−1, h ∈S} is the conjugation class of g. For prime orbifolds conjugationclasses are in one–to–one correspondence with the fixed points of P.For non–primeorbifolds the situation is a bit more involved since two different fixed points under θn maybe connected by θm, m < n. Then both of them correspond to the same conjugationclass.The paper is organized as follows. In Section 2 we expound the various steps neces-sary to obtain the final spectrum of Yukawa couplings for each orbifold, taking the Z6–Ias a guide example.

These steps include: determination of the geometrical structure ofthe orbifold and deformation parameters, physical states, calculation of explicit Yukawacouplings, and counting of different couplings. Section 3 is devoted to a comparative studyof the [SO(4)]3 and [SU(4)]2 Z4 orbifolds.

This shows which properties of the couplingsdepend on the lattice chosen and which do not. Furthermore, the [SO(4)]3 case providesan example of a non–Coxeter orbifold.

Besides this, the Z4 orbifold allows one to see thephysical meaning of a (1,2) modulus (absent in the Z6–I case) and its effect in the Yukawacoupling values. The complete results for the rest of Zn orbifolds are given in Appendix1 and summarized in Table 1.2The methodSeveral steps are necessary in order to obtain the final spectrum of Yukawa couplings foreach orbifold.

We explain these steps in the present section, taking the Z6–I orbifold asan illustrative example. The reason for this choice is that prime orbifolds (Z3 and Z7)have already been studied in depth in references [10, 9, 11].

A complete exposition of themethod followed here can be found in ref. [11].

It is, however, convenient to discuss thepresent example in some detail, since non–prime orbifolds exhibit certain features whichare absent in the prime ones.2.1Geometrical structureThe twist θ of the Z6–I orbifold has the form (see Table 1)θ = diag(eiα, eiα, e−2iα),α = 2π6(1)3

in the complex orthogonal basis {(˜e1, ˜e2), (˜e3, ˜e4), (˜e5, ˜e6) }. Very often it is more suitableto work in the lattice basis {e1, ..., e6}, which in this case is simply a set of simple rootsof G22 × SU(3).

Then θ acts as the Coxeter elementθe1 = −e1 −e2,θe2 = 3e1 + 2e2,θe3 = −e3 −e4,θe4 = 3e3 + 2e4,θe5 = e6,θe6 = −e5 −e6. (2)Note that we have labelled as e5, e6 the simple roots of SU(3).

As mentioned above Λcan get deformations compatible with the point group. These degrees of freedom corre-spond to the Hermitian part of the five untwisted (1,1) moduli surviving compactificationN1¯1, N2¯2, N3¯3, N1¯2, N2¯1, where Ni¯j = |i >R ⊗α−1¯jL|0 >L;|0 >L (|0 >R) being the left(right) vacuum, αL 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:|e2| =√3 |e1|,|e4| =√3 |e3|,|e5| = |e6|,α12 = −√3/2,α34 = −√3/2,α56 = −1/2,α24 = α13,α23 = −√3α13 −α14,αij = 0i = 1, 2, 3, 4 j = 5, 6(3)where αij = cos θij and eiej = |ei||ej| cos θij. Therefore we can take the 5 deformationdegrees of freedomRi = |ei|;i = 1, 3, 5α13, α14(4)Ri are global scales of the three sublattices (G2, G2, SU(3)), and for α13 = α14 = 0 werecover the rigid G2 × G2 × SU(3) lattice.The connection between the lattice basis(e1, ..., e6) and the orthogonal basis (˜e1, ..., ˜e6) in which θ takes the form (1) is given byei=Ai [cos(ϕi1)˜e1 + sin(ϕi1)˜e2] + Bi [cos(ϕi2)˜e3 + sin(ϕi2)˜e4]ei+1=−√3[Ai (sin(π3 −ϕi1)˜e1 + cos(π3 −ϕi1)˜e2) + Bi (sin(π3 −ϕi2)˜e3 + cos(π3 −ϕi2)˜e4)]ej=R5 [cos((j −5)2π3 + φ)˜e5 + sin((j −5)2π3 + φ)˜e6] ; i = 1, 3 j = 5, 6(5)where ϕ11, ϕ31, ϕ12, ϕ32, φ are arbitrary angles that are irrelevant for our results andA1 = R11√2[∆1 ± ∆3]1/2B1 = R11√2[∆2 ∓∆3]1/2A3 = k2R3√2[∆1 ± ∆3]−1/2B3 = k1R3√2[∆2 ∓∆3]−1/24

withki = 2R1R3(−1)i[α13 sin( π3 + ϕ1i −ϕ3i ) + α14 cos(ϕ1i −ϕ3i )][sin(ϕ12 −ϕ32 −ϕ11 + ϕ31)]−1 i = 1, 2∆1 = 1 + k22 −k21∆2 = 1 −k22 + k21∆3 = [(1 −k21 −k22)2 −4k21k22]1/2.Let us consider now the fixed points under the action of the point group. fn is afixed point under θn if it satisfies fn = θnfn + u, u ∈Λ.

As Z6–I is a non–prime orbifold,a point fixed by θn (n ̸= 1) may be not fixed by θm (m ̸= n). Consequently, the fixedpoints under θ, θ2 and θ3 must be considered separately (θ4, θ5 are simply the antitwistsof θ2 and θ).

It is easy to check from (2) that there are three different fixed points underθ. Working in the lattice basis their coordinates (up to lattice translations) aref (1)1=g(0)1⊗g(0)1⊗ˆg(0)1 ,f (2)1=g(0)1⊗g(0)1⊗ˆg(1)1 ,f (3)1=g(0)1⊗g(0)1⊗ˆg(2)1(6)withg(0)1= (0, 0),ˆg(0)1= (0, 0),ˆg(1)1= ( 13, 23),ˆg(2)1= ( 23, 13).

(7)Similarly under θ2 there are 27 fixed points. 12 of them are connected to the others by aθ rotationf (1)2= g(0)2⊗g(0)2⊗ˆg(0)2 ,f (2)2= g(0)2⊗g(0)2⊗ˆg(1)2 ,f (3)2= g(0)2⊗g(0)2⊗ˆg(2)2 ,f (4)2= g(0)2⊗g(1)2⊗ˆg(0)2∼g(0)2⊗g(2)2⊗ˆg(0)2 ,f (10)2= g(1)2⊗g(1)2⊗ˆg(0)2∼g(2)2⊗g(2)2⊗ˆg(0)2 ,f (5)2= g(0)2⊗g(1)2⊗ˆg(1)2∼g(0)2⊗g(2)2⊗ˆg(1)2 ,f (11)2= g(1)2⊗g(1)2⊗ˆg(1)2∼g(2)2⊗g(2)2⊗ˆg(1)2 ,f (6)2= g(0)2⊗g(1)2⊗ˆg(2)2∼g(0)2⊗g(2)2⊗ˆg(2)2 ,f (12)2= g(1)2⊗g(1)2⊗ˆg(2)2∼g(2)2⊗g(2)2⊗ˆg(2)2 ,f (7)2= g(1)2⊗g(0)2⊗ˆg(0)2∼g(2)2⊗g(0)2⊗ˆg(0)2 ,f (13)2= g(1)2⊗g(2)2⊗ˆg(0)2∼g(2)2⊗g(1)2⊗ˆg(0)2 ,f (8)2= g(1)2⊗g(0)2⊗ˆg(1)2∼g(2)2⊗g(0)2⊗ˆg(1)2 ,f (14)2= g(1)2⊗g(2)2⊗ˆg(1)2∼g(2)2⊗g(1)2⊗ˆg(1)2 ,f (9)2= g(1)2⊗g(0)2⊗ˆg(2)2∼g(2)2⊗g(0)2⊗ˆg(2)2 ,f (15)2= g(1)2⊗g(2)2⊗ˆg(2)2∼g(2)2⊗g(1)2⊗ˆg(2)2(8)withg(0)2= (0, 0),g(1)2= (0, 13),g(2)2= (0, 23),ˆg(0)2= (0, 0),ˆg(1)2= ( 13, 23),ˆg(2)2= ( 23, 13).

(9)Consequently, there are 15 conjugation classes under θ2. Finally, under θ3, there are 16fixed tori that are the direct product of 16 fixed points in the sublattice (e1, ..., e4) by5

the 2–torus defined by the sublattice (e5, e6). (Notice that θ3 is trivial in the SU(3) rootlattice.) 15 of these fixed points are connected between themselves by θ rotationsf (1)3= g(0)3⊗g(0)3 ,f (2)3= g(1)3⊗g(0)3∼g(2)3⊗g(0)3∼g(3)3⊗g(0)3 ,f (3)3= g(0)3⊗g(1)3∼g(0)3⊗g(2)3∼g(0)3⊗g(3)3 ,f (4)3= g(1)3⊗g(1)3∼g(2)3⊗g(2)3∼g(3)3⊗g(3)3 ,f (5)3= g(2)3⊗g(1)3∼g(3)3⊗g(2)3∼g(1)3⊗g(3)3 ,f (6)3= g(3)3⊗g(1)3∼g(1)3⊗g(2)3∼g(2)3⊗g(3)3(10)withg(0)3= (0, 0),g(1)3= (0, 12),g(2)3= ( 12, 0),g(3)3= ( 12, 12).

(11)The direct product under the (e5, e6) torus has been understood. Consequently, thereare 6 conjugation classes under θ3.

A similar analysis for other orbifolds can be found inAppendix 1.2.2Physical statesThe next step is to determine which are the physical states. These must be invariant undera total Z6 transformation which, besides the twist θ in the 6–dimensional space, includes aZ6 gauge transformation, usually represented by a shift V I (the so–called embedding) onΛE8×E8 and a shift vt on ΛSO(10).

Accordingly one has to construct for each θk sector linearcombinations of states, associated with θk fixed points, that are eigenstates of θ [19, 12].If fk is a fixed point of θk such that l is the smallest number giving θlfk = fk + u, u ∈Λ,then the eigenstates of θ have the form|fk > + e−iγ|θfk > +... + e−i(l−1)γ|θl−1fk >γ =2πpl,p = 1, 2, ..., l(12)with eigenvalue e−iγ (obviously, if k = 1, then l = 1 and eq. (12) is trivial).

Under a Z6transformation the complete state gets a phase [19, 12]∆(k, eiγ)=exp(2πi[−12k(XI(V I)2 −Xt(vt)2)++XI(P I + kV I)V I −Xt(pt + kvt)vt])exp{iγ},(13)where pt is the NSR part momentum put on the SO(8) weight lattice and P I is thetransverse 8–dim. momentum (E8 × E8 root momentum) fulfilling the right–mover andleft–mover massless conditions respectively.

Then ∆(k, eiγ) = 1 for physical states.6

Let us apply this to the E8 × E8 heterotic string compactified on the Z6–I orbifoldwith V I = 16(1, 1, −2, 0, ..., 0)(0, ..., 0), i.e. the standard embedding.

The unbroken gaugegroup is (E6 × SU(2) × U(1)) × E8.In the θ sector there are three physical statestransforming as (27)’s of E6 corresponding to |f (1)1>, |f (2)1>, |f (3)1> respectively (see eq.(6)). In the θ2 sector we can construct 27 eigenstates of θ (see eq.

(8))|f (1)2>, |f (2)2>, |f (3)2>,n|f (j)2> +e−iγ|θf (j)2>oj=4,...,15 ,γ = π, 2π. (14)After some algebra, only the symmetric combinations survive (i.e.

∆(k, eiγ)=1 forthem), giving rise to 15 (27)’s under E6. Similarly in the θ3 sector we can construct 16eigenstates of θ (see eq.

(10))|f (1)3>,n|f (j)3> +e−iγ|θf (j)3> +e−i2γ|θ2f (j)3>oj=2,...,6 ,γ = 2π3 , 4π3 , 2π. (15)In this case there survive 6 (27)’s, corresponding to the symmetric combinations, and 5(27)’s, corresponding to the γ = 2π/3 combinations.

We have performed a similar analysisfor each orbifold.2.3Allowed Yukawa couplingsLet us now analyse the allowed Yukawa couplings between physical states. A twistedstring associated with a fixed point f and a rotation θj is closed due to the action ofg = (θj, (I −θj)f), so the corresponding conjugation class is given by {(θk, u)(θj, (I −θj)f)(θ−k, −θ−ku)}, with k ∈Z, u ∈Λ.

After some algebra, the general expression of theconjugation class of g isθj, (I −θj)h(f + Λ) ∪(θf + Λ) ∪.... ∪(θj−1f + Λ)i. (16)The set of translations (1 −θj) {S(θkf + Λ), k = 0, ..., j −1} is called the coset associatedwith θj and f (note that the cosets associated with f and θkf are obviously the same).

Fora trilinear coupling of twisted fields T1T2T3 to be allowed, the product of the respectiveconjugation classes should contain the identity.This implies, in particular, that theproduct of the three point group elements θj1θj2θj3 should be 1 (this is the so–calledpoint group selection rule). For the Z6–I orbifold this implies that only θθ2θ3, θ2θ2θ2 andθθθ4 couplings have to be considered.

A straightforward application of the H–momentumconservation [10, 20, 21] shows that the θθθ4 couplings are also forbidden. Furthermorefor the θθ2θ3 couplings we must require(I, 0)∈(θ, (I −θ)(f1 + Λ))θ2, (I −θ2) [(f2 + Λ) ∪(θf2 + Λ)]θ3, (I −θ3)h(f3 + Λ) ∪(θf3 + Λ) ∪(θ2f3 + Λ)i,(17)7

which leads to the so–called space group selection rule for the coupling θθ2θ3 in the Z6–Iorbifoldf1 + (I + θ)f2 −(I + θ + θ2)f3 ∈Λ. (18)It should be noticed that if the space group selection rule (18) is satisfied for three fixedpoints f1, f2, f3, then it is also satisfied for θk1f1, θk2f2, θk3f3, and, consequently, for allthe physical combinationsPk1 e−ik1γ1|θk1f1 >,Pk2 e−ik2γ2|θk2f2 >,Pk3 e−ik3γ3|θk3f3 >,see eq.

(12). For the case at hand, i.e.

the Z6–I orbifold, one can consider 270 kinds ofcouplings f (j1)1f (j2)2f (j3)3of the θθ2θ3 type, from which only 90 are allowed: those for whichthe e5, e6 components (i.e. the SU(3) sublattice projection) of f1 −f2 are vanishing.

So,if we write the fixed pointsf1=g(0)1⊗g(0)1⊗ˆg(k1)1f2=g(i2)2⊗g(j2)2⊗ˆg(k2)2f3=g(i3)3⊗g(j3)3⊗[α(e5) + β(e6)]k1, k2, i2, j2 = 0, 1, 2,i3, j3 = 0, 1, 2, 3,α, β ∈R. (19)the selection rule isk1=k2.

(20)At this point it is important to stress the following fact: for two given f1, f2, the thirdfixed point f3 (corresponding to θ3) is not determined uniquely 1. We say that the spacegroup selection rule is not diagonal.

From the physical point of view this is extremelyimportant, since it allows for non–diagonal fermion mass matrices and, hence, a non–trivial Kobayashi–Maskawa matrix. This feature is absent for prime orbifolds.

On theother hand the space selection rule for the θ2θ2θ2 couplings simply readsf1 + f2 + f3 ∈Λ(21)which is diagonal. Note, however, that in this case the selection rule can be satisfied bysome representatives of the conjugation classes and not by others.

In this case there are3375 couplings to consider, from which only 369 are allowed. These arei1 + i2 + i3=0j1 + j2 + j3=0k1 + k2 + k3=0mod.

3. (22)denotingf1=g(i1)2⊗g(j1)2⊗ˆg(k1)2f2=g(i2)2⊗g(j2)2⊗ˆg(k2)2f3=g(i3)2⊗g(j3)2⊗ˆg(k3)2i1, i2, i3 = 0, 1, 2j1, j2, j3 = 0, 1, 2k1, k2, k3 = 0, 1, 2.

(23)1This also happens for given f1, f3 but not for f2, f3.8

Let us finally note that the product of the θ eigenvalues of the physical combinations offixed points involved in the coupling should be one, otherwise the coupling is vanishing.For example, in the Z6–I orbifold the following θθ2θ3 coupling|f (j1)1> (|f (j2)2> +|θf (j2)2>) (|f (j3)3> +e−2πi3 |θf (j3)3> +e−4πi3 |θ2f (j3)3>)(24)is forbidden on these grounds, since, due to θ invariance, it is equal to|f (j1)1> (|f (j2)2> +|θf (j2)2>) |f (j3)3>(1 + e−2πi3 + e−4πi3 ) = 0. (25)This result can also be obtained for the standard embedding case from gauge invariancesince the state considered in the θ3 sector corresponds to a (27), while the others are(27)’s.

In consequence all the couplings to be considered in the Z6–I involve symmetriccombinations of fixed points (i.e. θ eigenvalue = 1) exclusively.

We have performed theprevious analysis for all the Zn orbifolds. In all cases only couplings between symmetriccombinations of fixed points survive.

We do not really know what is the fundamentalprinciple behind this rule (if any), but it has important consequences. For instance, inref.

[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.2.4Calculation of Yukawa couplingsWe are interested in couplings of the type ψψφ (i.e. fermion–fermion–boson).

A trilinearstring scattering amplitude is given by the correlator < V1(z1)V2(z2)V3(z3) > of the vertexoperators creating the corresponding states. Complete expressions for the vertex operatorsof 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< σ1(z1)σ2(z2)σ3(z3) >, where σi represents a twist field creating the appropriate twistedground 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, sayσθj,f.

For example, for a physical state in the θj sector whose twist part is given byPk=0,...,l−1 e−ikγ|θkf > (the meaning of γ and l is given in eq. (12)) the corresponding twistfield is simplyPk=0,...,l−1 e−ikγσθj,θkf.

According to the result of the previous subsectiononly symmetric combinations (γ = 2π) are relevant for trilinear couplings, so the correlator< σ1(z1)σ2(z2)σ3(z3) > associated with a θj1θj2θj3 will take the form< σ1(z1) σ2(z2) σ3(z3) >=ql1l2l3−1l1−1Xk1=0l2−1Xk2=0l3−1Xk3=0< σθj1,θk1f(1)(z1) σθj2,θk2f(2)(z2) σθj3,θk3f(3)(z3) >, (26)9

where the square root is a normalization factor. The correlation functions on the right–hand side are evaluated following standard lines [10].

They are defined by< σθj1,θk1f(1)(z1)σθj2,θk2f(2)(z2)σθj3,θk3f(3)(z3) >=ZDX e−S σθj1,θk1f(1)(z1)σθj2,θk2f(2)(z2)σθj3,θk3f(3)(z3). (27)Owing to the Gaussian character of the action SS = 14πZd2z(∂X ¯∂¯X + ¯∂X∂¯X) ,(28)where X = X1 + iX2 and a sum over the three complex coordinates is understood, thescattering amplitude can be separated into a classical and a quantum part [10]Z = ZquXexp(−Scl).

(29)The quantum contribution represents a global factor for all the couplings with the sameθj1θj2θj3 pattern in a given orbifold; so the physical information mostly resides in theclassical contribution. Eventually, a total normalization factor, which depends on the sizeof the compactified space, has to be determined with the help of the four–point correlationfunction.

The final task lies in writing the couplings in terms of the physically significantparameters, i.e. those that parametrize the size and shape of the orbifold.Let us consider, for the sake of definiteness, a θθ2θ3 coupling in our guide example,the Z6–I orbifold.

The classical contribution, see eq. (29), to a < σθσθ2σθ−3 > correlatorhas 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 classicalaction, eq.

(28), for a three–point correlation on the sphere, isSicl=14π| sin(2πki/N)| | sin(3πki/N)|| sin(πki/N)||vi|2v∈(f2 −f3 + Λ)⊥(30)where i = 1, 2, 3 denotes the corresponding complex coordinate (and thus the projectionover the associated z–plane), the fields Xi are twisted by exp(2πki/N) (k1 = 1, k2 =1, k3 = 2 and N = 6 for the Z6–I orbifold) and (f2−f3+Λ)⊥selects only (f2−f3+Λ) shiftsthat are orthogonal to the invariant plane (this means that we can choose (f2 −f3)i=3 = 0and Λ = ΛG2×G2). 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 fori = 3 behaves much as an untwisted one.

In fact in the invariant plane the three stringsmust be attached to the same fixed point, i.e. (f1)i=3 = (f2)i=3 = (f3)i=3.

These facts are10

general for all the couplings, in any orbifold, when fixed tori are involved. Second, thev–coset in (30) does not depend on f1 since in the calculation of Xcl(z), z1 has been sentto infinity by using SL(2, C) invariance.

We call this the 2–3 picture (for more details seeref. [11]).

Equation (30) can be expressed in the 1–2 and 1–3 pictures as wellSicl(1−2)=116π| sin(πki/N)|| sin(2πki/N)| | sin(3πki/N)| |v(12)i|2v(12)∈(I −θ2)(f1 −f2 + Λ12)⊥,(31)Sicl(1−3)=116π| sin(πki/N)|| sin(2πki/N)| | sin(3πki/N)| |v(13)i|2v(13)∈(I −θ3)(f1 −f3 + Λ13)⊥,(32)whereΛ12 = (I + θ + θ2)Λ + ω , Λ13 = (I + θ)Λ + ω ,ω = (I + θ + θ2)f3 −(θ + θ2)f2 −f1. (33)We can check by using the space group selection rule (18) that there is a one–to–onecorrespondence between Scl(1−2), Scl(1−3), Scl(2−3).

The 2–3 picture, eq. (30) is the mostconvenient one since Λ12, Λ13 are subsets of the original lattice Λ.

Furthermore Scl(1−2)and Scl(1−3) depend on the three fixed points considered f1, f2, f3; while Scl(2−3) dependsonly on f2, f3. 2We can now write the complete form of the correlator using eqs.

(29,30)< σθσθ2σθ3 >= Nql2 l3Xu∈Λ⊥exp−12π sin(π3 )h(f23 + u)21 + (f23 + u)22i,(34)where (f23 + u)i is the i–plane projection of (f2 −f3 + u), Λ⊥= ΛG2×G2, and N is theproperly normalized quantum part [13]N =qV⊥12πΓ( 56)Γ( 23)Γ( 16)Γ( 13),(35)with V⊥the volume of the G2 × G2 unit cell.General expressions for the couplingssimilar to eq. (34) can be found in ref.

[13] for all the Zn orbifolds. We have performedthe calculation in all the cases, checking that the results of the mentioned reference arecorrect.2This difference can be understood recalling that for f2, f3 given, the space group selection rule (18)determines f1 uniquely, which does not hold for the other two possibilities.11

Expression (34) is not explicit enough for most purposes.For example it doesnot allow examination of the transformation properties of the Yukawa couplings undertarget–space modular transformations (e.g. R →1/R).

From a phenomenological pointof view eq. (34) does not exhibit the dependence of the value of the coupling on physicalquantities, i.e.

those that parametrize the size and shape of the compactified space. In facteq.

(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 towrite (f23 + u)i in terms of (e1, ..., e6), i.e. the lattice basis.

This can be done with thehelp of the results of subsection 2.1, see eq. (5).

Then the correlator (34) appears as anexplicit function of the deformation parameters of the compactified space. Substitutingthe resulting expression in (26) we obtain the final Yukawa coupling, which can be writenin a quite compact wayCθθ2θ3=Nql2 l3X⃗u∈Z4exp"−√34π ( ⃗f23 + ⃗u)⊤M( ⃗f23 + ⃗u)#=qV⊥ql2 l312πΓ( 56)Γ( 23)Γ( 16)Γ( 13) ϑ⃗f230[0, Ω](36)where ⃗f23 represents the first four components of (f2 −f3) (i.e.

those corresponding tothe G2 × G2 sublattice basis (e1, ..., e4) ), andϑ⃗f230[0, Ω] =X⃗u∈Z4exphiπ( ⃗f23 + ⃗u)⊤Ω( ⃗f23 + ⃗u)i,Ω= i√34π2 M(37)withΩ= i√34π2R21−32R21R1R3α13√3R1R3α14−32R213R21−R1R3(3α13 +√3α14)3R1R3α13R1R3α13−R1R3(3α13 +√3α14)R23−32R23√3R1R3α143R1R3α13−32R23R23(38)where the deformation parameters R2i , αij have been defined in eqs. (3, 4).It is worthwhile to have a look at eq.

(36) to realize which are the physical quantitieson which the value of the coupling depends. First Cθθ2θ3 depends on the relative posi-tions in the lattice of the relevant fixed points to which the physical fields are attached.This information is condensed in ⃗f23.

Second Cθθ2θ3 depends on the size and shape ofthe compactified space, which is reflected in the orbifold compactification parameters(R2i , α13, α14) appearing in Ωand (implicitely) in V⊥. Note that both pieces of informa-tion appear in a completely distinguishable way from each other in eq.

(36). Notice also12

that the deformation parameter R25 does not appear in Ω. This is due to the fact that R5parametrizes the size of the i = 3 sublattice, i.e.

the fixed torus, and we have learnt thatfor i = 3 the coupling is equivalent to an untwisted one. This is a general fact for all theorbifold couplings in which fixed tori are involved (e.g.

it does not occur for the θ2θ2θ2coupling of the Z6–I orbifold, see below). We say that R5 is not an effective deformationparameter for the θθ2θ3 couplings.

The number of effective deformation parameters (4in this case) is physically relevant since it is strongly related to the number of differentYukawa couplings and their corresponding sizes.For the other twisted coupling θ2θ2θ2 in the Z6–I orbifold, the expression of thecoupling can be calculated in the same way as in the θθ2θ3 case, and is given byCθ2θ2θ2=F(l1, l2, l3) NXv∈(f3−f2+Λ)exp[−√38π |v|2]=F(l1, l2, l3) NX⃗u∈Z6exp[−√38π ( ⃗f23 + ⃗u)⊤M( ⃗f23 + ⃗u)]=F(l1, l2, l3) Nϑ⃗f230[0, Ω],(39)where F = 1 for l1 = 1 or l2 = 1 or l3 = 1 and F =1√2 for l1 = l2 = l3 = 2. li is the numberof elements in the conjugation class associated with fi, see eq. (12).⃗f23 represents thecomponents of (f2 −f3) in the lattice basis (e1, ..., e6).

The global normalization factorand the Ωmatrix are given byN = √VΛ33/48π3Γ6( 23 )Γ3( 13 )Ω= i√38π2a−32abc00−32a3a−3b −c3b00b−3b −cd−32d00c3b−32d3d000000e−12e0000−12eea = R21b = R1R3α13c =√3R1R3α14d = R23e = R25. (40)Clearly the number of effective parameters is 5.We have performed a similar analysis for all the trilinear twisted couplings in all theZn orbifolds, giving the number of effective deformation parameters in each case.

Theresults are expounded in Appendix 1.13

2.5Accidental symmetries and the number of different cou-plingsWe 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 stringconstruction, since it is directly related to the possibility of reproducing the observedpattern of fermion masses and mixing angles. Unfortunately, this task is probably themost tedious part of the work presented here.

Again, we expound in some detail theanalysis for the two possible couplings in the Z6–I orbifold. Let us begin with the twistedcoupling θθ2θ3.The corresponding results for other orbifolds can be found in Appendix 1.The first point is that the Ωmatrix appearing in the Jacobi theta function of thecoupling, eqs.

(36–38), is universal for all the θθ2θ3 couplings.This means that thedifferences between the Yukawa couplings come exclusively from the sum in (f23 + u) inthe classical part of the correlation. Hence, two couplingsC ∼ϑ⃗f230[ 0, Ω]andC′ ∼ϑ⃗f23′0[ 0, Ω](41)will have the same value if there exists an integer unimodular transformation U (i.e.

U ∈GL(4, Z), | U |= ±1) such thatU⊤ΩU=Ω(42)U ⃗f23=⃗f23′ + ⃗v ,⃗v ∈Z4. (43)Then, if the previous equations are true for some U, there is a one–to–one correspondencebetween the terms of the series defining ϑ⃗f230, see eq.

(37), and those of ϑ⃗f23′0. Sowe have to look for U–matrices satisfying (43).

There is a set of U–matrices that alwaysfulfil (43). These are {I, −I} and {θn, n ∈Z}.

To check the latter, note that when thesum (37) is expressed in the complex orthogonal basis, the exponent takes a diagonal formXu∈Λ⊥exp { a1(f23 + u)21 + a2(f23 + u)22 }(44)as can be seen from (34). Then the terms multiplying the coefficients ai are unchangedunder θn twists, since these correspond to make rotations in each i–plane.

This argumentis always valid because the factorization (44) is a consequence of the fact that the classicalcontributions can be computed in each i–plane separately.In addition to the groupgenerated by {−I, θ} , there can be ”accidental U–symmetries” leaving Ωunchanged in14

eq. (43).

Some of these symmetries con be spontaneously broken when deformationsare taken into account. After inspection it turns out that, for the case at hand, theseaccidental symmetries are generated byU1 =eiα1U2 =1eiαU3 =0110(45)(when expressed in the complex orthogonal basis) plus products of these matrices by{−I, θ}.

U1, U2, U3 are broken when deformations are considered.Now, two Yukawa couplings C, C′ are equal (in the non–deformed case) if ⃗f23 and⃗f23′ are connected as in (43) by one of the U–matrices mentioned above. The analysis hasto be performed for the 90 θθ2θ3 allowed couplings, see subsection 2.3.

The result is thatfor the rigid G2 × G2 × SU(3) lattice ( i.e. R21 = R23 = R25 , α13 = α14 = 0 ) there are 10different couplings, corresponding to the following set of ⃗f23 shifts (in G2 × G2)l3 = 1 l2 = 1 : (0, 0) ⊗(0, 0),l3 = 3 l2 = 1 : (0, 0) ⊗(0, 12),(0, 12) ⊗(0, 12),l3 = 1 l2 = 2 : (0, 0) ⊗(0, 13),(0, 13) ⊗(0, 13),l3 = 3 l2 = 2(0, 0) ⊗(0, 16),(0, 16) ⊗(0, 16),(0, 12) ⊗(0, 13),(0, 13) ⊗(0, 16),(0, 12) ⊗(0, 16).

(46)The meaning of li and its influence in the couplings are given in eqs. (12),(26).

Withdeformations the symmetry of the Ωmatrix is smaller, as explained above, and it turnsout to be 30 different couplingsl3 = 1 l2 = 1 : (0, 0) ⊗(0, 0),l3 = 1 l2 = 2 : (0, 0) ⊗(0, 13),(0, 13) ⊗(0, 0),(0, 13) ⊗(0, 13),(0, 13) ⊗(0, 23),l3 = 3 l2 = 1(0, 0) ⊗(0, 12),(0, 12) ⊗(0, 0),(0, 12) ⊗(0, 12),(0, 12) ⊗( 12, 0),(0, 12) ⊗( 12, 12),l3 = 3 l2 = 2(0, 0) ⊗(0, 16)(0, 16) ⊗(0, 0),(0, 16) ⊗(0, 16),(0, 16) ⊗(0, 56),(0, 16) ⊗( 12, 13),(0, 16) ⊗( 12, 23),(0, 16) ⊗( 12, 16),(0, 16) ⊗( 12, 56),(0, 13) ⊗(0, 12),(0, 12) ⊗(0, 13),(0, 12) ⊗(0, 16),(0, 12) ⊗( 12, 13),(0, 12) ⊗( 12, 16),(0, 16) ⊗(0, 12),( 12, 13) ⊗(0, 12),( 12, 16) ⊗(0, 12),(0, 16) ⊗(0, 13),(0, 16) ⊗(0, 23),(0, 13) ⊗(0, 16),(0, 23) ⊗(0, 16)(47)The absolute and relative size of these 30 couplings obviously depend on the value of thedeformation parameters, as reflected in eqs. (36–38).Performing an analysis similar to the θθ2θ3 case, we find out the number of in-equivalent shifts for the θ2θ2θ2 coupling.

For the non–deformed case there are 8 differentcouplings, namely15

⃗f23 =l1 = 1 or l2 = 1 or l3 = 1g(0)2⊗g(0)2⊗ˆg(0)2 ,g(0)2⊗g(0)2⊗ˆg(1)2 ,g(0)2⊗g(1)2⊗ˆg(0)2 ,g(0)2⊗g(1)2⊗ˆg(1)2 ,g(1)2⊗g(1)2⊗ˆg(0)2 ,g(1)2⊗g(1)2⊗ˆg(1)2l1 = l2 = l3 = 2 :g(0)2⊗g(0)2⊗ˆg(0)2 ,g(0)2⊗g(0)2⊗ˆg(1)2. (48)For deformations the number is increased to 12 different couplings given by thefollowing shifts⃗f23 =l1 = 1 or l2 = 1 or l3 = 1g(0)2⊗g(0)2⊗ˆg(0)2 ,g(0)2⊗g(0)2⊗ˆg(1)2 ,g(0)2⊗g(1)2⊗ˆg(0)2 ,g(0)2⊗g(1)2⊗ˆg(1)2 ,g(1)2⊗g(1)2⊗ˆg(0)2 ,g(1)2⊗g(1)2⊗ˆg(1)2 ,g(1)2⊗g(0)2⊗ˆg(0)2 ,g(1)2⊗g(0)2⊗ˆg(1)2 ,g(1)2⊗g(2)2⊗ˆg(0)2 ,g(1)2⊗g(2)2⊗ˆg(1)2l1 = l2 = l3 = 2 :g(0)2⊗g(0)2⊗ˆg(0)2 ,g(0)2⊗g(0)2⊗ˆg(1)2.

(49)We have performed a similar analysis for all the Zn orbifolds, the results are in Appendix1. In all cases we have checked by computer that the number of different Yukawa couplingsis correct.3A comparative study of the [SO(4)]3 and [SU(4)]2 Z4orbifoldsAlthough most of the aspects of orbifold Yukawa couplings have been adequately illus-trated in the previous section by the Z6–I example, there are still some interesting featuresthat can be exhibited in the framework of a Z4 orbifold.

In particular we will see the phys-ical meaning of a (1,2) modulus (absent in the Z6–I case) and its effect in the Yukawacoupling values. Furthermore, the comparison of the Yukawa couplings of a Z4 orbifold,formulated in an [SO(4)]3 lattice, with those of a Z4 orbifold, formulated in an [SU(4)]2lattice, will show us which properties of the couplings depend on the chosen lattice andwhich do not.

Moreover, the [SO(4)]3 case provides an example of a non–Coxeter orbifold.The twist of a Z4 orbifold in an orthogonal complex basis has the form (see Table 1)θ = diag(eiα, eiα, e−2iα),α = 2π4 . (50)Again, the lattice Λ can get deformations compatible with the twist θ.

These degrees offreedom correspond to the Hermitian part of the five (1,1) moduli surviving compactifi-cation, N1¯1, N2¯2, N3¯3, N1¯2, N2¯1 with Ni¯j = |i >R ⊗α−1¯jL |0 >L, and the (1,2) modulus16

N33 = |3 >R ⊗α−13L|0 >L. (Notice that no Nij moduli appeared in the Z6–I case.) Un-twisted moduli can be easily expressed in terms of gmn, bmn ( m, n = 1,...,6), i.e.

theinternal metric and torsion respectively. It is easy to check, however, that N33 containsonly gmn degrees of freedom, more precisely (g55 −g66) and g56.

Therefore both Re(N33)and Im(N33) correspond to deformation parameters. In order to see what these parame-ters are, let us choose first an [SO(4)]3 root lattice, with basis (e1, ..., e6), as a lattice onwhich the twist θ, see eq.

(50), acts crystallographically asθe1 = e2,θe3 = e4,θe5 = −e5,θe2 = −e1,θe4 = −e3,θe6 = −e6. (51)Then, as in subsection 2.1, P invariance impose the following relations|e1| = |e2|,|e3| = |e4|,αij = 0i = 1, 2, 3, 4 j = 5, 6α14 = −α23,α12 = α34 = 0,α13 = α24(52)where αij = cos θij and eiej = |ei||ej| cos θij.

Therefore we can take the seven deformationdegrees of freedom asRi = |ei|i = 1, 3, 5, 6,α13, α14, α56. (53)Now it is easy to see that the two deformation parameters coming from N33 corre-spond to a variation of the relative size of |e5| and |e6| and to the θ56 angle; thus allowingfor a rhomboid-like lattice from the original third SO(4) sublattice.

It is remarkable how-ever that, as will be seen shortly, the deformation parameters coming from N33 are notinvolved in the Yukawa couplings.Let us briefly summarize the main results of the [SO(4)]3 Z4 orbifold. They havebeen obtained performing an analysis similar to that followed in the previous section forthe Z6–I one.

There are 16 fixed points under θ in this orbifold, given byf (ijk)1= g(i)1 ⊗g(j)1⊗g(k)1;i, j = 0, 2 ; k = 0, 1, 2, 3(54)whereg(0)1= (0, 0) ,g(1)1= ( 12, 0) ,g(2)1= ( 12, 12) ,g(3)1= (0, 12).Under θ each fixed point is associated with a conjugation class in a one–to–one corre-spondence. Under θ2 there are 16 fixed tori that are the product of 16 fixed points in the17

sublattice (e1, e2, e3, e4) by the 2–torus defined by the sublattice (e5, e6) (with or with-out deformations). Six of these 16 fixed points are connected to the others through θrotations.

The fixed points are (in the first two SO(4)’s)f (ij)2= g(i)2 ⊗g(j)2;i, j = 0, 1, 2, 3(55)with g(i)2 = g(i)1 . And we can see thatg(0)2⊗g(1)2∼g(0)2⊗g(3)2 ,g(1)2⊗g(0)2∼g(3)2⊗g(0)2 ,g(2)2⊗g(1)2∼g(2)2⊗g(3)2 ,g(1)2⊗g(2)2∼g(3)2⊗g(2)2 ,g(1)2⊗g(1)2∼g(3)2⊗g(3)2 ,g(1)2⊗g(3)2∼g(3)2⊗g(1)2 .Note that in the two first SO(4)’s g(0)2and g(2)2are fixed points under θ while θg(1)2→g(3)2 .Consequently there are 10 θ2 conjugation classes and, as was explained in subsection 2.3,only symmetric combinations of fixed points (for the conjugation classes with more thanone fixed point) take part in the Yukawa couplings.For this orbifold all the twistedcouplings are of the θθθ2 type and the selection rule readsf1 + f2 −(I + θ)f3 ∈Λ,(56)where f3 is the θ2 fixed point.

Denoting the fixed points byf1=g(i1)1⊗g(j1)1⊗g(k1)1f2=g(i2)1⊗g(j2)1⊗g(k2)1f3=g(i3)2⊗g(j3)2⊗[α(e5) + β(e6)]i1, i2, j1, j2 = 0, 2,k1, k2, i3, j3 = 0, 1, 2, 3α, β ∈R,(57)see eqs. (54–55), the selection rule is simplyi1 + i2 + 2i3 = 0j1 + j2 + 2j3 = 0k1 = k2mod.

4. (58)The number of allowed couplings is 160.

It is clear now that the third SO(4) latticealways enters in the couplings as the fixed torus associated with the θ2 field. Then thecoupling in this invariant plane is of the untwisted type and, consequently, the deformationparameters for the third SO(4) sublattice (i.e.

R5, R6, α56; see eq. (53)) do not affectthe value of the coupling.

Two of these parameters are precisely those coming from N33.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 asthough there is a selection rule (unknown to us) forbidding this kind of dependences.18

For the case where f3 is also a θ fixed point, the value of the coupling in the 2–3picture isCθθθ2=NXv∈(f2−f3+Λ)⊥exp[−14π(|v1|2 + |v2|2)]=NXv∈(f2−f3+Λ)⊥exp[−14π⃗v⊤M⃗v](59)=N ϑ⃗f230[0, Ω],where (f2 −f3 + Λ)⊥selects only (f2 −f3 + Λ) shifts that are orthogonal to the invariantsublattice, i.e. the third SO(4) lattice.

Thus, (f2 −f3 + Λ)⊥has non–zero componentsin the first two SO(4)’s only. Similarly ⃗f23 represents the four components of (f2 −f3) inthe basis (e1, ..., e4) of the first SO(4) lattices.

FinallyN = √V⊥12πΓ2( 34)Γ2( 14)M = (−4π2i)Ω=R210R1R3α13R1R3α140R21−R1R3α14R1R3α13R1R3α13−R1R3α14R230R1R3α14R1R3α130R23(60)where V⊥is the volume of the unit cell of the first two SO(4)×SO(4) sublattice orthogonalto the invariant plane.If f3 is not fixed by θ, see eq. (60), the result is exactly the same but multiplyingCθθθ2 by√2.

Clearly the number of effective deformation parameters is 4. The numberof different Yukawa couplings, from the 160 allowed ones, is 6 (without deformations),corresponding to⃗f23 =l3 = 1 :g(0)1⊗g(0)1 ,g(2)1⊗g(0)1 ,g(2)1⊗g(2)1 ,l3 = 2 :g(0)2⊗g(1)2 ,g(2)2⊗g(1)2 ,g(1)2⊗g(1)2(61)and 10 (when deformations are considered), namely⃗f23 =l3 = 1 :g(0)1⊗g(0)1 ,g(2)1⊗g(0)1 ,g(0)1⊗g(2)1 ,g(2)1⊗g(2)1 ,l3 = 2g(0)2⊗g(1)2 ,g(2)2⊗g(1)2 ,g(1)2⊗g(1)2 ,g(1)2⊗g(0)2 ,g(1)2⊗g(2)2 ,g(1)2⊗g(3)2 .

(62)19

We would like to compare all the previous results with those of the Z4 orbifold basedon a Coxeter twist acting on an [SU(4)]2 root lattice. This will illustrate what aspectsof the orbifold dynamics are independent of the chosen lattice and what aspects do not.Furthermore, for the [SU(4)]2 Z4 orbifold, the lattice cannot be decomposed as the directproduct of an invariant sublattice under θ2 times an orthogonal sublattice, as happenedin the [SO(4)]3 case.This peculiarity, which is shared by other orbifolds, introducessome additional complications which we would like to show.

The Coxeter element in the[SU(4)]2 root lattice is of the formθe1 = e2,θe2 = e3,θe3 = −e1 −e2 −e3,θe4 = e5,θe5 = e6,θe6 = −e4 −e5 −e6. (63)The 7 deformation parameters coming from (N1¯1, N2¯2, N3¯3, N1¯2, N2¯1, N33) areRi = |ei|, i = 1, 4α12, α14, α15, α16, α45,(64)where (e1, e2, e3) is the basis of the first SU(4), and (e4, e5, e6) the basis of the second one.Equation (64) should be compared with eq.

(53), i.e. its analogue in the [SO(4)]3 case.Clearly the geometrical interpretation of the deformation parameters is different for eachone.

Other parameters of the SU(4)2 lattice are related to the previous ones by|e1| = |e2| = |e3|,|e4| = |e5| = |e6|,α23 = α12,α34 = α16,α13 = −1 −2α12,α24 = α35 = −α14 −α15 −α16,α25 = α36 = α14,α56 = α45,α26 = α15,α46 = −1 −2α45. (65)It is important to point out that the [SU(4)]2 lattice cannot be consistently deformedinto an [SO(4)]3 one.

To see this, note that the invariant sublattice under the action ofthe Coxeter element (63) is generated by (e1 + e3) and (e4 + e6). If such a deformationexisted, these vectors could be identified with the basis of the invariant SO(4) sublatticein the [SO(4)]3 case.

Now, it can be shown that we cannot construct a basis of [SU(4)]2with (e1 + e3) , (e4 + e6) and four additional lattice vectors orthogonal to these (with orwithout deformations). In fact, it is easy to check that the [SU(4)]2 Coxeter element (63)has the same form as the twist θ of the [SO(4)]3 case, i.e.

eq. (51), when acting in thefollowing set of lattice vectors˜e1 = e1 + e2,˜e3 = e1 + e3,˜e5 = e5 + e6,˜e2 = e2 + e3,˜e4 = e4 + e5,˜e6 = e4 + e6.

(66)20

Notice that when deformations are included, (˜e1, ˜e2, ˜e4, ˜e5) remain orthogonal to (˜e3, ˜e6).Actually, (˜e1, ˜e2, ˜e3, ˜e4, ˜e5, ˜e6) generate an [SO(4)]3 sublattice of the [SU(4)]2 lattice butthey are not a basis of the whole lattice. Anyway the ˜ei will be of help below.

The numberof fixed points is the same in both cases. For the case at hand, [SU(4)]2, there are 16fixed points under θ which can be expressed asf (ij)1= g(i)1 ⊗g(j)1;i, j = 0, 1, 2, 3(67)withg(0)1= (0, 0, 0),g(1)1= ( 14, 12, 34),g(2)1= ( 12, 0, 12),g(3)1= ( 34, 12, 14).Under θ2 there is a fixed torus generated by (e1 + e3) and (e4 + e6).

Then we can form 16fixed tori as products of this fixed torus by the following 16 θ2 fixed points (six of themconnected to the others by θ rotations)g(0)2⊗g(0)2 ,g(2)2⊗g(2)2 ,g(0)2⊗g(2)2 ,g(2)2⊗g(0)2 ,g(0)2⊗g(1)2∼g(0)2⊗g(3)2 ,g(2)2⊗g(1)2∼g(2)2⊗g(3)2 ,g(1)2⊗g(0)2∼g(3)2⊗g(0)2 ,g(1)2⊗g(2)2∼g(3)2⊗g(2)2 ,g(1)2⊗g(1)2∼g(3)2⊗g(3)2 ,g(1)2⊗g(3)2∼g(3)2⊗g(1)2(68)whereg(0)2= (0, 0, 0), g(1)2= 12(1, 1, 0), g(2)2= 12(1, 0, 1), g(3)2= 12(0, 1, 1).Note that g(0)2and g(2)2are fixed under θ but g(1)2and g(3)2are connected by a θ rotation,θg(1)2→g(3)2 . As in the [SO(4)]3 case there are 10 conjugation classes.

The space groupselection rule also has the same formf1 + f2 −(I + θ)f3 ∈Λ. (69)Denoting the fixed points byf1 = g(i1)1⊗g(j1)1, f2 = g(i2)1⊗g(j2)1, f3 = [g(i3)2⊗g(j3)2] ⊗[α(e1 + e3) + β(e4 + e6)],i1, i2, i3, j1, j2, j3 = 0, 1, 2, 3, α, β ∈R(70)eq.

(69) can be expressed asi1 + i2 + 2i3=0j1 + j2 + 2j3=0mod. 4.

(71)(Note that in spite of eqs. (69–71) being formally identical with (56–58) the meaning ofthe vectors implicitely involved is quite different.) The number of allowed couplings isagain the same, 160.

The Yukawa coupling, if f3 is fixed by θ, is given by21

Cθθθ2=¯NXv∈(f2−f3+Λ)⊥exp[−14π⃗v⊤¯M⃗v]. (72)As usual, the arrows denote components in the lattice basis (e1, ...e6).The subscript⊥means that only v shifts orthogonal to the invariant plane (defined by (e1 + e3) and(e4 + e6)) have to be considered.

If f3 is not fixed by θ the previous expression has to bemultiplied by a√2 factor. ¯N and ¯M are given by¯N = √V⊥12πΓ2( 34)Γ2( 14)¯M =ab−a −2befgbab−e −f −gef−a −2bbag−e −f −gee−e −f −ggcd−c −2dfe−e −f −gdcdgfe−c −2ddca = R21b = R21α12c = R24d = R24α45e = R1R4α14f = R1R4α15g = R1R4α16.

(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 f2 and f3 in (72) such that f2 −f3 isorthogonal to the invariant plane. Then f2 −f3 can be expressed in the ”basis” (66) asf2 −f3 = x1˜e1 + x2˜e2 + x4˜e4 + x5˜e5 .

(74)We can check that xi = 0, 12 (up to lattice vectors) for all the choices of f2, f3. Howeverit is amusing to see that many of the possibilities are in fact equivalent.

Consider, fordefiniteness, the case f2 −f3 = 0, i.e. xi = 0 in (74).

Now we can add to f2 −f3 any shiftcontained in the invariant plane,f2 −f3 = α(e1 + e3) + β(e4 + e6) ,α, β ∈R. (75)Demanding v = f2 −f3 +Λ to be orthogonal to the invariant plane we find constraints forα and β.

A shift Pi aiei is orthogonal to the invariant plane if it satisfies the conditiona1 −a2 + a3 = 0 and a4 −a5 + a6 = 0 (with or without deformations). Thenv = α(e1 + e3) + β(e4 + e6) +6Xi=1niei ,ni ∈Z(76)22

is orthogonal if(α, β) = (0, 0), (0, 12), (12, 0), (12, 12),(77)up to lattice vectors. Then v can be expressed asv=(n1 + α)˜e1 + (n3 + α)˜e2 + (n4 + β)˜e4 + (n6 + β)˜e5.

(78)Therefore we have to sum up four possibilities for (x1, x2, x4, x5), namely(0, 0, 0, 0), (12, 12, 0, 0), (0, 0, 12, 12), (12, 12, 12, 12). (79)This is characteristic of the lattices that cannot be decomposed as the direct product ofan invariant sublattice times an orthogonal sublattice.

In particular it did not happen inthe [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 formu⊥= n1˜e1 + n2˜e2 + n4˜e4 + n5˜e5,(80)as is reflected in (78). Now we can express the coupling (72), which contained a 6 × 6 ¯Mmatrix, as a sum of four ϑ functions defined in the four-dimensional lattice (˜e1, ˜e2, ˜e4, ˜e5)Cθθθ2=¯NX˜f23X˜v∈( ˜f23+Λ⊥)exp[−14π⃗˜v⊤¯M′⃗˜v]=¯NX˜f23ϑ⃗˜f230[0, Ω′](81)where ⃗˜v and ⃗˜f23 are the components in (˜e1, ˜e2, ˜e4, ˜e5) of v and (f2 −f3) respectively.⃗˜f23runs over the possibilities displayed in (79), and Ω′ is given byΩ′ = i 14π2 ¯M′ = i 14π2¯a0¯b¯c0¯a−¯c¯b¯b−¯c¯d0¯c¯b0¯d¯a=2R21(1 + α12)¯b=R1R4(α14 −α16)¯c=R1R4(α14 + 2α15 + α16)¯d=2R24(1 + α45).

(82)Note that there are 4 effective deformation parameters, as in the [SO(4)]3 case. Besides(79), there are three other inequivalent possibilities for f3 −f2, namely{ (0, 0, 0, 12),(0, 0, 12, 0),( 12, 12, 0, 12),( 12, 12, 12, 0) },{ (0, 12, 0, 0),(0, 12, 12, 12),( 12, 0, 0, 0),( 12, 0, 12, 12) },{ (0, 12, 0, 12),(0, 12, 12, 0),( 12, 0, 0, 12),( 12, 0, 12, 0) }.

(83)23

Taking into account that the coupling gets a factor√2 if f3 is not fixed by θ, this gives 8different 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, wherethere were 10 and 6 respectively.

Note that the matrix Ω′, eq. (82), appearing in thecoupling 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 effectivedeformation parameters is the same for both cases.4ConclusionsWe have calculated the complete twisted Yukawa couplings for all the Zn orbifold con-structions in the most general case, i.e.

when deformations of the compactified space areconsidered. This includes a certain number of tasks.

Namely, determination of the al-lowed couplings, calculation of the explicit dependence of the Yukawa couplings values onthe moduli expectation values (i.e. the parameters determining the size and shape of thecompactified space), etc.

Some progress in this direction has recently been made but with-out arriving at such explicit expressions as those given in this paper. This is an essentialingredient in order to relate theory and observation.

In particular it allows a counting ofthe different Yukawa couplings for each orbifold (with and without deformations), whichis crucial to determine the phenomenological viability of the different schemes, since it isdirectly related to the fermion mass hierarchy. In this sense some orbifolds (e.g.

Z3, Z4,Z6–I, Z8–I, Z12–I) have much better phenomenological prospects than others (e.g. Z7,Z6–II, Z8-II, Z12–II).

The results for the whole set of Coxeter orbifolds are summarizedin Table 1. Other facts concerning the phenomenological profile of Zn orbifolds are alsodiscussed, e.g.

the existence of non–diagonal entries in the fermion mass matrices, whichis related to a non–trivial structure of the Kobayashi–Maskawa matrix.In this sensenon–prime orbifolds are favoured over prime ones which do not have off–diagonal entriesin the mass matrices at this fundamental level.The results of this paper give the precise form in which moduli fields are coupledto twisted matter. This is essential in order to study in detail other important issues.Namely, the supersymmetry breaking mechanism by gaugino condensation (in which themoduli develop an additional non–perturbative superpotential), and cosmological impli-cations (note that the moduli are also coupled to gravity in a Jordan-Brans-Dicke–likeway).

The level of explicitness given in the paper is also necessary for more theoreticalmatters (e.g. the study of the transformation properties of the Yukawa couplings undertarget–space modular transformations like R →1/R ).

Concerning the last aspect we24

have found some appealing results, such as the fact that (1,2) moduli never appear inthe expressions of the Yukawa couplings. Likewise, (1,1) moduli associated with fixedtori 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 whichcontribute to the string loop corrections to gauge coupling constants [22].ACKNOWLEDGEMENTSThe work of J.A.C.

was supported in part by the C.I.C.Y.T., Spain. The work of F.G. wassupported by an F.P.I.

grant, Spain. C.M.

is grateful to the members of the Departamentode F´ısica de Part´ıculas, Universidad de Santiago de Compostela, Spain, for their kindhospitality. F.G. thanks J. Mas for very useful discussions.25

APPENDIX 1We follow a notation as compact as possible. The precise meaning of all theconcepts appearing here is explained in detail in the text for the Z6–I and Z4examples.ORBIFOLD Z3Twistθ = diag(eiα, eiα, e−2iα),α = 2π3Lattice[SU(3)]3Coxeter elementθei = ei+1,θei+1 = −ei −ei+1,i = 1, 3, 5Deformation parametersRelations|ei|2 = |ei+1|2,αi,i+1 = −12,αi,j = αi+1,j+1,αi,j + αi,j+1 + αi+1,j = 0,i, j = 1, 3, 5i < jαij ≡cos(θij)Degrees of freedom (9)Ri = |ei|, αi,j, αi,j+1, i, j = 1, 3, 5 i < jLattice basis (ei) in terms of orthogonal basis (˜ei)Not necessary in this case.Fixed points of θ (27)f (ijk)1= g(i)1 ⊗g(j)1⊗g(k)1, i, j, k = 0, 1, 2,g(0)1= (0, 0) , g(1)1= ( 13, 23) , g(2)1= ( 23, 13)Coupling θθθSelection rulef1 + f2 + f3 ∈ΛDenotingf1 = g(i1)1⊗g(j1)1⊗g(k1)1f2 = g(i2)1⊗g(j2)1⊗g(k2)1f3 = g(i3)1⊗g(j3)1⊗g(k3)1i1, i2, i3 = 0, 1, 2j1, j2, j3 = 0, 1, 2k1, k2, k3 = 0, 1, 226

the selection rule readsi1 + i2 + i3=0j1 + j2 + j3=0k1 + k2 + k3=0mod. 3Number of allowed couplings: 729Expression of the couplingCθθθ=NXv∈(f3−f2+Λ)exp[−14π sin(2π3 ) |v|2]=NX⃗u∈Z6exp[−√38π ( ⃗f23 + ⃗u)⊤M( ⃗f23 + ⃗u)]=N ϑ⃗f230[0, Ω]withΩ=i√38π2M,N =qVΛ33/48π3Γ6( 23)Γ3( 13)Ω=i√38π2R21−R212R1R3α13R1R3α14R1R5α15R1R5α16−R212R21R1R3α23R1R3α13R1R5α25R1R5α15R1R3α13R1R3α23R23−R232R3R5α35R3R5α36R1R3α14R1R3α13−R232R23R3R5α45R3R5α35R1R5α15R1R5α25R3R5α35R3R5α45R25−R252R1R5α16R1R5α15R3R5α36R3R5α35−R252R25α23 = −(α13 + α14) , α25 = −(α15 + α16) , α45 = −(α35 + α36)Number of effective parameters: 9Number of different couplings without deformations: 4corresponding to the following ⃗f23 shifts⃗f23 = g(0)1⊗g(0)1⊗g(0)1 , g(1)1⊗g(0)1⊗g(0)1 , g(1)1⊗g(1)1⊗g(0)1 , g(1)1⊗g(1)1⊗g(1)1Number of different couplings with deformations: 14corresponding to the following ⃗f23 shifts⃗f23 =g(0)1⊗g(0)1⊗g(0)1 , g(1)1⊗g(0)1⊗g(0)1 , g(0)1⊗g(1)1⊗g(0)1 , g(0)1⊗g(0)1⊗g(1)1 ,g(1)1⊗g(1)1⊗g(0)1 , g(1)1⊗g(0)1⊗g(1)1 , g(0)1⊗g(1)1⊗g(1)1 , g(1)1⊗g(2)1⊗g(0)1 ,g(1)1⊗g(0)1⊗g(2)1 , g(0)1⊗g(1)1⊗g(2)1 , g(1)1⊗g(1)1⊗g(1)1 , g(1)1⊗g(1)1⊗g(2)1 ,g(1)1⊗g(2)1⊗g(2)1 , g(1)1⊗g(2)1⊗g(1)127

ORBIFOLD Z4See Section 3ORBIFOLD Z6–ISee Section 2ORBIFOLD Z6–IITwist θ = diag(eiα, e2iα, e−3iα),α = 2π6Lattice SU(6) ⊗SU(2)Coxeter elementθei = ei+1,i = 1, ..., 4,θe5 = −e1 −e2 −e3 −e4 −e5,θe6 = −e6Deformation parametersRelations|e1| = |e2| = |e3| = |e4| = |e5|,α12 = α23 = α34 = α45 = −12(1 + α14 + 2α15),α15 = α13 = α24 = α35,α16 = −α26 = α36 = −α46 = α56,α14 = α25αij ≡cos(θij)Degrees of freedom (5)R1 = |e1|,R6 = |e6|,α14,α15,α16Lattice basis (ei) in terms of orthogonal basis (˜ei)ei=Xj=1,3,5Aj[cos(ϕj + (i −1)bjα)˜ej + sin(ϕj + (i −1)bjα)˜ej+1]i = 1, ..., 5e6=R6[cos(ϕ3 + ∆)˜e5 + sin(ϕ3 + ∆)˜e6]with α = π3 and b1 = 1, b2 = 2, b3 = 3cos(∆) =√3α16√1+2α15 ,A1 = R1√6√1 −3α14 −4α15,A3 = R1√2√1 + α14,A5 = R1√3√1 + 2α15ϕ1, ϕ2, ϕ3 are free parameters.28

Fixed points of θ (12)f (ij)1= g(i)1 ⊗ˆg(j)1,i = 0, 1, ..., 5,j = 0, 1g(0)1= (0, 0, 0, 0, 0) ,g(1)1= 16(5, 4, 3, 2, 1) ,g(2)1= 16(4, 8, 6, 4, 2) ,g(3)1= 16(3, 6, 9, 6, 3) ,g(4)1= 16(2, 4, 6, 8, 4) ,g(5)1= 16(1, 2, 3, 4, 5) ,ˆg(0)1= (0) ,ˆg(1)1= ( 12)Fixed points of θ2 (9)Fixed torus: α(e1 + e3 + e5) + β(e6) ,α, β ∈Rf (i)2= g(i)2 ⊗[α(e1 + e3 + e5) + β(e6)] ,i = 0, 1, ..., 8, α, β ∈Rg(0)2= (0, 0, 0, 0, 0),˜g(1)2= 13(0, 1, 1, 2, 2),g(2)2= 13(0, 2, 2, 1, 1),g(3)2= 13(1, 0, −2, 0, 1),g(4)2= 13(1, 1, 2, 2, 0),g(5)2= 13(2, 2, 1, 1, 0),g(6)2= 13(−1, 0, 2, 0, −1),g(7)2= 13(2, −2, 0, 2, −2),g(8)2= 13(1, −1, 0, 1, −1)Note that θ : g(1)2→g(4)2 , g(2)2→g(5)2 , θ : g(3)2→g(6)2Number of conjugation classes: 6Fixed points of θ3 (14)Fixed torus: α(e1 + e4) + β(e2 + e5) , α, β ∈Rf (ij)3= g(i)3 ⊗ˆg(j)3⊗[α(e1 + e4) + β(e2 + e5)] ,i = 0, 1, ..., 7,j = 0, 1,α, β ∈Rg(0)3= (0, 0, 0, 0, 0) ,g(1)3= 12(1, 1, 1, 0, 0) ,g(2)3= 12(1, 1, 0, −1, −1) ,g(3)3= 12(0, 1, 1, 1, 0) ,g(4)3= 12(1, 0, 0, 1, 0) ,g(5)3= 12(0, 0, 1, 1, 1) ,g(6)3= 12(0, 1, 0, 0, 1) ,g(7)3= 12(1, 0, 1, 0, 1) ,ˆg(0)3= (0) ,ˆg(1)3= ( 12)Note that in the SU(6) lattice θ : g(1)3→g(3)3→g(5)3and θ : g(2)3→g(4)3→g(6)3Number of conjugation classes: 8Coupling θθ2θ3Selection rulef1 + (I + θ)f2 −(I + θ + θ2)f3 ∈Λ29

Denotingf1=g(i1)1⊗ˆg(j1)1f2=g(i2)2⊗[α(e1 + e3 + e5) + β(e6)]f3=g(i3)3⊗ˆg(j3)3⊗[γ(e1 + e4) + δ(e2 + e5)]i1 = 0, 1, ..., 5 ,j1, j3 = 0, 1 ,i2 = 0, 1, ..., 8 ,i3 = 0, 1, ..., 7 ,α, β, γ, δ ∈Rthe selection rule readsi1 + 2i2 + 3i3 = 0j1 = j3mod. 6Number of allowed couplings: 48Expression of the couplingCθθ2θ3=ql2 l3 NXv∈(f3−f2+Λ)⊥exp[−√34π |v1|2]where li is the number of elements in the fi conjugation class and (f3−f2+Λ)⊥denotes elements orthogonal to the two invariant planes(f3 −f2 + Λ)⊥=6Xi=1(hi1 + ni1)(12e1 + e2 + e3 + 12e4) + (hi2 + ni2)(−e1 −e2 + e4 + e5)where denoting ⃗¯f i23 ≡(hi1, hi2), ⃗¯f i23 is always⃗¯f 123 = (0, 0)⃗¯f 223 = (0, 12)⃗¯f 323 = ( 13, 13)⃗¯f 423 = ( 13, 56)⃗¯f 523 = ( 23, 23)⃗¯f 623 = ( 23, 56)with ni1, ni2 ∈Z.

The coupling takes the final formCθθ2θ3=ql2 l3 NXiXu∈Z2exp[−√34π ( ⃗¯f i23 + ⃗u)⊤M( ⃗¯f i23 + ⃗u)]=ql2 l3 NXiϑ⃗¯f i230[0, Ω]withΩ= i√34π2M = i√32π2R21(1 −3α14 −4α15)14−14−141,N =qV⊥12πvuutΓ( 56)Γ( 23)Γ( 13)Γ( 16)30

with V⊥the volume of the unit cell generated by { 12e1+e2+e3+ 12e4, e1+e2−e4−e5}Number of effective parameters: 1Number of different couplings without deformations: 4Number of different couplings with deformations: 4Coupling θθθ4Selection rulef1 + f2 −(I + θ)f3 ∈ΛDenotingf1=g(i1)1⊗ˆg(j1)1f2=g(i2)1⊗ˆg(j2)1f3=g(i3)2⊗[α(e1 + e3 + e5) + β(e6)]i1, i2 = 0, 1, ..., 5 ,j1, j2 = 0, 1 ,i3 = 0, 1, ..., 8 ,α, β ∈Rthe selection rule readsi1 + i2 + 4i3 = 0j1 = j2mod. 6Number of allowed couplings: 72Expression of the couplingCθθθ4=ql3 NXv∈(f3−f2+Λ)⊥exp[−√38π (|v1|2 + |v2|2)]where (f3 −f2 + Λ)⊥denotes that the coset elements must be orthogonal tothe (e1 + e3 + e5, e6) plane(f3−f2+Λ)⊥=3Xi=1[(hi1+ni1)(e1−e3)+(hi2+ni2)(e2+e3)+(hi3+ni3)(e3+e4)+(hi4+ni4)(e5−e3)]where denoting ⃗¯f i23 = (hi1, ..., hi4) there are two possible tri–plets of values for⃗¯f i23 depending on the values of f2, f3⃗¯f 123 = (0, 0, 0, 0)⃗¯f 223 = ( 13, 0, 0, 13)⃗¯f 323 = ( 23, 0, 0, 23)and⃗¯f 123 = ( 13, 23, 13, 23)⃗¯f 223 = ( 23, 23, 13, 0)⃗¯f 323 = (0, 23, 13, 13)31

with ni1, ni2, ni3, ni4 ∈Z. Finally the coupling takes the formCθθθ4=ql3 NXiX⃗u∈Z4exp[−√38π ( ⃗¯f i23 + ⃗u)⊤M( ⃗¯f i23 + ⃗u)]=ql3 NXiϑ⃗¯f i230[0, Ω]withN =qV⊥12πvuutΓ( 56)Γ( 23)Γ( 13)Γ( 16) ,Ω= i√38π2MV⊥is the unit cell volume of the sublattice orthogonal to the invariant planeΩ= i√38π22a−aa+c−2b2−a−abca+c−2b2a+c−2b2cb−a−aa+c−2b2−a2aa = R21(1 −α15)b = R21(1 −α14 −2α15)c = R21(α14 + α15)Number of effective parameters: 3Number of different couplings without deformations: 4Number of different couplings with deformations: 4ORBIFOLD Z7Twist θ = diag(eiα, e2iα, e−3iα),α = 2π7Lattice SU(7)Coxeter elementθei = ei+1,i = 1, ..., 5,θe6 = −e1 −e2 −e3 −e4 −e5 −e6Deformation parametersRelations|e1| = |e2| = |e3| = |e4| = |e5| = |e6|,α12 = α23 = α34 = α45 = α56,α13 = α24 = α35 = α46 = α16,α14 = α25 = α36 = α15 = α26 = −12 −α12 −α13αij ≡cos(θij)Degrees of freedom (3)R = |e1|,α12,α1332

Lattice basis (ei) in terms of orthogonal basis (˜ei)ei =Xj=1,3,5Rj[cos((i −1)bjα + ϕj)˜ej + sin((i −1)bjα + ϕj)˜ej+1]i = 1, ..., 6with α = 2π7 and b1 = 1, b3 = 2, b5 = 4R21 = R2[α12(α25 −α21) + α13(α25 −α23) + 12α25]R23 = R2[α12(α21 −α23) + α13(α21 −α25) + 12α21]R25 = R2[α12(α23 −α25) + α13(α23 −α21) + 12α23]α2i = 47[1 −cos(biα)] ,i = 1, 3, 5ϕ1, ϕ2, ϕ3 are free parameters.Fixed points of θ (7)f (0)1= (0, 0, 0, 0, 0, 0),f (1)1= 17(6, 5, 4, 3, 2, 1),f (2)1= 17(5, 3, 1, 6, 4, 2),f (3)1= 17(4, 1, 5, 2, 6, 3),f (4)1= 17(3, 6, 2, 5, 1, 4),f (5)1= 17(2, 4, 6, 1, 3, 5),f (6)1= 17(1, 2, 3, 4, 5, 6)Coupling θθ2θ4Selection rulef1 + 2f2 −3f3 ∈ΛDenotingf1 = f (i1)1f2 = f (i2)1f3 = f (i3)1i1, i2, i3 = 0, 1, ..., 6 ,the selection rule readsi1 + 2i2 −3i3 = 0mod. 7Number of allowed couplings: 49Expression of the couplingCθθ2θ4=NXv∈(f3−f2+Λ)exp"−14π sin(α) sin(2α) sin(3α) |v1|2sin2(3α) +|v2|2sin2(α) +|v3|2sin2(2α)!#=NX⃗u∈Z6exp[−14π ( ⃗f23 + ⃗u)⊤M( ⃗f23 + ⃗u)]=N ϑ⃗f230[0, Ω]33

Ω= i 14π2MN =qVΛ 12π3/2 "Γ( 37)Γ( 57)Γ( 67)Γ( 17)Γ( 27)Γ( 47)#3/2Ω= i 14π2 sin(α) sin(2α) sin(3α)abcddcbabcddcbabcddcbabcddcbabcddcbaa =R21sin2(3α) +R22sin2(α) +R23sin2(2α)b = R21 cos(α)sin2(3α) + R22 cos(2α)sin2(α)+ R23 cos(3α)sin2(2α)c = R21 cos(2α)sin2(3α) + R22 cos(3α)sin2(α)+ R23 cos(α)sin2(2α)d = R21 cos(3α)sin2(3α) + R22 cos(α)sin2(α) + R23 cos(2α)sin2(2α)Number of effective parameters: 3Number of different couplings without deformations: 2corresponding to the following ⃗f23 shifts⃗f23 = {f (0)1 , f (1)1 }Number of different couplings with deformations: 4corresponding to the following ⃗f23 shifts⃗f23 = {f (0)1 , f (1)1 , f (2)1 , f (3)1 }ORBIFOLD Z8–ITwist θ = diag(eiα, e2iα, e−3iα),α = 2π8Lattice SO(5) ⊗SO(9)Coxeter elementθe1 = e1 + 2e2,θe2 = −e1 −e2,θe3 = e4,θe4 = e5,θe5 = e3 + e4 + e5 + 2e6,θe6 = −e3 −e4 −e5 −e6Deformation parametersRelations|e1| =√2|e2|,|e3| = |e4| = |e5|,−2α56|e6| = |e3|,α12 = −1√2,α35 = 0,α34 = α45,α36 = α46,α36 =12α56 −α56,α34 =14α256 −1,αij = 0i = 1, 2j = 3, 4, 5, 634

αij ≡cos(θij)Degrees of freedom (3)R1 = |e1|,R3 = |e3|,α56Lattice basis (ei) in terms of orthogonal basis (˜ei)e1=R12 {[(2 +√2)1/2 cos(ϕ1) + (2 −√2)1/2 sin(ϕ1)]˜e1++[−(2 +√2)1/2 sin(ϕ1) + (2 −√2)1/2 cos(ϕ1)]˜e2}e2=R12 {−[(2 +√2)1/2 cos(ϕ1) + (2 −√2)1/2 sin(ϕ1)]˜e1++[(2 +√2)1/2 sin(ϕ1) + (2 −√2)1/2 cos(ϕ1)]˜e2}e3=A[cos(ϕ2)˜e3 + sin(ϕ2)˜e4] + B[cos(ϕ3)˜e5 + sin(ϕ3)˜e6]e4=A[cos(α + ϕ2)˜e3 + sin(α + ϕ2)˜e4] −B[cos(α + ϕ3)˜e5 + sin(α + ϕ3)˜e6]e5=−A[sin(ϕ2)˜e3 −cos(ϕ2)˜e4] −B[sin(ϕ3)˜e5 −cos(ϕ3)˜e6]e6=A2 [−cos(ϕ2) + sin(ϕ2) −2 cos(ϕ2) cos(α)]˜e3++ A2 [−cos(ϕ2) −sin(ϕ2) −2 sin(ϕ2) cos(α)]˜e4++ B2 [−cos(ϕ3) + sin(ϕ3) + 2 cos(ϕ3) cos(α)]˜e5++ B2 [−cos(ϕ3) −sin(ϕ3) + 2 sin(ϕ3) cos(α)]˜e6with α = 2π8 andA = R31+√22−14√2α2561/2,B = R31−√22+14√2α2561/2ϕ1, ϕ2, ϕ3 are free parameters.Fixed points of θ (4)f (ij)1= g(i)1 ⊗ˆg(j)1,i = 0, 1,j = 0, 1g(0)1= (0, 0) ,g(1)1= 12(1, 0) ,ˆg(0)1= (0, 0, 0, 0) ,ˆg(1)1= 12(1, 0, 1, 0)Fixed points of θ2 (16)f (ij)2= g(i)2 ⊗ˆg(j)2,i, j = 0, 1, 2, 3g(0)2= (0, 0),g(1)2= 12(0, 1),g(2)2= 12(1, 0),g(3)2= 12(1, 1),ˆg(0)2= (0, 0, 0, 0),ˆg(1)2= 12(0, 1, 1, 0),ˆg(2)2= 12(1, 0, 1, 0),ˆg(3)2= 12(1, 1, 0, 0)35

Note that θ : g(1)2→g(3)2and θ : ˆg(1)2→ˆg(3)2 .Number of conjugation classes: 10Fixed points of θ3 (4)The same as for θ.f (ij)3= g(i)3 ⊗ˆg(j)3,i = 0, 1,j = 0, 1g(0)3= (0, 0) ,g(1)3= 12(1, 0) ,ˆg(0)3= (0, 0, 0, 0) ,ˆg(1)3= 12(1, 0, 1, 0)Fixed points of θ4 (16)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)4= [α(e1) + β(e2)] ⊗ˆg(i)4 ,i = 0, 1, ..., 15,α, β ∈Rˆg(0)4= (0, 0, 0, 0),ˆg(1)4= 12(1, 0, 1, 1),ˆg(2)4= 12(1, 0, 0, 0),ˆg(3)4= 12(1, 0, 0, 1),ˆg(4)4= 12(1, 1, 0, 0),ˆg(5)4= 12(0, 0, 0, 1),ˆg(6)4= 12(0, 1, 0, 0),ˆg(7)4= 12(0, 0, 1, 1),ˆg(8)4= 12(1, 0, 1, 0),ˆg(9)4= 12(1, 1, 0, 1),ˆg(10)4= 12(0, 0, 1, 0),ˆg(11)4= 12(0, 1, 0, 1),ˆg(12)4= 12(0, 1, 1, 0),ˆg(13)4= 12(0, 1, 1, 1)ˆg(14)4= 12(1, 1, 1, 0),ˆg(15)4= 12(1, 1, 1, 1)Note that in the SO(9) latticeθ : ˆg(4)4→ˆg(12)4,θ : ˆg(1)4→ˆg(3)4→ˆg(9)4→ˆg(11)4,θ : ˆg(2)4→ˆg(6)4→ˆg(10)4→ˆg(14)4,θ : ˆg(5)4→ˆg(7)4→ˆg(13)4→ˆg(15)4Number of conjugation classes: 6Coupling θ2θ2θ4Selection rulef1 + f2 −(I + θ2)f3 ∈ΛDenotingf1=g(i1)2⊗ˆg(j1)2f2=g(i2)2⊗ˆg(j2)2f3=[α(e1) + β(e2)] ⊗ˆg(j3)4i1, i2, j1, j2 = 0, 1, 2, 3,j3 = 0, 1, ..., 15 ,α, β ∈Rthe selection rule readsi1 = i2j1 + (−1)(j3+1)j2 = j3mod. 436

Number of allowed couplings: 84Expression of the couplingCθ2θ2θ4=F(l1, l2, l3)2N {Xv∈(f3−f2+Λ)⊥exp[−14π |v|2] +Xv∈(θf3−f2+Λ)⊥exp[−14π |v|2]}=F(l1, l2, l3)2N{ϑ⃗f230[0, Ω] + ϑ⃗f ′230[0, Ω]}where (f3−f2+Λ)⊥denotes that only coset elements belonging to SO(9) latticeare considered; f23 = f2 −f3, f ′23 = θf2 −f3, the arrows denote components inthe SO(9) lattice. li is the number of elements in the fi conjugation class (inall the cases, except l1, l2, l3 = 2, f23 = f ′23).

Finally the values of F(l1, l2, l3)arel1 = l2 = l3 = 1 :F = 1l1 = l2 = 1 l3 = 2 :F =√2l1 = l2 = 1 l3 = 4 :F = 2l1 = l2 = 2 l3 = 1 :F = 1l1 = l2 = l3 = 2 :F =√2l1 = l2 = 2 l3 = 4 :F = 1l1 = 1(2) l2 = 2(1) l3 = 4 :F =√2N =qV⊥12πΓ2( 34)Γ2( 14)Ω= i 14π2ab0cbabc0badccdea = R23b = R23[14α256 −1]c = R23[12α56 −α56]d = −R232e =R234α256where V⊥is the volume of the SO(9) latticeNumber of effective parameters: 2Number of different couplings without deformations: 8corresponding to the following ⃗f23 shifts⃗f23 =F = 1n(0, 0, 0, 0),( 12, 0, 12, 0),( 12, 12, 0, 0),F =√2( 12, 12, 0, 0),( 12, 0, 0, 12),(0, 0, 12, 12),(0, 0, 0, 0) ∪( 12, 0, 12, 0),F = 2 ( 12, 0, 0, 0)Number of different couplings with deformations: 937

corresponding to the following ⃗f23 shifts⃗f23 =F = 1(0, 0, 0, 0),( 12, 0, 12, 0),( 12, 12, 0, 0),(0, 12, 0, 0),F =√2( 12, 12, 0, 0),( 12, 0, 0, 12),(0, 0, 12, 12),(0, 0, 0, 0) ∪( 12, 0, 12, 0),F = 2 ( 12, 0, 0, 0)Coupling θθ2θ5Selection rulef1 + (I + θ)f2 −(I + θ + θ2)f3 ∈ΛDenotingf1=g(i1)1⊗ˆg(j1)1f2=g(i2)2⊗ˆg(j2)2f3=g(i3)1⊗ˆg(j3)1i1, i3, j1, j3 = 0, 1 ,i2, j2 = 0, 1, 2, 3 ,the selection rule readsi1 + i2 + i3=0j1 + j2 + j3=0mod. 2Number of allowed couplings: 40Expression of the couplingCθθ2θ5=ql2 NXv∈(f3−f2+Λ)⊥exp[−14π (√2 + 1√2|v1|2 + |v2|2 +√2 −1√2|v3|2)]=ql2 NX⃗u∈Z6exp[−14π ( ⃗f23 + ⃗u)⊤M( ⃗f23 + ⃗u)]=ql2 N ϑ⃗f230[0, Ω]with l2 the number of elements in the f2 conjugation classΩ= i 14π2MN =qVΛ 12π3/2 Γ( 78)Γ( 38)Γ( 18)Γ( 58)Γ2( 34)Γ2( 14)38

Ω= i 14π2a−a0000−a2a000000bc0e00cbcd000cbe00edefa = R21b =1√2[(√2 + 1)A2 + (√2 −1)B2]c = 12[(√2 + 1)A2 −(√2 −1)B2]d = −12[(√2 + 1)2A2 + (√2 −1)2B2]e = −12√2[(√2 + 1)2A2 −(√2 −1)2B2]f =12√2[(√2 + 1)3A2 + (√2 −1)3B2]Number of effective parameters: 3Number of different couplings without deformations: 8corresponding to the following ⃗f23 shifts⃗f23 =l2 = 1ng(0)2⊗ˆg(0)2 ,g(0)2⊗ˆg(2)2 ,g(2)2⊗ˆg(0)2 ,g(2)2⊗ˆg(2)2 ,l2 = 2g(0)2⊗ˆg(1)2 ,g(1)2⊗ˆg(0)2 ,g(2)2⊗ˆg(1)2 ,g(1)2⊗ˆg(2)2Number of different couplings with deformations: 9corresponding to the following ⃗f23 shifts⃗f23 =l2 = 1ng(0)2⊗ˆg(0)2 ,g(0)2⊗ˆg(2)2 ,g(2)2⊗ˆg(0)2 ,g(2)2⊗ˆg(2)2 ,l2 = 2g(0)2⊗ˆg(1)2 ,g(1)2⊗ˆg(0)2 ,g(1)2⊗ˆg(1)2 ,g(2)2⊗ˆg(1)2 ,g(1)2⊗ˆg(2)2ORBIFOLD Z8–IITwist θ = diag(eiα, e3iα, e−4iα),α = 2π8Lattice SO(4) ⊗SO(8)Twist in the lattice basisθe1 = −e1,θe2 = −e2,θe3 = e4 + e5,θe4 = e3 + e4 + e6,θe5 = −e3 −e4 −e5 −e6,θe6 = −e3 −e4Deformation parametersRelations|e3| = |e5|,|e4| =1√2[|e3|2 + |e6|2]1/2,α35 = 0,α56 = 0,α34 =1√2[ 12 |e6|2−32|e3|2]|e3|[|e3|2+|e6|2]1/2,α36 = [|e3|2−|e6|2]2|e3||e6| ,α45 =1√2[ 12|e3|2−32|e6|2]|e3|[|e3|2+|e6|2]1/2,α46 = −1√2[|e3|2+|e6|2]1/2|e6|,αij = 0i = 1, 2 j = 3, 4, 5, 639

αij ≡cos(θij)Degrees of freedom (5)R1 = |e1|,R2 = |e2|,R3 = |e3|,R6 = |e6|,α12Lattice basis (ei) in terms of orthogonal basis (˜ei)e1=R1[sin(ϕ1 + θ12)˜e1 + cos(ϕ1 + θ12)˜e2]e2=R2[sin(ϕ1)˜e1 + cos(ϕ1)˜e2]e3=A[cos(ϕ2)˜e3 + sin(ϕ2)˜e4] + ρ2[cos(ϕ3)˜e5 + sin(ϕ3)˜e6]e4=A√2[(cos(ϕ2) + (1 +√2) sin(ϕ2))˜e3 + (−(1 +√2) cos(ϕ2) + sin(ϕ2)˜e4]−−ρ2√2[(cos(ϕ3) −(1 −√2) sin(ϕ3))˜e5 + ((1 −√2) cos(ϕ3) + sin(ϕ3)˜e6]e5=−A[sin(ϕ2)¯e3 −cos(ϕ2)˜e4] + ρ2[sin(ϕ3)˜e5 −cos(ϕ3)˜e6]e6=−(1 +√2)A[cos(ϕ2)˜e3 + sin(ϕ2)˜e4] + (√2 −1)ρ2[cos(ϕ3)˜e5 + sin(ϕ3)˜e6]A =R325/4R6R32 −(1 −√2)21/2,ρ2 =R625/4R3R62 (1 +√2)2 −11/2ϕ1, ϕ2, ϕ3 are free parameters.Fixed points of θ (8)f (ij)1= g(i)1 ⊗ˆg(j)1,i = 0, 1, 2, 3,j = 0, 1g(0)1= (0, 0) ,g(1)1= 12(1, 0) ,g(2)1= 12(1, 1) ,g(3)1= 12(0, 1) ,ˆg(0)1= (0, 0, 0, 0) ,ˆg(1)1= 12(0, 0, 1, 1)Fixed points of θ2 (4)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)2= [α(e1) + β(e2)] ⊗ˆg(i)2 ,i = 0, 1, 2, 3, α, β ∈Rˆg(0)2= (0, 0, 0, 0),ˆg(1)2= 12(1, 0, 1, 0),ˆg(2)2= 12(0, 0, 1, 1),ˆg(3)2= 12(1, 0, 0, 1)Note that in the SO(8) lattice θ : ˆg(1)2→ˆg(3)2 .Number of conjugation classes: 3Fixed points of θ3 (8)40

The same as for θ.f (ij)3= g(i)3 ⊗ˆg(j)3,i = 0, 1, 2, 3,j = 0, 1g(0)3= (0, 0) ,g(1)3= 12(1, 0) ,g(2)3= 12(1, 1) ,g(3)3= 12(0, 1) ,ˆg(0)3= (0, 0, 0, 0) ,ˆg(1)3= 12(0, 0, 1, 1)Fixed points of θ4 (16)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)4= [α(e1) + β(e2)] ⊗ˆg(i)4 ,j = 0, 1, ..., 15, α, β ∈Rˆg(0)4= (0, 0, 0, 0),ˆg(1)4= 12(1, 0, 0, 0),ˆg(2)4= 12(0, 1, 0, 0),ˆg(3)4= 12(0, 0, 0, 1),ˆg(4)4= 12(1, 0, 1, 0),ˆg(5)4= 12(0, 1, 1, 0),ˆg(6)4= 12(1, 1, 0, 1),ˆg(7)4= 12(1, 1, 0, 0),ˆg(8)4= 12(0, 0, 1, 1),ˆg(9)4= 12(0, 0, 1, 0),ˆg(10)4= 12(0, 1, 1, 1),ˆg(11)4= 12(1, 0, 1, 1),ˆg(12)4= 12(1, 0, 0, 1),ˆg(13)4= 12(1, 1, 1, 1),ˆg(14)4= 12(1, 1, 1, 0),ˆg(15)4= 12(0, 1, 0, 0)Note that in the SO(8) latticeθ : ˆg(4)4→ˆg(12)4,θ : ˆg(1)4→ˆg(5)4→ˆg(9)4→ˆg(13)4,θ : ˆg(2)4→ˆg(6)4→ˆg(10)4→ˆg(14)4,θ : ˆg(3)4→ˆg(7)4→ˆg(11)4→ˆg(15)4Number of conjugation classes: 6Coupling θθθ6Selection rulef1 + f2 −(I + θ)f3 ∈ΛDenotingf1=g(i1)1⊗ˆg(j1)1f2=g(i2)1⊗ˆg(j2)1f3=[α(e1) + β(e2)] ⊗ˆg(j3)2i1, i2, j3 = 0, 1, 2, 3 ,j1, j2 = 0, 1 ,α, β ∈Rthe selection rule readsi1 = i2j1 + j2 + j3 = 0mod. 2Number of allowed couplings: 24Expression of the coupling41

Cθθθ6=ql3 NXv∈(f3−f2+Λ)⊥exp[−√28π (|v2|2 + |v3|2)]=ql3 NX⃗u∈Z4exp[−√28π ( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=ql3 N ϑ⃗¯f230[0, Ω]where the expression (f3 −f2 + Λ)⊥indicates that the coset elements mustbelong to SO(8) and¯f23 is the restriction of f23 to SO(8), l3 denotes thenumber of elements in the f3 conjugation class, and the arrows denote thecomponents in the SO(8) latticeN =qV⊥12πΓ( 78)Γ( 58)Γ( 18)Γ( 38) , Ω= i√28π2MΩ= i√28π2ab0cbde−d20ea0c−d20fa = R23b = −34R23 + 14R26c = 12R23 −12R26d = 12R23 + 12R26e = −34R26 + 14R23f = R26where V⊥is the volume of the SO(8) latticeNumber of effective parameters: 2Number of different couplings without deformations: 3Number of different couplings with deformations: 3corresponding to the following ⃗¯f23 shifts⃗¯f23 =l3 = 1g(0)2 ,g(2)2 ,l3 = 2g(1)2Coupling θθ3θ4Selection rulef1 + f2 −(I + θ + θ2 + θ3)f3 ∈Λ42

Denotingf1=g(i1)1⊗ˆg(j1)1f2=g(i2)1⊗ˆg(j2)1f3=[α(e1) + β(e2)] ⊗ˆg(j3)4i1, i2 = 0, 1, 2, 3 ,j1, j2 = 0, 1 ,j3 = 0, 1, ..., 15 ,α, β ∈Rthe selection rule readsi1 = i2j1 + j2 + j3 = 0mod. 2Number of allowed couplings: 48Expression of the couplingCθθ3θ4=ql3 NXv∈(f3−f2+Λ)⊥exp[−14π ((√2 + 1)|v2|2 + (√2 −1)|v3|2)]=ql3 NX⃗u∈Z4exp[−14π ( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=ql3 N ϑ⃗¯f230[0, Ω]where (f3 −f2 + Λ)⊥indicates that the coset elements must belong to SO(8),and V⊥is the volume of the SO(8) latticeN =qV⊥12πΓ( 78)Γ( 58)Γ( 18)Γ( 38)Ω= i 14π2MΩ= i 14π2ab0ebcdd0da0ed0fa = [(√2 + 1)A2 + (√2 −1)B2]b =1√2[(√2 + 1)A2 −(√2 −1)B2]c = [(√2 + 2)(√2 + 1)A2 −(√2 −1)(√2 −2)B2]d = −1√2[(√2 + 1)2A2 + (√2 −1)2B2]e = −[(√2 + 1)2A2 −(√2 −1)2B2]f = [(√2 + 1)3A2 + (√2 −1)3B2]Number of effective parameters: 2Number of different couplings without deformations: 6Number of different couplings with deformations: 643

corresponding to the following ⃗¯f23 shifts⃗¯f23 =l3 = 1g(0)4 ,g(8)4 ,l3 = 2g(4)4 ,l3 = 4g(1)4 ,g(2)4 ,g(3)4ORBIFOLD Z12–ITwist θ = diag(eiα, e4iα, e−5iα),α = 2π12Lattice SU(3) ⊗F4Coxeter elementθe1 = e2,θe2 = −e1 −e2,θe3 = e4,θe4 = e3 + e4 + 2e5,θe5 = e6,θe6 = −e3 −e4 −e5 −e6Deformation parametersRelations|e1| = |e2|,|e3| = |e4| =√2|e5| =√2|e6|,α12 = −12,α45 = −1√2,α34 = α56,α35 = α46 = α36,α35 = −12√2[1 + 2α34],αij = 0i = 1, 2, j = 1, 2, 3, 4αij ≡cos(θij)Degrees of freedom (3)R1 = |e1|,R3 = |e3|,α34Lattice basis (ei) in terms of orthogonal basis (˜ei)e1 = R1 cos(φ1)˜e1 + R1 sin(φ1)˜e2e2 = R1 cos(φ1 + α)˜e1 + R1 sin(φ1 + α)˜e2e3 = A cos(φ2)˜e3 + A sin(φ2)˜e4 + B cos(φ3)˜e5 + B cos(φ3)˜e6e4 = A cos(φ2 + β)˜e3 + A sin(φ2 + β)˜e4 + B cos(φ3 + 7β)˜e5 + B cos(φ3 + 7β)˜e6e5 =A√2[−sin(φ2 + 52β)˜e3 + cos(φ2 + 52β)˜e4] + B√2[sin(φ3 + 112 β)˜e5 −cos(φ3 + 112 β)˜e6]e6 =A√2[−sin(φ2 + 72β)˜e3 + cos(φ2 + 72β)˜e4] + B√2[sin(φ3 + 12β)˜e5 −cos(φ3 + 12β)˜e6]44

α = 2π3 ,β = π6,A = R3h12 + α34√3i1/2 ,B = R3h12 −α34√3i1/2φ1, φ2, φ3 are free parametersFixed points of θ (3)f (i)1= g(i)1 ⊗ˆg(0)1,i = 0, 1, 2g(0)1= (0, 0) ,g(1)1= 13(1, 2) ,g(2)1= 13(2, 1) ,ˆg(0)1= (0, 0, 0, 0)Fixed points of θ2 (3)The same as for θf (i)2= g(i)2 ⊗ˆg(0)2,i = 0, 1, 2g(0)2= (0, 0) ,g(1)2= 13(1, 2) ,g(2)2= 13(2, 1) ,ˆg(0)2= (0, 0, 0, 0)Fixed points of θ3 (4)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)3= [α(e1) + β(e2)] ⊗ˆg(i)3 ,i = 0, 1, 2, 3, α, β ∈Rˆg(0)3= (0, 0, 0, 0) ,ˆg(1)3= 12(1, 0, 0, 0) ,ˆg(2)3= 12(0, 1, 0, 0) ,ˆg(3)3= 12(1, 1, 0, 0)Note that in the F4 lattice θ : ˆg(1)3→ˆg(2)3→ˆg(3)3Number of conjugation classes: 2Fixed points of θ4 (27)f (ij)4= g(i)4 ⊗ˆg(j)4,i = 0, 1, 2,j = 0, 1, ..., 8g(0)4= (0, 0),g(1)4= 13(1, 2),g(2)4= 13(2, 1),ˆg(0)4= (0, 0, 0, 0),ˆg(1)4= 13(2, 1, 2, 0),ˆg(2)4= 13(2, 2, 0, 2),ˆg(3)4= 13(1, 0, 2, 2),ˆg(4)4= 13(0, 2, 2, 1),ˆg(5)4= 13(1, 2, 1, 0),ˆg(6)4= 13(1, 1, 0, 1),ˆg(7)4= 13(2, 0, 1, 1),ˆg(8)4= 13(0, 1, 1, 2)Note that in the F4 lattice θ : ˆg(1)4→ˆg(3)4→ˆg(5)4→ˆg(7)4and θ : ˆg(2)4→ˆg(4)4→ˆg(6)4→ˆg(8)445

Number of conjugation classes: 9Fixed points of θ5 (3)The same as for θf (i)5= g(i)5 ⊗ˆg(0)5,i = 0, 1, 2g(0)5= (0, 0) ,g(1)5= 13(1, 2) ,g(2)5= 13(2, 1) ,ˆg(0)5= (0, 0, 0, 0)Fixed points of θ6 (16)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)6= [α(e1) + β(e2)] ⊗ˆg(i)6 ,i = 0, 1, ..., 15, α, β ∈Rˆg(0)6= (0, 0, 0, 0) ,ˆg(1)6= 12(1, 1, 1, 1) ,ˆg(2)6= 12(0, 0, 0, 1) ,ˆg(3)6= 12(0, 0, 1, 0) ,ˆg(4)6= 12(1, 0, 0, 0) ,ˆg(5)6= 12(0, 0, 1, 1) ,ˆg(6)6= 12(0, 1, 0, 1) ,ˆg(7)6= 12(1, 0, 1, 0) ,ˆg(8)6= 12(0, 1, 0, 0) ,ˆg(9)6= 12(0, 1, 1, 1) ,ˆg(10)6= 12(1, 1, 0, 1) ,ˆg(11)6= 12(0, 1, 1, 0) ,ˆg(12)6= 12(1, 1, 0, 0) ,ˆg(13)6= 12(1, 0, 1, 1) ,ˆg(14)6= 12(1, 0, 0, 1) ,ˆg(15)6= 12(1, 1, 1, 0)Note that in the F4 lattice θ : ˆg(3)6→ˆg(2)6→ˆg(1)6→ˆg(11)6→ˆg(10)6→ˆg(9)6,θ : ˆg(7)6→ˆg(6)6→ˆg(5)6→ˆg(15)6→ˆg(14)6→ˆg(13)6and θ : ˆg(4)6→ˆg(8)6→ˆg(12)6Number of conjugation classes: 4Coupling θθ2θ9Selection rulef1 + (I + θ)f2 −(I + θ + θ2)f3 ∈ΛDenotingf1=g(i1)1⊗ˆg(0)1f2=g(i2)2⊗ˆg(0)2f3=[α(e1) + β(e2)] ⊗ˆg(j3)3i1, i2, j3 = 0, 1, 2, 3 ,α, β ∈Rthe selection rule readsi1=i2Number of allowed couplings: 6Expression of the coupling46

Cθθ2θ9=ql3 NXv∈(f3−f2+Λ)⊥exp[−14π sin(π6 sin(π4 ) ( |v2|2sin( π12) +|v3|2cos( π12))]=ql3 NXu∈Z4exp[−√24π ( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=ql3 N ϑ⃗¯f230[0, Ω]where (f3 −f2 + Λ)⊥indicates that the coset elements must belong to F4,l3 is the number of elements in the f3 conjugation class, the arrows denotecomponents in the F4 lattice, and V⊥is the volume of the F4 lattice unit cellN =qV⊥12πΓ( 56)Γ( 16)"Γ( 1112)Γ( 512)Γ( 112)Γ( 712)#1/2,Ω= i√24π2MΩ= i√24π2ab√32−b√2−b√2b√32a−a2b√32−b√2−a2a2b√34−b√2b√32b√34a2a = [A2 cos( π12) + B2 sin( π12)]b = [A2 cos( π12) −B2 sin( π12)]Number of effective parameters: 2Number of different couplings without deformations: 2Number of different couplings with deformations: 2corresponding to the following ⃗¯f23 shifts⃗¯f23 = ˆg(0)3 , ˆg(1)3Note that this coupling is the same as θ2θ3θ7Coupling θθ4θ7Selection rulef1 + (I + θ + θ2 + θ3)f2 −(I + θ + θ2 + θ3 + θ4)f3 ∈ΛDenotingf1=g(i1)1⊗ˆg(0)1f2=g(i2)4⊗ˆg(j2)4f3=g(i3)5⊗ˆg(0)5i1, i2, i3 = 0, 1, 2,j2 = 0, 1, ..., 847

the selection rule readsi1 + i2 + i3 = 0 ,mod. 3Number of allowed couplings: 6Expression of the couplingCθθ4θ7=ql2 NXv∈(f3−f2+Λ)exp[−14π (√32 |v1|2 + 2√3(|v2|2cos2( π12) +|v3|2sin2( π12)))]=ql2 NX⃗u∈Z6exp[−√38π ( ⃗f23 + ⃗u)⊤M( ⃗f23 + ⃗u)]=ql2 N ϑ⃗f230[0, Ω]l2 is the number of elements in the f2 conjugation class, and the arrows denotecomponents in the SU(3) ⊗F4 latticeN =qVΛ31/44π2Γ3( 23)Γ2( 13)"Γ( 1112)Γ( 512)Γ( 112)Γ( 712)#,Ω= i√24π2MΩ= i√38π2R21−R2120000−R212R21000000ab√32−b√2−b√200b√32a−a2b√3200−b√2−a2a2b√3400−b√2b√32b√34a2a = 4[A2 cos( π12) + B2 sin( π12)]b = 4[A2 cos( π12) −B2 sin( π12)]Number of effective parameters: 3Number of different couplings without deformations: 4Number of different couplings with deformations: 4corresponding to the following ⃗f23 shifts⃗f23 =l2 = 1g(0)4⊗ˆg(0)4 ,g(1)4⊗ˆg(0)4 ,l2 = 2g(0)4⊗ˆg(1)4 ,g(1)4⊗ˆg(1)448

Coupling θ2θ4θ6Selection rulef1 + (I + θ2)f2 −(I + θ2 + θ4)f3 ∈ΛDenotingf1=g(i1)2⊗ˆg(0)2f2=g(i2)4⊗ˆg(j2)4f3=[α(e1) + β(e2)] ⊗ˆg(j3)6i1, i2 = 0, 1, 2,j2 = 0, 1, ..., 8,j3 = 0, 1, ..., 15,α, β ∈Rthe selection rule readsi1 = i2 ,Number of allowed couplings: 36Expression of the coupling (in all the cases except l3 = 6, l2 = 4)Cθ2θ4θ6=Nql2 l3Xv∈(f23+Λ)⊥exp[−14πsin( π3)sin( π6) |v|2]=Nql2 l3X⃗u∈Z4exp[−√34π ( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=Nql2 l3 ϑ⃗¯f230[0, Ω]where f23 = f2 −f3, ¯f23 is the restriction of f23 to the F4 lattice; (f23 + Λ)⊥indicates that the coset must belong to F4 and li is the number of elements inthe fi conjugation class. V⊥is the volume of the F4 unit cellIn the l3 = 6, l2 = 4 caseCθ2θ4θ6=N√6Xv∈(f2−f3+Λ)⊥∪(θf2−f3+Λ)⊥exp[−14πsin( π3)sin( π6) |v|2]=N√6 {ϑ⃗¯f230[0, Ω] + ϑ⃗¯f ′230[0, Ω]}N =qV⊥12π"Γ( 23)Γ( 56)Γ( 16)Γ( 13)#,Ω= i√34π2M49

Ω= i√34π2R231α34−14[1 + 2α34]−14[1 + 2α34]α341−12−14[1 + 2α34]−14[1 + 2α34]−1212α342−14[1 + 2α34]−14[1 + 2α34]α34212Number of effective parameters: 2Number of different couplings without deformations: 7Number of different couplings with deformations: 7corresponding to the following ⃗¯f23 shifts⃗¯f23 =l2 = l3 = 1 (0, 0, 0, 0),l2 = 1 l3 = 3 ( 12, 0, 0, 0),l2 = 1 l3 = 6 (0, 0, 12, 0),l2 = 4 l3 = 1 ( 23, 13, 23, 0),l2 = 4 l3 = 3 ( 16, 13, 23, 0),l2 = 4 l3 = 6( 23, 13, 16, 0) ∪( 23, 13, 23, 12),( 16, 13, 16, 0) ∪( 23, 56, 23, 12)Coupling θ4θ4θ4Selection rulef1 + f2 + f3 ∈ΛDenotingf1=g(i1)4⊗ˆg(j1)4f2=g(i2)4⊗ˆg(j2)4f3=g(i3)4⊗ˆg(j3)4i1, i2, i3 = 0, 1, 2,j1, j2, j3 = 0, 1, ..., 8,The selection rule readsi1 + i2 + i3 = 0mod. 3jσ(1) = 0 jσ(2), jσ(3) ̸= 0jσ(2) −jσ(3) = 4jσ(1) even jσ(2), jσ(3) oddjσ(3) −5 = jσ(2) + 1 = jσ(1)jσ(1) odd jσ(2), jσ(3) evenjσ(3) −7 = jσ(2) −5 = jσ(1)jσ(1), jσ(2), jσ(3) odd or evenjσ(3) = jσ(2) = jσ(1)mod.

8σ ≡permutation of {1, 2, 3}Number of allowed couplings: 189Expression of the coupling50

Cθ4θ4θ4=N F(l1, l2, l3)Xv∈(f3−f2+Λ)exp[−√38π |v|2]=N F(l1, l2, l3)X⃗u∈Z6exp[−√38π ( ⃗f23 + ⃗u)topM( ⃗f23 + ⃗u)]=N F(l1, l2, l3) [ϑ⃗¯f230[0, Ω]where F = 1 except for the case f1 = f2 = f3 and l1 = l2 = l3 = 4 in which F = 12Ω= i√38π MN =qVΛ31/44π2"Γ5( 23)Γ4( 16)#Ω= i√38πR21−R2120000−R212R21000000R23R23α34−R231+2α344−R231+2α34400R23α34R23−R2312−R231+2α34400−R231+2α344−R2312R2312R23α34200−R231+2α344−R231+2α344R23α342R2312Number of effective parameters: 3Number of different couplings without deformations: 6Number of different couplings with deformations: 6corresponding to the following ⃗f23 shifts⃗f23 =F = 12(0, 0, 23, 13, 23, 0),( 13, 23, 23, 13, 23, 0),F = 1(0, 0, 0, 0, 0, 0),( 13, 23, 0, 0, 0, 0),(0, 0, 23, 13, 23, 0),( 13, 23, 23, 13, 23, 0)ORBIFOLD Z12–IITwist θ = diag(eiα, e5iα, e−6iα),α = 2π12Lattice SO(4) ⊗F451

Coxeter elementθe1 = −e1,θe2 = −e2,θe3 = e4,θe4 = e3 + e4 + 2e5,θe5 = e6,θe6 = −e3 −e4 −e5 −e6Deformation parametersRelations|e3| = |e4| =√2|e5| =√2|e6|,α45 = −1√2,α34 = α56,α35 = α46 = α36,α35 = −12√2[1 + 2α34]αij = 0i = 1, 2, j = 3, 4, 5, 6αij ≡cos(θij)Degrees of freedom (5)R1 = |e1|,R2 = |e2|,R3 = |e3|,α12,α34Lattice basis (ei) in terms of orthogonal basis (˜ei)Not necessary in this case.Fixed points of θ (4)f (i)1= g(i)1 ⊗ˆg(0)1,i = 0, 1, 2, 3g(0)1= (0, 0) ,g(1)1= 12(0, 1) ,g(2)1= 12(1, 1) ,g(3)1= 12(1, 0) ,ˆg(0)1= (0, 0, 0, 0)Fixed points of θ2 (1)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (2)i= [α(e1) + β(e2)] ⊗ˆg(0)2,α, β ∈Rˆg(0)2= (0, 0, 0, 0)Fixed points of θ3 (16)f (ij)3= g(i)3 ⊗ˆg(j)3,i, j = 0, 1, 2, 3g(0)3= (0, 0) ,g(1)3= 12(0, 1) ,g(2)3= 12(1, 1) ,g(3)3= 12(1, 0) ,ˆg(0)3= (0, 0, 0, 0) ,ˆg(1)3= 12(1, 0, 0, 0) ,ˆg(2)3= 12(1, 1, 0, 0) ,ˆg(3)3= 12(0, 1, 0, 0)Note that, in F4 , θ : ˆg(1)3→ˆg(3)3→ˆg(2)352

Number of conjugation classes: 8Fixed points of θ4 (9)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)4= [α(e1) + β(e2)] ⊗ˆg(i)4 ,i = 0, 1, ..., 8ˆg(0)4= (0, 0, 0, 0),ˆg(1)4= 13(2, 1, 2, 0),ˆg(2)4= 13(2, 2, 0, 2),ˆg(3)4= 13(1, 0, 2, 2),ˆg(4)4= 13(0, 2, 2, 1),ˆg(5)4= 13(1, 2, 1, 0),ˆg(6)4= 13(1, 1, 0, 1),ˆg(7)4= 13(2, 0, 1, 1),ˆg(8)4= 13(0, 1, 1, 2)Note that, in F4, θ : ˆg(1)4→ˆg(3)4→ˆg(5)4→ˆg(7)4and θ : ˆg(2)4→ˆg(4)4→ˆg(6)4→ˆg(8)4Number of conjugation classes: 3Fixed points of θ5 (4)The same as for θ.f (i)5= g(i)5 ⊗ˆg(0)5,i = 0, 1, 2, 3g(0)5= (0, 0) ,g(1)5= 12(0, 1) ,g(2)5= 12(1, 1) ,g(3)5= 12(1, 0) ,ˆg(0)5= (0, 0, 0, 0)Fixed points of θ6 (16)Fixed torus: α(e1) + β(e2) , α, β ∈Rf (i)6= [α(e1) + β(e2)] ⊗ˆg(i)6 ,i = 0, 1, ..., 15, α, β ∈Rˆg(0)6= (0, 0, 0, 0) ,ˆg(1)6= 12(1, 1, 1, 1) ,ˆg(2)6= 12(0, 0, 0, 1) ,ˆg(3)6= 12(0, 0, 1, 0) ,ˆg(4)6= 12(1, 0, 0, 0) ,ˆg(5)6= 12(0, 0, 1, 1) ,ˆg(6)6= 12(0, 1, 0, 1) ,ˆg(7)6= 12(1, 0, 1, 0) ,ˆg(8)6= 12(0, 1, 0, 0) ,ˆg(9)6= 12(0, 1, 1, 1) ,ˆg(10)6= 12(1, 1, 0, 1) ,ˆg(11)6= 12(0, 1, 1, 0) ,ˆg(12)6= 12(1, 1, 0, 0) ,ˆg(13)6= 12(1, 0, 1, 1) ,ˆg(14)6= 12(1, 0, 0, 1) ,ˆg(15)6= 12(1, 1, 1, 0)Note that in F4 θ : ˆg(3)6→ˆg(2)6→ˆg(1)6→ˆg(11)6→ˆg(10)6→ˆg(9)6, θ : ˆg(7)6→ˆg(6)6→ˆg(5)6→ˆg(15)6→ˆg(14)6→ˆg(13)6and θ : ˆg(4)6→ˆg(8)6→ˆg(12)6Number of conjugation classes: 4Coupling θθθ10Selection rulef1 + f2 −(I + θ)f3 ∈Λ53

Denotingf1=g(i1)1⊗ˆg(0)1f2=g(i2)1⊗ˆg(0)1f3=[α(e1) + β(e2)] ⊗ˆg(0)2i1, i2 = 0, 1, 2, 3 , α, β ∈RThe selection rule readsi1 = i2 ,Number of allowed couplings: 4Expression of the couplingCθθθ10=NXv∈(f3−f2+Λ)⊥exp[−14π sin(π6 |v|2]=NX⃗u∈Z4exp[−14π sin(π6 ( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=N ϑ⃗¯f230[0, Ω]where ¯f23 is the restriction of f23 to the F4 lattice, (f3 −f2 + Λ)⊥indicatesthat the coset elements must belong to F4 and V⊥is the volume of the F4 unitcell. In all cases ¯f23 = 0.

FinallyΩ= i 14π2 sin(π6 )M ,N =qV⊥12π"Γ( 1112)Γ( 712)Γ( 112)Γ( 512)#Ω= i 14π2 sin( π6) R231α34−14[1 + 2α34]−14[1 + 2α34]α341−12−14[1 + 2α34]−14[1 + 2α34]−1212α342−14[1 + 2α34]−14[1 + 2α34]α34212Number of effective parameters: 2Number of different couplings without deformations: 1Number of different couplings with deformations: 1Note that this coupling is the same as θ2θ5θ5Coupling θθ3θ854

Selection rulef1 + (I + θ + θ2)f2 −(I + θ + θ2 + θ3)f3 ∈ΛDenotingf1=g(i1)1⊗ˆg(0)1f2=g(i2)3⊗ˆg(j2)3f3=[α(e1) + β(e2)] ⊗ˆg(j3)4i1, i2, j2 = 0, 1, 2, 3 ,j3 = 0, 1, ..., 8 ,α, β ∈Rthe selection rule readsi1 = i2Number of allowed couplings: 6Expression of the couplingCθθ3θ8=Nql2l3Xv∈(f3−f2+Λ)⊥exp[−14πsin( π3) sin( π4)sin( π12)|v|2]=Nql2l3X⃗u∈Z4exp[−14πsin( π3) sin( π4)sin( π12)( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=Nql2l3 ϑ⃗¯f230[0, Ω]with the same notation as in the previous coupling, li is the number of elementsof the fi conjugation class and V⊥is the volume of the F4 unit cell. FinallyΩ= i 14π2sin( π3) sin( π4)sin( π12)M ,N =qV⊥12πΓ( 34)Γ( 14)"Γ( 1112)Γ( 712)Γ( 112)Γ( 512)#1/2Ω= i 14π2sin( π3 ) sin( π4 )sin( π12 )R231α34−14[1 + 2α34]−14[1 + 2α34]α341−12−14[1 + 2α34]−14[1 + 2α34]−1212α342−14[1 + 2α34]−14[1 + 2α34]α34212Number of effective parameters: 2Number of different couplings without deformations: 4Number of different couplings with deformations: 4corresponding to the following ⃗¯f23 shifts55

⃗¯f23 =l2 = 1 l3 = 1(0, 0, 0, 0),l2 = 3 l3 = 1( 12, 0, 0, 0),l2 = 1 l3 = 4( 23, 23, 0, 23),l2 = 3 l3 = 4( 16, 16, 0, 23)Coupling θ3θ3θ6Selection rulef1 + f2 −(I + θ3)f3 ∈ΛDenotingf1=g(i1)3⊗ˆg(j1)3f2=g(i2)3⊗ˆg(j2)3f3=[α(e1) + β(e2)] ⊗ˆg(j3)6i1, i2, j1, j2 = 0, 1, 2, 3 ,j3 = 0, 1, ..., 15 ,α, β ∈Rthe selection rule readsi1 = i2j1 + (−1)(j3+1)j2 = j3mod. 4Number of allowed couplings: 56Expression of the couplingIn all the cases, except for the case l1 = l2 = l3 = 3Cθ3θ3θ6=N F(l1, l2, l3)Xv∈(f3−f2+Λ)⊥exp[−14π |v|2]=N F(l1, l2, l3)X⃗u∈Z4exp[−14π ( ⃗¯f23 + ⃗u)⊤M( ⃗¯f23 + ⃗u)]=N F(l1, l2, l3) ϑ⃗¯f230[0, Ω]with the same notation as in the previous coupling.

li is the number of elementsof the fi conjugation class. F(l1, l2, l3) is given byl1 = l2 = l3 = 1 and l1 = l2 = 3 l3 = 1 :F = 1l1 = l2 = 1 l3 = 3 :F =√3l1 = 1(3) l2 = 3(1) l3 = 6 :F =√2l1 = l2 = 3 l3 = 6 :F = 2√2In the case l1 = l2 = l3 = 356

Cθ3θ3θ6=N1√3Xv∈∪2p=0(θpf3−f2+Λ)⊥exp[−14π |v|2]=N1√3 {ϑ⃗¯f230[0, Ω] + ϑ⃗¯f ′230[0, Ω] + ϑ⃗¯f ′′230[0, Ω]}f ′23 = θf2 −f3 and f ′′23 = θ2f2 −f3Ω= i 14π2MN =qV⊥12π"Γ( 34)Γ( 14)#2V⊥is the volume of the F4 unit cellΩ= i 14π2 R231α34−14[1 + 2α34]−14[1 + 2α34]α341−12−14[1 + 2α34]−14[1 + 2α34]−1212α342−14[1 + 2α34]−14[1 + 2α34]α34212Number of effective parameters: 2Number of different couplings without deformations: 6Number of different couplings with deformations: 6corresponding to the following ⃗¯f23 shifts⃗¯f23 =l1 = l2 = l3 = 1(0, 0, 0, 0),l1 = l2 = 3 l3 = 1( 12, 0, 0, 0),l1 = l2 = 1 l3 = 3( 12, 0, 0, 0),l1 = l2 = l3 = 3(0, 0, 0, 0) ∪( 12, 0, 0, 0) ∪(0, 12, 0, 0),l1 = 1(3) l2 = 3(1) l3 = 6(0, 12, 12, 0),l1 = l2 = 3 l3 = 6( 12, 0, 12, 0)57

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TABLE 1Orb.Twist θLattice#DPCoupling#AC#EDP#DCR#DCDZ3(1, 1, −2)/3SU(3)39θθθ7299414Z4(1, 1, −2)/4SU(4)27θθθ21604610SO(4)37θθθ2160468Z6 −I(1, 1, −2)/6G22 × SU(3)5θθ2θ39041030θ2θ2θ23695812Z6 −II(1, 2, −3)/6SU(6) × SU(2)5θθ2θ348144θθθ472344Z7(1, 2, −3)/7SU(7)3θθ2θ449324Z8 −I(1, 2, −3)/8SO(5) × SO(9)3θ2θ2θ484289θθ2θ540389Z8 −II(1, 3, −4)/8SO(4) × SO(8)5θθθ624233θθ3θ448266Z12 −I(1, 4, −5)/12SU(3) × F43θθ2θ96222θθ4θ76344θ2θ4θ636277θ4θ4θ4189366Z12 −II(1, 5, −6)/12SO(4) × F45θθθ104211θθ3θ86244θ3θ3θ656266Table 1: Characteristics of twisted Yukawa couplings for Zn Coxeter orbifolds (the non–Coxeter Z4 one with SO(4)3 lattice is given for comparison). The twist θ is specified by thethree ci parameters (one for each complex plane rotation) appearing in θ = exp(P ciJi).#DP ≡# of deformation parameters, #AC ≡# of allowed couplings, #EDP ≡# ofeffective deformation parameters, #DCR ≡# of different Yukawa couplings for the non–deformed (rigid) orbifold, #DCD ≡# of different Yukawa couplings when deformationsare considered.60

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