Number of zero-modes on magnetized $T^4/Z_N$ orbifolds analyzed by modular transformation

We study fermion zero-mode wavefunctions on $T^4/Z_N$ orbifold with background magnetic fluxes. The number of zero-modes is analyzed by use of $Sp(4,\mathbb{Z})$ modular transformation. Conditions needed to realize three generation models are clarified. We also study parity transformation in the compact space which leads to better understanding of relationship between positive and negative chirality wavefunctions.


Introduction
The Standard Model (SM) of particle physics succeeds in describing particle interactions: electromagnetic, weak, and strong forces. There can be seen good agreements between its predictions and numerous experimental results. However, there are still unsolved puzzles. For example, the theory is incapable of explaining the origin of the flavor structure. This includes the generation number of fermions, hierarchical structures in Yukawa couplings, and so on. Therefore, we are driven to study their origin from a viewpoint of high-energy underlying theory. Superstring theory is a promising candidate of unified theory including quantum gravity. It requires 10 dimensional (10D) space-time which implies that extra 6 dimensional space is compactified. Therefore, we study higher-dimensional theory in the hope of realizing the flavor structure of the SM through suitable compactification. See ref. [1] for phenomenological aspects of superstring theory.
Motivated by superstring theory, we start with N = 1 super Yang-Mills theory(SYM) in 10 dimensions. Then we consider Kaluza-Klein decomposition of 10D fields. The degeneracy of fermion zero-modes on the compact space is equal to the generation number in the 4D field theory resulting from the dimensional reduction. Yukawa coupling constants can be evaluated by the overlap integral of wavefunctions corresponding to those zero-modes and lightest boson field over the compact space. Toroidal compactification with background magnetic fluxes is quite interesting. Thanks to the magnetic field, 4D chiral theory is realized. The degeneracy of the zero-modes is determined by the magnitude of the magnetic flux. Zero-mode wavefunctions are analytically obtained [2]. In magnetized T 2 models, they are written in terms of Jacobi theta-function. Since their profiles are quasi-localized, nontrivial overlaps may generate hierarchical structure in Yukawa couplings.
It is intriguing to extend such analysis of zero-mode number to magnetized T 2n /Z N orbifolds. Magnetized T 4 and T 4 /Z 2 models were previously studied [2,[26][27][28]. Zero-mode wavefunctions are written in terms of Riemann-theta function. In T 4 , there is Sp(4, Z) modular symmetry. Here, we apply this symmetry to zero-mode counting in magnetized T 4 /Z N , (N = 3, 4, 5, 6, etc.) orbifold models. We perform systematic analyses and clarify conditions needed to realize three generations of fermions. We assume that the 6 dimensional compact space is T 4 /Z N × M 2 where M 2 is a two dimensional compact space such as two-dimensional orbifolds on which there is a single zero-mode, making no contribution to the overall zero-mode numbers. The D-term condition preserving 4D N = 1 supersymmetry (SUSY) depends on both T 4 /Z N and M 2 sectors. This paper is organized as follows. In section 2, we give a review of magnetized T 4 models. In section 3, we introduce magnetized T 4 /Z N orbifold models. In section 4, Sp(4, Z) modular transformation of magnetic fluxes as well as zero-mode wavefunctions are explicitly presented. In section 5, we show some algebraic relations satisfied by the generators of Sp(4, Z). In section 6, by use of these algebraic relations, we construct T 4 /Z N orbifold models and systematically analyze zero-mode numbers. In Section 7, we show that a parity transformation connects two different magnetized T 4 /Z 6 orbifold models. There is a duality in the sense that positive chirality modes on one side correspond to negative modes on the other. Section 8 is our conclusion. In appendix A, we give a derivation of modular transformation properties of zeromode wavefunctions. In Appendix B, we provide some details of magnetized T 4 /Z 4 models. In Appendix C, we define T 4 /Z 2 permutation orbifolds and show some basic facts. In Appendix D, we show negative chirality wavefunction and its relation to positive chirality mode via parity transformation. In Appendix E, we review the F-term SUSY condition. Some useful properties of the Riemann-theta functions are shown in Appendix F.

Dirac operator on magnetized T 4
We begin with constructing the Dirac operator on the magnetized four dimensional torus T 4 ≃ C 2 /Λ, where Λ is a lattice spanned by four independent lattice vectors e i , (i = 1, 2, 3, 4) in C 2 . In this paper, we are interested in cases when at least two of the vectors have the same length and perpendicular to each other. This means by a suitable orthogonal rotation of the coordinates of C 2 ≃ R 4 , we can write e i 's of the form where τ i , (i = 1, 2, 3, 4) ∈ C and R > 0. We may further restrict the situation by demanding τ 3 = τ 4 because we will treat only such cases in later discussions about orbifold models. We introduce x i , y i , (i = 1, 2) as real coordinates along the lattice vectors of the torus. This means the complex coordinates (Z 1 , Z 2 ) of C 2 are related as where ⃗ z = ⃗ x + Ω⃗ y denotes the complex coordinates of the torus. When T 4 /Z N orbifolds are considered, ImΩ can be made positive definite 2 . We parameterize Ω as The set of such symmetric complex 2 × 2 matrices Ω with positive definite imaginary part is known as Siegel upper-half-space of genus 2, which is denoted by H 2 [32]. The metric of C 2 is where Therefore, the metric of our T 4 is given by Gamma matrices on the complex coordinates of T 4 are satisfying {Γ z i , Γ z j } = 2h ij , where Here, σ a (a = 1, 2, 3) denote the Pauli matrices, and σ Z = σ 1 + iσ 2 , σZ = σ 1 − iσ 2 . The chirality matrix Γ 5 is given by Then, the Dirac operator on T 4 is written by, The covariant derivatives are written by where we assumed that the fermion field coupled to the U (1) vector potential A = A z j dz j + Azj dz j is unit charged.
We constrain the form of flux to conserve N = 1 supersymmetry in the 4D space-time field theory resulting from the dimensional reduction. That is the flux needs to be a (1, 1)-form corresponding to the F-flat condition [2], which we review in Appendix E. This can be satisfied if, Ωp xx Ω + p yy + p T xy Ω − Ωp xy = 0.
As a result, F can be written as From this, we see the hermiticity of the flux: F z izj = F † z izj . The expression is further simplified by assuming p xx = p yy = 0, The F-flat SUSY condition is also simplified The flux must be quantized due to the consistency with the boundary conditions we take. Thus, we can write p xy in terms of a 2 × 2 real integer matrix N , This is referred to as Dirac's quantization condition. Then we obtain with the F-flat condition rewritten as The corresponding gauge potential is given by where ⃗ ζ is a complex constant 2-component vector known as the Wilson lines. The boundary conditions are where Here, ⃗ e k , (k = 1, 2) denote standard unit vectors. In order to simplify our discussion, we take vanishing Wilson lines ⃗ ζ = 0.

Zero-mode wavefunction
Fermion zero-modes satisfy the Dirac equation, where Here, ψ j + and ψ j − denote the positive and negative chirality components respectively. Writing eq.(25) in components, we getDz where Boundary conditions are which allow us to solve Dirac equations consistently on T 4 .

Solving the Dirac equation
Commutation relations of covariant derivatives under the SUSY condition are obtained as [Dzi,Dzj ] = Fzizj = 0, We define the Laplace operator ∆ as Then the positive chirality components satisfy Eigenvalues of the Laplace operator in eq.(36) are non-negative. This can be checked as follows. Firstly, consider the following integration of some wavefunctions ψ a and ψ b satisfying the same boundary conditions eq.(32). We find because the surface term vanishes. This shows that iD z j and iDzj are Hermitian conjugate with each other. Therefore, ∆ is a Hermitian operator. Secondly, let us denote an eigenvalue of the Laplacian by λ ∈ R, ∆ψ = λψ, and consider The left-hand side of eq.(40) is Consequently, we find λ ≥ 0. Thus, if F z 1z1 + F z 2z2 = tr(N T (ImΩ) −1 ) > 0, the second positive chirality component is zero, ψ 2 + = 0. This leads to simpler equations for ψ 1 + , On the other hand, if F z 1z1 + F z 2z2 < 0, the first positive chirality component is zero, where is known as the Riemann theta-function. ⃗ J is a two dimensional integer column vector which lies inside the lattice cell Λ N spanned by where ⃗ e n 's are unit vectors. 3 N J is a normalization constant, which follows from normalization condition, By noting ψ the degeneracy of zero-modes is detN . In general, ψ 1 + is written by their linear combinations, Note that wavefunctions in eq.(44) are normalizable when the following positive definite condition, is satisfied [2]. 4 This is equivalent to are the solutions of eq.(43). If detN < 0, negative chirality components become non-zero. In Appendix D, we discuss negative chirality wavefunctions.

Magnetized T 4 /Z N model
Here, we briefly summarize how to construct magnetized T 4 /Z N twisted orbifold models. Our discussion is along ref. [4] where magnetized T 2 /Z N twisted orbifold models are studied. T 4 /Z N orbifolds are defined by imposing following identification, on T 4 ≃ C 2 /Λ. Here, we take Ω twist as a 2 × 2 unitary matrix satisfying (Ω twist ) N = I 2 , representing a Z N -twist. We define the boundary condition on T 4 /Z N as in addition to eq.(32). Here, S denotes the spinor representation of the Z N -twist and V (⃗ z, ⃗ z) is a transformation function. As in the preceding study ref. [4], we conventionally define V (⃗ z, ⃗ z) so that S acts trivially on ψ 1 + . For example, if we consider the twist we take so that ψ 1 + is unchanged. In some other Z N -twists, Ω twist may not be diagonal, so z 1 and z 2 are mixed. Then, ψ 1 + and ψ 2 + will be mixed by the action of S, so the situation seems more complicated. However, it still makes sense to define V (⃗ z, ⃗ z) so that on-shell ψ 1 + is invariant under S. Suppose we start with conditions eq.(51), so that only ψ 1 + is non-zero, and ψ 2 + = 0. As we will see in the following sections, both N and Ω will be invariant under our twists, therefore ψ 2 + = 0 is maintained. Thus, S would act to on-shell ψ 1 + up to U (1) phase which can be removed by the redefinition of V (⃗ z, ⃗ z). The transformation function V (⃗ z, ⃗ z) is written as where β is a real number. Since N repeated Z N -twists need to be identity, we must have This shows that there are N different sectors with different Z N -twist charges, e 2πi k N , (k = 0, 1, ..., N − 1).
The eigenstates on T 4 /Z N are written by the linear combination of states on T 4 as in eq.(50). However, a ⃗ J are no longer arbitrary. They are chosen to satisfy the additional Z Ntwist boundary condition.

Modular transformation
Under the modular transformation γ ∈ Sp(4, Z), the complex coordinates and the complex structure Ω ∈ H 2 transform as [32], where γ is given by 2 × 2 matrices, A, B, C, and D as By the definition of Sp(4, Z), γ must satisfy γ T Jγ = J where We will use following generators of Sp(4, Z), as in ref. [33]. Here, B i 's are symmetric 2 × 2 integer matrices,

S transformation
Under the S transformation, we obtain where ⃗ z (S) = ⃗ x (S) − Ω −1 ⃗ y (S) denotes the coordinates after the S transformation. From eq.(65), we find,

S transformation of the magnetic flux
We study the S transformation of the magnetic flux, Thus, we find We clearly see that the condition p xx = p yy = 0 is consistent with S transformation. The flux matrix N is transformed as N At last, we should check the F-flatness SUSY condition eq. (21). It can be easily verified that holds if (N Ω) T = N Ω is satisfied. This shows that the F-flat condition is consistent with S transformation.

S transformation of zero-modes
We concentrate on the case when N T = N , so that it is invariant under S. As we will see later, this symmetry follows automatically from the F-flat SUSY condition in orbifolds we have studied. Then, the zero-modes in eq.(44) transform as where the branch of the square root is chosen so that it takes positive value when Ω is purely imaginary. We give a proof of eq.(71) in Appendix A.1.

T i transformation
Under the T i transformation, we obtain where ⃗ z (T i ) = ⃗ x (T i ) + (Ω + B i )⃗ y (T i ) denotes the coordinates after the T i transformation. From eq.(72), we find, Here, we omitted the subscript i(= 1, 2, 3). This is because when we consider the composite of T i , we still obtain the same form of transformations as in eq.(72). We refer to them by just saying T transformation. This means we may take B as any symmetric 2 × 2 integer matrices including B i .

T i transformation of the magnetic flux
We study the T transformation of the magnetic flux, Thus, we find We clearly see that the condition p xx = p yy = 0 is no longer maintained when T transformations are considered, unless (Bp xy ) T = Bp xy , is a symmetric matrix. Under this condition, the magnetic flux N is invariant, At last, we should check the F-flat SUSY condition eq. (21). It can be easily verified that holds if (N Ω) T = N Ω is satisfied. This shows that the F-flat condition is consistent with the T transformation given eq.(76).

T transformation of zero-modes
Here, we assume that the magnetic flux N satisfies so the corresponding T transformation is consistent with p xx = p yy = 0. Furthermore, we demand that the diagonal matrix element of N B be all even. Then the zero-modes in eq.(44) transform as, ψ We give a proof of eq.(80) in Appendix A.2.

A ∈ GL(2, Z) transformation
For later discussion, consider matrices of the form where A ∈ GL(2, Z). Then, γ is an element of Sp(4, Z). Under the A ∈ GL(2, Z) transformation, we obtain

A ∈ GL(2, Z) transformation of the magnetic flux
The flux matrix N is transformed as, It is straightforward to verify the consistency of A transformation with the condition p xx = p yy = 0.

A ∈ GL(2, Z) transformation of zero-modes
We concentrate on the case when N is invariant under a specific A ∈ GL(2, Z) transformation, Then, the zero-modes in eq.(44) transform as We give a proof of eq.(85) in Appendix A.3.

Algebraic relations
We defined T 4 /Z N orbifold by identifying different positions on T 4 related by the Z N -twist as shown in eq.(54). In fact, the twist can be written as a modular transformation as we will see. In ref. [13], magnetized T 2 /Z N orbifolds were studied based on this fact. Thus, we extend it to T 4 /Z N . Our starting point is to look for a modular transformation γ ∈ Sp(4, Z) satisfying the algebraic relation γ N = I 4 , so it may represent a Z N -twist. 5 Thus, we list up algebraic relations satisfied by the elements of Sp(4, Z) which we have used to study magnetized T 4 /Z N orbifold models, where γ P is an A ∈ GL(2, Z) transformation with 6 Degeneracy of zero-modes We focus on the following algebraic relation, This allows us to relate S to the Z 4 -twist which then is used for the construction of T 4 /Z 4 orbifold, as we will see. The complex coordinate and the complex structure moduli are transformed as eq.(65). We parameterize the moduli as eq.(3). Let us look for the S-invariant Ω ∈ H 2 . Then it must satisfy, From the off-diagonal elements, τ 3 = 0 or τ 1 = −τ 2 must hold. The latter case is discarded, because it contradicts to the requirement Ω ∈ H 2 . Therefore, we have τ 3 = 0. Then from the diagonal elements, the moduli parameters are fixed as, τ 2 1 = −1, τ 2 2 = −1. The relevant solution is unique, where we denoted the S-invariant moduli as Ω (S) . As a result, the complex coordinate of This is nothing but a Z 4 -twist. We are interested in the T 4 /Z 4 orbifold which is defined by imposing the following identification ⃗ z ∼ i⃗ z, on the complex coordinate of T 4 . This means that the moduli of T 4 must be S-invariant leading to the fixing of moduli as shown in eq.(90).

Lattice vectors
Here, we explicitly show the lattice vectors defining the T 4 /Z 4 orbifold. They are given by, Lattice vectors are perpendicular with each other, namely they form a root lattice which corresponds to SU (2) 4 ≃ SO(4) 2 . Both (e 1 , e 2 ) and (e 3 , e 4 ) are simple roots of SO (4). Under the Z 4 -twist realized by the S transformation, they behave as The Z 4 -twist is the generalized Coxeter element of SO(4) 2 including the Z 2 outer automorphism exchanging e 1 ↔ e 2 and e 3 ↔ e 4 [30,31].

Flux
When the complex structure moduli are Ω (S) , the F-flat SUSY condition shown in eq.(21) is satisfied only if N is a symmetric matrix, Note that N in eq.(95) is S-invariant according to eq.(69). We study the case when the positive definite condition eq.(51) is satisfied so that only the first positive chirality component ψ 1 + has non-zero solution. This means that we further restrict N by (96)

Zero-mode counting method
We have seen that zero-mode wavefunctions behave as eq.(71) under the S transformation or we could say Z 4 -twist instead. To analyze the degeneracy of zero-modes, it would be helpful to know the trace of the transformation matrix ρ(S), Let us denote the number of zero-modes as D (±1) , D (±i) which correspond to the Z 4 -twist eigenvalues ±1, ±i. Then we obtain following relations, Eqs.(98) and (99) together constrain three real degrees of freedom. One more independent constraint is needed. The information of zero-modes' number on magnetized T 4 /Z 2 provides it. This is due to the fact that two consecutive Z 4 -twists are equivalent to Z 2 -twist. Thus, Z 4 eigenstates with eigenvalues ±1 are Z 2 even. Z 4 eigenstates with eigenvalues ±i are Z 2 odd. The number of Z 2 eigenstates was studied in refs. [27,28]. We have 2 : if detN ≡ 0 (mod 2), gcd(n i , m) ≡ 1 (mod 2) for i = 1 or 2, 4 : gcd(n i , m) ≡ 0 (mod 2) for i = 1 and 2, where N Z 2 + and N Z 2 − denote the number of Z 2 even(+) and odd(−) modes respectively. As we saw in eq.(99), the trace of ρ(S) is important in determining the zero-mode degeneracy. Even when we fixed the value of detN , there are a large number of possible N . This means we have a variety of Λ N , characterizing the range of summation variables ⃗ K in eq.(97), so one might expect that we must treat individual N 's separately. However, it is possible to simplify the analysis greatly, if one makes rearrangements of ⃗ K. Firstly, consider when detN = p is a prime number and N is written by eq.(95). There are three cases, 1. gcd(n 1 , m) = p and gcd(n 2 , m) = 1, 2. gcd(n 1 , m) = 1 and gcd(n 2 , m) = p, 3. gcd(n 1 , m) = gcd(n 2 , m) = 1.
according to eq.(49). Let us take α and β to satisfy Since n 2 and m are relatively prime, general solutions are Then, we get If we choose l = 1, we obtain which is the smallest unit of identification. All of p integer points on this ⃗ v are not equivalent, so our claim is verified. For case 2, we may rearrange as ⃗ K T = (0, 0), (0, 1), ..., (0, p − 1). For case 3, ⃗ K T can be taken as (n, 0) or (0, n) where n = 0, 1, · · · , p − 1. Secondly, consider general cases, when detN is prime factorized as Here, p i are prime numbers and a i are non-negative integers. Then, gcd(n 2 , m) As before, ⃗ Ks' are identical if they differ by Eq.(101). Now, let us select α, β to satisfy v 2 = 0, so we have General solutions are given by If we take l = 1, we obtain We know that v 2 = αm + βn 2 is always 0 modulo n i=1 p b i i . Thus, following points are inequivalent with each other and not identified with any of those in Eq.(110), Altogether we classified alignment of all ⃗ K. Consequently, trρ(S) can be evaluated systematically for each value of detN without individual treatment of N .

Result
Here, we show the results showing how the number of zero-modes is dependent on the assignments of the flux N . They are summarized in Tables 1,2, and 3. It is convenient to use the Legendre and the Jacobi symbols for the presentation of our results. The Legendre symbol is defined for an odd prime p and an integer a as [34], : if a is a quadratic residue modulo p and a ̸ ≡ 0 (mod p), The Jacobi symbol is an extension of the Legendre symbol. For a positive integer a and a positive odd integer P , the Jacobi symbol a P J is defined in terms of the Legendre symbol, where P is prime factorized as P = p k 1 1 p k 2 2 · · · p kr r . It is worth noting that the Jacobi symbol has the following property [34],  Table 1:  Table 2: ≡ 0, (i = 1 and 2) 7 7 5 5 Table 3:

Derivation
Here, we show the derivation of some of our results. The reason of the appearance of Legendre and Jacobi symbols will become clear.
When detN = p (p in an odd prime): We have gcd(n i , m) = 1, (i = 1 or 2). Thus, the trace is given by This can be evaluated by use of results in number theory [34,35] as, Then, one might wonder why we have only trρ(S) = i when detN = 7 in Table1. It is simply explained because there is no positive definite 2 × 2 symmetric integer matrix N such that n i is non-quadratic residue modulo 7. We prove this fact in Appendix B.1. Due to the same reason, we do not have trρ(S) = −i when detN = 23 in Table 2.
where we transformed the summation variable as K = pL + qM, (L ∈ Z/q), (M ∈ Z/p) in the second equality. We have used the law of quadratic reciprocity, in the third equality. If gcd(n 1 , m) = p, gcd(n 2 , m) = q, we obtain We have only trρ(S) = i when detN = 15 in Table1. This is because there is no positive definite 2 × 2 symmetric integer matrix N which would correspond to trρ(S) = −i.
When detN ≡ 0 (mod 4): We do not obtain a general formula. However, the trace can be evaluated analytically if desired.
As an example, we show our derivation when detN = 12 in Appendix B.2.
When detN = p 2 (p is an odd prime): If gcd(n i , m) = 1, we obtain trρ(S) = 1 p as in ref. [35]. If gcd(n 1 , m) = gcd(n 2 , m) = p, we can show is a well-known result [34]. It may be interesting to construct a trace evaluating formula which is applicable when detN is an arbitrary integer.

Observation and discussion
We find that three generation models appear when the size of the flux is 8 ≤ detN ≤ 16. It may be questioned the possibility of three degeneracy in larger detN , so that we have not captured all three generation models within Tables 1, 2, and 3. Following analysis will clarify that our results are in fact enough. It is obvious from eq.(115) that holds. Firstly, let us assume that D (+i) is the smallest. From eqs. (98) and (100), From eqs. (99) and (124), one finds Then one obtains We find that the necessary condition to obtain D (+i) = 3 is detN ≤ 24. Similarly, we consider the cases when D (±1) or D (−i) is the smallest. It can be checked that detN ≤ 24 is still necessary to generate 3 degeneracy of zero-modes.
We focus on the following algebraic relation, This allows us to relate ST 1 T 2 to a Z 3 -twist which then is used to construct a T 4 /Z 3 orbifold. The complex structure moduli behave under the ST 1 T 2 as, Then the complex coordinate is transformed as, This is nothing but a Z 3 -twist. We define on the complex coordinate of T 4 . This means that the moduli of T 4 must be ST 1 T 2 invariant leading to the fixing of moduli as shown in eq.(130).

Lattice vectors
Here, we explicitly show the lattice vectors defining the T 4 /Z (a) 3 orbifold, Notice that this is identical to the root lattice of SU (3) × SU (3). Under the Z 3 -twist realized by the ST 1 T 2 transformation, they behave as This Z 3 -twist is the Coxeter element of SU (3) [30,31].

Flux
When the complex structure moduli are Ω (ST 1 T 2 ) , the F-flat condition shown in eq.(21) restricts N to be a symmetric matrix. Then we notice that T 1 T 2 transformation is consistent with the the condition p xx = p yy = 0 according to eq.(79). Moreover, in order to write down the T 1 T 2 transformation of the wavefunctions as in eq.(80), diagonal elements of N must be even. Thus, Note that N in eq.(135) is ST 1 T 2 invariant. We will consider the case when the positive definite condition eq.(51) is satisfied, so that N is positive definite.

Zero-mode counting method
In order to analyze the number of zero-modes, we evaluate the trace of the transformation matrix, ρ( Let us denote the number of degeneracy in zero-modes by D (ω n ) , (n = 0, 1, 2). Then we have Eqs. (137) and (138) constrain three real degrees of freedom. Thus, we have sufficient information to determine D (+1) , D (ω) , and D (ω 2 ) . Results are shown in Tables 4 and 5.  Table 4:  Table 5:

Observation and discussion
We find that three generation models appear when the size of the flux is detN = 7, 11, 12. It can be shown that three degeneracy of zero-modes is not produced for other values of detN . It is obvious that holds. Firstly, let us assume D (+1) ≥ D (ω) ≥ D (ω 2 ) and carry out our analysis. From eqs. (138) and (139), we get We substitute D (ω 2 ) = 3, and define x, y as x := D (+1) − 3, y := D (ω) − 3, (x ≥ y ≥ 0). Then, f (x, y) = x 2 − xy + y 2 satisfies From eq.(137), the variables x and y are related by x + y = detN − 9. Thus, we obtain If detN ≥ 9, the minimum of f (x(y), y), (y ≥ 0) is 1 4 (detN − 9) 2 . This means needs to be satisfied as a necessary condition for the realization of three generation models. Eq.(143) tells us that detN ≤ 17 is necessary. Similarly, we can consider cases when is not satisfied. In any cases, eq.(143) is obtained. Consequently, we do not need to search for three generation models in the region, detN > 17.
We focus on the following algebraic relation, This allows us to relate (ST 3 ) 2 to a Z 3 -twist which then is used to construct a T 4 /Z 3 orbifold. The complex structure moduli behave under (ST 3 ) 2 as.
To find a (ST 3 ) 2 invariant Ω ∈ H 2 , we search ST 3 invariant moduli, Ω (ST 3 ) . It is shown in eq.(176). Then, the complex coordinate is transformed as, We define T 4 /Z

Zero-mode counting method
To determine the number of zero-modes, we evaluate the transformation matrix ρ((ST 3 ) 2 ) as, Then we solve equations of the form eqs. (137) and (138). Results are shown in Table 6.

T 4 /Z 5
We focus on the following algebraic relation, This allows us to relate ST 1 T 3 to a Z 5 -twist which then is used to construct a T 4 /Z 5 orbifold. The complex structure moduli behave under the ST 1 T 3 as, Then the complex coordinate is transformed as Since Ω 5 (ST 1 T 3 ) = I 2 , this is nothing but a Z 5 -twist. We define a T 4 /Z 5 orbifold by imposing the following on the complex coordinate of T 4 . This means that the moduli of T 4 must be ST 1 T 3 invariant leading to the fixing of moduli as shown in eq.(150).

Lattice vectors
Here, we explicitly show the lattice vectors defining the T 4 /Z 5 orbifold, Notice that this is identical to the root lattice of SU (5). Under the Z 5 -twist realized by the ST 1 T 3 transformation, they behave as This Z 5 -twist is the Coxeter element of SU (5) root lattice.

Flux
When the complex structure moduli are Ω (ST 1 T 3 ) , the F-flat condition shown in eq.(21) restricts N to be of the form Then, we notice that T 1 T 3 transformation is consistent with the the condition p xx = p yy = 0 according to eq.(79). Moreover, in order to write down the T 1 T 3 transformation of the wavefunctions as in eq.(80), diagonal elements of N (B 1 + B 3 ) must be even. This further restricts N to satisfy n 1 ≡ n 2 ≡ 0 (mod 2). Thus, we obtain As a result, detN ≡ 0 (mod 4) is necessary. We will consider the case when the positive definite condition eq.(51) is satisfied.
Once the trace is evaluated, Eq.(159) constrain D (ζ k ) up to D (1) = D (ζ) = · · · = D (ζ 4 ) = 1. 6 This arbitrariness can be eliminated once we fixed detN . Therefore, eqs.(158) and (159) provide sufficient information to determine D (ζ k ) . The results are shown in Table 7.  We find that three generation models appear when the size of the flux is detN = 16, 20. It is obvious that even if we used the algebraic relation (ST 2 T 3 ) 5 = I 4 instead of eq.(148), we obtain essentially the same result. This simply corresponds to the interchange of two complex coordinates z 1 and z 2 .
We focus on the following algebraic relation, Note that Ω (ST 1 T −1 2 ) is connected to Ω (ST 1 T 2 ) by T 2 transformation. This means that lattice points are identical between T 4 /Z (1) 6 and T 4 /Z (a) 3 . However, the complex coordinate is transformed differently This is a Z 6 -twist. We define T 4 /Z (1) 6 orbifold by identifying this twist. The Z 6 -twist is the generalized Coxeter element of SU (3) including the outer automorphism of two simple roots [30,31].

T 4 /Z (2) 6
We focus on the following algebraic relation, (T 1 T 2 ) −1 S invariant Ω ∈ H 2 is uniquely determined as, which is identical to Ω (ST 1 T 2 ) in eq.(130). The complex coordinate is transformed as This is nothing but a Z 6 -twist which we identify to define T 4 /Z (2) 6 orbifold. Note that two consecutive Z 6 -twists are equivalent to the Z 3 -twist in eq.(131). Three consecutive Z 6 -twists are equivalent to the Z 2 -twist.

Lattice vectors and flux
The lattice vectors defining the T 4 /Z (2) 6 orbifold are shown in eq.(133). Under the Z 6 -twist realized by (T 1 T 2 ) −1 S transformation, they behave as The consistent flux N on this T 4 /Z 6 orbifold is shown in eq.(135). Then (T 1 T 2 ) −1 S invariance of the flux is satisfied.

Zero-mode counting method
To analyze the number of zero-modes, we evaluate the trace of ρ((T 1 T 2 ) −1 S) as, Let us denote the number of degeneracy in zero-modes by D (κ k ) , (k = 0, 1, .., 5). We have the following information, Eqs. (171) and (172) constrain only three real degrees of freedom. Thus, we need more constraint equations to determine D's. From the fact that Z 6 eigenstates with eigenvalues 1, κ 2 , and κ 4 are Z 2 even, we have Here, N Z 2 ,+ denotes the number of even modes on T 4 /Z 2 . Likewise from the relationship between Z 6 and Z 3 eigenvalues, we have In the above, D Z 3 ,(ω k ) represent the number of zero-modes with Z 3 charges ω k on the T 4 /Z    We find that three generation models appear when 12 ≤ detN ≤ 27. Although we can not give a complete proof, it is observed that the trace, trρ((T 1 T 2 ) −1 S) is always equal to ω = e

T 4 /Z (3) 6
We focus on the following algebraic relation, ST 3 invariant Ω ∈ H 2 is uniquely determined as, Since Ω 6 (ST 3 ) = I 2 , this is a Z 6 -twist which we identify to define T 4 /Z 6 orbifold. Note that the composition of three Z 6 -twists is equivalent to which is the Z 2 permutation. Composition of two Z 6 -twists is equivalent to the Z 3 -twist in eq.(146).

Lattice vectors
Here, we explicitly show the lattice vectors defining the T 4 /Z 6 orbifold, Notice that this is identical to the root lattice of SU (3) × SU (3) with the Dynkin diagram shown in Fig.1. Under the Z 6 -twist realized by ST 3 transformation, they behave as (180)

Flux
Magnetic fluxes on the T 4 /Z We see that the F-flat condition shown in eq.(21) is satisfied. We also notice that T 3 transformation is consistent with p xx = p yy = 0 according to eq.(79). Moreover, since off-diagonal elements of N are even, T 3 transformation of the wavefunctions can be written by eq.(80). Note that N in eq.(181) is ST 3 invariant. We will study the case when positive definite condition eq.(51) is satisfied, so that N is positive definite.

Zero-mode counting method
To analyze the number of zero-modes, we evaluate the trace of ρ(ST 3 ) as, Let us denote the number of zero-modes by D (κ k ) , (k = 0, 1, ..., 5). We have equations of the form eqs. (171) and (172). From the fact that Z 6 eigenstates with eigenvalues 1, κ 2 and κ 4 are even-modes of the Z 2 -permutation, we have For details of T 4 /Z 2 permutation orbifold models, see Appendix C. Moreover, from the relationship between Z 6 and Z 3 eigenvalues, we have In the above, D Z 3 ,(ω k ) represent the number of zero-modes with Z 3 charges ω k on the T 4 /Z    We find that three generation models appear when 9 ≤ detN ≤ 28 or detN = 33, 36, 44.
We focus on the following algebraic relation, where γ P ∈ Sp(4, Z) is defined as This is a Z 6 -twist which we identify to define T 4 /Z (4) 6 orbifold. From eq.(82), we understand γ P as Z 2 permutation which interchanges the two complex coordinates z 1 and z 2 . The complex coordinates are transformed as, Note that the composition of three Z 6 -twists is equivalent to the Z 2 permutation. Composition of two Z 6 -twists is equivalent to the inverse of the Z 3 -twist we studied in eq.(131).

Lattice vectors
The lattice vectors defining the T 4 /Z  Under the Z 6 -twist realized by ST 1 T 2 γ P transformation, they behave as (188)

Flux
Magnetic fluxes on the T 4 /Z orbifold via the Z 2 permutation, γ P ∈ Sp(4, Z). From eq.(83), we find fluxes N of the form in eq.(135) are transformed as, Thus, further identification by Z 2 permutation requires n ′ 1 = n ′ 2 .

Zero-mode counting method
To analyze the number of zero-modes, we evaluate the trace of ρ(ST 1 T 2 γ P ) as, Let us denote the number of zero-modes by D (κ k ) , (k = 0, 1, ..., 5). We have equations of the form eqs. (171) and (172). From the fact that Z 6 eigenstates with eigenvalues 1, κ 2 and κ 4 are even-modes of the Z 2 -permutation, we have For details of T 4 /Z 2 permutation orbifold models, see Appendix C. Moreover, from the relationship between Z 6 and the inverse of Z 3 eigenvalues, we have In the above, D Z 3 ,(+ω k ) represent the number of zero-modes with Z 3 charges ω k on the T 4 /Z  We find that three generation models appear when 7 ≤ detN ≤ 28 or detN = 36, 44.

T 4 /Z 8
We focus on the following algebraic relation, This suggests that we can construct a T 4 /Z 8 orbifold with ST 1 T −1 2 T −1 3 invariant complex structure moduli Ω ∈ H 2 which is uniquely determined as, (195) Since Ω 8 this is a Z 8 -twist. By identifying coordinates of T 4 by this twist, we obtain the T 4 /Z 8 orbifold.

Lattice vectors
Here, we study the lattice vectors defining the T 4 /Z 8 orbifold. For this purpose, it is convenient to consider moduli Ω related to Ω (ST 1 T −1 Although basis vectors of the lattice are different between the two, lattice points are invariant under the Sp(4, Z) transformation. Thus, it makes sense to move to Lattice vectors which correspond to Ω ′ are Their orientation corresponds to the root lattice of SO(8) as in Fig.3.

T 4 /Z 10
We focus on the following algebraic relation, This suggests that we can construct a T 4 /Z 10 orbifold with ( The complex coordinates are transformed as, under the Z 10 -twist. This is the general Coxeter element of SU (5) including the Z 2 outer automorphism. Note that the composition of five Z 10 -twists is equivalent to the Z 2 -twist. Composition of four Z 10 -twists is equivalent to the Z 5 -twist by the ST 1 T 3 transformation as in eq.(151) 8 . Degeneracy of zero-modes D (η 0 ) , ..., D (η 9 ) (η = e πi/5 ) can be analyzed as before by use of modular transformation. It seems clear that we obtain essentially the same result when we consider the algebraic relation ((T 2 T 3 ) −1 S) 10 = I 4 instead of eq.(200) as this corresponds to the interchange of two complex coordinates z 1 and z 2 .

T 4 /Z 12
We focus on the following algebraic relation, This suggests that we can construct a T 4 /Z 12 orbifold with ST 1 invariant complex structure moduli Ω ∈ H 2 which is uniquely determined as, The complex coordinates are transformed as, which is clearly a Z 12 ( ∼ = Z 3 × Z 4 )-twist. By identifying coordinates of T 4 by this twist, we obtain the T 4 /Z 12 orbifold. The lattice vectors are equivalent to the root lattice of SU (3) × SU (2) × SU (2).

Flux
Magnetic fluxes on the T 4 /Z 12 orbifold can be consistently introduced if N is of the form, The F-flat condition shown in eq.(21) is satisfied only if N is diagonal. Then, it follows that flux is ST 1 invariant. The (1, 1)-component of N needs to be even for the T 1 transformation of zero-modes written by eq.(80) 9 .

Discussion
Both complex structure Ω and flux N are diagonal as shown in eqs.(203) and (205). Thus, we are led to analyze magnetized T 2 /Z 3 × T 2 /Z 4 where the flux size is M 1 = 2n ′ 1 and M 2 = n 2 respectively. Zero-modes on them are already studied in ref. [13]. Thus, zero-mode number can be analyzed easily by just noting the fact that zero-mode wavefunctions on magnetized T 2 /Z 3 × T 2 /Z 4 are given by the product of those on each orbifold.
It is obvious that even if we used the algebraic relation (ST 2 ) 2 = I 4 instead of eq.(202), we obtain essentially the same result. This simply corresponds to the interchange of two complex coordinates z 1 and z 2 .  correspond to negative chirality wavefunctions on T 4 /Z (4) 6 and vice versa. 9 It was also pointed out that the flux needs to be even for the invariance of boundary conditions under the T transformation in magnetized T 2 models [17,24].
A parity transformation, relates the two orbifolds. If ⃗ z denotes complex coordinate of T 4 /Z We observe that e are given by eqs.(181) and (189) respectively. We find where showing that N (p) is consistently identified with N (ST 1 T 2 γ P ) in eq.(189). Notice that the determinant changes sign as detN (ST 3 ) = −detN (p) implying that positive (negative) chirality on 6 and negative (positive) chirality on T 4 /Z 6 are related. For more detailed discussions, see Appendix D.
Lastly, we confirm that Z 6 -twist generated by ST 3 on ⃗ z coordinate system is identified as Z 6 -twist by ST 1 T 2 γ P on ⃗ z (p) system. It is straightforward to check that is equivalent to This guarantees that Z 6 eigenstates on T 4 /Z 6 side are Z 6 eigenstates on T 4 /Z 6 side as well. In short, magnetized T 4 /Z are connected by the parity transformation. Positive chirality on one side correspond to negative chirality on the other side. This kind of duality will be useful to analyze the number of negative chirality zero-modes by modular transformation since positive chirality wavefunctions are written by the Riemann theta function whose behaviour under Sp(4, Z) is well known.

Conclusion
We have studied T 4 /Z N orbifold models with background magnetic fluxes. We saw that Sp(4, Z) modular transformations can be related to Z N -twists defining the orbifolds. Behaviour of zeromode wavefunctions under the modular transformation was studied. This enabled systematic analysis of zero-mode numbers. We used results of number theory in mathematics for this. Our results explicitly show the condition needed to realize three generations of fermions in the effective field theory. Moreover, we saw that the parity transformation on the compact space can elegantly relate positive and negative chirality wavefunctions. Obtained negative chirality modes are consistent with ref. [26] when real part of the complex structure ReΩ is vanishing. What is more, our result is relevant even when ReΩ ̸ = 0. This revealed a kind of duality between two different T 4 /Z 6 orbifolds as we studied in Section 7.
This work will activate the study of magnetized T 2n /Z N models. Extension to orbifolds of T 6 will be our future work. Also, it will be helpful to develop mathematical formula to evaluate the trace of transformation matrix of zero-modes under modular transformation with arbitrary flux size detN .
When we embed our T 4 /Z N orbifold models to superstring theory, some models may include tachyonic modes in closed string sector, e.g. T 4 /Z 5 . Such configurations may be unstable. Some moduli develop their vacuum expectation values deforming geometry without changing topology. If that is a stable manifold, our results on T 4 /Z 5 could be realized, because the number of zero-modes is determined by topology. Such a study on embedding to string theory is beyond our scope.

A Modular transformation of zero-modes A.1 S transformation
Here we give a proof of eq.(71), where it is assumed that N is a symmetric positive definite integer matrix. It is known that the Riemann theta function satisfies the following relation [32], Note that the branch of the square root is chosen so that it gives positive value when Ω is purely imaginary. Firstly, replace the complex coordinates as ⃗ z → ⃗ z + N −1 ⃗ J and by use of eqs.(304), (305), we obtain Secondly, we replace the complex structure moduli as Ω → N −1 Ω to get, The Riemann theta function on the right-hand side of eq.(214) can be written as We have changed the summation variable as ⃗ l = N⃗ s + ⃗ K, where ⃗ K's are integer points inside the lattice Λ N spanned by N⃗ e n . Thus, we get Thirdly, we consider the S transformation of the ⃗ z dependent factor, When this factor is multiplied by the factor e πi⃗ z T (N −1 Ω) −1 ⃗ z which appears in the right-hand side of eq.(216), we get Consequently, we obtain the S transformation of the zero-mode wavefunctions, Note that the normalization constant shown in eq.(47) is invariant under the S transformation.

A.2 T transformation
Here we give a proof of eq.(80). It is known that the Riemann theta function satisfies the following relation provided diagonal matrix elements of N B i are all even [32], Firstly, replace the complex coordinates as ⃗ z → ⃗ z + (Ω + B i )N −1 T ⃗ J and by use of eq.(305), we obtain Secondly, by use of eqs.(304), (306), we get ϑ A.3 A ∈ GL(2, Z) transformation Here we give a proof of eq.(85). It is known that the Riemann theta function satisfies the following relation for A ∈ GL(2, Z) [32], Firstly, replace the complex coordinates as ⃗ z → ⃗ z + ΩA T N −1 T ⃗ J and use eq.(305) to get Secondly, replace the complex coordinates and moduli as ⃗ z → N⃗ z, Ω → N Ω to get Note that the ⃗ z dependent factor e πi[N⃗ z] T ·(ImΩ) −1 ·Im⃗ z is invariant under the A transformation. The normalization constant eq.(47) is also unchanged. Consequently, we obtain Proof To prove it, we use the Kronecker symbol [36]. We denote arbitrary non-zero integer by n, with prime factorization, where u is a unit (±1), and p i are odd primes. Let a be an integer. Then the Kronecker symbol a n K is given by a The factor a u K is just 1 when u = 1. When u = −1, we define it by Notice that when n is a positive odd integer, the Kronecker symbol is identical to the Jacobi symbol. For even n, we define a 2 K by The Kronecker symbol has the following property [36]. For a ̸ ≡ 3 (mod 4), a ̸ = 0, we have Next, we note the following basic fact regarding the quadratic residue.
• a is a quadratic residue modulo n ↔ a is a quadratic residue modulo p k for every prime power dividing n.
We show there is no solution of eq.(235) for any positive integer, s. When s = 1, there is no solution since −7 ≡ −1 (mod 6) is not a quadratic residue modulo 6. Now for arbitrary s ∈ Z + , we have because of the property Eq.(232). Suppose 7s − 1 is prime factorized as 10 (→) : m 2 − a = nr = p e0 0 p e1 1 · · · p e k k r for some r ∈ Z. Then, taking p αj j , (0 ≤ j ≤ k, 1 ≤ α j ≤ e j ) gives us m 2 ≡ a (mod p where p i , (i = 1, ..., k) denotes odd primes. Then it is immediate holds because −7 2 K = 1. When the Jacobi symbol is −1, we must have at least one p e l l , (1 ≤ l ≤ k) such that −7 is its non-quadratic residue, According to the aforementioned basic property of the quadratic residue, this is sufficient to conclude that eq.(235) has no solution.
By the same argument, we conclude that have no integer solution for any s ∈ Z + .

B.2 Number of zero-modes when detN = 12
Here, we analyze the number of positive chirality zero-modes when detN = 12. We only need to consider cases when gcd(n 1 , m) ≥ gcd(n 2 , m) because trρ(S) is invariant under the interchange of n 1 and n 2 .
When n ′ i ̸ ≡ 0 (mod p), (i = 1 or 2) : It suffices to consider the case n ′ 1 ̸ ≡ 0 (mod p). We transform the variable as where g ∈ Z. Then we can write eq.(252) as We can always find g ∈ Z such that n ′ 1 g − m ′ ≡ 0 (mod p). 12 Having chosen such a g, we obtain trρ(S) = 1 p When n ′ i ≡ 0 (mod p), (i = 1 and 2) : We consider the case n ′ 1 ≡ n ′ 2 ≡ 0 (mod p). 13 In this case, we have Let us define L 1 and L 2 as Even after this transformation, the new variables L 1 , L 2 move from 0 to p − 1. To understand it, consider where r, s ∈ Z. We verify our claim if we could show that all solutions of above equations satisfy L 1 ≡ L ′ 1 (mod p) and L 2 ≡ L ′ 2 (mod p) simultaneously. Eq.(258)×(p − 1) + Eq.(259) gives us Since we have gcd(p, 2) = 1, the relation L 1 ≡ L ′ 1 (mod p) must be satisfied. Substituting this result into Eq.(258) yields L 2 ≡ L ′ 2 (mod p). Thus, we verified our claim. Consequently, we get C T 4 /Z 2 permutation orbifold Here, we study the T 4 /Z 2 permutation orbifold and zero-mode number on it. (T 2 × T 2 )/Z 2 permutation orbifold was studied in Ref. [24]. Our results are consistent with the previous works and more general in the sense that we took into account oblique components of fluxes and complex structure moduli. Moreover, we will see that orbifolding of T 4 by the Z 2 permutation is related to the modular transformation.

C.1 Z 2 permutation
We define Z 2 permutation as z 1 ↔ z 2 . This can be realized by the modular transformation shown in eq.(185). The two complex coordinates are interchanged Now, let us look at the conditions of N and Ω for being invariant under the γ P transformation.
For Ω, therefore we need τ 1 = τ 2 . (266) The algebraic relation, γ 2 P = I suggests that we can construct a Z 2 permutation orbifold where two complex coordinates z 1 and z 2 are identified. Then the flux must be of the form for the Z (per) 2 invariance. The complex structure needs to be for the Z The above results can be understood by referring to eq.(85). Z

D Negative chirality zero-mode wavefunction
Here we study the negative chirality zero-mode wavefunctions which become non-zero when detN < 0. Our results are consistent with ref. [26] when the complex structure moduli are purely imaginary. We have also studied when ReΩ is non-vanishing, and obtained solutions satisfying both Dirac equation and proper boundary conditions even in such a case.

D.2 Solution
Let us solve the Dirac equations under the boundary conditions in eq.(32). Our argument is based on the basic fact that a parity transformation flips the chirality of spinors. Here, we study when N and Ω are of the form, N = n m m n , detN < 0, for simplicity. Note that our choice of N and Ω is consistent with the F-flat condition in eq.(21). Firstly, eq.(272) gives us two solutions for q, Secondly, consider the parity transformation of the complex coordinates as shown in eq.(206). In the new coordinate system ⃗ z (p) , the flux is N (p) = m n n m , detN (p) > 0.
This shows that we obtain positive chirality solutions in the new coordinate system. We should also note that the F-flat condition is maintained even after the parity transformation. If N (p) ImΩ (p) > 0, we obtain ψ ⃗ j (p) , Ω (p) ) = N ⃗ j (p) · e πi[N (p) ⃗ z (p) ] T ·(ImΩ (p) ) −1 ·Im⃗ z (p) · ϑ ⃗ j T (p) 0 (N (p) ⃗ z (p) , N (p) Ω (p) ), N T in the first positive chirality component, ψ 1 + as we have reviewed in Section 2. 15 Maintaining the profile of wavefunctions, we describe positions on T 4 in terms of the initial complex coordinates ⃗ z. Then we expect that negative chirality wavefunctions we sought will be obtained. The results are as follows, When ReΩ = 0, our solution is consistent with the result obtained in ref. [26]. Thirdly, we should confirm that eq.(279) indeed satisfies the Dirac equation in eq.(271). We can check that the obtained solutions are correct with the choice q = β α = −1 through direct substitutions. On the other hand, another solution, q = +1 is selected when N (p) ImΩ (p) < 0. In this case, the negative chirality solution is modified from eq.(279). However, it can be easily obtained by noting the fact that eq.(53) is the right choice instead of eq.(278) if N (p) ImΩ (p) is negative definite.
Thus, we just need to verify which is straightforward.

E F-flatness supersymmetry condition
We review the F-flatness supersymmetry condition in the 10D N = 1 super Yang-Mills theory (SYM).

E.1 10D N = 1 SYM
The action of 10D N = 1 SYM with U (N ) gauge symmetry is given by where the trace is taken over the gauge space of the adjoint representation. The gauge coupling constant is denoted by g. We first consider T 4 × T 2 ≃ R 4 /Λ 4 × R 2 /Λ 2 compact space, where Λ 4 and Λ 2 represent 4D and 2D lattices respectively. Thus, we can take real orthogonal coordinates over the 10D space-time and they are denoted by X M , (M = 0, 1, ..., 9). Metric tensor in this basis is given by G M N = η M N = diag(−1, 1, · · · , 1). 10D gauge field is denoted by A M and 10D Majorana-Weyl spinor is denoted by λ. Its field strength and the covariant derivatives are given by Suppose that Λ 4 is spanned by e i ∈ C 2 , (i = 1, 2, 3, 4) where e 1 and e 3 have the same norm and perpendicular to each other. Then, by a SO(4) rotation in R 4 , we can write their components as eq.(1). Similarly, Λ 2 is spanned by e 5 = 2πr and e 6 = (2πr)τ where τ ∈ C and r is a scale factor. For convenience, we define complex coordinates on T 6 ≃ T 4 × T 2 as, (X 4 + iX 5 ), z 2 = 1 2πR (X 6 + iX 7 ), z 3 = 1 2πr (X 8 + iX 9 ).
It follows that corresponding gauge fields are given by Note that z 1 and z 2 are identified as those defined on the right-hand side of eq.(2). We have shift identifications of the form ⃗ z ∼ ⃗ z + m a ⃗ e a + n b Ω⃗ e b , (m a(=1,2,3) , n b(=1,2,3) ∈ Z), where The metric corresponding to the complex coordinates is written as, where h ij = 1 2 (2πR i ) 2 δ ij , (i, j = 1, 2, 3) and R 1 = R 2 = R and R 3 = r. Then we find where vielbeins are given by therefore Roman indices correspond to the local Lorentz frame. We write the inverse and the complex conjugates of the vielbeins as e i i e j i = δ j i , e i i e j i = δ j i ,ē¯ī i = (e i i ) * .
We raise and lower Italic indices by h¯i j and h ij respectively. Similarly, Roman indices are raised and lowered by δ¯i j and δ ij respectively. Orbifolds of tori are obtained by identifying positions related by rotational symmetries. Then the moduli are usually fixed. We still have the same metric as eqs.(292) and (293) in the bulk.