Supersymmetric SU(5) GUT with Stabilized Moduli
CCERN-PH-TH/2007-163
Supersymmetric SU(5) GUT with Stabilized
Moduli
Ignatios Antoniadis ∗† , Alok Kumar ‡ , Binata Panda , §¶ Department of Physics, CERN - Theory Division, CH-1211 Geneva 23, Switzerland Institute of Physics, Bhubaneswar 751 005, India
November 2, 2018
Abstract
We construct a minimal example of a supersymmetric grand unified model in atoroidal compactification of type I string theory with magnetized D SU (5) and the gauge non-singlet chiral spectrum contains only three families ofquarks and leptons transforming in the + ¯5 representations. ∗ On leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique, F-91128 Palaiseau Cedex † [email protected] ‡ [email protected] § [email protected], [email protected] ¶ CERN Marie Curie fellow a r X i v : . [ h e p - t h ] S e p Introduction
Closed string moduli stabilization has been intensively studied during the last years forits 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 particlemodels derived from string theory. Such stabilizations employ various supergravity [1,3], non-perturbative [2] and string theory [4–6] techniques to generate potentials for themoduli fields. However, very few examples are known so far of a complete stabilization ofclosed string moduli in any specific model, while the known ones are too constrained toaccommodate interesting models from physical point of view. Hence, there have been veryfew attempts to construct a concrete model of particle physics even with partially stabilizedmoduli. Nevertheless, in view of the importance of the task at hand, we revisit the typeI string constructions [7, 8] with moduli stabilizations [4–6], to explore the possibility ofincorporating particle physics models, such as the Standard Model or GUT models basedon grand unified groups, in such a framework.A new calculable method of moduli stabilization was recently proposed, using constantinternal magnetic fields in four-dimensional (4d) type I string compactifications [4, 5]. Inthe generic Calabi-Yau case, this method can stabilize mainly K¨ahler moduli [4, 9] and isthus complementary to 3-form closed string fluxes that stabilize the complex structure andthe 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 D D T by the supersymmetry conditions [13, 14]. Moreover, at least six of them must have2blique fluxes given by mutually non-commuting matrices, in order to fix all off-diagonalcomponents of the metric. On the other hand, all nine U (1) brane factors become massiveby absorbing the RR partners of the K¨ahler class moduli [14]. (2) Some extra branes areneeded to satisfy the RR tadpole cancellation conditions, with non-trivial charged scalarVEVs turned on in order to maintain supersymmetry.In this work, we apply the above method to construct phenomenologically interestingmodels. In the minimal case, three stacks of branes are needed to embed locally theStandard Model (SM) gauge group and the quantum numbers of quarks and leptons intheir intersections [15]. They give rise to the gauge group U (3) × U (2) × U (1), with thehypercharge being a linear combination of the three U (1)’s. Three different models canthen be obtained, one of which corresponds to an SU (5) Grand Unified Theory (GUT)when U (3) and U (2) are coincident. Here, we focus precisely on this U (5) × U (1) modelemploying two magnetized D U (5) with its orientifold image give rise to 3 chiral generations in the antisymmetricrepresentation of SU (5), while the intersection of U (5) with the orientifold image of U (1) gives 3 chiral states transforming as ¯5 . Finally, the intersection of U (5) with the U (1) is non chiral, giving rise to Higgs pairs + ¯5 .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 generalizationof the minimal set of branes with oblique magnetic fluxes that generate only diagonal5-brane tadpoles on the three orthogonal tori of T = (cid:81) i =1 T i . We find indeed a set ofeight such “oblique” branes which combined with U (5) can fix all geometric moduli by thesupersymmetry conditions. The metric is fixed in a diagonal form, depending on six radiigiven in terms of the magnetic fluxes. At the same time, all nine corresponding U (1)’sbecome massive yielding an SU (5) × U (1) gauge symmetry. This U (1) factor cannot bemade supersymmetric without the presence of charged scalar VEVs. Moreover, two extrabranes are needed for RR tadpole cancellation, which also require non-vanishing VEVs tobe made supersymmetric. As a result, all extra U (1)’s are broken and the only leftovergauge symmetry is an SU (5) GUT. Furthermore, the intersections of the U (5) stack withany additional brane used for moduli stabilization are non-chiral, yielding the three families3f quarks and leptons in the + ¯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 , U , O . . . , O , A , and B . The SU (5) gauge group arises from the open string states of stack- U containing five magnetized branes. The remaining eleven stacks contain only a singlemagnetized brane. Also, the stack- U containing the GUT gauge sector, contributes to theGUT particle spectrum through open string states which either start and end on itself oron the stack- U , having only a single brane and therefore contributing an extra U (1). Forthis reason we will also refer to these stacks as U and U stacks.The matter sector of the SU (5) GUT is specified by 3 generations of fermions in thegroup representations ¯5 and of SU (5), both of left-handed helicity. In the magnetizedbranes construction, the dimensional (antisymmetric) representation of left-handedfermions: ≡ u c u c u d u c u d u d e + L (1.1)arises from the doubly charged open string states starting on the stack- U and ending atits orientifold image: U ∗ and vice verse. They transform as ( , ) of SU (5) × U (1) × U (1),where the first U (1) refers to stack- U and the second one to stack- U , while the subscriptdenotes the corresponding U (1) charges. The ¯5 of SU (5) containing left-handed chiralfermions, or alternatively the with right-handed fermions: ≡ d d d e + ν c R (1.2) For simplicity, we do not distinguish a brane stack with its orientifold image, unless is explicitly stated. U (with five magnetized branes)and ending on stack- U ∗ (i.e. the orientifold image of stack- U ) and vice verse. Themagnetic fluxes along the various branes are constrained by the fact that the chiral fermionspectrum, 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 incorporatesthe above features of SU (5) GUT while stabilizing all the K¨ahler and complex struc-ture moduli. More precisely, for fluxes to be supersymmetric, one demands that theirholomorphic (2 ,
0) part vanishes. This condition then leads to complex structure modulistabilization [4]. In our case we show that, for the fluxes we turn on, the complex structure τ of T is fixed to τ = i , (1.3)with 11 being the 3 × T , as well as for the general definitions of the K¨ahlerand complex structure moduli. In particular, the coordinates of three factorized tori:( T ) ∈ T are given by x i , y i i = 1 , , (cid:90) dx ∧ dy ∧ dx ∧ dy ∧ dx ∧ dy = 1 . (1.4)For K¨ahler moduli stabilization, we make use of the mechanism based on the magne-tized D -branes supersymmetry conditions as discussed in [4, 5, 13]. Physically this corre-sponds to the requirement of vanishing of the potential which is generated for the modulifields from the Fayet-Iliopoulos (FI) D-terms associated with the various branes. Even inthis simplified scenario, the mammothness of the exercise is realized by noting that everymagnetic flux that is introduced along any brane also induces charges corresponding tolower dimensional branes, giving rise to new tadpoles that need to be canceled. In particu-lar, for the type I string that we are discussing, there are induced D D U and the extra branestacks (except with the U ), but in order to implement the mechanism of complex struc-ture and K¨ahler moduli stabilization, as well. Specifically, for stabilizing the non-diagonal5omponents of the metric, one is forced to introduce ‘oblique’ fluxes along the D D O , . . . , O , withdifferent oblique fluxes, such that the combined net induced D x i , y i ]. The holomorphicity conditions of fluxes, namelythe vanishing of field strengths with purely holomorphic indices, for these brane stacksstabilizes the complex structure moduli to the value (1.3). These fluxes also introduceD-term potential for the K¨ahler moduli. Once the complex structure is fixed as in (1.3),the fluxes in the nine stacks U , O , . . . , O generate potential in such a a way that all thenine K¨ahler moduli, J i ¯ j , ( i, j = 1 , ,
3) are completely fixed by the D-flatness conditions,imposing the vanishing of the FI terms. The residual diagonal tadpoles of the branes inthe stacks U , U , O , . . . , O are then canceled by introducing the last two brane stacks A and B . D-flatness conditions for the brane stacks U , A and B are also satisfied,provided some VEVs of charged scalars living on these branes are turned on to cancelthe corresponding FI parameters. Magnetized D -branes provide exact CFT (conformalfield theory) construction of the GUT model. However, in the presence of the these non-vanishing scalar VEVs, exact CFT description is lost. The validity of the approximationthen requires these VEVs to be smaller than unity in string units, a condition which ismet in our case. We explicitly determine the charged scalar VEVs and verify that they alltake values v a <<
1. Our model therefore corresponds to the Higgsing of a magnetized D U , A and B .At this point we would like to point out that, our strategy in this paper is to startwith a suitable ansatz for both the complex structure (1.3) and K¨ahler moduli leading todiagonal internal metric. Using this ansatz, we then determine fluxes along the branessatisfying all the constraints we elaborated upon earlier. We then use the flux solutions,to show explicitly that the moduli are indeed completely fixed, consistent with our ansatz.The rest of the paper is organized as follows. In Section 2, we give the necessaryconstraints needed for building the model. This includes the discussion on moduli sta-bilization in subsection 2.2, the tadpole constraints in subsection 2.3 and the fermion6egeneracies in subsection 2.4. Since a crucial step in a three generation model buildingis the introduction of a NS-NS (Neveu-Schwarz) B -field background without which onlyeven generation models can be built, the effect of non-zero B on the chirality and tadpolesis summarized in subsection 2.5. In Section 3, we obtain general solutions for fluxes alongmagnetized D U , A and B are determined. Our conclusions are presented in Section 6. In Appendix A, the fluxesalong branes are written explicitly for the stacks O , . . . , O and the associated D We now briefly review the string construction using magnetized branes, and in particularthe chiral spectrum that follows for such stacks of branes due to the presence of magneticfluxes.
We first briefly describe the construction based on D -branes with magnetic fluxes in typeI string theory, or equivalently type IIB with orientifold O D T compactification. Later on, in subsection 2.5, we study the introductionof constant NS-NS B -field background in this setup.The stacks of D N a , (b) winding matrices W ˆ I, aI and (c) 1st Chern numbers m a ˆ I ˆ J 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, ˆ I 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 gauge7roup U ( N a ) on each D W ˆ I, aI is the covering of the world-volume of each stack of D W ˆ I, aI defined as W ˆ IJ = ∂ξ ˆ I ∂X J for ˆ I, J = 0 , . . . , , (2.1)where the coordinates on the world-volume are denoted by ξ ˆ I , while the coordinates ofthe space-time M are X I . Since space-time is assumed to be a direct product of afour-dimensional Minkowski manifold with a six-dimensional torus, the covering matrix isof the form: W ˆ I, aJ = δ ˆ µµ W ˆ α ,aα for µ, ˆ µ = 0 , . . . , α, ˆ α = 1 , . . . , , (2.2)with the upper block corresponding to the covering of Σ a on the four-dimensional space-time M . Since these are assumed to be identical, the associated covering map W ˆ µµ isthe identity, W ˆ µµ = δ ˆ µµ . The entries of the lower block, on the other hand, describe thewrapping numbers of the D T whichare therefore restricted to be integers W ˆ αα ∈ Z , ∀ α, ˆ α = 1 , . . . , W ˆ αα in the internaldirections is also chosen to be a six-dimensional diagonal matrix, implying an embeddingsuch that the six compact D T , up to a winding multiplicity factor n aα , for a brane stack- a : n aα ≡ W ˆ α,aα . (2.3)We will also use the notationˆ n a ≡ n a n a , ˆ n a ≡ n a n a , ˆ n a ≡ n a n a , (no sum on a) (2.4)to define the diagonal wrapping of the D T ’s inside T , givenby: x i ≡ X α , α = 1 , , y i ≡ X α , α = 2 , , , (2.5)8ith periodicities: x i = x i + 1, y i ≡ y i + 1: T = ⊗ i =1 T i , (2.6)and coordinates of the orthogonal 2-tori ( T i ) being ( x i , y i ) for i = 1 , , n aα ≡ W ˆ α,aα = 1 , for α = 1 , .., , a = U , U , O · · · O , A, B. (2.7)However in this section, in order to describe the formalism, we keep still general windingmatrices W ˆ α,aα .(c) First Chern numbers: The parameters m a ˆ I ˆ J of the brane data given above are the 1stChern numbers of the U (1) ⊂ U ( N a ) background on the world-volume of the D U ( N a ) = U (1) a × SU ( N a ), the U (1) a has a constant field strength onthe covering of the internal space. These are subject to the Dirac quantization conditionwhich implies that all internal magnetic fluxes F a ˆ α ˆ β , on the world-volume of each stack of D F a ˆ α ˆ β and the corresponding target space inducedfluxes p aαβ are quantized as F a ˆ α ˆ β = m a ˆ α ˆ β ∈ Z ∀ ˆ α, ˆ β = 1 , . . . , ∀ a = 1 , . . . , K .p aαβ = ( W − ) ˆ α, aα ( W − ) ˆ β, aβ m a ˆ α ˆ β ∈ Q , ∀ α, β = 1 , . . . , T ’s of eq. (2.6), aswill be the case for some of our brane stacks, we make use of the following convenientnotation: ˆ m a ≡ m a ≡ m ax y , ˆ m a ≡ m a ≡ m ax y , ˆ m a ≡ m a ≡ m ax y . (2.9)The magnetized D U (1) flux associated with the gaugefields 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 D q L ≡ q i and q R ≡ − q j , while the Cartan generator h is given by h =diag( h N , . . . , h N N K ), with 11 N a being the N a × N a identity matrix. In addition, in9ype I string theory, the number of magnetized D 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, themagnetized D We now write down the supersymmetry conditions for magnetized D T decompose in a complex structure variation which isparametrized by the matrix τ ij entering in the definition of the complex coordinates z i = x i + τ ij y j , (2.10)and in the K¨ahler variation of the mixed part of the metric described by the real (1 , J = iδg i ¯ j dz i ∧ d ¯ z j . The supersymmetry conditions then read [4, 5]: F a (2 , = 0 ; F a ∧ F a ∧ F a = F a ∧ J ∧ J ; det W a ( J ∧ J ∧ J − F a ∧ F a ∧ J ) > , (2.11)for each a = 1 , . . . , K . The complexified fluxes can be written as F a (2 , = ( τ − ¯ τ ) − T (cid:2) τ T p axx τ − τ T p axy − p ayx τ + p ayy (cid:3) ( τ − ¯ τ ) − , (2.12) F a (1 , = ( τ − ¯ τ ) − T (cid:2) − τ T p axx ¯ τ + τ T p axy + p ayx ¯ τ − p ayy (cid:3) ( τ − ¯ τ ) − , (2.13)where the matrices ( p ax i x j ), ( p ax i y j ) and ( p ay i y j ) are the quantized field strengths in targetspace, 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 , and F a (1 , are 3 × F a : F a ≡ − (2 π ) iα (cid:48) F a (2 , F a (1 , − F a † (1 , F a ∗ (2 , , (2.14)which is the total field strength in the cohomology basis e i ¯ j = idz i ∧ d ¯ z j [4, 5].10he 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 complexstructure τ . Using eq. (2.12), the supersymmetry conditions for each stack can first beseen as a restriction on the parameters of the complex structure matrix elements τ : F a (2 , = 0 → τ T p axx τ − τ T p axy − p ayx τ + p ayy = 0 , (2.15)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 theK¨ahler moduli. This can be understood as a D-flatness condition. In the four-dimensionaleffective action, the magnetic fluxes give rise to topological couplings for the differentaxions of the compactified field theory. These arise from the dimensional reduction of theWess Zumino action. In addition to the topological coupling, the N = 1 supersymmetricaction yields a Fayet-Iliopoulos (FI) term of the form: ξ a g a = 1(4 π α (cid:48) ) (cid:90) T (cid:0) F a ∧ F a ∧ F a − F a ∧ J ∧ J (cid:1) . (2.16)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 inequalityin eq. (2.11) may also be understood from a four-dimensional viewpoint as the positivityof the U (1) a gauge coupling g a , since its expression in terms of the fluxes and moduli reads1 g a = 1(4 π α (cid:48) ) (cid:90) T (cid:0) J ∧ J ∧ J − F a ∧ F a ∧ J (cid:1) . (2.17)The above supersymmetry conditions, get modified in the presence of VEVs for scalarscharged under the U (1) gauge groups of the branes. The D-flatness condition, in the lowenergy field theory approximation, then reads:D a = − (cid:32)(cid:88) φ q φa | φ | G φ + M s ξ a (cid:33) = 0 , (2.18)where M s = α (cid:48)− / is the string scale , and the sum is extended over all scalars φ chargedunder the a -th U (1) a with charge q φa and metric G φ . Such scalars arise in the compacti-fication of magnetized D When mass scales are absent, string units are implicit throughout the paper.
11f open strings stretched between the a -th brane and its image a (cid:63) , or between the stack- a and another stack- b or its image b ∗ . When one of these scalars acquire a non-vanishingVEV (cid:104)| φ |(cid:105) = v φ , the calibration condition of eq. (2.11) is modified to: q a v a (cid:90) T (cid:0) J ∧ J ∧ J − F a ∧ F a ∧ J (cid:1) = − M s G (cid:90) T (cid:0) F a ∧ F a ∧ F a − F a ∧ J ∧ J (cid:1) (2.19)det W a ( J ∧ J ∧ J − F a ∧ F a ∧ J ) > , ∀ a = 1 , . . . , K . (2.20)Note that our computation is valid for small values of v a (in string units), since theinclusion 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 (cid:112) G φ .However, to estimate the validity of the perturbative approach, it is more appropriateto keep v φ instead of v φ (cid:112) G φ . The reason is that the next to leading correction to theD-term involves a quartic term of the type | φ | , proportional to a new coefficient K , andthe condition of validity of perturbation theory is K v φ /G φ <<
1. A rough estimate is thenobtained 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 thecase of diagonal fluxes [18]. In this special case, the fluxes are denoted by three angles θ ai ,( i = 1 , , Then suppressing index- a , we have:tan πθ i = p x i y i J i ≡ ( F (1 , ) z i ¯ z i J i , (2.21)and G = e γ E ( θ + θ + θ ) × (cid:115) Γ( θ )Γ( θ )Γ( θ )Γ(1 − θ )Γ(1 − θ )Γ(1 − θ ) , (2.22)with γ E being the Euler constant. The above results will be applied in section 5 to findout the FI parameters and charged scalar VEVs along three of the twelve brane stacks: U , A and B . The other nine stacks, U , O , . . . , O , stabilizing all the geometric moduli,will satisfy the calibration condition ξ a = 0 in the absence of open string scalar VEVs.Moreover, the RR moduli that appear in the same chiral multiplets as the geometricK¨ahler 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. See examples in Appendix A for the precise map between p x i y i and ( F (1 , ) z i ¯ z i . .3 Tadpoles In toroidal compactifications of type I string theory, the magnetized D K (cid:88) a =1 N a det W a ≡ K (cid:88) a =1 Q , a , (2.23)0 = K (cid:88) a =1 N a det W a Q a, αβ ≡ K (cid:88) a =1 Q , aαβ , ∀ α, β = 1 , . . . , Q a, αβ = (cid:15) αβδγστ p aδγ p aστ . The l.h.s. of eq. (2.23) arises from the contribution of the O O W ˆ ii = 1, the D N a . The D F a (2 , = 0 condition (2.11). The fluxesare then represented by three-dimensional Hermitian matrices ( F a (1 , ) which appeared ineq. (2.14) and the D Q , ai ¯ j are the Cofactors of the i ¯ j matrix elements ( F a (1 , ) i ¯ j .Fluxes and tadpoles in such a form are given in Appendix A. The gauge sector of the spectrum follows from the open string states corresponding tostrings starting and ending on the same brane stack. The gauge symmetry group is givenby a product of unitary groups ⊗ a U ( N a ), upon identification of the associated open stringsattached on a given stack with the ones attached on the mirror (under the orientifoldtransformation) stack. In addition to these vector bosons, the massless spectrum containsadjoint scalars and fermions forming N = 4, d = 4 supermultiplets.In the matter sector, the massless spectrum is obtained from the following open stringstates [14, 19]: 13. Open strings stretched between the a -th and b -th stack give rise to chiral spinors inthe bifundamental representation ( N a , ¯ N b ) of U ( N a ) × U ( N b ). Their multiplicity I ab is given by [6]: I ab = det W a det W b (2 π ) (cid:90) T (cid:0) q a F a (1 , + q b F b (1 , (cid:1) , (2.25)where F a (1 , (given in eqs. (2.13) and (2.14)) is the pullback of the integrally quan-tized world-volume flux m a ˆ α ˆ β 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(anti-fundamental representation). The transformation under the gauge group andtheir multiplicities are thus determined in terms of the data ( N a , W ˆ I, aI , m ˆ I ˆ J ).For factorized toroidal compactifications ( T ) (2.6) with only diagonal fluxes p x i y i ( i = 1 , , a and ending at b or vice versa, take the simple form (using notations ofeqs. (2.4) and (2.9)): ( N a , N b ) : I ab = (cid:89) i ( ˆ m ai ˆ n bi − ˆ n ai ˆ m bi ) , ( N a , N b ) : I ab ∗ = (cid:89) i ( ˆ m ai ˆ n bi + ˆ n ai ˆ m bi ) . (2.26)where i is the label of the i -th two-tori T i , and the integers ˆ m ai , ˆ n ai enter in themultiplicity expressions through the magnetic field as in eq. (2.8).In the model that we construct, however, we need stacks with fluxes which containboth diagonal and oblique flux components, for the purpose of complete K¨ahler andcomplex structure moduli stabilization.2. Open strings stretched between the a -th brane and its mirror a (cid:63) give rise to masslessmodes associated to I aa (cid:63) chiral fermions. These transform either in the antisymmetricor symmetric representation of U ( N a ). For factorized toroidal compactifications( T ) , the multiplicities of chiral fermions are given by;Antisymmetric : 12 (cid:32)(cid:89) i m ai (cid:33) (cid:32)(cid:89) j ˆ n aj + 1 (cid:33) , Symmetric : 12 (cid:32)(cid:89) i m ai (cid:33) (cid:32)(cid:89) j ˆ n aj − (cid:33) . (2.27)14n generic configurations, where supersymmetry is broken by the magnetic fluxes, thescalar partners of the massless chiral spinors in twisted open string sectors ( i.e. from non-trivial brane intersections) are massive (or tachyonic). Moreover, when a chiral index I ab vanishes, the corresponding intersection of stacks a and b is non-chiral. The multiplicityof the non-chiral spectrum is then determined by extracting the vanishing factor andcalculating the corresponding chiral index in higher dimensions. This is done explicitlyfor our model below, in section 3.7. In toroidal models with vanishing B -field, the net generation number of chiral fermionsis in general even [17]. Thus, it is necessary to turn on a constant B -field backgroundin order to obtain a Standard Model like spectrum with three generations. Due to theworld-sheet parity projection Ω, the NS-NS two-index field B αβ is projected out from thephysical spectrum and constrained to take the discrete values 0 or 1 / αβ ) of T [16].For branes at angles, B αβ = 1 / D B is turned on only along the threediagonal 2-tori: B x i y i ≡ b i = 12 , i = 1 , , , (2.28)the effect is accounted for by introducing an effective world-volume magnetic flux quantum,defined by ˜ˆ m aj = ˆ m aj + ˆ n aj , while the first Chern numbers along all other 2-cycles remainunchanged (and integral). Thus, the modification can be summarized by( ˆ m aj , ˆ n aj ) for b j = 0 → ( ˆ m aj + 12 ˆ n aj , ˆ n aj ) ≡ ( ˜ˆ m aj , ˆ n aj ) , for b j = 12 , (2.29)along the particular 2-cycles where the NS-NS B -field is turned on. This transformationalso takes into account the changes in the fermion degeneracies given in eqs. (2.26) and(2.27) (as well as in (2.33), (2.34) below), due to the presence of a non-zero B :( N a , N b ) : I ab = (cid:89) i ( ˜ˆ m ai ˆ n bi − ˆ n ai ˜ˆ m bi ) , N a , N b ) : I ab ∗ = (cid:89) i ( ˜ˆ m ai ˆ n bi + ˆ n ai ˜ˆ m bi ) , (2.30)Antisymmetric : I Aaa ∗ = 12 (cid:32)(cid:89) i m ai (cid:33) (cid:32)(cid:89) j ˆ n aj + 1 (cid:33) , (2.31)Symmetric : I Saa ∗ = 12 (cid:32)(cid:89) i m ai (cid:33) (cid:32)(cid:89) j ˆ n aj − (cid:33) . (2.32)In addition, similar modifications take place in the tadpole cancellation conditions, as well.Note that for non trivial B , if ˆ n ai is odd ˜ˆ m ai is half-integer, while if ˆ n ai is even ˜ˆ m ai must beinteger.When restricting to the trivial windings of eq. (2.7) that we use in this paper, ˆ n ai = 1,the degeneracy formula (2.25) simplifies to:( N a , N b ) : I ab = det (cid:16) ˜ F a (1 , − ˜ F b (1 , (cid:17) , (2.33)( N a , N b ) : I ab ∗ = det (cid:16) ˜ F a (1 , + ˜ F b (1 , (cid:17) , (2.34)where ˜ F = F + B and we have assumed the canonical volume normalization (1.4) on T .Similarly, the multiplicity of chiral antisymmetric representations is given by:Antisymmetric : I Aaa ∗ = (cid:89) i (cid:16) m ai (cid:17) , (2.35)while there are no states in symmetric representations. Finally, the tadpole cancellationconditions (2.23) and (2.24) become: K (cid:88) a =1 N a = 16 ; K (cid:88) a =1 N a Co( ˜ F a (1 , ) i ¯ j = 0 ∀ i, j = 1 , . . . , . (2.36) SU (5) GUT model
In this section, we first present in subsection 3.1 the brane stacks U and U , on which the SU (5) GUT, with three generations of chiral fermions, lives. Then, in subsection 3.2, wewrite down the conditions which any extra stacks, called O a have to satisfy, so that thereare no net SU (5) non-singlet chiral fermions corresponding to open strings of the type: U − O a and U − O ∗ a . In other words: I U O a + I U O ∗ a = 0 . (3.1)16n addition, we also write down, in subsection 3.3, the condition that such stacks aremutually supersymmetric with the stack U , without turning on any charged scalar VEVson these branes. The solution of these conditions giving eight branes O , ..., O is presentedin subsections 3.4 and 3.5. They are all supersymmetric, stabilize all K¨ahler moduli(together with stack- U ) and cancel all tadpoles along the oblique directions, x i x j , x i y j , y i y j for i (cid:54) = j . Finally in subsection 3.6, two more stacks A and B are found which cancelthe overall D D U stack).As stated earlier, our strategy to find solutions for branes and fluxes is to first assumea canonical complex structure and K¨ahler moduli which have non-zero components onlyalong the three factorized orthogonal 2-tori. In other words, we look for solutions whereK¨ahler moduli are eventually stabilized such that J i ¯ j = 0 , i (cid:54) = j, ( i, j = 1 , , . (3.2)By assuming the complex structure and K¨ahler moduli as in eqs. (1.3) and (3.2), we thenfind fluxes needed to be turned on in order to cancel tadpoles. These fluxes are also usedin the stabilization equations, in section 4 and Appendices B and C, to show that moduliare indeed completely fixed in a way that the six-torus metric becomes diagonal. We now present the two brane stacks U and U which give the particle spectrum of SU (5)GUT. For this purpose, we consider diagonally magnetized D B -field turned on according toeq. (2.28). The stacks of D N U = 5 and N U = 1, so that an SU (5) gauge group can be accommodated on the first one. Next, we impose a constrainton the windings ˆ n U i (defined in eq.(2.4)) of this stack by demanding that chiral fermionmultiplicities in the symmetric representation of SU (5) is zero. Then from eqs. (2.32), weobtain the constraint: (cid:89) j ˆ n U j = 1 . (3.3)We solve eq. (3.3) by making the choice (2.7): n U α ≡ W ˆ α,U α = 1 for the stack U . This alsoimplies ˆ n U i = 1 for i = 1 , ,
3. Moreover, since from (2.23) the total D D to n ai = 1 so that a maximum number of branestacks can be accommodated (with Q = 16), in view of fulfilling the task of stabilization.Indeed, the stack U already saturates five units of D D U stack, respon-sible for producing the chiral fermions in the representation ¯5 of SU (5) at its intersectionwith U . Moreover, it cannot be made supersymmetric in the absence of charged scalarVEVs, as we will see below. Thus, stabilization of the eight remaining K¨ahler moduli,apart from the one stabilized by the U stack, needs eight additional branes O , . . . , O ,contributing at least that many units of D D D U , U and O , . . . , O . We find thatthis is achieved by two stacks A and B, also of windings one, so that the total D Q = 16 and all D Q αβ = 0.Now, after having imposed the condition that symmetric doubly charged representa-tions of SU (5) are absent, we find solutions for the first Chern numbers and fluxes, so thatthe the degeneracy of chiral fermions in the antisymmetric representation is equal tothree. These multiplicities are given in eqs. (2.31), (2.35), and when applied to the stack U give the constraint: (2 ˆ m U + 1)(2 ˆ m U + 1)(2 ˆ m U + 1) = 3 , (3.4)with a solution: ˆ m U = − , ˆ m U = − , ˆ m U = 0 . (3.5)The corresponding flux components are: p U x y = − , p U x y = − , p U x y = 12 , (3.6)associated to the total (target space) flux matrix˜ F U (1 , = − −
12 12 . (3.7) detW is restricted to be positive definite in order to avoid the presence of anti-branes.
18t 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 fermionstransforming in ¯5 of SU (5). These come from singly charged open string states startingfrom the U stack and ending on the U stack or its image. In other words, we use thecondition: I U U + I U U ∗ = − . (3.8)To solve this condition for diagonal fluxes, one can use the formulae (2.30), or alternativelyeqs. (2.33) and (2.34). In the presence of the NS-NS B αβ -field of our choice (2.28), andusing the fluxes along the U stack (3.6) or (3.7), the formulae take a form:( N U , N U ) : I U U = ( − − F U )( − − F U )( 12 − F U ) , (3.9)( N U , N U ) : I U U ∗ = ( −
32 + F U )( −
12 + F U )( 12 + F U ) , (3.10)where we have used the notation F ai ≡ ( ˜ F a (1 , ) i ¯ i for a given stack-a. We will also demandthat all components F U , F U , F U are half-integers, due to the shift in 1st Chern numbersˆ m U i by half a unit, in the presence of a non-zero NS-NS B -field along the three T ’s (2.6).We then get a solution of eq. (3.8): I U U = 0 , I U U ∗ = − , (3.11)for flux components on the stack U : F U = − , F U = 32 , F U = 12 . (3.12)One can ask whether solutions other than (3.12) are possible for the U stack. Forinstance, instead of the choice (0 , −
3) of eq. (3.11) for the intersections U − U and U − U ∗ subject to the condition (3.8), one could try ( − ,
0) or in general ( n, − n − n anyinteger. Note that n (for n >
0) or − n − n < −
3) is the number of electroweak Higgspairs contained in + ¯5 of SU (5). Thus, the cases ( − , −
2) and ( − , −
1) were excludedbecause of the absence of higgses, but other cases such as n = 1 or n = − m i , ˆ n i ) for both U and U stacks, are sum-marized in Table 1. Moreover, the (chiral) massless spectrum under the resulting gauge19tack no. No. of Windings Chern no. Fluxesa branes: N a (ˆ n a , ˆ n a , ˆ n a ) ( ˆ m a , ˆ m a , ˆ m a ) [ ( ˆ m a +ˆ n a / n a , ( ˆ m a +ˆ n a / n a , ( ˆ m a +ˆ n a / n a ]Stack- U , ,
1) ( − , − ,
0) [- , - , ]Stack- U , ,
1) ( − , ,
0) [ − , , ]Table 1: Basic branes for the SU (5) modelgroup U (5) × U (1) is summarized in Table 2. The intersection of U with U is non-chiralsince I U U vanishes. The corresponding non-chiral massless spectrum shown in the tableconsists of four pairs of + ¯5 and will be discussed in section 3.7. SU (5) × U (1) number( ; 2 ,
0) 3( ; 1 , − ; − ,
1) 4 − So far, we have obtained the gauge and matter chiral spectrum of the SU (5) GUT usingtwo stacks of magnetized branes. However, in order to complete the model and stabilize allmoduli, one needs to add additional stacks of magnetized branes. This has to be done ina manner such that the supersymmetries of all the brane stacks are mutually compatible.To this end, we first examine whether the first two stacks U and U can have mutuallycompatible supersymmetry in a way suitable for moduli stabilization. The K¨ahler modulistabilization conditions are written in eqs. (2.11) and (2.19), corresponding to the caseswhere charged scalar VEVs are respectively zero or non-zero.Since the VEV of any charged scalar on the U stack is required to be zero, in orderto preserve the gauge symmetry, the supersymmetry conditions for the U stack read:38 −
12 ( J J − J J − J J ) = 0 , (3.13)20 J J −
14 ( − J − J + 3 J ) > , (3.14)where we have used the fact that all windings are equal to unity and that eventually theK¨ahler moduli are stabilized according to our ansatz (3.2), such that J i ¯ j = 0 for i (cid:54) = j ,and we have also defined J i ¯ i ≡ J i . (3.15)For the U stack on the other hand, one has the option of turning on a charged scalarVEV without breaking SU (5) gauge invariance. However, since all windings are equal tounity, there are no charged states under U (1) which are SU (5) singlets. Indeed, thereis no antisymmetric representation for U (1), while symmetric representations are absentbecause of our winding choice. The only charged states then come from the intersectionof U with U (or its image). Thus, the supersymmetry condition for the U stack followsfrom eq. (2.11), with the fluxes given in eq. (3.12) and Table 1: − −
12 ( J J − J J + 3 J J ) = 0 , (3.16) J J J −
14 (3 J − J − J ) > . (3.17)Subtracting eq. (3.16) from eq. (3.13) one obtains: J J = − which is clearly not allowed.We then conclude that the U stack is not suitable for closed string moduli stabilizationwithout charged scalar VEVs from its intersection with other brane stacks (besides U ).We therefore need eight new U (1) stacks for stabilizing all the nine geometric K¨ahlermoduli, in the absence of open string VEVs.In order to find such new stacks, one needs to impose the condition that any chiralfermions, other than those discussed in section 3.1, are SU (5) singlets and thus belong tothe ‘hidden sector’, satisfying: I U a + I U a ∗ = 0 , for a = 1 , .., . (3.18)We then introduce eight new stacks O , . . . , O , which carry in general both oblique anddiagonal fluxes in order to stabilize eight of the geometric K¨ahler moduli, using the su-persymmetry constraints (2.11). The remaining one is stabilized by the stack U . Moreprecisely, to determine the brane stacks O , . . . , O , we start with our ansatz for both21¨ahler and complex structure moduli, and use them to find out the allowed fluxes, con-sistent with zero net chirality and supersymmetry. Later on, we use the resulting fluxesto 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 × F a (1 , = f a ba ∗ f cb ∗ c ∗ f , (3.19)with f i ’s being real numbers, and we have suppressed the superscript ‘ a ’ on the ma-trix components in the rhs of eq. (3.19). The relationships between the matrix elements( F a (1 , ) i ¯ j and the flux components p ax i x j , p ax i y j , p ay i y j are: f i = p x i y i , a = p x y + ip x x , b = p x y + ip x x , c = p x y + ip x x . (3.20)The subscript (1 ,
1) will also sometimes be suppressed for notational simplicity. We nowsolve the non-chirality condition (3.18) that a general flux of the type (3.19) must satisfy: I U a + I U a ∗ = det( F U − F a ) + det( F U + F a ) = 0 . (3.21)The general solution for the flux (3.19) is:34 + ( f f − f f − f f ) + (3 cc ∗ − aa ∗ + bb ∗ ) = 0 . (3.22)All additional stacks, including O , . . . , O , are required to satisfy this condition. We now impose an additional requirement on the fluxes along the stacks O , . . . , O , thattogether with the stack U they should satisfy the supersymmetry conditions (2.11), in theabsence of charged scalar VEVs. Using F a of eq. (3.19), the supersymmetry equationsanalogous to (3.13) and (3.14) for a stack O a read:( f f f − cc ∗ f − bb ∗ f − aa ∗ f + a ∗ bc ∗ + ab ∗ c ) − ( J J f + J J f + J J f ) = 0 , (3.23) J J J − [ J ( f f − cc ∗ ) + J ( f f − bb ∗ ) + J ( f f − aa ∗ )] > . (3.24)22ext, we obtain two sets of fluxes of the form (3.19) which satisfy eqs. (3.22) and(3.23). The two sets, O , . . . , O and O , . . . , O , are characterized by the diagonal entriesin the matrix F a (3.19), which will be the same for the branes of each set. The motivationbehind such choices is dictated by the fact that once the off diagonal components of J i ¯ j are fixed to zero, these two sets of fluxes along the diagonal, together with the flux of U stack, determine the three diagonal elements J i (3.15), completely. O , . . . , O In order to find a constraint on the flux components f , f , f and a, b, c arising out of therequirement that equations (3.13) and (3.23) should be satisfied simultaneously, we startwith a particular one-parameter solution of eq. (3.13): J = 34 (cid:15) , J = 12 (cid:15) + 12 , J = 12 (cid:15) −
12 (3.25)for arbitrary parameter (cid:15) ∈ (0 , Then, by inserting (3.25) into eq. (3.23), one obtainsthe relation:34 (cid:15) ( f + f (cid:15) [ 32 ( f − f ) + f ]= ( f f f − cc ∗ f − bb ∗ f − aa ∗ f + a ∗ bc ∗ + ab ∗ c ) + f . (3.26)In solving eqs. (3.22) and (3.26), satisfying also the positivity condition (3.24), we haveto keep in mind that f i ’s take half-integer values due to the NS-NS B -field background(2.28). On the other hand the parameters a, b, c must be integers, since the windings areall one and there is no B -field turned on along any off-diagonal 2-cycle. Our approach isthen to first look for a solution of eq. (3.22) and then examine whether such a solutiongives an (cid:15) from eq. (3.26) such that all the J i ’s in eq. (3.25) are positive. In addition,both positivity conditions (3.14) and (3.24) have to be satisfied.To solve eq. (3.22), we impose the relation f = − f . The two equations (3.22) and One can also write down a full two-parameter solution of eq. (3.13), however we prefer to use twodifferent one-parameter families with appropriate parametrization for convenience in model building. Thesecond one-parameter solution will be used in section 3.5. f f + 3 f + 3 cc ∗ + bb ∗ − aa ∗ = 0 , (3.27)and 14 (cid:15) ( − f + f ) = − f f − cc ∗ f − bb ∗ f + aa ∗ f + a ∗ bc ∗ + ab ∗ c + f . (3.28)A solution of eq. (3.27) with purely real flux components is found to be: f = 52 , f = 12 , f = − , a = 4 , b = 3 , c = 1 . (3.29)Moreover, we notice from eqs. (3.27), (3.28) and the identity: a ∗ bc ∗ + ab ∗ c = 2 a ( b c + b c ) + 2 a ( b c − b c ) , (3.30)with a = a + ia , b = b + ib , c = c + ic , that other solutions can be found simplyby replacing some of the real components of a, b, c by imaginary ones modulo signs, aslong as the values of the products aa ∗ , bb ∗ , cc ∗ , as well as that of ( a ∗ bc ∗ + ab ∗ c ) remainunchanged. We make use of such choices for canceling off-diagonal D Q ,a = ( f f − cc ∗ ) , Q ,a = ( f f − bb ∗ ) , Q ,a = ( f f − aa ∗ ) ,Q ,a = ( b ∗ c − a ∗ f ) , Q ,a = ( b ∗ a − c ∗ f ) , Q ,a = ( ac − bf ) . (3.31)Here we have used the complex coordinates z i , ¯ z i and the assumption that complex struc-ture is eventually stabilized as in eq. (1.3).The result of our analysis above, giving fluxes for the brane stacks O , . . . , O , (includ-ing the solution (3.29)) is presented in Appendix A, in eqs. (A.2), (A.7), (A.12), (A.17).In this Appendix, we also show that the net chiral fermion contribution from the inter-section of each of the four stacks O , . . . , O with U (and its image) is zero, as shown ineqs. (A.3), (A.8), (A.13), (A.18). Oblique tadpoles Q , Q , Q are given in eqs. (A.4),(A.9), (A.14), (A.19) and their cancellations among these branes is also apparent. Thisleaves only diagonal D O , . . . , O , as well as those for the stacks O , . . . , O appearing in the next subsection.From eqs. (3.23) and (3.28), the stacks O , . . . , O satisfy the supersymmetry condition:1958 −
12 [ − J J + 5 J J + J J ] = 0 , (3.32)for (cid:15) = in eq. (3.25). The positivity condition (3.24) for all of them has the followingfinal form: J J J + 54 J + 414 J + 594 J > , (3.33)which is obviously satisfied for the solution (3.25) with (cid:15) = . Also, the chiral fermiondegeneracies on the intersections U − O a and U − O ∗ a are equal to I U O a = 23 , I U O ∗ a = − , a = 1 , . . . , , (3.34)giving vanishing net chirality for all of them individually. The non-trivial tadpole contri-butions from the four stacks are: Q = 4 , Q x y = − , Q x y = − , Q x y = − . (3.35) O , . . . , O In the last subsection we found four stacks O , . . . , O with oblique fluxes but diagonal5-brane charges. Clearly, in order to stabilize all the K¨ahler moduli, we need at least fouradditional stacks with oblique fluxes. The search for such branes is simplified by observingthat the supersymmetry condition (3.13) for the stack U has another one parameter familyof solutions, independent of (3.25), which solves also the condition (3.32) for the stacks O , . . . , O : J = 300 α α − , J = α , J = 994 α , with α > . (3.36)By inserting expressions (3.36) into the general supersymmetry condition (3.23), andfollowing steps similar to those of the last subsection, we find the set of stacks O , . . . , O given in Appendix A, with fluxes as in eqs. (A.22), (A.27), (A.32), (A.37). One of thesesolutions has flux components: f = − , f = 12 , f = 12 , a = − i , b = − i , c = 1 , (3.37)25hile the others can be obtained by trivial changes of the off-diagonal elements, as for thestacks O , . . . , O discussed in the previous subsection. Oblique D SU (5) non-singlet fermion chirality for these stacks is also zero, as shownin eqs. (A.23), (A.28), (A.33), (A.38). Once again, all off-diagonal D Q , Q and Q cancel among the contributions of the four brane stacks. In Table 3, wesummarize the Chern numbers and windings of the stacks O , . . . , O , as well.The four stacks O , . . . , O satisfy the supersymmetry condition:878 −
12 [ J J − J J + J J ] = 0 , (3.38)for α = 994 × , (3.39)consistently with the inequality (3.36). For this value of α , the positivity conditions (3.14)and (3.17) for the U and U stacks are also satisfied by J i ’s of the form (3.36). On theother hand, using the flux components (3.19) from Table 3, the positivity condition forthe four new stacks takes the following form: J J J + 34 J + 294 J + 414 J > , (3.40)and is again obviously satisfied, as is the positivity condition (3.33) for stacks O , . . . , O .The final uncanceled tadpoles from these stacks are: Q = 4 , Q x y = − , Q x y = − , Q x y = − , (3.41)while the chiral fermion degeneracy from the intersections U − O a and U − O ∗ a is givenby: I U O a = 14 , I U O ∗ a = − , a = 5 , . . . , . (3.42) We now collect the tadpole contribution from different stacks to find out how the total RRcharges cancel in our model by adding two extra stacks of single branes, A and B . The26tack No. of Windings Diag. Chern no. Diagonal Obliquebranes: ( n O a x , n O a x , n O a x ) ( m O a x y , m O a x y , m O a x y ) fluxes Chern no. N O a ( n O a y , n O a y , n O a y ) [ f a , f a , f a ] O , ,
1) (2,0,-1) [ , ,- ] m O x y = m O x y = 4(1 , , m O x y = m O x y = 3 m O x y = m O x y = 1 O , ,
1) (2,0,-1) [ , ,- ] m O x y = m O x y = 4(1 , , m O x y = m O x y = − m O x y = m O x y = − O , ,
1) (2,0,-1) [ , ,- ] m O x y = m O x y = − , , m O x x = m O y y = 3 m O x x = m O y y = 1 O , ,
1) (2,0,-1) [ , ,- ] m O x y = m O x y = − , , m O x x = m O y y = − m O x x = m O y y = − O , ,
1) (-13,0,0) [ − , , ] m O x x = m O y y = − , , m O x x = m O y y = 1 m O x y = m O x y = 1 O , ,
1) (-13,0,0) [ − , , ] m O x x = m O y y = − , , m O x x = m O y y = − m O x y = m O x y = − O , ,
1) (-13,0,0) [ − , , ] m O x x = m O y y = 2(1 , , m O x y = m O x y = − m O x x = m O y y = 1 O , ,
1) (-13,0,0) [ − , , ] m O x x = m O y y = 2(1 , , m O x y = m O x y = 1 m O x x = m O y y = − O , . . . , O N a (ˆ n a , ˆ n a , ˆ n a ) ( ˆ m a , ˆ m a , ˆ m a ) [ ( ˆ m a +ˆ n a / n a , ( ˆ m a +ˆ n a / n a , ( ˆ m a +ˆ n a / n a ]Stack-A 1 (1 , ,
1) (147 , ,
0) [ , , ]Stack-B 1 (1 , ,
1) (1 , ,
0) [ , , ]Table 4: A and B branestadpole contributions from stacks O , . . . , O with oblique fluxes, are given in eq. (3.35),while those from stacks O , . . . O are given in eq. (3.41). In addition, the stacks U and U together contribute: Q = 6 , Q x y = − , Q x y = − , Q x y = 32 , (3.43)where we used the flux components (3.6) and (3.12). These tadpoles are then saturatedby the brane stacks A and B of Table 4. Their contributions to the tadpoles are: Q = 2 , Q x y = 344 , Q x y = 2984 , Q x y = 3944 , (3.44)which precisely cancel the contributions from eqs. (3.35), (3.41) and (3.43). Moreover,chiral fermion multiplicities from the intersections of stacks A and B with U vanish, aswell: I U A = I U A ∗ = I U B = I U B ∗ = 0 . (3.45)We have thus obtained fluxes for the twelve stacks, saturating both D D U and O , . . . , O . The status of supersymmetry for thebrane stacks U , A and B will be studied later, in section 5.Before closing this section, we also examine briefly whether it would be possible to man-age tadpole cancellation without adding the extra stacks A and B , within the context ofour construction specified by the choice (3.11) of intersection numbers. Note that the ninestacks U and O , . . . , O were the minimal ones needed for K¨ahler moduli stabilization,since the use of the U brane for this purpose was ruled out, as we discussed in section 3.2.The U stack on the other hand is needed to get the right SU (5) particle spectrum. Thus,28n order to avoid the use of stacks A and B , one needs to examine whether there aresolutions, other than the one found in eq. (3.12), for fluxes along the stack- U such thattadpole cancellations are possible, while a scalar VEV charged under this U (1) may haveto be turned on in order to maintain supersymmetry. In such a situation, one needs awinding number three (det W = 3) for the stack U to saturate the D U stack have to vanish, otherwise they would give rise touncanceled tadpoles in oblique directions. Then, by writing the tadpole contributions ofthree diagonal fluxes f i satisfying the constraint (3.11), it can be easily seen that one is notable to cancel the combined tadpoles from stacks U and O , . . . , O . Such a possibility istherefore ruled out. Of course, one could try to find a solution that satisfies the constraint(3.11) but not necessarily (3.8), as we discussed already in section 3.1. Alternatively, onecan possibly attempt to manage with just two stacks U and A , by using winding numbertwo in one of them. These are straight-forward exercises for the interested reader whowould like to examine these cases. The degeneracies of non-chiral states coming from intersections of the stack U with O a and O ∗ a are already given in eqs. (3.34) and (3.42), leading to 4 × (23 + 14) = 148 pairs of( + ¯5 ) representations of SU (5). They follow from the degeneracy formulae (2.30), whenthe net numbers of left- and right-handed fermions are equal. In our case, this is insuredsince I U O a = − I U O ∗ a . However, non-chiral particle spectrum also follows from eqs. (2.30),(2.31) and (2.32), when any of I ab , I ab ∗ , I Aaa ∗ and I Saa ∗ are zero, as explained at the end ofsection 2.4. This occurs because for instance (cid:81) i ( ˜ˆ m ai ˆ n bi ± ˆ n ai ˜ˆ m bi ) vanishes along one or moreof the 2-tori, T j . Similarly for I Aaa ∗ or I Saa ∗ , this occurs because of the vanishing of fluxesalong one or more of the T ’s. Given the fluxes in stack U , which are non-zero along allthree T ’s, non-chiral states can come only from various intersections of the U stack withother branes.For example, the intersection numbers between stacks U and U are given in eq. (3.11).One sees that I U U is zero as ( ˜ˆ m U i ˆ n U i − ˆ n U i ˜ˆ m U i ) vanishes along T and T . However, inthis case there exists a non-zero intersection number in d = 8 dimensions corresponding29o the T compactification of the d = 10 theory, given by: I U U | T ,T = ( ˜ˆ m U ˆ n U − ˆ n U ˜ˆ m U ) = − , (3.46)with the subscripts T , T of I U U | standing for those tori along which the intersectionnumber vanishes. This implies two negative chirality (right-handed) fermions in d = 8,in the fundamental representation of SU (5). Under further compactification along T and T , we get four Dirac spinors in d = 4, or equivalently four pairs of ( + ¯5 ) Weylfermions, shown already in the massless spectrum of Table 2. They give rise to four pairs ofelectroweak higgses, having non-vanishing tree-level Yukawa couplings with the down-typequarks and leptons, as it can be easily seen.A similar analysis for the remaining stacks A and B gives chiral spectra in d = 6 withdegeneracies: I U A | T = ( ˜ˆ m U ˆ n A − ˆ n U ˜ˆ m A ) × ( ˜ˆ m U ˆ n A − ˆ n U ˜ˆ m A ) = 149 , (3.47)and I U A ∗ | T = ( ˜ˆ m U ˆ n A + ˆ n U ˜ˆ m A ) × ( ˜ˆ m U ˆ n A + ˆ n U ˜ˆ m A ) = 146 . (3.48)They give rise to 149 + 146 = 295 pairs of ( + ¯5 ). Similarly, we obtain for the stack B : I U B | T = ( ˜ˆ m U ˆ n B − ˆ n U ˜ˆ m B ) × ( ˜ˆ m U ˆ n B − ˆ n U ˜ˆ m B ) = 51 , (3.49)and I U B ∗ | T = ( ˜ˆ m U ˆ n B + ˆ n U ˜ˆ m B ) × ( ˜ˆ m U ˆ n B + ˆ n U ˜ˆ m B ) = 16 , (3.50)leading to 51 + 16 = 67 pairs of ( + ¯5 ). All these non chiral states become massive bydisplacing appropriately the branes A and B in directions along the tori T , T and T , T , respectively.In addition to the states above, there are several SU (5) singlets coming from theintersections among the branes O , . . . , O , U , A and B . Since they do not play anyparticular role in physics concerning our analysis, we do not discuss them explicitly here.However, such scalars from the non-chiral intersections among U , A and B will be usedin section 5 for supersymmetrizing these stacks, by cancelling the corresponding non-zeroFI parameters upon turning on non-trivial VEVs for these fields. The correspondingnon-chiral spectrum will be therefore discussed below, in section 5.30 Moduli stabilization
Earlier, we have found fluxes along the nine brane stacks U , O , . . . , O , given in Tables 1,2, 3, 4 and in Appendix A, consistent with our ansatz (1.3) for the complex structure and(3.2) for the geometric K¨ahler moduli. We now prove our ansatz by showing that both τ and J are uniquely fixed to the values (1.3), (3.2) and (3.36), (3.39). To show this, wemake use of the full supersymmetry conditions for the U stack as well as for the stacks O , . . . , O .For the complex structure moduli stabilization, we make use of the F a (2 , condition(2.15) implying that purely holomorphic components of fluxes vanish. Then, by insertingthe 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 inAppendix A, along the U and O , .., O stacks, we obtain a set of conditions on thecomplex structure matrix τ , given explicitly in Appendix B in eqs. (B.1) - (B.47). Theseequations imply the final answer (1.3). The details can be found in Appendix B.For K¨ahler moduli stabilization, we make use of the D-flatness condition in stacks U , O , . . . O which amounts to using the last two equations in (2.11). Explicit stabilization ofthe geometric K¨ahler moduli to the diagonal form, J i ¯ j = 0, ( i (cid:54) = j ) is given in eqs. (C.2) -(C.26) of Appendix C. For the stabilization of the diagonal components, the relevantequations are: (3.13), (3.14), (3.32), (3.33), (3.38), (3.40). The final solution for thestabilized moduli is given in eqs. (3.36) and (3.39). The numerical values of J i ’s can alsobe approximated as: J ∼ . , J ∼ . , J ∼ . . (4.1) U , A and B We now discuss the supersymmetry of the remaining stacks U , A and B by making useof the D-flatness conditions (2.18), (2.19) and (2.20). From these equations, suppressingthe superscript a , we obtain the FI parameters ξ as: ξ = F , − J F (1 , J − J F , , (5.1)31here 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 , A and B from Tables 1 and 4, theexplicit form of the FI parameters in terms of the moduli J i (that are already completelyfixed to the values (4.1)) is given by: ξ U = − − ( J J − J J + 3 J J ) J J J − (3 J − J − J ) , (5.2) ξ A = − ( J J + 295 J J + J J ) J J J − ( J + 295 J + 295 J ) , (5.3) ξ B = − ( J J + 3 J J + 33 J J ) J J J − (33 J + 3 J + 99 J ) , (5.4)leading to the numerical values: ξ U ∼ − . , ξ A ∼ − . , ξ B ∼ − . . (5.5)On the other hand, the charged scalar VEVs v φ entering in the modified D-flatnessconditions (2.18) and (2.19) are related to the modified FI parameters ξ a /G a , as it canbe easily seen from the expressions (2.16) and (2.17), that are also relevant for the per-turbativity criterion: v φ << G a on the branes U , A and B . For this purpose, we make use ofeq. (2.22) with the angles θ i defined in eq. (2.21). One finds the following values for themetric G in the three stacks: G U ∼ . , G A ∼ . , G B ∼ . , (5.6)that lead to the modified FI parameters: ξ U G U ∼ − . , ξ A G A ∼ − . , ξ B G B ∼ − . . (5.7)Note that the positivity conditions (2.20), giving positive gauge couplings through eq. (2.17)for the stacks U , A and B , hold as well. These expressions appear also in the FI param-eters ξ a as the denominators in the rhs of eqs. (5.2) - (5.4).The last part of the exercise is to cancel the FI parameters (5.7) with VEVs of specificcharged scalars living on the branes U , A and B , in order to satisfy the D-flatness condition322.18). For this we first compute the chiral fermion multiplicities on their intersections: I U A = ( F U − F A ) = 0 , I U B = ( F U − F B ) = 0 , I AB = ( F A − F B ) = 0 . (5.8)Since they all vanish, there are equal numbers of chiral and anti-chiral fields in each ofthese intersections. In order to determine separately their multiplicities, we follow themethod used in section 3.7 and compute: I U A | T = − , I U B | T = 45 , I AB | T = − . (5.9)These correspond to chiral fermion multiplicities in six dimensions generating upon com-pactification to d = 4 pairs of left- and right-handed fermions. We also have: I U A ∗ = ( F U + F A ) = 292 , I U B ∗ = ( F U + F B ) = 0 , I AB ∗ = ( F A + F B ) = 149 × , (5.10)which gives zero net chirality for the U − B ∗ intersection. Computing I U B ∗ | T = 18 , (5.11)one then finds 18 pairs of left- and right-handed fermions in d = 4 from this intersection.As a result, we have the following non-chiral fields, where the superscript refers tothe two stacks between which the open string is stretched and the subscript denotes thecharges under the respective U (1)’s : ( φ U A + − , φ U A − + ), ( φ U B + − , φ U B − + ), ( φ AB + − , φ AB − + ), ( φ U B ∗ ++ , φ U B ∗ −− ), 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 forthe stacks U , A and B : ξ U /G U + | φ U A + − | − | φ U A − + | + | φ U B + − | − | φ U B − + | + | φ U B ∗ ++ | − | φ U B ∗ −− | = 0 , (5.12) ξ A /G A + | φ U A − + | − | φ U A + − | + | φ AB + − | − | φ AB − + | = 0 , (5.13) ξ B /G B + | φ U B − + | − | φ U B + − | + | φ AB − + | − | φ AB + − | + | φ U B ∗ ++ | − | φ U B ∗ −− | = 0 . (5.14)These equations can also be written as: ξ U /G U + ( v U ) = 0 ⇒ ( v U ) = − ( ξ U /G U ) , (5.15) ξ A /G A + ( v A ) = 0 ⇒ ( v A ) = − ( ξ A /G A ) , (5.16) ξ B /G B + ( v B ) = 0 ⇒ ( v B ) = − ( ξ B /G B ) , (5.17)33ollowing the notation of eq. (2.19), where we defined:( v U ) = | φ U A + − | − | φ U A − + | + | φ U B + − | − | φ U B − + | + | φ U B ∗ ++ | − | φ U B ∗ −− | ≡ ( v U A ) + ( v U B ) + ( v U B ∗ ) , (5.18)( v A ) = | φ U A − + | − | φ U A + − | + | φ AB + − | − | φ AB − + | ≡ − ( v U A ) + ( v AB ) , (5.19)( v B ) = | φ U B − + | − | φ U B + − | + | φ AB − + | − | φ AB + − | + | φ U B ∗ ++ | − | φ U B ∗ −− | ≡ − ( v U B ) − ( v AB ) + ( v U B ∗ ) , (5.20)with for instance ( v AB ) = | φ AB + − | − | φ AB − + | and similarly for the others.Since we have three equations and four unknowns, we choose to obtain a special solutionby setting ( v U B ) = 0. Equations (5.18) - (5.20) then give:( v U A ) + ( v U B ∗ ) = − ( ξ U /G U ) ∼ . , (5.21) − ( v U A ) + ( v AB ) = − ( ξ A /G A ) ∼ . , (5.22) − ( v AB ) + ( v U B ∗ ) = − ( ξ B /G B ) ∼ . , (5.23)that can be solved to obtain:( v U A ) = − . , ( v U B ∗ ) = 0 . , ( v AB ) = 0 . . (5.24)Recalling from eqs. (5.18) - (5.20) that( v U A ) = | φ U A + − | − | φ U A − + | , ( v U B ∗ ) = | φ U B ∗ ++ | − | φ U B ∗ −− | , ( v AB ) = | φ AB + − | − | φ AB − + | , (5.25)and comparing with the results of eq. (5.24) (taking into account the different signs), VEVsfor the fields φ U A − + , φ U B ∗ ++ and φ AB + − are switched on. Moreover, as required by the validityof the approximation, the values of the charged scalar VEVs satisfy the condition v a << Conclusions
In conclusion, in this work, we have constructed a three generation SU (5) supersymmetricGUT in simple toroidal compactifications of type I string theory with magnetized D N = 1 supersymmetric vacuum, apartfrom the dilaton, in a way that the T -torus metric becomes diagonal with the six in-ternal radii given in terms of the integrally quantized magnetic fluxes. Supersymmetryrequirement and RR tadpole cancellation conditions impose some of the charged openstring scalars (but SU (5) singlets) to acquire non-vanishing VEVs, breaking part of the U (1) factors. The rest become massive by absorbing the RR scalars which are part of theK¨ahler moduli supermultiplets. Thus, the final gauge group is just SU (5) and the chiralgauge non-singlet spectrum consists of three families with the quantum numbers of quarksand leptons, transforming in the + ¯5 representations of SU (5). It is of course desirableto study the physics of this model in detail and perhaps to construct other more ‘realistic’variations, using the same framework which has an exact string description. Some of theobvious questions to examine are:1. Give a mass to the non-chiral gauge non-singlet states with the quantum numbersof higgses transforming in pairs of + ¯5 representations, keeping massless only onepair needed to break the electroweak symmetry. A first partial discussion was givenin section 3.7.2. Break the SU (5) GUT symmetry down to the Standard Model, which can be inprinciple realized at the string level separating the U (5) stack into U (3) × U (2) byparallel brane displacement. However, one would like to realize at the same timethe so-called doublet-triplet splitting for the Higgs + ¯5 pair, i.e. giving mass tothe unwanted triplets which can mediate fast proton decay and invalidate gaugecoupling unification, while keeping the doublets massless. One possibility would beto deform the model by introducing angles, in realizing the SU (5) breaking. In anycase, problems (1) and (2) may be related.3. Compute and study the Yukawa couplings. A general defect of the present construc-tion, already known in the literature, is the absence of up-type Yukawa couplings.35n this respect, some recent progress using D -brane instantons may be useful forup-quark mass generation [20–22].4. Study the question of supersymmetry breaking. An attractive direction wouldbe to start with a supersymmetry breaking vacuum in the absence of chargedscalar VEVs for the extra branes needed to satisfy the RR tadpole cancellation, U (1) × U (1) A × U (1) B in our construction. This ‘hidden sector’ could then medi-ate supersymmetry breaking, which is mainly of D-type, to the Standard Model viagauge interactions. Gauginos can then acquire Dirac masses at one loop withoutbreaking the R-symmetry, due to the extended supersymmetric nature of the gaugesector [23].Thus, this framework offers a possible self-consistent setup for string phenomenology, inwhich one can build simple calculable models of particle physics with stabilized moduliand implement low energy supersymmetry breaking that can be studied directly at thestring level. Acknowledgments
This research project has been supported in part by the European Commission under theRTN contract MRTN-CT-2004-503369, in part by a Marie Curie Early Stage ResearchTraining Fellowship of the European Community’s Sixth Framework Programme undercontract number (MEST-CT-2005-0020238 - EUROTHEPHY) and in part by the INTAScontract 03-51-6346.
A Explicit solutions for O , . . . , O z, ¯ z ) with z = x + iy . Then, for the windings and 1st Chern numbers of Table 1, we obtain: F U (1 , = − i (cid:16) dz dz dz (cid:17) − −
12 12 d ¯ z d ¯ z d ¯ z . (A.1)36elow, we sometimes suppress the subscript (1 ,
1) to keep the expressions simpler. Thefluxes of the 8 stacks O , . . . , O can also be written in the same coordinate basis: F O (1 , = − i (cid:16) dz dz dz (cid:17)
13 1 − d ¯ z d ¯ z d ¯ z . (A.2)From eq. (A.2) we get | F U + F O | = 23 , | F U − F O | = − , | F O | = 1958 , (A.3)where we have used the notation | F U + F O | ≡ det( F U + F O ) etc. The oblique D Q O = 3 + 2 , Q O = 12 − , Q O = 4 − , (A.4)while the diagonal ones are: Q O = − , Q O = − , Q O = − . (A.5)In real coordinates, the fluxes are: p O x y = 52 , p O x y = − p O x y = 12 , p O x y = p x y = 4 , p O x y = p O x y = 3 , p O x y = p O x y = 1 . (A.6)The 1st Chern numbers given in Table 4 can then be read directly from the values offluxes given above. We now give similar data for the stacks O , . . . , O : F O (1 , = − i (cid:16) dz dz dz (cid:17) − − − − − d ¯ z d ¯ z d ¯ z , (A.7)leading to: | F U + F O | = 23 , | F U − F O | = − , | F O | = 1958 . (A.8)The oblique tadpoles are: Q O = 3 + 2 , Q O = −
12 + 52 , Q O = − , (A.9)while the diagonal tadpoles are: Q O = − , Q O = − , Q O = − . (A.10)37he fluxes in the real basis are: p O x y = 52 , p O x y = − p O x y = 12 , p O x y = p O x y = 4 , p O x y = p O x y = − , p O x y = p O x y = − . (A.11) F O (1 , = − i (cid:16) dz dz dz (cid:17) − − i − i i − i − d ¯ z d ¯ z d ¯ z , (A.12)leading to | F U + F O | = 23 , | F U − F O | = − , | F O | = 1958 . (A.13)The oblique tadpoles are: Q O = − − , Q O = − i + 5 i , Q O = − i + 3 i , (A.14)and the diagonal ones are: Q O = − , Q O = − , Q O = − . (A.15)The fluxes in the real basis are: p O x y = 52 , p O x y = − p O x y = 12 , p O x y = p O x y = − , p O x x = p O y y = 3 , p O x x = p O y y = 1 . (A.16) F O (1 , = − i (cid:16) dz dz dz (cid:17) − i − − i − i i − d ¯ z d ¯ z d ¯ z , (A.17)leading to | F U + F O | = 23 , | F U − F O | = − , | F O | = 1958 . (A.18)The oblique tadpoles are: Q O = − − , Q O = 12 i − i , Q O = 4 i − i , (A.19)and the diagonal tadpoles are: Q O = − , Q O = − , Q O = − . (A.20)38he fluxes in the real basis are: p O x y = 52 , p O x y = − p O x y = 12 , p O x y = p O x y = − , p O x x = p O y y = − , p O x x = p O y y = − . (A.21)The stacks O , . . . , O , given above, satisfy the supersymmetry conditions (3.32). Wenow give the set of four stacks, O , . . . , O , which satisfy the supersymmetry condition(3.38) for the values of J i given in eqs. (3.36), (3.39): F O (1 , = − i (cid:16) dz dz dz (cid:17) − − i − i i i d ¯ z d ¯ z d ¯ z ; (A.22) | F U + F O | = 14 , | F U − F O | = − , | F O | = 878 ; (A.23) Q O = i − i , Q O = 2 + 252 , Q O = − i + i , (A.24) Q O = − , Q O = − , Q O = −
414 ; (A.25) p O x y = − , p O x y = p O x y = 12 , p O x x = p O y y = − , p O x x = p O y y = 1 , p O x y = p O x y = 1 . (A.26) F O (1 , = − i (cid:16) dz dz dz (cid:17) − − i i i − − i − d ¯ z d ¯ z d ¯ z ; (A.27) | F U + F O | = 14 , | F U − F O | = − , | F O | = 878 ; (A.28) Q O = i − i , Q O = − − , Q O = 2 i − i , (A.29) Q O = − , Q O = − , Q O = −
414 ; (A.30) p O x y = − , p O x y = p O x y = 12 , p O x x = p O y y = − , p O x x = p O y y = − , p O x y = p O x y = − . (A.31) F O (1 , = − i (cid:16) dz dz dz (cid:17) − i − − i i − − i d ¯ z d ¯ z d ¯ z ; (A.32)39 F U + F O | = 14 , | F U − F O | = − , | F O | = 878 ; (A.33) Q O = − i + i , Q O = − i − i , Q O = − , (A.34) Q O = − , Q O = − , Q O = −
414 ; (A.35) p O x y = − , p O x y = p O x y = 12 , p O x x = p O y y = 2 , p O x y = p O x y = − , p O x x = p O y y = 1 . (A.36) F O (1 , = − i (cid:16) dz dz dz (cid:17) − i − i − i i d ¯ z d ¯ z d ¯ z ; (A.37) | F U + F O | = 14 , | F U − F O | = − , | F O | = 878 ; (A.38) Q O = − i + i , Q O = 2 i + 25 i , Q O = 2 − , (A.39) Q O = − , Q O = − , Q O = −
414 ; (A.40) p O x y = − , p O x y = p O x y = 12 , p O x x = p O y y = 2 , p O x y = p O x y = 1 , p O x x = p O y y = − . (A.41) B Complex structure moduli stabilization
For each stack of magnetized D O : 4 τ + 12 τ + τ = 52 τ + 4 τ + 3 τ , (B.1)3 τ + τ − τ = 52 τ + 4 τ + 3 τ , (B.2)3 τ + τ − τ = 4 τ + 12 τ + τ . (B.3)40rom stack- O : 4 τ + 12 τ − τ = 52 τ + 4 τ − τ , (B.4) − τ − τ − τ = 52 τ + 4 τ − τ , (B.5) − τ − τ − τ = 4 τ + 12 τ − τ . (B.6)From stack- O : − τ τ + τ τ + 3 τ τ − τ τ + 4 τ − τ + 52 τ − τ = 0 , (B.7) − τ τ + τ τ + 3 τ τ − τ τ + 12 τ + 52 τ − τ − , (B.8) − τ τ + τ τ + 3 τ τ − τ τ + 12 τ − τ + 12 τ + 1 = 0 . (B.9)From stack- O :3 τ τ − τ τ − τ τ + τ τ + 4 τ − τ + 52 τ − τ = 0 , (B.10)3 τ τ − τ τ − τ τ + τ τ + 12 τ + 52 τ − τ + 3 = 0 , (B.11)3 τ τ − τ τ − τ τ + τ τ + 12 τ − τ + 12 τ − . (B.12)From stack- O : − τ τ − τ τ + 2 τ τ + τ τ − τ − τ − τ − , (B.13) − τ τ − τ τ + 2 τ τ + τ τ − τ − τ − τ − , (B.14) − τ τ − τ τ + 2 τ τ + τ τ − τ − τ + 12 τ + τ = 0 . (B.15)From stack- O : − τ τ + τ τ + 2 τ τ − τ τ − τ + τ − τ − , (B.16) − τ τ + τ τ + 2 τ τ − τ τ + τ − τ − τ + 1 = 0 , (B.17) − τ τ + τ τ + 2 τ τ − τ τ + τ − τ + 12 τ − τ = 0 . (B.18)41rom stack- O :2 τ τ − τ τ + τ τ − τ τ − τ − τ − τ + 2 = 0 , (B.19)2 τ τ − τ τ + τ τ − τ τ + τ − τ − τ − τ = 0 , (B.20)2 τ τ − τ τ + τ τ − τ τ + τ − τ + 12 τ + 1 = 0 . (B.21)From stack- O :2 τ τ − τ τ − τ τ + τ τ − τ − τ + τ + 2 = 0 , (B.22)2 τ τ − τ τ − τ τ + τ τ − τ − τ − τ + τ = 0 , (B.23)2 τ τ − τ τ − τ τ + τ τ − τ − τ + 12 τ − . (B.24)Now, from stack- O and stack- O one obtains from eqs. (B.1) and (B.4): τ = 3 τ , (B.25)and 4 τ + 12 τ = 52 τ + 4 τ ; (B.26)from eqs. (B.2) and (B.5): 3 τ + τ = 3 τ , (B.27)and − τ = 52 τ + 4 τ ; (B.28)and from eqs. (B.3) and (B.6): 3 τ + τ = τ , (B.29)and − τ = 4 τ + 12 τ ; (B.30)Similarly, from stack- O and stack- O one has, by adding eqs. (B.7) and (B.10):4 τ − τ + 52 τ − τ = 0 ; (B.31)42y adding eqs. (B.8) and (B.11): 12 τ + 52 τ − τ = 0 ; (B.32)and by adding eqs. (B.9) and (B.12):12 τ − τ + 12 τ = 0 . (B.33)Use of eqs. (B.30) and (B.33) gives: τ = 0 , (B.34)and τ + τ = 0 . (B.35)Moreover, one has from eqs. (B.34) and (B.32): τ = 8 τ ; (B.36)from eqs. (B.36) and (B.25): 3 τ = 8 τ ; (B.37)from eqs. (B.37) and (B.35): τ = τ = 0 ; (B.38)and from eqs. (B.38) and (B.36): τ = 0 . (B.39)Similarly, use of eqs. (B.26) and (B.31) implies: τ = 5 τ , (B.40)and τ = τ ; (B.41)while use of eq. (B.41) in eqs. (B.27) and (B.29) gives:3 τ + τ − τ = 0 , (B.42)43nd 3 τ + 9 τ − τ = 0 . (B.43)Eqs. (B.42) and (B.43) give: τ = 9 τ , (B.44)which comparing with eq. (B.40) implies: τ = τ = 0 . (B.45)Using the result of eq. (B.45) into eq. (B.42) then gives (using also eq. (B.41)), τ = τ = τ ≡ τ . (B.46)The value of τ is finally determined from any of the bilinear equations, such as eq. (B.8)or (B.9): τ = i . (B.47) C K¨ahler class moduli stabilization
For the stabilization of K¨ahler class, let us denote for definiteness the volume of the 4-cyclesassociated to J ∧ J as ( J ∧ J ) i ¯ j = V i ¯ j , (C.1)where the indices i, ¯ j correspond to the (1 , O using eq. (A.2):1958 − (cid:20) V + 12 V − V + 4 V + 4 V + 3 V + 3 V + V + V (cid:21) = 0 , (C.2)from stack- O using eq. (A.7):1958 − (cid:20) V + 12 V − V + 4 V + 4 V − V − V − V − V (cid:21) = 0 , (C.3)from stack- O using eq. (A.12):1958 − (cid:20) V + 12 V − V − V − V − iV + 3 iV + iV − iV (cid:21) = 0 , (C.4)44rom stack- O using eq. (A.17):1958 − (cid:20) V + 12 V − V − V − V + 3 iV − iV − iV + iV (cid:21) = 0 , (C.5)from stack- O using eq. (A.22):878 − (cid:20) − V + 12 V + 12 V − iV + 2 iV − iV + iV + V + V (cid:21) = 0 , (C.6)from stack- O using eq. (A.27):878 − (cid:20) − V + 12 V + 12 V − iV + 2 iV + iV − iV − V − V (cid:21) = 0 , (C.7)from stack- O using eq. (A.32):878 − (cid:20) − V + 12 V + 12 V + 2 iV − iV − V − V + iV − iV (cid:21) = 0 , (C.8)from stack- O using eq. (A.37):878 − (cid:20) − V + 12 V + 12 V + 2 iV − iV + V + V − iV + iV (cid:21) = 0 . (C.9)Now, from stacks- O and O , eqs. (C.2) and (C.3) give:3 ( V + V ) + ( V + V ) = 0; (C.10)from stacks- O and O , eqs. (C.4) and (C.5) give: − i ( V − V ) + i ( V − V ) = 0; (C.11)from stacks- O and O , eqs. (C.6) and (C.7) give: − i ( V − V ) + ( V + V ) = 0; (C.12)and from stacks- O and O , eqs. (C.8) and (C.9) give: − ( V + V ) + i ( V − V ) = 0 . (C.13)Eq. (C.13) implies i ( V − V ) = ( V + V ) , (C.14)45hich leads from eq. (C.10) 3 i ( V − V ) + ( V + V ) = 0 . (C.15)Similarly, eq.(C.12) implies i ( V − V ) = ( V + V ) , (C.16)which leads from eq. (C.11) − V + V ) + i ( V − V ) = 0 . (C.17)Now eqs. (C.15) and (C.17) can be solved to give V + V = 0 , (C.18)and V − V = 0 , (C.19)implying V = V = 0 . (C.20)Then one has from eq. (C.10) V + V = 0 , (C.21)and from eq. (C.11) V − V = 0 , (C.22)implying V = V = 0 . (C.23)Using the obtained values, eqs. (C.2) - (C.4) give V + V = 0 , (C.24)while eqs. (C.8) - eq. (C.6) give V − V = 0 , (C.25)implying V = V = 0 . (C.26)46 eferences [1] S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D (1997) 106006,[arXiv:hep-th/0105097].[2] R. Kallosh, S. Kachru, A. Linde and S. Trivedi, Phys. Rev. D (2003) 046005,[arXiv:hep-th/0301240].[3] S. Kachru, M.B. Schulz and S. Trivedi, JHEP (2003) 007, [arXiv: hep-th/0201028]; A. Frey and J. Polchinski, Phys. Rev. D (2002) 126009, [arXiv:hep-th/0201029].[4] I. Antoniadis and T. Maillard, Nucl. Phys. B (2005) 3, [arXiv:hep-th/0412008].[5] I. Antoniadis, A. Kumar, T. Maillard, arXiv: hep-th/0505260; Nucl. Phys. B (2007) 139, [arXiv:hep-th/0610246].[6] M. Bianchi and E. Trevigne, JHEP (2005) 034, [arXiv:hep-th/0502147] andJHEP (2006) 092, [arXiv:hep-th/0506080].[7] For a review on type I constructions, see e.g. C. Angelantonj and A. Sagnotti, Phys.Rept. (2002) 1 [Erratum-ibid. (2003) 339] [arXiv:hep-th/0204089]; and ref-erences therein.[8] For recent reviews on type I model building, see e.g. A. M. Uranga, Class. Quant.Grav. (2005) S41; R. Blumenhagen, M. Cvetic, P. Langacker and G. Shiu, Ann.Rev. Nucl. Part. Sci. (2005) 71 [arXiv:hep-th/0502005]; R. Blumenhagen, B. Kors,D. Lust and S. Stieberger, Phys. Rept. (2007) 1 [arXiv:hep-th/0610327]; andreferences therein.[9] R. Blumenhagen, D. Lust and T. R. Taylor, Nucl. Phys. B (2003) 319 [arXiv:hep-th/0303016]; J. F. G. Cascales and A. M. Uranga, JHEP (2003) 011 [arXiv:hep-th/0303024].[10] E. S. Fradkin and A. A. Tseytlin, Phys. Lett. B (1985) 123; A. Abouelsaood,C. G. . Callan, C. R. Nappi and S. A. Yost, Nucl. Phys. B (1987) 599;4711] C. Bachas, arXiv:hep-th/9503030.[12] I. Antoniadis, J.-P. Derendinger and T. Maillard, to appear.[13] M. Marino, R. Minasian, G. W. Moore and A. Strominger, JHEP (2000)005, [arXiv:hep-th/9911206].[14] C. Angelantonj, I. Antoniadis, E. Dudas and A. Sagnotti, Phys. Lett. B (2000)223 [arXiv:hep-th/0007090].[15] I. Antoniadis, S. Dimopoulos, Nucl. Phys. B (2005) 120 [arXiv:hep-th/0411032].[16] M. Bianchi, G. Pradisi and A. Sagnotti, Nucl. Phys. B (1992) 365; C. Ange-lantonj, Nucl. Phys. B (2000) 126 [arXiv:hep-th/9908064]; C. Angelantonj andA. Sagnotti, arXiv:hep-th/0010279;[17] R. Blumenhagen, B. Kors and D. Lust, JHEP (2001) 030 [arXiv:hep-th/0012156].[18] D. Lust, P. Mayr, R. Richter and S. Stieberger, Nucl. Phys. B (2004) 205,[arXiv: hep-th/0404134].[19] R. Blumenhagen, L. Goerlich, B. Kors and D. Lust, JHEP (2000) 006,[arXiv:hep-th/0007024]; G. Aldazabal, S. Franco, L. E. Ibanez, R. Rabadan andA. M. Uranga, J. Math. Phys. (2001) 3103, [arXiv:hep-th/0011073].[20] D. Cremades, L. E. Ibanez and F. Marchesano, JHEP (2003) 038, [arXiv:hep-th/0302105].[21] R. Blumenhagen, M. Cvetic, D. Lust, R. Richter and T. Weigand, arXiv:hep-th/0707.1871.[22] M. Billo, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta and I. Pesando,arXiv:0708.3806 [hep-th] and arXiv:0709.0245 [hep-th].[23] I. Antoniadis, K. Benakli, A. Delgado, M. Quiros and M. Tuckmantel, Nucl. Phys.B744