Supersymmetric SU(5) GUT with Stabilized Moduli

We construct a minimal example of a supersymmetric grand unified model in a toroidal compactification of type I string theory with magnetized D9-branes. All geometric moduli are stabilized in terms of the background internal magnetic fluxes which are of"oblique"type (mutually non-commuting). The gauge symmetry is just SU(5) and the gauge non-singlet chiral spectrum contains only three families of quarks and leptons transforming in the $10+{\bar 5}$ representations.


Introduction
Closed string moduli stabilization has been intensively studied during the last years for its implication towards a comprehensive understanding of the superstring vacua [1,2], as well as due to its significance in deriving definite low energy predictions for particle models derived from string theory. Such stabilizations employ various supergravity [1,3], non-perturbative [2] and string theory [4][5][6] techniques to generate potentials for the moduli fields. However, very few examples are known so far of a complete stabilization of closed string moduli in any specific model, while the known ones are too constrained to accommodate interesting models from physical point of view. Hence, there have been very few attempts to construct a concrete model of particle physics even with partially stabilized moduli. Nevertheless, in view of the importance of the task at hand, we revisit the type I string constructions [7,8] with moduli stabilizations [4][5][6], to explore the possibility of incorporating particle physics models, such as the Standard Model or GUT models based on grand unified groups, in such a framework.
A new calculable method of moduli stabilization was recently proposed, using constant internal magnetic fields in four-dimensional (4d) type I string compactifications [4,5]. In the generic Calabi-Yau case, this method can stabilize mainly Kähler moduli [4,9] and is thus complementary to 3-form closed string fluxes that stabilize the complex structure and the dilaton [3]. On the other hand, it can also be used in simple toroidal compactifications, stabilizing all geometric moduli in a supersymmetric vacuum using only magnetized D9branes that have an exact perturbative string description [10,11]. Ramond-Ramond (RR) tadpole cancellation requires then some charged scalar fields from the branes to acquire non-vanishing vacuum expectation values (VEVs), breaking partly the gauge symmetry in order to preserve supersymmetry [5]. Alternatively, one can break supersymmetry by D-terms and fix the dilaton at weak string coupling, by going "slightly" off-criticality and thus generating a tree-level bulk dilaton potential [12].
There are two main ingredients for this approach of moduli stabilization [4,5]: (1) A set of nine magnetized D9-branes is needed to stabilize all 36 moduli of the torus T 6 by the supersymmetry conditions [13,14]. Moreover, at least six of them must have oblique fluxes given by mutually non-commuting matrices, in order to fix all off-diagonal components of the metric. On the other hand, all nine U (1) brane factors become massive by absorbing the RR partners of the Kähler class moduli [14]. (2) Some extra branes are needed to satisfy the RR tadpole cancellation conditions, with non-trivial charged scalar VEVs turned on in order to maintain supersymmetry.
In this work, we apply the above method to construct phenomenologically interesting models. In the minimal case, three stacks of branes are needed to embed locally the Standard Model (SM) gauge group and the quantum numbers of quarks and leptons in their intersections [15]. They give rise to the gauge group U In order to obtain an odd number (3) of fermion generations, a NS-NS (Neveu-Schwarz) 2-form B-field background [16] must be turned on [17]. This requires the generalization of the minimal set of branes with oblique magnetic fluxes that generate only diagonal 5-brane tadpoles on the three orthogonal tori of T 6 = 3 i=1 T 2 i . We find indeed a set of eight such "oblique" branes which combined with U (5) can fix all geometric moduli by the supersymmetry conditions. The metric is fixed in a diagonal form, depending on six radii given in terms of the magnetic fluxes. At the same time, all nine corresponding U (1)'s become massive yielding an SU (5) × U (1) gauge symmetry. This U (1) factor cannot be made supersymmetric without the presence of charged scalar VEVs. Moreover, two extra branes are needed for RR tadpole cancellation, which also require non-vanishing VEVs to be made supersymmetric. As a result, all extra U (1)'s are broken and the only leftover gauge symmetry is an SU (5) GUT. Furthermore, the intersections of the U (5) stack with any additional brane used for moduli stabilization are non-chiral, yielding the three families of quarks and leptons in the 10+5 representations as the only chiral spectrum of the model (gauge non-singlet).
To elaborate further, the model is described by twelve stacks of branes, namely U 5 , U 1 , O 1 . . . , O 8 , A, and B. The SU (5) gauge group arises from the open string states of stack-U 5 containing five magnetized branes. The remaining eleven stacks contain only a single magnetized brane. Also, the stack-U 5 containing the GUT gauge sector, contributes to the GUT particle spectrum through open string states which either start and end on itself 1 or on the stack-U 1 , having only a single brane and therefore contributing an extra U (1). For this reason we will also refer to these stacks as U 5 and U 1 stacks.
The matter sector of the SU (5) GUT is specified by 3 generations of fermions in the group representations5 and 10 of SU (5), both of left-handed helicity. In the magnetized branes construction, the 10 dimensional (antisymmetric) representation of left-handed fermions: arises from the doubly charged open string states starting on the stack-U 5 and ending at its orientifold image: U * 5 and vice verse. They transform as 10 (2,0) of SU (5) × U (1) × U (1), where the first U (1) refers to stack-U 5 and the second one to stack-U 1 , while the subscript denotes the corresponding U (1) charges. The5 of SU (5) containing left-handed chiral fermions, or alternatively the 5 with right-handed fermions: 1 For simplicity, we do not distinguish a brane stack with its orientifold image, unless is explicitly stated.
are identified as states of open strings starting from stack-U 5 (with five magnetized branes) and ending on stack-U * 1 (i.e. the orientifold image of stack-U 1 ) and vice verse. The magnetic fluxes along the various branes are constrained by the fact that the chiral fermion spectrum, mentioned above, of the SU (5) GUT should arise from these two sectors only.
Our aim, in this paper, is to give a supersymmetric construction which incorporates the above features of SU (5) GUT while stabilizing all the Kähler and complex structure moduli. More precisely, for fluxes to be supersymmetric, one demands that their holomorphic (2, 0) part vanishes. This condition then leads to complex structure moduli stabilization [4]. In our case we show that, for the fluxes we turn on, the complex structure τ of T 6 is fixed to with 1 1 3 being the 3 × 3 identity matrix.
In this paper, we make use of the conventions given in Appendix A of Ref. [5], for the parametrization of the torus T 6 , as well as for the general definitions of the Kähler and complex structure moduli. In particular, the coordinates of three factorized tori: (T 2 ) 3 ∈ T 6 are given by x i , y i i = 1, 2, 3 with a volume normalization: For Kähler moduli stabilization, we make use of the mechanism based on the magnetized D-branes supersymmetry conditions as discussed in [4,5,13]. Physically this corresponds to the requirement of vanishing of the potential which is generated for the moduli fields from the Fayet-Iliopoulos (FI) D-terms associated with the various branes. Even in this simplified scenario, the mammothness of the exercise is realized by noting that every magnetic flux that is introduced along any brane also induces charges corresponding to lower dimensional branes, giving rise to new tadpoles that need to be canceled. In particular, for the type I string that we are discussing, there are induced D5 tadpoles from fluxes along the magnetized D9 branes. These fluxes, in turn, are forced to be non-zero not only in order to satisfy the condition of zero net chirality among the U 5 and the extra brane stacks (except with the U 1 ), but in order to implement the mechanism of complex structure and Kähler moduli stabilization, as well. Specifically, for stabilizing the non-diagonal components of the metric, one is forced to introduce 'oblique' fluxes along the D9-branes, thus generating 'oblique' D5-brane tadpoles, and all these need to be canceled.

Preliminaries
We now briefly review the string construction using magnetized branes, and in particular the chiral spectrum that follows for such stacks of branes due to the presence of magnetic fluxes.

Fluxes and windings
We first briefly describe the construction based on D-branes with magnetic fluxes in type I string theory, or equivalently type IIB with orientifold O9-planes and magnetized D9branes, in a T 6 compactification. Later on, in subsection 2.5, we study the introduction of constant NS-NS B-field background in this setup.
The stacks of D9-branes are characterized by three independent sets of data: (a) their multiplicities N a , (b) winding matrices WÎ , a I and (c) 1st Chern numbers m â IĴ of the U (1) background on their world-volume Σ a , a = 1, . . . , K. In our case, as already stated earlier, we have K = 12 stacks. Also, I,Î run over the target space and world-volume indices, respectively. These parameters are described below: (a) Multiplicities: The first quantity N a describes the rank of the the unitary gauge group U (N a ) on each D9 stack. with the upper block corresponding to the covering of Σ a 4 on the four-dimensional spacetime M 4 . Since these are assumed to be identical, the associated covering map Wμ µ is the identity, Wμ µ = δμ µ . The entries of the lower block, on the other hand, describe the wrapping numbers of the D9-branes around the different 1-cycles of the torus T 6 which are therefore restricted to be integers Wα α ∈ Z, ∀ α,α = 1, . . . , 6 [6].
For simplicity, in the examples we consider here, the winding matrix Wα α in the internal directions is also chosen to be a six-dimensional diagonal matrix, implying an embedding such that the six compact D9 world-volume coordinates are identified with those of the internal target space T 6 , up to a winding multiplicity factor n a α , for a brane stack-a: We will also use the notation n a 1 ≡ n a 1 n a 2 ,n a 2 ≡ n a 3 n a 4 ,n a 3 ≡ n a 5 n a 6 , (no sum on a) (2.4) to define the diagonal wrapping of the D9's on the three orthogonal T 2 's inside T 6 , given by: x i ≡ X α , α = 1, 3, 5 ; y i ≡ X α , α = 2, 4, 6 , (2.5) with periodicities: x i = x i + 1, y i ≡ y i + 1: and coordinates of the orthogonal 2-tori (T 2 i ) being (x i , y i ) for i = 1, 2, 3. For further simplification, in our example, we will choose for all stacks trivial windings: However in this section, in order to describe the formalism, we keep still general winding matrices Wα ,a α . (c) First Chern numbers: The parameters m â IĴ of the brane data given above are the 1st Chern numbers of the U (1) ⊂ U (N a ) background on the world-volume of the D9-branes. For later use, when fluxes are turned on only along three factorized T 2 's of eq. (2.6), as will be the case for some of our brane stacks, we make use of the following convenient notation:m The magnetized D9-branes couple only to the U (1) flux associated with the gauge fields located on their own world-volume. In other words, the charges of the endpoints q R and q L of the open strings stretched between the i-th and the j-th D9-brane can be written as q L ≡ q i and q R ≡ −q j , while the Cartan generator h is given by h = diag(h 1 1 1 N 1 , . . . , h N 1 1 N K ), with 1 1 Na being the N a × N a identity matrix. In addition, in type I string theory, the number of magnetized D9-branes must be doubled. Since the orientifold projection O = Ω p is defined by the world-sheet parity, it maps the field strength F a = dA a of the U (1) a gauge potential A a to its opposite, O : F a → −F a . Therefore, the magnetized D9-branes are not an invariant configuration and for each stack a mirror stack must be added with opposite flux on its world-volume.

Stabilization
We now write down the supersymmetry conditions for magnetized D9-branes in the context of type I toroidal compactifications and discuss the stabilization of complex structure and Kähler class moduli using such conditions.
The geometric moduli of T 6 decompose in a complex structure variation which is parametrized by the matrix τ ij entering in the definition of the complex coordinates and in the Kähler variation of the mixed part of the metric described by the real (1, 1)-form The supersymmetry conditions then read [4,5]: for each a = 1, . . . , K. The complexified fluxes can be written as where the matrices (p a x i x j ), (p a x i y j ) and (p a y i y j ) are the quantized field strengths in target space, given in eq. (2.8). For our choice (2.7), they coincide with the Chern numbers m a along the corresponding cycles. The field strengths F a (2,0) and F a (1,1) are 3 × 3 matrices that correspond to the upper half of the matrix F a : which is the total field strength in the cohomology basis e ij = idz i ∧ dz j [4,5].
The first set of conditions of eq. (2.11) states that the purely holomorphic flux vanishes.
For given flux quanta and winding numbers, this matrix equation restricts the complex structure τ . Using eq. (2.12), the supersymmetry conditions for each stack can first be seen as a restriction on the parameters of the complex structure matrix elements τ : 15) giving rise to at most six complex equations for each brane stack a.
The second set of conditions of eq. (2.11) gives rise to a real equation and restricts the Kähler moduli. This can be understood as a D-flatness condition. In the four-dimensional effective action, the magnetic fluxes give rise to topological couplings for the different axions of the compactified field theory. These arise from the dimensional reduction of the Wess Zumino action. In addition to the topological coupling, the N = 1 supersymmetric action yields a Fayet-Iliopoulos (FI) term of the form: The D-flatness condition in the absence of charged scalars requires then that < D a >= ξ a = 0, which is equivalent to the second equation of eq. (2.11). Finally, the last inequality in eq. (2.11) may also be understood from a four-dimensional viewpoint as the positivity of the U (1) a gauge coupling g 2 a , since its expression in terms of the fluxes and moduli reads (2.17) The above supersymmetry conditions, get modified in the presence of VEVs for scalars charged under the U (1) gauge groups of the branes. The D-flatness condition, in the low energy field theory approximation, then reads: where M s = α −1/2 is the string scale 2 , and the sum is extended over all scalars φ charged under the a-th U (1) a with charge q φ a and metric G φ . Such scalars arise in the compactification of magnetized D9-branes in type I string theory, for instance from the NS sector of open strings stretched between the a-th brane and its image a , or between the stack-a and another stack-b or its image b * . When one of these scalars acquire a non-vanishing VEV |φ| 2 = v 2 φ , the calibration condition of eq. (2.11) is modified to: Note that our computation is valid for small values of v a (in string units), since the inclusion of the charged scalars in the D-term is in principle valid perturbatively.
Actually, the fields appearing in (2.18) are not canonically normalized since the metric G φ appears explicitly also in their kinetic terms. Thus, the physical VEV is v φ G φ .
However, to estimate the validity of the perturbative approach, it is more appropriate The reason is that the next to leading correction to the D-term involves a quartic term of the type |φ| 4 , proportional to a new coefficient K, and the condition of validity of perturbation theory is Kv 2 φ /G φ << 1. A rough estimate is then obtained by approximating K ∼ G φ , which gives our condition.
The metric G φ of the scalars living on the brane has been computed explicitly for the case of diagonal fluxes [18]. In this special case, the fluxes are denoted by three angles θ a i , (i = 1, 2, 3). 3 Then suppressing index-a, we have: and with γ E being the Euler constant. The above results will be applied in section 5 to find out the FI parameters and charged scalar VEVs along three of the twelve brane stacks: Moreover, the RR moduli that appear in the same chiral multiplets as the geometric Kähler moduli, become Goldstone modes which get absorbed by the U (1) gauge bosons [4] corresponding to each of the D-terms that stabilize the relevant geometric moduli.

Tadpoles
In toroidal compactifications of type I string theory, the magnetized D9-branes induce 5brane charges as well, while the 3-brane and 7-brane charges automatically vanish due to the presence of mirror branes with opposite flux. For general magnetic fluxes, RR tadpole conditions can be written in terms of the Chern numbers and winding matrix [5,6] as: The l.h.s. of eq. (2.23) arises from the contribution of the O9-plane. On the other hand, in toroidal compactifications there are no O5-planes and thus the l.h.s. of eq. (2.24) vanishes.
For our choice of windings (2.7), Wˆi i = 1, the D9 tadpole contribution from a given stack-a of branes is simply equal to the number of branes, N a . The D5 tadpole expression also takes a simple form for the fluxes satisfying the F a (2,0) = 0 condition (2.11). The fluxes are then represented by three-dimensional Hermitian matrices (F a (1,1) ) which appeared in eq. (2.14) and the D5 tadpoles Q 5, a ij are the Cofactors of the ij matrix elements (F a (1,1) ) ij . Fluxes and tadpoles in such a form are given in Appendix A. In the matter sector, the massless spectrum is obtained from the following open string states [14,19]:

Spectrum
1. Open strings stretched between the a-th and b-th stack give rise to chiral spinors in the bifundamental representation (N a ,N b ) of U (N a ) × U (N b ). Their multiplicity I ab is given by [6]: where F a (1,1) (given in eqs. (2.13) and (2.14)) is the pullback of the integrally quantized world-volume flux m â αβ on the target torus in the complex basis (2.10), and q a is the corresponding U (1) a charge; in our case q a = +1 (−1) for the fundamental where i is the label of the i-th two-tori T 2 i , and the integersm a i ,n a i enter in the multiplicity expressions through the magnetic field as in eq. (2.8).
In the model that we construct, however, we need stacks with fluxes which contain both diagonal and oblique flux components, for the purpose of complete Kähler and complex structure moduli stabilization. calculating the corresponding chiral index in higher dimensions. This is done explicitly for our model below, in section 3.7.

Constant NS-NS B-field backround
In toroidal models with vanishing B-field, the net generation number of chiral fermions is in general even [17]. Thus, it is necessary to turn on a constant B-field background in order to obtain a Standard Model like spectrum with three generations. Due to the world-sheet parity projection Ω, the NS-NS two-index field B αβ is projected out from the physical spectrum and constrained to take the discrete values 0 or 1/2 (in string units) along a 2-cycle (αβ) of T 6 [16].
For branes at angles, B αβ = 1/2 changes the number of intersection points of the two branes. For the case of magnetized D9-branes, if B is turned on only along the three diagonal 2-tori: the effect is accounted for by introducing an effective world-volume magnetic flux quantum, defined bym a j =m a j + 1 2n a j , while the first Chern numbers along all other 2-cycles remain unchanged (and integral). Thus, the modification can be summarized by In addition, similar modifications take place in the tadpole cancellation conditions, as well.
Note that for non trivial B, ifn a i is oddm a i is half-integer, while ifn a i is evenm a i must be integer.
When restricting to the trivial windings of eq. (2.7) that we use in this paper,n a i = 1, the degeneracy formula (2.25) simplifies to: Similarly, the multiplicity of chiral antisymmetric representations is given by: In other words: In addition, we also write down, in subsection 3.3, the condition that such stacks are mutually supersymmetric with the stack U 5 , without turning on any charged scalar VEVs on these branes. The solution of these conditions giving eight branes O 1 , ..., O 8 is presented in subsections 3.4 and 3.5. They are all supersymmetric, stabilize all Kähler moduli (together with stack-U 5 ) and cancel all tadpoles along the oblique directions, x i x j , x i y j , y i y j for i = j. Finally in subsection 3.6, two more stacks A and B are found which cancel the overall D9 and D5-brane tadpoles (together with the U 1 stack).
As stated earlier, our strategy to find solutions for branes and fluxes is to first assume a canonical complex structure and Kähler moduli which have non-zero components only along the three factorized orthogonal 2-tori. In other words, we look for solutions where Kähler moduli are eventually stabilized such that

SU(5) GUT brane stacks
We now present the two brane stacks U 5 and U 1 which give the particle spectrum of SU (5) GUT. For this purpose, we consider diagonally magnetized D9-branes on a factorized sixdimensional internal torus (2.6), in the presence of a NS-NS B-field turned on according to eq. (2.28). The stacks of D9-branes have multiplicities N U 5 = 5 and N U 1 = 1, so that an SU (5) gauge group can be accommodated on the first one. Next, we impose a constraint on the windingsn U 5 i (defined in eq.(2.4)) of this stack by demanding that chiral fermion multiplicities in the symmetric representation of SU (5) is zero. Then from eqs. (2.32), we obtain the constraint: We solve eq. (3.3) by making the choice (2.7): n U 5 α ≡ Wα ,U 5 α = 1 for the stack U 5 . This also impliesn U 5 i = 1 for i = 1, 2, 3. Moreover, since from (2.23) the total D9-brane charge has to be sixteen and higher winding numbers give larger contributions to the D9 tadpole, the windings in all stacks will be restricted 4 to n a i = 1 so that a maximum number of brane stacks can be accommodated (with Q 9 = 16), in view of fulfilling the task of stabilization.
Indeed, the stack U 5 already saturates five units of D9 charge while stabilizing only a single Kähler modulus. One more unit of D9 charge is saturated by the U 1 stack, responsible for producing the chiral fermions in the representation5 of SU (5) at its intersection with U 5 . Moreover, it cannot be made supersymmetric in the absence of charged scalar VEVs, as we will see below. Thus, stabilization of the eight remaining Kähler moduli, apart from the one stabilized by the U 5 stack, needs eight additional branes O 1 , . . . , O 8 , contributing at least that many units of D9 charge (when windings are all one). These leave only two units of D9 charge yet to be saturated, which are also required to cancel any D5-brane tadpoles generated by the ten stacks, U 5 , U 1 and O 1 , . . . , O 8 . We find that this is achieved by two stacks A and B, also of windings one, so that the total D9 charge is Q 9 = 16 and all D5 tadpoles vanish Q 5 αβ = 0. Now, after having imposed the condition that symmetric doubly charged representations of SU (5) are absent, we find solutions for the first Chern numbers and fluxes, so that the the degeneracy of chiral fermions in the antisymmetric representation 10 is equal to three. These multiplicities are given in eqs. (2.31), (2.35), and when applied to the stack U 5 give the constraint: with a solution:m The corresponding flux components are: associated to the total (target space) flux matrix At this level, the choice of signs is arbitrary and is taken for convenience.
Next, we solve the condition for the presence of three generations of chiral fermions transforming in5 of SU (5). These come from singly charged open string states starting from the U 5 stack and ending on the U 1 stack or its image. In other words, we use the condition:  .7), the formulae take a form: where we have used the notation F a i ≡ (F a (1,1) ) iī for a given stack-a. We will also demand that all components F U 1 1 , F U 1 2 , F U 1 3 are half-integers, due to the shift in 1st Chern numberŝ m U 1 i by half a unit, in the presence of a non-zero NS-NS B-field along the three T 2 's (2.6). We then get a solution of eq. (3.8): for flux components on the stack U 1 : One can ask whether solutions other than ( The present results, including the quanta (m i ,n i ) for both U 5 and U 1 stacks, are summarized in Table 1  Since the VEV of any charged scalar on the U 5 stack is required to be zero, in order to preserve the gauge symmetry, the supersymmetry conditions for the U 5 stack read: (3.14) where we have used the fact that all windings are equal to unity and that eventually the Kähler moduli are stabilized according to our ansatz (3.2), such that J ij = 0 for i = j, and we have also defined  Table 1: Subtracting eq. (3.16) from eq. (3.13) one obtains: J 1 J 3 = − 3 4 which is clearly not allowed. We then conclude that the U 1 stack is not suitable for closed string moduli stabilization without charged scalar VEVs from its intersection with other brane stacks (besides U 5 ).
We therefore need eight new U (1) stacks for stabilizing all the nine geometric Kähler moduli, in the absence of open string VEVs.
In order to find such new stacks, one needs to impose the condition that any chiral fermions, other than those discussed in section 3.1, are SU (5) singlets and thus belong to the 'hidden sector', satisfying: Kähler and complex structure moduli, and use them to find out the allowed fluxes, consistent with zero net chirality and supersymmetry. Later on, we use the resulting fluxes to show the complete stabilization of moduli, and thus prove the validity of our ansatz.
In general, along a stack-a, the fluxes can be denoted by 3 × 3 Hermitian matrices, with f i 's being real numbers, and we have suppressed the superscript 'a' on the matrix components in the rhs of eq. (3.19). The relationships between the matrix elements (F a (1,1) ) ij and the flux components p a x i x j , p a x i y j , p a y i y j are: The subscript (1, 1) will also sometimes be suppressed for notational simplicity. We now solve the non-chirality condition (3.18) that a general flux of the type (3.19) must satisfy: The general solution for the flux (3.19) is: All additional stacks, including O 1 , . . . , O 8 , are required to satisfy this condition.

Supersymmetry constraint
We now impose an additional requirement on the fluxes along the stacks O 1 , . . . , O 8 , that together with the stack U 5 they should satisfy the supersymmetry conditions (2.11), in the absence of charged scalar VEVs. Using F a of eq. (3.19), the supersymmetry equations analogous to (3.13) and (3.14) for a stack O a read: In order to find a constraint on the flux components f 1 , f 2 , f 3 and a, b, c arising out of the requirement that equations (3.13) and (3.23) should be satisfied simultaneously, we start with a particular one-parameter solution of eq. (3.13): for arbitrary parameter ∈ (0, 1). 5 Then, by inserting (3.25) into eq. (3.23), one obtains the relation: In solving eqs.
A solution of eq. (3.27) with purely real flux components is found to be: Moreover, we notice from eqs. (3.27), (3.28) and the identity: with a = a 1 + ia 2 , b = b 1 + ib 2 , c = c 1 + ic 2 , that other solutions can be found simply by replacing some of the real components of a, b, c by imaginary ones modulo signs, as long as the values of the products aa * , bb * , cc * , as well as that of (a * bc * + ab * c) remain unchanged. We make use of such choices for canceling off-diagonal D5-brane tadpoles which for a general flux matrix (3.19) read (using eq. (2.24)): for = 1 10 in eq. (3.25). The positivity condition (3.24) for all of them has the following final form:     Table 3, we summarize the Chern numbers and windings of the stacks O 5 , . . . , O 8 , as well.
The four stacks O 5 , . . . , O 8 satisfy the supersymmetry condition:  Table 3, the positivity condition for the four new stacks takes the following form: (3.40) and is again obviously satisfied, as is the positivity condition (3.33) for stacks O 1 , . . . , O 4 .
The final uncanceled tadpoles from these stacks are: while the chiral fermion degeneracy from the intersections U 5 − O a and U 5 − O * a is given by: where we used the flux components (3.6) and (3.12). These tadpoles are then saturated by the brane stacks A and B of Table 4. Their contributions to the tadpoles are:  The U 1 stack on the other hand is needed to get the right SU (5)  can possibly attempt to manage with just two stacks U 1 and A, by using winding number two in one of them. These are straight-forward exercises for the interested reader who would like to examine these cases.

Non-chiral spectrum
The For example, the intersection numbers between stacks U 5 and U 1 are given in eq. (3.11).
One sees that I U 5 U 1 is zero as (m U 5 in U 1 i −n U 5 im U 1 i ) vanishes along T 2 1 and T 2 3 . However, in this case there exists a non-zero intersection number in d = 8 dimensions corresponding to the T 2 2 compactification of the d = 10 theory, given by: with the subscripts T 2 1 , T 2 3 of I U 5 U 1 | standing for those tori along which the intersection number vanishes. This implies two negative chirality (right-handed) fermions in d = 8, in the fundamental representation of SU (5). Under further compactification along T 2 1 and T 2 3 , we get four Dirac spinors in d = 4, or equivalently four pairs of (5 +5) Weyl fermions, shown already in the massless spectrum of Table 2. They give rise to four pairs of electroweak higgses, having non-vanishing tree-level Yukawa couplings with the down-type quarks and leptons, as it can be easily seen.
A similar analysis for the remaining stacks A and B gives chiral spectra in d = 6 with degeneracies: and They give rise to 149 + 146 = 295 pairs of (5 +5). Similarly, we obtain for the stack B: and leading to 51 + 16 = 67 pairs of (5 +5). All these non chiral states become massive by displacing appropriately the branes A and B in directions along the tori T 2 3 , T 2 2 and T 2 3 , T 2 1 , respectively. In addition to the states above, there are several SU (5) singlets coming from the intersections among the branes O 1 , . . . , O 8 , U 1 , A and B. Since they do not play any particular role in physics concerning our analysis, we do not discuss them explicitly here.
However, such scalars from the non-chiral intersections among U 1 , A and B will be used in section 5 for supersymmetrizing these stacks, by cancelling the corresponding non-zero FI parameters upon turning on non-trivial VEVs for these fields. The corresponding non-chiral spectrum will be therefore discussed below, in section 5.

Moduli stabilization
Earlier, we have found fluxes along the nine brane stacks U 5 , O 1 , . . . , O 8 , given in Tables 1,   2 For the complex structure moduli stabilization, we make use of the F a (2,0) condition (2.15) implying that purely holomorphic components of fluxes vanish. Then, by inserting the flux components p x i x j , p x i y j p y i y j , as given in Table 1 and Table 3, as well as in the superscript a, we obtain the FI parameters ξ as: where we have made use of eq. (2.14) and the canonical volume normalization (1.4). Then, using the values of the magnetic fluxes in stacks U 1 , A and B from Tables 1 and 4, the explicit form of the FI parameters in terms of the moduli J i (that are already completely fixed to the values (4.1)) is given by:  The last part of the exercise is to cancel the FI parameters (5.7) with VEVs of specific charged scalars living on the branes U 1 , A and B, in order to satisfy the D-flatness condition (2.18). For this we first compute the chiral fermion multiplicities on their intersections: Since they all vanish, there are equal numbers of chiral and anti-chiral fields in each of these intersections. In order to determine separately their multiplicities, we follow the method used in section 3.7 and compute: These correspond to chiral fermion multiplicities in six dimensions generating upon compactification to d = 4 pairs of left-and right-handed fermions. We also have: which gives zero net chirality for the U 1 − B * intersection. Computing , with fields in the brackets having multiplicities 149, 45, 2336 and 18, respectively. Restricting only to possible VEVs for these fields, eq. (2.18) takes the following form for the stacks U 1 , A and B: These equations can also be written as: following the notation of eq. (2.19), where we defined: with for instance (v AB ) 2 = |φ AB +− | 2 − |φ AB −+ | 2 and similarly for the others. Since we have three equations and four unknowns, we choose to obtain a special solution by setting (v U 1 B ) 2 = 0. Equations (5.18) -(5.20) then give: that can be solved to obtain:

Conclusions
In conclusion, in this work, we have constructed a three generation SU (5)  In this respect, some recent progress using D-brane instantons may be useful for up-quark mass generation [20][21][22]. 4. Study the question of supersymmetry breaking. An attractive direction would be to start with a supersymmetry breaking vacuum in the absence of charged scalar VEVs for the extra branes needed to satisfy the RR tadpole cancellation, In this Appendix, we write all the fluxes in the complex coordinate basis (z,z) with z = x + iy. Then, for the windings and 1st Chern numbers of Table 1, we obtain: Below, we sometimes suppress the subscript (1, 1) to keep the expressions simpler. The fluxes of the 8 stacks O 1 , . . . , O 8 can also be written in the same coordinate basis: From eq. (A.2) we get where we have used the notation tadpoles are: while the diagonal ones are: In real coordinates, the fluxes are: The 1st Chern numbers given in Table 4 can then be read directly from the values of fluxes given above. We now give similar data for the stacks O 2 , . . . , O 8 : leading to: The oblique tadpoles are: while the diagonal tadpoles are: leading to The oblique tadpoles are: and the diagonal ones are: The fluxes in the real basis are: leading to The oblique tadpoles are: and the diagonal tadpoles are: The stacks O 1 , . . . , O 4 , given above, satisfy the supersymmetry conditions (3.32). We now give the set of four stacks, O 5 , . . . , O 8 , which satisfy the supersymmetry condition