SU(5) D-brane realizations, Yukawa couplings and proton stability

We discuss SU(5) Grand Unified Theories in the context of orientifold compactifications. Specifically, we investigate two and three D-brane stack realizations of the Georgi-Glashow and the flipped SU(5) model and analyze them with respect to their Yukawa couplings. As pointed out in arXiv:0909.0271 the most economical Georgi-Glashow realization based on two stacks generically suffers from a disastrous large proton decay rate. We show that allowing for an additional U(1) D-brane stack this as well as other phenomenological problems can be resolved. We exemplify with globally consistent Georgi-Glashow models based on RCFT that these D-brane quivers can be indeed embedded in a global setting. These globally consistent realizations admit rigid O(1) instantons inducing the perturbatively missing coupling 10105^H. Finally we show that flipped SU(5) D-brane realizations even with multiple U(1) D-brane stacks are plagued by severe phenomenological drawbacks which generically cannot be overcome.


Introduction
D-brane compactifications have been proven to be a promising framework for realistic string model building. The basic building blocks of such constructions are D-branes which fill out the four-dimensional space-time and wrap submanifolds in the internal manifold. The gauge bosons live on the world volume of the respective D-brane while chiral matter appears at intersections of different stacks of D-branes. The multiplicity of the latter is given by the number of intersections of the respective submanifolds in the internal space. Over the last decade many globally consistent semi-realistic D-brane models have been constructed (for recent reviews, see [2][3][4]) .
In this work we are investigating how supersymmetric SU(5) GUT's can be realized in this framework with the emphasis on the realization of the superpotential 1 . The SU(5) gauge symmetry arises from a stack a of 5 D-branes giving rise to the gauge symmetry U(5) a which further splits into U(5) a = SU(5) × U(1) a . Here the abelian part generically acquires a mass via the Green-Schwarz mechanism and survives only as global symmetry. The 10 is localized at intersections of the U(5) stack a and its orientifold image a ′ . To accommodate for the other matter fields 5, as well as the Higgs pair is perturbatively forbidden since it violates the global selection rules. Here the subscripts denote the charge of the respective matter fields under the global U(1)'s namely the one originating from the U(5) a and the U(1) under which the Higgs field 5 H is charged. Obviously, this coupling is not neutral under these two global U(1)'s and therefore perturbatively forbidden.
Recently, it has been realized that D-instantons carry charge under these global U(1)'s [15][16][17] (for recent reviews, see [18,19]). For a specific product of matter fields they can compensate for the overshoot in the global U(1) charge and induce the perturbatively missing couplings. For a rigid O(1) instanton, which satisfies the severe constraints on the uncharged zero mode structure [20][21][22][23], the charge under the global U(1) x arising from a stack of N x branes wrapping the cycle π X in the internal manifold is given by 2 (1.3) Here π E denotes the orientifold invariant cycle wrapped by the D-instanton. The nonperturbative generation of Yukawa couplings via a rigid O(1) instanton has been explicitly discussed in [33].
In this work we extend the analysis of supersymmetric SU(5) GUT's in a global context performed in [1], in which the authors pointed out that the most economical realization based on two stacks of D-branes poses severe phenomenological drawbacks. We investigate whether these drawbacks can be resolved by allowing additional D-brane stacks and find that the Georgi-Glashow model can be accommodated via three D-brane stacks, overcoming all problems encountered in the two D-brane stack realization. We present a global realization based on rational conformal field theory (RCFT), that admits a rigid O(1) instanton inducing the perturbatively missing coupling 10105 H . In contrast to Georgi-Glashow D-brane realizations, flipped SU(5) D-brane realizations, even with multiple U(1) stacks have severe phenomenological problems. Specifically, the intriguing flipped SU(5) breaking mechanism of the GUT gauge symmetry down to the standard model gauge symmetry cannot lead to a consistent low energy theory without requiring the presence of additional geometric symmetries.
The paper is organized as follows. In section 2 we analyze the Georgi-Glashow realization via 2 and 3 D-brane stacks in a bottom-up fashion. At the end we present a global three stack realization based on RCFT that exhibits a rigid O(1) instanton that induces the desired coupling 10105 H . In section 3 we perform an analogous analysis for the flipped SU(5) model. We conclude with some final remarks in section 4. In the appendices A and B we lay out the basic ingredients for our systematic bottomup analysis and present the results for the three-stack Georgi-Glashow realization. In appendix C we present all details of the global realizations including the spectrum of the hidden sector.

Georgi-Glashow model
Before turning to D-brane realizations of the Georgi-Glashow model let us briefly introduce the usual supersymmetric SU(5)-GUT model. Later on we discuss potential D-brane realizations of it. The embedding of the standard model fields is displayed in Table 1.

Representation SM matter embedding Multiplicity
where v is of the order 10 16 GeV . The hypercharge U(1) Y is embedded in the SU(5) and given by which remains unbroken once the adjoint 24 acquires a vev of the type (2.2). After this brief introduction of the Georgi-Glashow model we turn to the D-brane realization of it.

D-brane realization
The most economical way to embed the Georgi-Glashow SU(5) GUT in a D-brane configuration is via two stacks of D-branes a, b. Stack a contains 5 D-branes while stack b is just a single D-brane, giving rise to the gauge symmetry U(5) a × U(1) b . The abelian U(1) a and U(1) b are generically anomalous and become massive via the Green-Schwarz mechanism. Thus the resulting gauge symmetry is the desired SU(5).
The massive U(1) a and U(1) b survive as global symmetries in the low energy effective theory and have to be preserved by all perturbative couplings. In Table 2 we display the origin of the respective matter fields for the realization of the Georgi-Glashow SU(5) model based on two stacks of D-branes. This chiral spectrum satisfies the string consistency conditions laid out in appendix A. Note that the hypercharge U(1) Y is a subgroup of U(5) and thus is guaranteed to remain massless. Therefore tadpole cancellation is the only constraint one has to ensure. However, for chiral matter with the transformation property displayed in Table 2 there is potentially a massless combination, satisfying the constraints (A.6) and (A.7), given by Let us stress that the conditions on the transformation properties of the chiral matter fields arising from tadpole cancellation and masslessness of a U(1) derived in the Appendix A are just necessary constraints. Whether tadpoles are really cancelled and whether an abelian symmetry remains massless or not, depends crucially on the concrete global realization. Table 2: Chiral spectrum for a D-brane realization of the SU(5) model.

Sector Matter All Transformation Multiplicity
The perturbative realized Yukawa couplings are where the indices indicate the charge under the global U(1) a and U(1) b symmetries, respectively. The Yukawa coupling which contains the up-flavour quark coupling is perturbatively forbidden. An instanton with global U(1) charge (−5, −1) under U(1) a and U(1) b can induce the missing coupling. As shown in [33] one needs three different instantons with global U(1) charge (−5, −1) to give masses to all three families. Note though that the nonperturbative generation of the Yukawa coupling 10 10 5 H suggests that the bottom quark is heavier than the top quark, which is in contrast to experimental observations.
The perturbative realization of the Dirac neutrino mass term suggests that the neutrino masses are of the same order as the other matter field masses. However, experiments show that the neutrinos masses are 10 −10 to 10 −16 times smaller than the other matter field masses. The see-saw mechanism gives a natural explanation for the smallness of the neutrino masses. A necessary ingredient for the seesaw mechanism is a large Majorana mass term which can be induced non-perturbatively by an instanton [15,16,23,[34][35][36][37] with global U(1) charges (0, 4). If the Majorana mass term is in the range (10 12 − 10 15 ) GeV one obtains neutrino masses of the observed order (10 −2 − 1) eV . Let us comment on potential phenomenological drawbacks of this 2-stack quiver.
(1) The perturbatively realized coupling 10 5 5 H contains the Yukawa coupling giving masses to the down-flavour quarks. On the other hand the coupling 10 10 5 H , which contains the Yukawa coupling that gives masses to the upflavour quarks is perturbatively forbidden. It is induced by an instanton and thus suppressed compared to the coupling 10 5 5 H . This suggests that for this quiver the bottom quark is heavier than the top quark which is in contrast to experimental observations.
(2) The instanton inducing the perturbatively missing Yukawa coupling 10 10 5 H also generates the dangerous dimension 5 operator 10 10 10 5 [1]. For the Georgi-Glashow SU(5) model the dimension 5 operator 10 10 10 5 includes q L q L q L L and u R u R d R E R , which if not sufficiently suppressed lead to a disastrous proton decay rate. Since the Yukawa coupling 10 10 5 H is responsible for the up-flavour quark coupling we expect only a minor suppression from the instanton, which is not enough to saturate the bounds on the proton lifetime.
(3) The chiral spectrum displayed in table 2 allows for a massless U(1) X given in equation (2.4). A massless U(1) X directly contradicts observations, and furthermore its charge explicitly forbids Majorana neutrino masses, both perturbatively and non-perturbatively. This model may be viable if the U(1) X photon acquires a sufficiently large mass, and the mechanism responsible for that might also generate neutrino masses, but it would be preferable to achieve all that directly in string theory. This is possible if the U(1) X acquires a mass from axion mixing. Note that this can happen even though the conditions (A.6) and (A.7) are satisfied, since the latter are just necessary, but not sufficient, conditions to have a massless U(1) X 3 .
As we will show momentarily all these problems can be overcome if one allows for an additional U(1) stack c. The U(1) c becomes again massive via the Green-Schwarz mechanism and survives only as a global symmetry. The second problem, namely that the instanton that induces the desired Yukawa coupling 10 10 5 H generically also generates dimension five proton decay operators, can be avoided if the matter fields 5 carry some global charge U(1) b while the Higgs field 5 H is rather charged under the global U(1) c . As we show in appendix B if we furthermore want to avoid the presence of R-parity violating couplings 10 5 5 and 5 5 H there are only two different types of choices for the origins of the fields transforming non-trivially under the SU(5). Within each choice there are different realizations of the Georgi-Glashow model, depending on the transformation behaviour of the right-handed neutrinos which are singlets under the SU(5). In appendix B we derive all three-stack quivers that mimic the Georgi-Glashow model, pass all string consistency conditions as well as some minimal phenomenological requirements. In the following we will discuss for each type one representative. We first discuss the configuration in which the µ-term is realized perturbatively 2.1.1 Three-stack quiver with perturbatively realized µ-term In table 3 we display the origin of the respective matter fields for a realization of the Georgi-Glashow SU(5) model based on three stacks of D-branes. It corresponds to configuration 6 in table 9 in appendix B. In contrast to the realization based on two stacks of D-branes here we have only two perturbatively realized couplings, namely Note that the neutrino Dirac mass term is realized perturbatively. Thus in order to obtain the observed small neutrino masses we expect the presence of a large Majorana mass term for the right-handed neutrinos, which can be induced by an instanton E 3 carrying global U(1) charge (0, 2, −2). With the string scale of the order 10 18 GeV and a suppression factor e −S E 3 ins ≃ 10 −5 we obtain via the seesaw mechanism neutrino masses in the observed range.

Sector Matter All Transformation Multiplicity
Let us now discuss if this setup indeed overcomes all the issues encountered for the 2-stack realization. First note that for the three-stack setup displayed in table 3 both couplings, the 10 10 5 H as well as 10 5 5 H are perturbatively forbidden.
In case the suppression factor of the instanton inducing the latter one is larger than the suppression factor of the instanton generating the coupling 10 10 5 H one gets the desired hierarchy between top and bottom quark masses. Furthermore, the instanton inducing the coupling 10 10 5 H does not carry the right global charge to induce the dangerous dimension 5 operator, which could lead to a disastrous proton decay rate. Finally, this setup does not satisfy the necessary conditions on having a massless U(1), thus all linear combinations of U(1) a , U(1) b and U(1) c are massive. That allows the presence of the Majorana mass term for the right-handed neutrino induced by a D-instanton, which was potentially forbidden in the 2-stack realization.
2.1.2 Three-stack quiver with non-perturbative µ-term Again we discuss here only one representative of all the possible solutions displayed in appendix B. The chiral spectrum of this quiver is displayed in table 4 and corresponds to configuration 9 in table 10 in appendix B.
Sector Matter All Transformation Multiplicity For this quiver the only perturbatively realized Yukawa coupling is which gives masses to the down-flavour quarks. The other desired couplings The first term in (2.12), induced by the instanton E 1 , gives masses to the up-flavour quarks as well as to the charged leptons. Since the top-quark is the heaviest Standard model field particle the suppression of the instanton must be very small. The suppression factor of E 2 on the other hand should be rather large to account for a µ-term of the order 100GeV . The instantons E 3 and E 4 induce the Dirac neutrino masses. Together with D-instantons that generate Majorana masses for the righthanded neutrinos they give via the see-saw mechanism the observed small neutrino masses 5 . Let us again discuss whether this quiver indeed overcomes all the issues encountered for the two stack quiver. The major drawback of the two-stack quiver, namely that the instanton that induces the desired Yukawa coupling 10 10 5 H also generates the dangerous dimension five operator 10 10 10 5, is not a problem for this quiver. Also this quiver like the 3-stack quiver discussed before does not exhibit any abelian symmetry that remains massless. Thus all perturbatively missing terms can be generated via D-instantons. However the perturbative realization of down flavour quark masses compared to the non-perturbative up flavour quark mass suggests exactly the opposite mass hierarchy compared to the observed one. Let us point out though that both couplings are further suppressed via world-sheet instantons which after all can in principle suppress the down-flavour quark masses such that the top-quark is indeed as observed the heaviest standard model particle. The latter is however not the generic case and usually requires some amount of fine-tuning.

Global realization of a three-stack quiver
Here we present a global realization of a three-stack quiver which is similar to the ones we discussed above. We will see that one can indeed find a rigid O(1) instanton that exhibits the correct zero mode structure to induce the perturbatively forbidden Yukawa coupling 10 10 5 H . The model is based on RCFT which are called Gepner orientifolds.
Gepner orientifolds are constructed by replacing the geometric notion of curled extra dimensions to form a compact manifold, by an algebraic procedure where the internal sector consists of tensor products of N = 2 minimal superconformal models with total central charge c = 9 [39][40][41][42][43][44] 6 . In this context, there has been an extensive search for all possible embeddings of the standard model gauge theory in D-brane configurations [48][49][50][51].
Before presenting a concrete example let us describe our search for realistic threestack quivers, which is based on a previous study performed in [50]. There the authors searched first for local configurations of D-brane boundary states that reproduce the chiral spectrum of the MSSM or extensions of it, such as SU(5) GUT's. These local D-brane boundary state configurations were required to not saturate the tadpole constraints and moreover give rise to a hypercharge embedding that is compatible with the MSSM hypercharge assignment and does not become massive via the Green-Schwarz mechanism. In a few cases the D-brane boundary state configuration that gives rise to the MSSM, called visible sector in the following, was enough to satisfy 5 Large suppression factors of E 3 and E 4 can account for the smallness of the neutrino mass [38]. In that case no Majorana masses for the right-handed neutrinos should be generated. 6 For some initial studies on closed Gepner constructions see [45][46][47].
the tadpole constraints. However, generically one needs additional boundary states to cancel the tadpoles. In the search performed in [50] the authors required that these additional boundary states, usually called hidden sector, are added in such a way that one does not have any chiral matter fields charged with respect to gauge groups in the visible and hidden sector, simultaneously. With this approach the authors found many globally consistent configurations, that give realistic chiral spectra, where the latter include also SU(5) GUT and Pati-Salam-realizations. In this work we follow a similar path. We take the subset of local configurations that give rise to a SU(5) GUT-like spectrum and analyze the superpotential by looking at the global U(1) charges of the respective matter fields. In case a desired Yukawa coupling is missing we are looking for a rigid O(1) instanton that has the correct zero mode structure to induce the missing coupling. To be more precise we identify all instanton boundary states that are orientifold invariant and do not exhibit any additional neutral fermionic zero modes apart from the two universal θ α modes [20][21][22][23]. Then we further require that the intersection pattern of this instanton boundary state with the visible D-brane boundary state configuration is in such a way that it gives the correct charged zero mode structure to induce the perturbatively forbidden, but desired coupling.
Once such an instanton is found we are looking for a hidden sector that cancels the tadpoles in such a way that it does not intersect with the instanton. This way it is ensured that the instanton does not exhibit any additional zero modes charged with respect to the hidden sector, which would kill the instanton contribution to the perturbatively missing coupling. In general, an already known solution to the tadpole cancellation condition is not very likely to satisfy this criterion, so one usually has to perform a new search for hidden sectors, imposing the instanton zero-mode constraint. Note that this constraint is rather strong, since it demands complete absence of zero-modes, even vector-like ones.
In addition to intersecting the instanton brane, the hidden sector branes may also intersect the observable matter brane. This is usually indeed what happens, although there are rare examples where there is no massless observable-hidden matter at all [50], or where no hidden sector is needed to cancel all tadpoles [49]. In the latter case, the problem is of course already solved, but then one looses the possibility of using the hidden sector for supersymmetry breaking. Apart from these rare cases, the common procedure is to allow observable-hidden matter, provided it is completely vector-like with respect to the entire gauge group. In that case, the additional matter may acquire a mass without any gauge symmetry breaking. In principle, one may relax the above restriction further and even allow observable-hidden matter that is chiral, but that becomes vector-like if the hidden sector gauge symmetry is removed. Then it depends on further details of the hidden sector dynamics what ultimately happens to the exotic observable-hidden matter, but it is not difficult to think of scenarios where it becomes sufficiently massive. In explicit examples, the number of tadpole solutions increases by several orders of magnitude under these relaxed conditions in comparison to the strict ones (i.e. those allowing only vector-like observable-hidden matter). Our attitude here is that chiral observable-hidden matter of this kind is a lesser evil than superfluous instanton zero modes, and therefore we use the relaxed condition, after checking that the strict one does not generate any solutions. Previous searches have shown that under the strict observable-hidden conditions, instantons with the correct zero mode structure to generate desired interactions [1,23] are very rare.
In [50] the authors found 7 different semi-realistic three-stack realizations of the SU(5) GUT's. In this search they allowed also for O(1) gauge groups for the additional third D-brane stack. Let us also point out that this subset mainly contains setups in which the µ term is perturbatively forbidden 7 . Thus configurations of the type discussed 2.1.1 are not contained in the search we will perform. We leave it for future work to extend the search of three-stack quivers by also including quivers in which the µ-term is perturbatively realized.
Performing the analysis in the fashion described above we find one type of configuration that gives rise to a semi realistic model and exhibits a rigid O(1) instanton generating the perturbatively forbidden Yukawa coupling 10 10 5 H . The visible sector consists of three stacks of branes giving rise to the gauge symmetry (2.14) The spectrum of a specific model is displayed in table 5 , where any constituent D-brane boundary state of the hidden sector is denoted by h. For the sake of clarity we do not display any specifics of the hidden sector. For the details, such as the hidden sector gauge symmetry as well as the spectrum within the hidden sector we refer to appendix C.
Note that for this particular configuration there are three pairs of Higgs and no neutrinos. Moreover the hidden sector intersects chirally with the visible sector, giving rise to exotics. The net number of SU(5) exotics is as expected zero, thus they are non-chiral with respect to the GUT gauge symmetry. However, the exotics carry different charge with respect to the hidden gauge groups. They only acquire mass after a breakdown of the hidden gauge group or via D-instantons inducing mass terms for them.
Let us turn to the Yukawa couplings. The only perturbatively realized coupling is the down flavour coupling Sector Matter All Transformation Multiplicity For the up flavour Yukawa coupling which is perturbatively absent we find an instanton that wraps a rigid orientifold invariant cycle that has the following intersection pattern with the visible branes Thus it gives the correct uncharged and charged zero mode structure to induce the desired but perturbatively missing coupling. Note also that this instanton does not intersect with any of the hidden D-brane boundary states and thus does not exhibit any zero modes charged with respect to the hidden sector that would spoil the generation of 10105 H . The suppression of the instanton turns out to be too large to account for the observed masses of the standard model. Let us stress though that this analysis is performed at the exact RCFT point and moving away from this exact point in moduli space might improve the situation. Let us turn to the µ-term which is perturbatively forbidden and can be generated by an instanton with the intersection pattern In order to be compatible with phenomenology one expects the instanton to exhibit a large suppression factor. Unfortunately, for this specific example we do not find any rigid O(1) instantons with such intersection pattern.
Despite its phenomenological problems, such as absence of µ-terms and Dirac mass term for the neutrinos, a highly suppressed top-quark mass, as well as the presence of additional exotics that are chiral with respect to the hidden sector but not with respect to the SU(5), this configuration serves as global realization of the SU(5) quivers discussed above in which one can find an instanton that satisfies the severe zero mode constraints to induce a perturbatively missing coupling 10105 H . Let us stress again that we looked only in a small phenomenologically interesting subset of SU(5) quivers while other appealing quivers with a perturbatively realized µ-term were not covered by this search. Additionally, we only allowed for rigid O(1) instantons to give non-perturbative contributions to the superpotential. However, as shown in [24][25][26][27] also multi-instanton configurations and so called rigid U(1) instantons [25,[28][29][30][31][32] can generate some of the missing Yukawa couplings. We leave it for future work to extend the here performed analysis by extending the class of local configurations and by including additional effects for the non-perturbative generation of desired couplings.

Flipped SU (5) model
In this section we discuss the realization of supersymmetric flipped SU(5)-GUT models in orientifold models. Before we present and analyze specific D-brane configurations which give rise to flipped SU (5) In table 6 we present the embedding of the standard model fields into the flipped SU(5) multiplets. In addition to the electroweak Higgs fields 5 H and 5 H the flipped SU(5) model also contains the Higgs fields 10 H and 10 H , whose presence is crucial for the breaking mechanism of the GUT gauge symmetry down to the Standard model gauge symmetry. Note that the spectrum Representation SM matter embedding Multiplicity U(1) X

10
(q L , d R , ν R ) 3 The hypercharge is a subgroup of SU(5) × U(1) X , given by In addition to the gauge symmetries there is a discrete Z 2 symmetry 10 H → −10 H . Then the superpotential takes the form Here the first three terms give masses to the standard fields after the electroweak Higgses acquire a vev, the fourth term is the µ-term, and the last two terms are crucial for the doublet-triplet splitting after the component ∆ 45 and ∆ 45 of 10 H and 10 H , respectively, acquire a vev of the GUT scale.

D-brane realization
Again the most economical way to embed the flipped SU (5)   Let us again discuss the superpotential terms, beginning with the terms that give eventually masses to the standard model fields. The perturbatively realized couplings are whose presence is crucial are perturbatively forbidden. While the first one will be generated by the same instanton which also generates the Yukawa coupling 10 10 5 H8 an instanton with charge (5, 1) under the global U(1) charges can induce the perturbatively missing coupling 10 H 10 H 5 H .
While this D-brane quiver after taking into account the non-perturbative effects can in principle mimic the flipped SU(5) model it has some phenomenological flaws, which we will discuss below.
(1) The perturbatively realized Yukawa coupling 10 5 5 H contains the Yukawa couplings giving masses to the up-flavour quarks and the neutrinos. Thus they are expected to be of the same order which is in contradiction to experiments that observe a hierarchy of 10 −16 between the top-quark mass and the neutrino masses. Note though that this is not a problem due to the D-brane realization but rather a problem within the flipped SU(5) model. In [52] the authors present a flipped SU(5) model which allows for additional singlets Φ, which are uncharged under the SU(5) as well as under the U(1) X . These singlets couple to the left-handed neutrinos via the coupling 10 H 10 Φ. After the 10 H gets a vev on the order of the GUT-scale the Yukawa coupling effectively becomes a large Majorana neutrino mass which via the seesaw mechanism may explain the smallness of the neutrino masses. However for the D-brane realization of the flipped SU(5) model with only two D-brane stacks one cannot accommodate a matter field which is not charged under the SU(5) and the U(1) X 9 .
(2) As already discussed in [1] an instanton inducing the coupling 10 10 5 H also generates the dimension 5 operator 10 10 10 5. The latter contains the dangerous dimension 5 operator q L q L q L L which if not sufficiently suppressed leads to a disastrous proton decay rate. To match the observed hierarchy between the top-quark and bottom-quark mass we expect the instanton suppression on the order of 10 −2 , which is not enough to saturate the bounds on the proton lifetime. Moreover, in the quiver displayed in table 7 the dimension 5 operator 10 5 5 1 is perturbatively realized. This operator includes the dimension five operator u R u R d R E R , which also has to be highly suppressed to saturate the bounds on the proton lifetime. Since it is perturbatively realized and thus only suppressed by the string scale M s it poses a serious phenomenological problem and predicts a proton lifetime not compatible with experimental observations. Note, however that the mass term 10 H 10 H is induced via the three-point where Φ 5 and Φ 1 denote the scalar fields transforming in the adjoint of the overall U(1) of the U(5) D-brane stack and of the U(1) D-brane stack. These vevs are related to the position in the internal space and in case they take the same value the mass is zero and the problematic term (3.8) is absent.
Generically it requires a large amount of fine-tuning to avoid the presence of the mass terms of the form (3.8).
(4) In the quiver displayed in While the problem (3) can be avoided with some amount of fine-tuning the issues (1) and (2)  For specific string compactifications there may exist additional symmetries which emerge from the compactification manifold. In case such a symmetry forbids the undesired couplings the quiver displayed in table 7 is a viable D-brane configuration. However, let us emphasize that such symmetries may also forbid some of the desired couplings. Moreover, for a generic compactification we do not expect such symmetries to appear.
Summarizing we have shown that D-brane realization of the flipped SU(5) has serious phenomenological problems. Some of the problems can be overcome by allowing additional D-brane stacks. However for a generic string embedding the D-brane quivers mimicking the flipped SU(5) model exhibit the superpotential term 10 H 5 5 H , that gives rise to an R-parity violating term of the order of M GU T .

Conclusions
In this work we discussed the realization of SU(5) GUT's in the framework of orientifold compactifications. We analyze how in such compactifications the superpotential can be accommodated, where we assume that perturbatively non-realized couplings are generated via D-instanton effects. Often times the D-instanton that induces a desired Yukawa coupling also generates a coupling that poses phenomenological problems. For the SU(5) orientifold realizations the coupling 10105 H is perturbatively forbidden, and thus needs to be realized non-perturbatively. However, in the most economical SU(5) realization the D-instanton giving rise to the 10105 H induces also the dangerous dimension 5 operator 1010105. The presence of the latter would lead to a disastrous proton decay rate. We show that this problem can be overcome by allowing for an additional Dbrane stack. We display viable SU(5) quivers based on three stacks of D-branes and investigate them with respect to their phenomenology. Furthermore, we present global Gepner model realizations of these quivers. These models exhibit D-instantons that satisfy the severe constraints on the zero mode structure to induce the coupling 10105 H . Unfortunately the instanton suppressions are too high to be phenomenologically viable. Nevertheless, these examples serve as global realization of the phenomenological viable SU(5) quivers. The performed search of global realizations contained only a small subset of viable quivers and it would be interesting to extend the search by also allowing quivers with a perturbatively realized µ-term.
Finally, we perform an analogous analysis for flipped SU(5) models. In the absence of any additional geometric symmetries of the compactification manifold Dbrane quivers mimicking the flipped SU(5) model exhibit severe phenomenological problems. The 10 H required for the intriguing SU(5) breaking mechanism generically couples to the standard model fields. After acquiring a vev of the GUT scale it induces large masses for the standard model fields not compatible with observations.

A. String consistency conditions and phenomenological constraints
In this appendix, we briefly summarize string consistency conditions that D-brane quivers have to satisfy. The latter contain constraints arising from tadpole cancellation and constraints that have to be fulfilled in case a linear combination x q x U(1) x should remain massless and thus survive as abelian gauge symmetry in the low energy effective action. For a more detailed description, we refer the reader to [53] (for an analogous analysis see [49,54]) 10 .

A.1 Tadpole cancellation
The tadpole cancellation condition, given by is a condition on the cycles that the D-branes wrap. Here π x , π ′ x and π O denote the homology class of the cycles the brane x, its orientifold image x ′ and the orientifold O wrap. Moreover, N x is the number of D-brane for stack x. Multiplying the tadpole cancellation condition with the homology class π a corresponding to the cycle wrapped by the D-brane stack a and using the chiral spectrum displayed in table 8 one derives constraints on the transformation behaviour of the chiral matter given by Note that for N a > 2 this condition is the usual anomaly cancellation condition for non-abelian SU(N a ) gauge symmetries. For N a = 2 it is a string-theoretic condition for anti-symmetric U(2) tensors that does not correspond to any field-theoretic anomaly condition. However, since these anti-symmetric tensors carry a charge under the phase symmetry of U(2), they can be distinguished from SU(2) singlets. Therefore this condition can be imposed on the field theory spectrum, and it must be imposed to have any chance to find a string theory embedding. For N a = 1 the anti-symmetric tensor cannot even be detected in the massless spectrum, and hence a given field theory spectrum may correspond to a string theory spectrum with any number of chiral anti-symmetric tensors (where "chiral" is defined as for N a > 2), which are infinite towers with a vanishing ground state dimension. However, since that number must be an integer, this still imposes a condition #( a ) + 5#( a ) = 0 mod 3 .

A.2 Massless U(1)'s
In order to have a massless linear combination U(1) 11 Table 8: Chiral spectrum the cycles that the D-brane stacks x wrap have to satisfy [63] x Analogously to the tadpole cancellation, multiplying both sides with the homology class π a and using the relations displayed in table 8 one obtains constraints on the transformation properties of the chiral matter. They take the form for N a > 1. The case N a = 1 requires a little more care due to the fact that in massless spectrum the antisymmetric tensor is absent. Using (A.2) to express the "would be" antisymmetrics in terms of the fundamentals and symmetrics one obtains Since the flipped SU(5) model requires an additional abelian gauge symmetry, namely U(1) X we require these constraints on the chiral spectrum of the flipped SU(5) model to be satisfied by the linear combination U(1) X .

A.3 Derivation for RCFT models
The foregoing derivations were made using the language of D-branes wrapping cycles on a manifold. Here we will show how the same formulas can be derived using boundary and crosscap states on RCFT orientifolds. For equation (A.2) it suffices to refer to [54]. In [48] this was worked out for the simple current boundary state formalism developed in [69]. Equation (A.6) can be derived as follows. The condition that a massless U(1) boson does not couple to an RR-axion is [48] x q x N x (R x(m,J) − R x c (m,J) ) = 0 , where q x and N x are as above, and R x(m,J) are the boundary coefficients as defined in [69]. Here x labels distinct boundary state, and (m, J) labels Ishibashi states, closed string states that can propagate in the transverse channel of an annulus, where m refers to a state in the bulk theory, and J is a degeneracy label. Which m's appear and with which degeneracy is determined by the modular invariant partition function. Eqn. (A.8) must be satisfied for every Ishibashi state which contains massless spinors. 12 This is determined only by m and not by J.
In order to derive (A.6) it turns out that we need only a subset of these conditions, namely where w m is the Witten index, counting the difference of spinors and anti-spinors in a character. No sum over m is implied. This condition is a subset of (A.8) because in general there are some Ishibashi states with an equal number of spinors and antispinors, which would contribute to U(1) mass, but not to (A.9). Hence the condition we will derive is a necessary, but not sufficient condition for a vanishing U(1) mass. We now perform a transformation from the transverse channel to the direct channel of the annulus, in a completely analogous way as the derivation of cubic anomaly cancellation from tadpole cancellation. The Witten indices transform exactly like characters, but are constants. Hence under this transformation we get w m = i w i S im , where S is the modular transformation matrix. Now we multiply the equations with a factor where g Ω,m is the Ishibashi metric [69] on each degeneracy space. Finally we sum over m and J to obtain The last sum is precisely the expression for the annulus coefficients, and hence we get The contraction with the chiral characters w i turns this into the chiral intersection, i.e the first term is precisely #( a , x ) as defined above. This expression should hold for any boundary state label a. If one chooses a label a that coincides with one of the labels x which participates in the U(1) symmetry of interests (i.e. q a N a = 0), then one may write Note that the sum in (A.8) is over pairs (x, x c ) of conjugate boundary labels, labelled by x. Furthermore, if (A.12) holds for a label a it automatically holds for its conjugate Hence we can use the same basis of pairs (a, a c ) for all boundary labels, and then only the case x = a can occur. The final result is then indeed precisely (A.6).
Note that using the completeness condition for boundaries [70], a R * a m,J R * a m ′ ,J ′ = δ m,m ′ δ J,J ′ , one can invert the derivation, so that (A.9) can be derived from (A.6). However (A.8) does not follow, and hence, as already stated above, (A.6) is in general only a necessary condition for masslessness of a U(1). We have examined in a few cases how close it is to being sufficient. It turns out that very often the quantity R x(m,J) − R x c (m,J) vanishes for all x if the Witten index of m is zero. This is true, for example, for all modular invariant partition functions and all orientifolds of the tensor products (3,3,3,3,3), (3,8,8,8), (6,6,6,6) and (2,2,2,2,2,2). Hence in all these cases (A.6) is actually sufficient, provided all boundary labels a are taken into account. The tensor product (4,4,10,10)  The practical use of (A.6) is in determining if a postulated brane configuration has any chance of having a massless U(1) boson (for example Y ) in an explicit realization in string theory. Once one has found such a realization, one might as well check (A.8) directly. Hence in practice the set of labels a for which one uses it is just the set of branes appearing in the postulated brane configuration. Then certainly it is just a necessary, and not a sufficient condition.
This condition also plays a rôle in the discussion of charge violation by instantons. Then a is a candidate instanton brane, and the left-hand side of (A.12) must be non-zero to get the required charge violation. Clearly a non-vanishing charge violation implies that the corresponding U(1) must be massive, but the converse is not necessarily true: for a massive U(1) it may happen that there are no branes that violate conservation of the charge. This was pointed out already in [16] (in particular footnote 16 in that paper). We see now that this can only happen if there are contributions to the vector boson mass from Ishibashi states with a vanishing Witten index.

A.4 Phenomenological requirements
There are various phenomenological constraints which arise from experiments. We list them below.
• All the Yukawa couplings that give masses to the three families are realized, either perturbatively or non-perturbatively via D-instantons. Thus we require the presence of the terms 10 10 5 H , 10 5 5 H and 5 5 H 1.
• For the flipped SU(5) model we require the presence of terms 10 H 10 H 5 H and 10 H 10 H 5 H which are crucial for the breaking pattern of the flipped SU (5) down to the standard model gauge symmetry.
• We forbid any R-parity violating couplings 10 5 5 or 5 5 H on perturbative or non-perturbative level. Specifically that implies that none of the instantons whose presence is required to induce some of the missing but desired couplings induces also the R-parity violating couplings.
• We forbid the presence of the dimension five operator 10 10 10 5 on perturbative or non-perturbative level. For the flipped SU(5) model we also require the absence of the dimension five operator 10 5 5 1, again on perturbative and nonperturbative level. As before that implies that none of the instantons whose presence is required to induce some of the missing but desired couplings induces also these dimension five operators.
• For the Georgi-Glashow D-brane realization often times an instanton which is required to generate a desired Yukawa coupling also induces a tadpole 1 and thus an instability for the setup. We rule out any setup which requires the presence of such an instanton.

B. Georgi-Glashow realizations based on three stacks
In this appendix we present all 3-stack realizations of the Georgi-Glashow model that satisfy all the string consistency conditions as well as all the phenomenological constraints laid out in appendix A. We distinguish between two different types of setups, for the first type the µ-term is perturbatively realized and for the second type the µ-term is perturbatively forbidden and must be generated non-perturbatively.
In table 9 we display all possible solutions with exactly three right-handed neutrinos. In the second line we display all possible origins for the matter fields 13 . We find 12 different D-brane configurations, where solutions marked with a * potentially exhibit a massless U(1). In section 2.1.1 we discuss in detail the phenomenology of the configuration 6.
In table 10 we display all possible 3 D-brane-stack realizations of the Georgi-Glashow model in which the µ-term is not perturbatively realized. These satisfy the Solution # c  c   1  3  3  1  1  0  0  0  0  0  0  0  3  2  3  3  1  1  0  0  0  1  0  1  0  1  3  3  3  1  1  severe string consistency constraints as well as the phenomenological conditions laid out in the previous appendix. Again solutions marked with a * potentially exhibit a massless U(1). In section 2.1.2 we discuss the configuration 9 as a representative with respect to their phenomenology in detail.

C. Globally consistent 3-stack model
Here we present globally consistent Gepner configurations that give rise to a Georgi-Glashow-like structure and exhibit an instanton that induces the perturbatively missing Yukawa coupling 10 10 5 H . We find two different types of global realizations and we present a representative of each here. Before doing so let us however explain the notation in the tables to come.
In the first column the table displays the whole number of states for a particular sector. The last column gives the net chirality of these states. The gauge groups are displayed in the first row, where a V stands for fundamental,V for antifundamental, S for the symmetric, A for the anti-symmetric and Ad for the adjoint under the respective gauge group. The column denoted by Ins represents the rigid O(1)-instanton which will induce the Yukawa coupling 10 10 5 H . Fields charged with respect to it denote the charged zero modes. One can easily see that the representatives below indeed exhibit the correct charged zero mode structure to induce the coupling 10 10 5 H .

C.1 Gepner Orientifold of Type I
We find 6 global realizations with gauge group U(5) where the first 3 gauge groups (the highlighted ones) denote the visible sector. The 6 different global realizations differentiate only in their massless spectrum in the hidden sector, the visible sector is for all 6 realizations the same.
Let us specify the Gepner orientifold. The internal sector of these models consists of a tensor product of four copies of N = 2 superconformal minimal models with levels k i = {1, 10, 22, 22}. This tensor product has 50 symmetric modular invariant partition functions. The one of our interest yields a closed string spectrum characterized by Hodge numbers h 11 = 32, h 12 = 20 and 237 singlets. These numbers identify it uniquely. This MIPF allows 4 different orientifold choices, according to the prescription given in [69]. The results below were obtained for one of these four (according to the labelling conventions used in [50] this case corresponds to MIPF nr. 26, orientifold nr. 1). A representative of these 6 realizations is displayed in table 11.