GUTs and Exceptional Branes in F-theory - II: Experimental Predictions
aa r X i v : . [ h e p - t h ] J a n arXiv:0806.0102 GUTs and Exceptional Branes inF-theory - II:Experimental Predictions
Chris Beasley ∗ , Jonathan J. Heckman † and Cumrun Vafa ‡ Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USAMay, 2008 ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] bstract We consider realizations of GUT models in F-theory. Adopting a bottom upapproach, the assumption that the dynamics of the GUT model can in principledecouple from Planck scale physics leads to a surprisingly predictive framework. Aninternal U (1) hypercharge flux Higgses the GUT group directly to the MSSM orto a flipped GUT model, a mechanism unavailable in heterotic models. This newingredient automatically addresses a number of puzzles present in traditional GUTmodels. The internal U (1) hyperflux allows us to solve the doublet-triplet splittingproblem, and explains the qualitative features of the distorted GUT mass relationsfor lighter generations due to the Aharanov-Bohm effect. These models typicallycome with nearly exact global symmetries which prevent bare µ terms and alsoforbid dangerous baryon number violating operators. Strong curvature around ourbrane leads to a repulsion mechanism for Landau wave functions for neutral fields.This leads to large hierarchies of the form exp( − c/ε γ ) where c and γ are orderone parameters and ε ∼ α − GUT M GUT /M pl . This effect can simultaneously generate aviably small µ term as well as an acceptable Dirac neutrino mass on the order of 0 . × − ± . eV. In another scenario, we find a modified seesaw mechanism which predictsthat the light neutrinos have masses in the expected range while the Majorana massterm for the heavy neutrinos is ∼ × ± . GeV. Communicating supersymmetrybreaking to the MSSM can be elegantly realized through gauge mediation. In onescenario, the same repulsion mechanism also leads to messenger masses which arenaturally much lighter than the GUT scale. ontents U (1) Fluxes 53
10 Avoiding Exotica 57 G S = SO (10) . . . . . . . . . . . . . . . . . . 6010.3 MSSM Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.4 Candidates For Dark Matter . . . . . . . . . . . . . . . . . . . . . . . 66
11 Geometry and Matter Parity 6712 Proton Decay and Doublet-Triplet Splitting 68 U (1) ’s and Higher Dimension Operators 7114 Towards Realistic Yukawa Couplings 74
15 Suppression Factors From Singlet Wave Functions 86 µ Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9515.4 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9615.4.1 Majorana Masses and a Seesaw . . . . . . . . . . . . . . . . . 9715.4.2 Suppressed Dirac Masses . . . . . . . . . . . . . . . . . . . . . 9815.5 Relating µ and ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
16 Supersymmetry Breaking 100 SU (5) Model 10418 Evading the No Go Theorem and Flipped Models 109 SU (5) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
19 Numerology 114 A.1 A × A Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.2 A × D Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.3 E Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B Hypersurfaces in P C.1 Rank Four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132C.1.1 SU (5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133C.1.2 SO (8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133C.1.3 SO (9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.1.4 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.2 Rank Five . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.2.1 SU (6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.2.2 SO (10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138C.2.3 SO (11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142C.3 Rank Six . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148C.3.1 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149C.3.2 SU (7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165C.3.3 SO (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703 Introduction
Despite many theoretical advances in our understanding of string theory, this progresshas not produced a single verifiable prediction which can be tested against availableexperiments. Part of the problem is that in its current formulation, string theoryadmits a vast landscape of consistent low energy vacua which look more or less likethe real world.Reinforcing this gloomy state of affairs is the fact that the particle content of theStandard Model is generically of the type encountered in string theory. Indeed, thegauge group of the Standard Model is of the form Q i U ( N i ) and the chiral mattercontent corresponds to bi-fundamental fields transforming in representations such as( N i , N j ). While this may reinforce the idea that string theory is on the right track,precisely because this appears to be such a generic feature of string constructions,this also unfortunately limits the predictivity of the theory. To rectify this situation,we must impose additional criteria to narrow down the search in the vast landscape.From a top down approach, one idea is to further incorporate some specificallystringy principles. For instance, we have learned that the large N limit of many U ( N ) gauge theories causes the gauge system to ‘melt’ into a dual gravitationalbackground [1]. Moreover, this large N gauge theory can undergo a duality cascadeto a small N gauge theory [2]. Indeed, the Standard Model could potentially emergeat the end of such a process. In the string theory literature, this idea has beenexplored in [3–5]. Interesting as this idea is, it does not incorporate the idea of grandunification of the gauge forces into one gauge factor in any way.From a bottom up approach, it is natural to ask whether there is some way toincorporate the important fact that the gauge coupling constants of: G std ≡ SU (3) C × SU (2) L × U (1) Y (1.1)seem to unify in the minimal supersymmetric extension of the Standard Model(MSSM). This not only supports the idea that supersymmetry is realized at lowenergies, but also suggests that the multiple gauge group factors of the StandardModel unify into a single simple group such as SU (5) or SO (10). Moreover, thefact that the matter content of the Standard Model economically organizes into rep-resentations of the groups SU (5) and SO (10) provides a strong hint that the basicidea of grand unified theories (GUTs) is correct. For example, it is quite intriguing4hat all of the chiral matter of a single generation precisely organizes into the spinorrepresentation 16 of SO (10). Hence, we ask whether the principle of grand unifica-tion can narrow down the large list of candidate vacua in the landscape to a moretractable, and predictive subset.Despite the many attractive features of the basic GUT framework, the simplestimplementations of this idea in four-dimensional models suffer from some seriousdrawbacks. For example, the minimal four-dimensional supersymmetric SU (5) GUTwith standard Higgs content seems to be inconsistent with present bounds on protondecay [6]. In the absence of higher dimensional representations of SU (5) or somewhatelaborate higher dimension operator contributions to the effective superpotential,this model also leads to mass relations and over-simplified mixing matrices whichare generically too strong to be correct. This presents an opportunity for stringtheory to intervene: Can string theory preserve the nice features of GUT modelswhile avoiding their drawbacks?Indeed, the E × E heterotic string seems very successful in this regard becausethe usual GUT groups SU (5) and SO (10) can naturally embed in one of the E factors. See [7] for an early review on how GUT models could potentially origi-nate from compactifying the heterotic string on a Calabi-Yau threefold. Moreover,because no appropriate four-dimensional GUT Higgs field is typically available tobreak the GUT group to the Standard Model gauge group, it is necessary to employa higher-dimensional breaking mechanism. When the internal space has non-trivialfundamental group, the gauge group can break via a discrete Wilson line. In thisway, the gauge group in four dimensions is always the Standard Model gauge groupbut the matter content and gauge couplings still unify. Moreover, such higher di-mensional GUTs provide natural mechanisms to suppress proton decay and avoidunwanted mass relations. See [8–12] for some recent attempts in this direction.However, the heterotic string has its own drawbacks simply because it is ratherdifficult to break the gauge symmetry down to G std . One popular method isto use internal Wilson lines to directly break the gauge symmetry to that of theMSSM. This requires that the fundamental group of the Calabi-Yau must be non-trivial. Although this can certainly be arranged, the generic Calabi-Yau threefold At a pragmatic level, the perturbative regime of the heterotic string also seems to be inconsistentwith the relation between the GUT scale M GUT and the four-dimensional Planck scale M pl . Adiscussion of this discrepancy and related issues may be found in [13]. One potential way to bypassthis problem requires going to the regime of strong coupling [14].
5s simply connected and this mechanism is unavailable. Moreover, when the GUTgroup has rank five or higher, gauge group breaking by Wilson lines can also leavebehind additional massless U (1) gauge bosons besides U (1) hypercharge. Presentconstraints on additional long rang forces are quite stringent, and in many cases it isnot always clear how to remove these unwanted states from the low energy spectrum.In the absence of a basic principle which naturally favors a non-trivial fundamentalgroup, it therefore seems reasonable to look for other potential realizations of theGUT paradigm in string theory.There are two other natural ways that GUTs can appear in string theory. Thesepossibilities correspond to non-perturbatively realized four-dimensional N = 1 com-pactifications of type IIA and IIB string theory. In the type IIA case, the GUTmodels originate from the compactification of M-theory on manifolds with G holon-omy. For type IIB theories, the corresponding vacua are realized as compactificationsof F-theory on Calabi-Yau fourfolds. In the latter case, the gauge theory degrees offreedom of the GUT localize on the worldvolume of a non-perturbative seven-brane.The ADE gauge group on the seven-brane corresponds to the discriminant locus ofthe elliptic model where the degeneration is locally of
ADE type. Of these two possi-bilities, the holomorphic geometry of Calabi-Yau manifolds provides a more tractablestarting point for addressing detailed model building issues. It was with this aimthat we initiated an analysis of how GUT models can be realized in F-theory [15].See [16, 17] for related discussions in the context of F-theory/heterotic duality.Even so, there is a certain tension between string theory and the GUT paradigm.From a top down perspective, it is a priori unclear why there should be any distinctionbetween the Planck scale M pl and the GUT scale M GUT . In the bottom up approach,the situation is completely reversed. Indeed, insofar as effective field theory is validat the GUT scale, it is quite important that M GUT /M pl is small and not an orderone number. For example, in the extreme situation where the only chiral mattercontent of a four-dimensional GUT model originates from the MSSM, the resultingtheory is asymptotically free.In geometrically engineered gauge theories in string theory, asymptotic freedomtranslates to the existence of a consistent decompactification limit. It is thereforequite natural to ask if at least in principle we could have decoupled the two scales M GUT and M pl . This is also in accord with the bottom up approach to stringphenomenology [18–21]. In the present paper our main focus will therefore be to6earch for vacua which at least in principle admit a limit where M pl → ∞ while M GUT remains finite. Of course, in realistic applications M pl should also remain finite. Forcompleteness, we shall also present some examples of models where M GUT and M pl cannot be decoupled. In such cases, we note that it is not a priori clear whether thecorrect value of M GUT can be achieved.Nevertheless, the mere existence of a decoupling limit turns out to endow theresulting candidate models with surprising predictive power. It turns out that theonly way to achieve such a decoupling limit requires that the spacetime filling seven-brane must wrap a del Pezzo surface. The fact that the relevant part of the internalgeometry in this setup is limited to just ten distinct topological types is very welcome!In a certain sense, there is a unique choice corresponding to the del Pezzo 8 surfacebecause all of the other del Pezzo surfaces can be obtained from this one by blowingdown various two-cycles.At the next level of analysis, we must determine what kind of seven-brane shouldwrap the del Pezzo surface. As explained in [15], realizing the primary ingredientsof GUT models requires that the singularity type associated with the seven-braneshould correspond to a subgroup of the exceptional group E . Because the StandardModel gauge group has rank four, this determines a lower bound on the rank of anyputative GUT group. At rank four, SU (5) is the only available GUT group. Hence,the most ‘minimal’ choice is to have an SU (5) seven-brane wrapping the del Pezzo8 surface. We will indeed find that this minimal scenario is viable. The upperbound on the rank of a candidate GUT group is six. This bound comes about fromthe fact that if the rank is any higher, the model will generically contain localizedlight degrees of freedom at points on the del Pezzo surface which do not appearto admit a standard interpretation in gauge theory [15, 22]. This is because oncomplex codimension one subspaces, the rank of the gauge group goes up by one,and on complex codimension two subspaces, i.e. points, the rank goes up by two.Hence, if the rank is greater than six, the compactification contains points on thedel Pezzo with singularities of rank nine and higher which do not admit a standardgauge theoretic interpretation because E is the maximal compact exceptional group.In the minimal scenario where the seven-brane has gauge group SU (5), we findthat there is an essentially unique mechanism by which the GUT group can break toa four-dimensional model with gauge group G std . This breaking pattern occurs invacua where the U (1) hypercharge flux in the internal directions of the seven-brane7s non-trivial. This mechanism is unavailable in heterotic compactifications becausethe U (1) hypercharge always develops a string scale mass via the Green-Schwarzmechanism [23]. As noted for example in [23], in order to preserve a massless U (1)hypercharge gauge boson, additional U (1) factors must mix non-trivially with thisdirection, which runs somewhat counter to the idea of grand unification. Neverthe-less, for suitable values of the gauge coupling constants for these other factors, asemblance of unification can be maintained. See [24–27] for further discussion onvacua of this type.In F-theory, we show that there is no such generic obstruction. This is a conse-quence of the fact that while the cohomology class of the flux on the seven-brane canbe non-trivial, it can nevertheless represent a trivial class in the base of the F-theorycompactification. This topological condition is necessary and also sufficient for thecorresponding four-dimensional U (1) gauge boson to remain massless. An impor-tant consequence of this fact is that these F-theory vacua do not possess a heteroticdual.The particular choice of internal U (1) flux which breaks the GUT group is alsounique. To see how this comes about, we first recall that the middle cohomologyof the del Pezzo 8 surface splits as the span of the canonical class and the collec-tion of two-cycles orthogonal to this one-dimensional lattice. With respect to theintersection form on two-cycles, this orthogonal subspace corresponds to the rootlattice of E . Moreover, the admissible fluxes of the U (1) hypercharge are in one toone correspondence with the roots of E . This restriction occurs because for moregeneric choices of U (1) flux, the low energy spectrum contains exotic matter whichif present would ruin the unification of the gauge coupling constants. In keepingwith the general philosophy outlined in [15], we always specify the appropriate linebundle first and only then determine whether an appropriate K¨ahler class exists sothat the vacuum is supersymmetric. In this sense, there is a unique choice of flux be-cause the Weyl group of E acts transitively on the roots of E . On general grounds,this internal flux will also induce a small threshold correction near the GUT scale.Determining the size and sign of this correction would clearly be of interest to study. The matter and Higgs fields localize on Riemann surfaces in the del Pezzo surface.In F-theory, these Riemann surfaces are located at the intersection between theGUT model seven-brane and additional seven-branes in the full compactification. After our work appeared, this question has been studied in [28, 29]. SU (6) and the 10 or 10 for localenhancement to SO (10).The internal hypercharge flux automatically distinguishes the Higgs fields fromthe other chiral matter content of the MSSM. The Higgs fields localize on mattercurves where the U (1) hypercharge flux is non-vanishing, and the chiral matter of theMSSM localizes on Riemann surfaces where the net flux vanishes. In other words, thetwo-cycles for the Higgs curves intersect the root corresponding to this internal fluxwhile all the other chiral matter of the MSSM localizes on two-cycles orthogonal tothis choice of flux. This internal choice of flux implies that the chiral matter contentwill always fill out complete representations of SU (5), while the Higgs doublets cannever complete to full GUT multiplets. Moreover, by a suitable choice of flux onthe other seven-branes, the spectrum will contain no extraneous Higgs triplets,thus solving the doublet-triplet splitting problem. In certain cases, superheavyHiggs triplets can still cause the proton to decay too quickly. In traditional four-dimensional GUT models the missing partner mechanism is often invoked to avoidgenerating dangerous dimension five operators which violate baryon number. Here,this condition translates into the simple geometric condition that the Higgs up anddown fields must localize on distinct matter curves.In our study of Yukawa couplings, we shall occasionally encounter situations in-volving two fields charged under the GUT group and one neutral field (for examplea 1 × × R ∼ M GUT ), the normal geometryis negatively curved. Moreover, this leads to the wave function being either attractedto, or repelled away from our brane, depending on the choice of the gauge flux on thenormal intersecting seven-branes. In one case the wave function is attracted to ourseven-brane, making it behave as if the wave function is localized inside the brane.In another case the wave function is repelled away from our brane, leading to anexponentially small amplitude at our brane. The exponential hierarchy is given byexp( − cR ⊥ /R GUT ) where c is a positive order one constant, R ⊥ is the radius of the9ormal geometry to the brane, and R GUT is the length associated to GUT. The esti-mate for R ⊥ depends on assumptions about how the geometry normal to our branelooks, and in particular to what extent it is tubular. We find that: R ⊥ R GUT = ε − γ (1.2)where 1 / . γ . ε is a small parameter: ε ∼ M GUT α GUT M pl ∼ . × − . (1.3)This leads to a natural hierarchy given byexp (cid:18) − c R ⊥ R GUT (cid:19) ∼ exp (cid:18) − c ε γ (cid:19) . (1.4)There are various vector-like pairs which can only develop a mass through a cubicYukawa coupling with a third field coming from a neutral normal wave function.This suppression mechanism will be useful in many such cases, including solving the µ problem and also obtaining a small Dirac neutrino mass leading to realistic lightneutrino masses without using the seesaw mechanism.There are two ways we can solve the µ problem. Perhaps most simply, we canconsider geometries where the Higgs up and down fields localize on distinct mattercurves which do not intersect. In this case, the µ term is identically zero. When thesecurves do intersect, the value of the µ term depends on the details of a gauge singletwave function which localizes on a matter curve normal to the del Pezzo surface. Inthe case of attraction, the µ term is near the GUT scale, which is untenable. In therepulsive case, the µ term is suppressed to a much lower value: µM GUT ∼ exp (cid:18) − c ε γ (cid:19) , (1.5)so that the resulting value of µ can then naturally fall in a phenomenologically viablerange.In fact, a similar exponential suppression in the wave functions of the right-10anded neutrinos can generate small Dirac neutrino masses of the form: m Dν ∼ µε − γ h H u i × h H u i M GUT ∼ . × − ± . eV (1.6)which differs by a factor of µε − γ / h H u i from the value predicted by the simplesttype of seesaw mechanisms with Majorana masses at the GUT scale. We note thatthe value we obtain is in reasonable agreement with recent experimental results onneutrino oscillations. In this case, the Majorana mass term must identically vanishto remain in accord with observation.A variant of the standard seesaw mechanism is also available when the right-handed neutrino wave functions are attracted to the del Pezzo surface. In thiscase, the Majorana mass terms in the neutrino sector are suppressed by some overallvolume factors. Although the standard seesaw mechanism again generates naturallylight neutrino masses ∼ × − ± . eV, we find that the Majorana mass term isnaturally somewhat lighter than the GUT scale and is on the order of ∼ × ± . GeV. It is interesting that the numerical values we obtain in either scenario areboth in a range of values consistent with leptogenesis, as well as the observed lightneutrino masses.Non-trivial flavor structures can potentially arise in a number of ways in this classof models. For example, one common approach in the model building literature isto use a discrete symmetry to induce additional structure in the form of the Yukawacouplings. The Weyl group symmetries of the exceptional groups naturally act onthe del Pezzo surfaces. This symmetry can be partially broken by the choice of theK¨ahler classes of two-cycles. This may potentially lead to a model of flavor based onthe discrete symmetry groups S , A or S . Indeed, these are all subgroups of theWeyl group of E .One of the main conceptual issues with the usual GUT framework is to explainwhy m b ∼ m τ at the GUT scale while the lighter generations do not satisfy sucha simple mass relation. At a qualitative level, the behavior of the omnipresentinternal U (1) hypercharge flux again plays a central role in the resolution of thisissue. Although the net hypercharge flux vanishes on curves which support fullGUT multiplets, in general it will not vanish pointwise. Hence, the hyperchargeflux can still leave behind an important imprint on the wave functions of the fieldsin the MSSM. Indeed, because the individual components of a GUT multiplet have11ifferent hypercharge, the Aharonov-Bohm effect will alter the distinct components ofa GUT multiplet differently, leading to violations in the most naive mass relations.In fact, because the mass of a generation is higher the smaller the volume of thematter curve, the amount of flux which can pierce the curve also decreases. Inthis way, the most naive mass relations remain approximately intact for the heaviestgeneration but will in general receive corrections for the lighter generations.In the next to minimal GUT scenario, we can consider seven-branes where thebulk gauge group has rank five. In this case there are three choices correspondingto SO (10), SO (11) and SU (6). In this paper we mainly focus on the SO (10) = E case because it fits most closely with our general philosophy that the exceptionalgroups play a distinguished role in GUT models. It turns out that this modelcan only descend to the MSSM by a sequence of breaking patterns where the eight-dimensional theory first breaks to a four-dimensional flipped SU (5) model with gaugegroup SU (5) × U (1). The model then operates as a traditional four-dimensionalflipped SU (5) GUT which breaks to the Standard Model gauge group when a fieldin the 10 − of SU (5) × U (1) develops a suitable vev. Indeed, direct breaking of SO (10) to the Standard Model gauge group via fluxes taking values in a U (1) × U (1)subgroup always generates exotic matter which would ruin the unification of thegauge coupling constants. Many of the more refined features of these models suchas textures and our solution to the µ problem share a common origin to those studiedin the minimal SU (5) model.Even though our main emphasis in this paper is on models which admit a de-coupling limit, we also consider models where such a limit does not exist. In suchcases the problem of engineering a GUT model becomes more flexible because thelocal model is incomplete. We study examples of this situation because there arewell-known difficulties in heterotic models in realizing traditional four-dimensionalGUT group breaking via fields in the adjoint representation. This is due to the factthat in many cases, the requisite adjoint-valued fields do not exist. Indeed, gaugegroup breaking by Wilson lines is not so much an elegant ingredient in heterotic con-structions as much as it is a necessary element of any construction. Gauge groupbreaking via Wilson lines can also occur in F-theory when the surface wrapped by the It is also possible to avoid this constraint in heterotic models which descend to a four-dimensional flipped SU (5) GUT. See [25, 30, 31] for further details on this approach. We alsonote that in certain cases, chiral superfields transforming in other representations can arise fromhigher Kac-Moody levels of the heterotic string. π ( S ) = 0 is the Enriques surface which can be viewed as the Z quotient of a K h , ( S ) = 0. But in contrast to the usual approach tofour-dimensional effective field theories where it is common to assume that Planckscale physics can in principle be decoupled, here we see that the traditional four-dimensional GUT cannot be decoupled from Planck scale physics.We also briefly consider supersymmetry breaking in our setup. This is surpris-ingly simple to accommodate because extra messenger fields can naturally arise fromadditional matter curves which do not intersect any of the other curves on which thematter content of the MSSM localizes. Supersymmetry breaking can then communi-cate to the MSSM via the usual gauge mediation mechanism. We note that becausethe µ term naturally develops a value around the electroweak scale independently ofany supersymmetry breaking mechanism, we can retain many of the best features ofgauge mediation such as the absence of additional flavor changing neutral currents(FCNCs) while avoiding some of the problematic elements of this scenario which arerelated to generating appropriate values for the µ and Bµ terms. Depending on thelocal behavior of the wave functions which propagate in directions normal to the delPezzo surface, the messenger scale can quite flexibly range from values slightly belowthe GUT scale to much lower but still phenomenologically viable mass scales.The organization of this paper is as follows. In Section 2, we formulate whatwe wish to achieve in our GUT constructions. In Section 3 we review and slightlyextend our previous work on realizing GUT models in F-theory. To this end, wedescribe many of the necessary ingredients for an analysis of the matter content andinteraction terms of any potential model. Before proceeding to any particular classof models, in Section 4 we discuss the various mass scales which will genericallyappear throughout this paper. In Section 5, we give a general overview of theclass of GUT models in F-theory we shall study. These models intrinsically dividebased on how the GUT breaks to the MSSM. We first study models where theGUT scale cannot be decoupled from the Planck scale. In Section 6 we discuss13odels where GUT breaking proceeds just as in four-dimensional models. Next, wediscuss GUT breaking via discrete Wilson lines in Section 7. In the remainder ofthe paper we focus on the primary case of interest where a decoupling limit exists.Section 8 reviews some relevant geometrical facts about del Pezzo surfaces. This isfollowed in Section 9 by a study of GUT breaking to the MSSM via an internal U (1)hypercharge flux. In Section 10 we determine which bulk gauge groups can breakdirectly to the Standard Model gauge group via internal fluxes. We also explainin greater detail how to obtain the exact spectrum of the MSSM from such models.In Section 11 we discuss a geometric realization of matter parity, and in Section12 we study the interrelation between proton decay and doublet triplet splitting inour models. After giving a simple criterion for avoiding the simplest dimension fiveoperators responsible for proton decay, in Section 13 we explain how extra global U (1)symmetries in the low energy effective theory are encoded geometrically in F-theory,and in particular, how these symmetries can forbid potentially dangerous higherdimension operators. In Section 14 we discuss some coarse properties of Yukawacouplings and also speculate on how further details of flavor physics could in principlebe incorporated. In this same Section we also provide a qualitative explanationfor why the usual mass relations of GUT models become increasingly distorted asthe mass of a generation decreases. In Section 15 we show that interaction termsinvolving matter fields which localize on Riemann surfaces outside of the surface cangenerate hierarchically small values for both the µ term as well as Dirac neutrinomasses. We also study a variant on the usual seesaw mechanism which generates theexpected mass scale for the light neutrinos. Intriguingly, the Majorana mass of theright-handed neutrinos is somewhat lower than the value expected in typical GUTmodels. In Section 16 we propose how supersymmetry breaking could communicateto the MSSM, and in Section 17 we present an SU (5) model which incorporates some(but not all!) of the mechanisms developed in previous sections. Our expectationis that further refinements are possible which are potentially more realistic. In asimilar vein, in Section 18 we present a flipped SU (5) model. Section 19 collectsvarious numerical estimates obtained throughout the paper, and Section 20 presentsour conclusions. The Appendices contain further background material used in themain body of the paper and which may also be of use in future model building efforts.14 −theory/CY
4d MSSM 4d Flipped SU(5)4d MSSM10 = 0E Root N U(1) Flux NoYesS = del Pezzo S = del Pezzo8d SO(10)8d SU(5) 4d GUTs, 8d GUTs, ...Decoupling Limit?
Figure 1: General overview of how GUT breaking constrains the type of GUT model.15
Constraints From Low Energy Physics
In this Section we define the criteria by which we shall evaluate how successfully ourmodels reproduce features of low energy physics obtained by a minimal extrapola-tion of experimental data to the MSSM. There are a number of open questions inboth phenomenology and string theory which must ultimately be addressed in anyapproach. See [32, 33] for an expanded discussion of some of the issues we brieflyaddress here.At the crudest level, we require that any viable model contain precisely threegenerations of chiral matter. It is an experimental fact that the chiral mattercontent of the Standard Model organizes into SU (5) and SO (10) GUT multiplets.Coupled with the fact that the gauge couplings of the MSSM appear to unify at anenergy scale M GUT ∼ × GeV, we shall aim to reproduce these features in all ofthe models we shall consider. For all of these reasons, we require that the low energycontent of all of our models must match to the matter content of the MSSM. By thiswe mean that in addition to achieving the correct chiral matter content and Higgscontent of the MSSM, all additional matter charged under the gauge groups must atthe very least fit into vector-like pairs of complete GUT multiplets in order to retaingauge coupling unification. In the minimal incarnation of GUT models consideredhere, we shall further require that the low energy spectrum of particles charged underthe Standard Model gauge group must exactly match to the matter content of theMSSM. We note that historically, even this qualitative requirement has been difficultto achieve in Calabi-Yau compactifications of the perturbative heterotic string.Although the correct particle content is a necessary step in achieving a realisticmodel, it is certainly not sufficient because we must also reproduce the superpotentialof the MSSM: W = µH u H d + λ uij Q i U j H u + λ dij Q i D j H d + λ lij L i E j H d + λ νij L i N jR H u + ... (2.1)where the indices i and j label the three generations. While the precise form of theYukawa matrices labeled by the λ ’s will lead to masses and mixing terms between thegenerations, a necessary first step is that there are in principle non-zero contributions While it is in principle possible to consider models where vector-like exotics preserve gaugecoupling unification, we believe this runs contrary to the spirit of GUT models. Although we shallnot entertain this possibility here, see [34, 35] for further discussion of this possibility.
16o the above superpotential! As a first approximation, we require that the tree levelsuperpotential of the theory at high energy scales generate a non-trivial interactionterm for the third generation so that there is a rough hierarchy in mass scales. Inthe context of GUT models, it is well-known that because the particle content ofthe Standard Model organizes into complete GUT multiplets, the Yukawa couplingscouple universally to fields organized in such multiplets. One attractive featureof the tree level superpotential in most GUT models is that the third generationobeys a simple mass relation of the form m b /m τ ∼ m b /m τ ∼ ≥ − yrs [36]). Thisrequires that certain operators must be absent or sufficiently suppressed in the lowenergy superpotential. Indeed, note that in equation (2.1), we have implicitly onlyincluded renormalizable R-parity invariant couplings because if present, the interac-tion terms λ ijk U i D j D k and λ ′ ijk L i L j E k will cause the proton to decay too rapidly.We shall consider models with and without R-parity. In the latter case, we thereforemust present alternative reasons to expect renormalizable operators responsible forR-parity to vanish.Proton decay is a hallmark of GUT models. Aside from renormalizable interac-tion terms, the dominant contribution to proton decay in the simplest GUT modelscomes from the dimension five operator [37, 38]: O = c M GUT Z d θQQQL (2.2)and the dimension six operator: O = c M GUT Z d θQQU † E † . (2.3) There is an additional contribution to the superpotential given by
U U DE . At the level ofdiscussion in this paper, it is sufficient to only deal with the term
QQQL . O can originate from the exchange of heavy Higgs triplets and cancause the decay p → K + ν . The operator O can originate from the exchange ofheavy off-diagonal GUT group gauge bosons and can cause the decay p → e + π . Toremain in accord with current bounds on nucleon decay, c can typically be an orderone coefficient whereas c must be suppressed at least to the order of 10 − . See [39]for further discussion on proton decay in GUT models.In four-dimensional GUT models, this issue is closely related to the mechanismresponsible for removing the Higgs triplets from the low energy spectrum. Onecommon approach is to invoke some continuous or discrete symmetry to sufficientlysuppress this operator. The use of discrete symmetries in compactifications of M-theory on manifolds with G holonomy has been studied in [40]. Note that whilethe Higgs triplet must develop a sufficiently large mass in order to reproduce theparticle content of the MSSM, we must also require that the supersymmetric Higgsmass µ should be on the order of the weak scale.While the above problems are necessary requirements for any potentially viablemodel, there are many additional phenomenological constraints which must be sat-isfied in a fully realistic compactification. In principle, a complete model should alsonaturally accommodate hierarchical masses for the quarks and leptons. For exam-ple, in conventional GUT models, the seesaw mechanism allows the neutrino massesin the Standard Model to be much lighter than the electroweak scale. At a morerefined level, a full model should explain why the CKM matrix is nearly equal to theidentity matrix whereas the MNS matrix contains nearly maximal mixing betweenthe neutrinos.A fully realistic model must of course specify how supersymmetry is broken andprovide a mechanism for communicating this breaking to the MSSM. Our expecta-tion is that this issue can be treated independently from the supersymmetric modelswhich shall be our primary focus here. We note that for general string compactifica-tions, supersymmetry breaking is closely entangled with moduli stabilization. Whilewe will not specify a method for stabilizing moduli, we note that F-theory providesa natural arena for further study of this issue. See [41] for a particular example ofmoduli stabilization in F-theory and [42] for a review of this active area of research.18 Basic Setup
In this Section we review the basic properties of exceptional seven-branes in F-theory.In particular, we explain how to compute the low energy matter spectrum as well asthe effective superpotential of the four-dimensional theory. Further details may befound in [15].F-theory compactified on an elliptically fibered Calabi-Yau fourfold preserves N = 1 supersymmetry in the four uncompactified spacetime dimensions. Letting B denote the base of the Calabi-Yau fourfold, the discriminant locus of the ellip-tic fibration determines a subvariety ∆ of complex codimension one in the base B .Denoting by S the K¨ahler surface defined by an irreducible component of ∆, whenthis degeneration locus is a singularity of ADE type, the resulting eight-dimensionaltheory defines the worldvolume of an exceptional seven-brane with gauge group G S of ADE type. This singularity type can enhance along complex codimension onecurves in S to a singularity of type G Σ and can further enhance at complex codi-mension two points in S to a singularity of type G p . Such points correspond to thetriple intersection of three matter curves. Because the Cartan subalgebra of eachsingularity type is visible to the geometry [43, 44], these enhancements satisfy thecontainment relations: G S × U (1) × U (1) ⊂ G Σ × U (1) ⊂ G p . (3.1)As argued in [15], many necessary features of even semi-realistic GUT models requirethat G p ⊂ E . In particular, this implies that the rank of the bulk gauge group G S is at most six. This significantly limits the available bulk gauge groups becausethe rank of G S must be at least four in order to contain the Standard Model gaugegroup.In this paper we shall assume that given a choice of matter curves, there existsa Calabi-Yau fourfold which contains the corresponding local enhancement in singu-larity type. While this assumption is clearly not fully justified for compact models,in the context of local models this can always be done. As an example, we now en-gineer a local model where the bulk gauge group E enhances along a matter curveΣ in S to an E singularity. A local elliptic model of this type is: y = x + f xz + q z . (3.2)19n the above, q is a section of O S (Σ), f is a section of L ⊗ K − S and the coordinates( x, y, z ) transform as a section of [15]: L ⊕ L ⊕ L ⊗ K S (3.3)where K S denotes the canonical bundle on S and L is a line bundle which can beexpressed in terms of K S and O S (Σ). The essential point of this example is that ina local model, there always exists a line bundle L such that the resulting local modelis well-defined. For example, in this case we have: L = O S (Σ) ⊗ K S . (3.4)Further, we shall make the additional assumption that there is no mathematicalobstruction to various twofold enhancements in the rank of the singularity at pointsof the surface S . It would certainly be of interest to study this issue.We now describe in greater detail the effective action of exceptional seven-branes.In terms of four-dimensional N = 1 superfields, the matter content of the theoryconsists of an N = 1 vector multiplet which transforms as a scalar on S , a collectionof chiral superfields A i which transform as a (0 ,
1) form on S (the bulk gauge bosons)and a collection of chiral superfields Φ which transform as a holomorphic (2 ,
0) formon S . The bulk modes couple through the superpotential term: W S = Z S T r (cid:2)(cid:0) ∂ A + A ∧ A (cid:1) ∧ Φ (cid:3) . (3.5)When two irreducible components S and S ′ of ∆ intersect on a Riemann surfaceΣ, the singularity type enhances further. In this case, additional six-dimensionalhypermultiplets localize along Σ. As in [44], the representation content of thesefields is given by decomposing the adjoint representation of the enhanced singularityto the product G S × G S ′ associated with the gauge groups on S and S ′ . In terms offour-dimensional N = 1 superfields, the matter content localized on a curve consistsof chiral superfields Λ and Λ c which transform as spinors on Σ. The bulk modes20ouple to matter fields localized on the curve via the superpotential term: W Σ = Z Σ (cid:10) Λ c , ( ∂ + A + A ′ )Λ (cid:11) (3.6)where h· , ·i denotes the natural pairing which is independent of any metric data.Finally, when three irreducible components of ∆ intersect at a point p , the sin-gularity type can enhance even further. Evaluating the overlap of three Λ’s forthree matter curves yields a further contribution to the four-dimensional effectivesuperpotential: W p = Λ Λ Λ | p . (3.7)An analysis similar to that given below equation (3.2) shows that given three mattercurves which form a triple intersection, so long as the resulting interaction termis consistent with group theoretic considerations, there exists a local Calabi-Yaufourfold with the desired twofold enhancement in singularity type.Having specified the individual contributions to the quasi-topological eight-dimensionaltheory, the superpotential is: W [Φ , A, Λ] = W S + ... + W S l + W Σ + ... + W Σ m + W p + ... + W p n + W flux + W np . (3.8)In the above, the corresponding fields entering the above expression are to be viewedas a large collection of four-dimensional chiral superfields labeled by points of thecomplex surfaces S i and the Riemann surfaces Σ i . We have also included the contri-bution from the flux-induced superpotential which couples to the various (2 ,
0) formsof the seven-branes and indirectly to matter fields localized on curves. As explainedin [15], the vevs for the (2 ,
0) form and fields localized on matter curves correspond tocomplex deformations of the Calabi-Yau fourfold. Because the flux-induced super-potential couples to the complex structure moduli of the Calabi-Yau fourfold, suchterms will generically be present. In equation (3.8), we have also included the term W np which denotes all non-perturbative contributions from wrapped Euclidean three-branes. These terms are proportional to exp( − aV ol ( S )) ∼ exp( − c/α GUT ) where c is an order one positive constant. In a GUT model where the gauge couplingconstants unify perturbatively, such contributions are negligible.The fields of the four-dimensional effective theory correspond to zero mode solu-21ions in the presence of a background field configuration. As in [15], we shall confineour analysis of the matter spectrum to backgrounds where all fields other than thebulk gauge field are expanded about zero. In the presence of a non-trivial back-ground gauge field configuration, the chiral matter content of the four-dimensionaleffective theory descends from bulk modes on S and Riemann surfaces which wedenote by the generic label Σ. An instanton taking values in a subgroup H S willbreak G S to the commutant subgroup. Decomposing the adjoint representation of G S to the maximal subgroup of the form Γ S × H S , the chiral matter transforming ina representation τ of Γ S descends from the bundle-valued cohomology groups: τ ∈ H ∂ ( S, T ∗ ) ∗ ⊕ H ∂ ( S, T ) ⊕ H ∂ ( S, T ∗ ) ∗ (3.9)where T denotes a bundle transforming in the representation T of H S obtained bythe decomposition of the adjoint representation of the associated principle G S bundleon S . When S is a del Pezzo surface, the cohomology groups H ∂ and H ∂ vanishfor supersymmetric gauge field configurations so that the number of zero modestransforming in the representation τ is given by an index: n τ = χ ( S, T ) = − (cid:18) c ( S ) · c ( T ) + 12 c ( T ) · c ( T ) (cid:19) . (3.10)An analogous computation holds for the zero mode content localized on a Rie-mann surface transforming in a representation ν × ν ′ of H S × H S ′ : ν × ν ′ ∈ H ∂ (Σ , K / ⊗ V ⊗ V ′ ) (3.11)so that the net number of zero modes is given by the index: n ν × ν ′ − n ν × ν ′ = deg ( V ⊗ V ′ ) . (3.12)In many cases we shall compute the relevant cohomology groups in equation (3.11)by assuming a canonical choice of spin structure. As argued in [15], this can alwaysbe done when the curve corresponds to the vanishing locus of the holomorphic (2 , π ( S ) = 0, it is also possible to consider vacua with non-trivial Wilson lines.In order to avoid complications from the reduction of additional supergravity modes,22e shall always assume that π ( S ) is a finite group. The discussion closely parallelsa similar analysis in heterotic compactifications (see for example [45]). Recall thatadmissible Wilson lines are specified by a choice of element ρ S ∈ Hom ( π ( S ) , G S ).In order to maintain continuity with the discussion reviewed above, we shall requirethat the non-trivial portion of the discrete Wilson line takes values in the subgroupΓ S ⊂ G S defined above. More generally, this restriction can be lifted and mayallow additional possibilities for projecting out phenomenologically unviable repre-sentations from the low energy spectrum. Under these restrictions, the unbrokenfour-dimensional gauge group is given by the commutant subgroup of ρ S ( π ( S )) × H S in G S .We now determine the zero mode content of the theory in the presence of a non-trivial discrete Wilson line. As in Calabi-Yau compactifications of the heteroticstring, our strategy will be to lift all computations to a covering theory. Because π ( S ) is finite, the universal cover of S denoted by e S is a compact K¨ahler surface.Letting p : e S → S denote the covering map, the bundle T on S now lifts to a bundle e T = p ∗ ( T ) on e S . Under the present restrictions, the Wilson line corresponds to aflat Γ S -bundle induced from the covering map from e S to S . The deck transformationdefined by the action of π ( S ) on e S also determines a group action of π ( S ) on thecohomology groups H i∂ ( e S, e T ). Treating H i∂ ( e S, e T ) as a complex vector space, theeigenspace decomposition of H i∂ ( e S, e T ) is of the form: H i∂ ( e S, e T ) ≃ ⊕ λ C λ (3.13)in the obvious notation. The irreducible representation of Γ S defined by τ decom-poses into irreducible representations of the maximal subgroup Γ × ρ S ( π ( S )) ⊂ Γ S as: τ ≃ ⊕ i τ i ⊗ R i . (3.14)The zero modes transforming in the representation τ i are therefore specified by the ρ S invariant subspaces: τ i : h H ∂ ( e S, e T ∗ ) ∗ ⊗ R i i ρ S ⊕ h H ∂ ( e S, e T ∗ ) ∗ ⊗ R i i ρ S ⊕ h H ∂ ( e S, e T ∗ ) ∗ ⊗ R i i ρ S . (3.15)Having specified the zero mode content of the theory, we can now in principledetermine the full superpotential of the low energy effective theory by integrating23ut all Kaluza-Klein modes from equation (3.8). This is similar to the treatmentof Chern-Simons gauge theory as a string theory [46]. For quiver gauge theoriesdefined by D-brane probes of Calabi-Yau threefolds, the higher order terms of theeffective superpotential are given by integrating out all higher Kaluza-Klein modesfrom the associated holomorphic Chern-Simons theory for B-branes [47].In the present context, we can follow the procedure outlined in [48] to determinethe full expression for the effective superpotential. This is given by a bosonicpartition function with action given by the superpotential of equation (3.8). Viewingthe higher-dimensional fields as a collection of four-dimensional chiral superfieldslabeled by points of the internal space, the effective superpotential is now given bythe bosonic path integral:exp ( − W eff [Φ , A , Λ ]) = Z P I [ d Φ][ dA ][ d Λ] exp ( − W [Φ + Φ , A + A , Λ + Λ ])(3.16)where the zero subscript denotes the zero mode, and the path integral is over all oneparticle irreducible Feynman diagrams. In this expression, W tree should be viewedas a bosonic action with functional dependence identical to that of equation (3.8).The complete four-dimensional effective superpotential for the zero modes is thendetermined by the partition function of the quasi-topological theory. We emphasizethat this partition function is well-defined without any reference to metric data. Avery similar procedure for extracting the superpotential by integrating out Kaluza-Klein modes in heterotic compactifications has been given in [23]. Some examples ofsimilar computations for quiver gauge theories can be found in [49]. To conclude thisSection, we note that any symmetry of the full eight-dimensional theory descendsto the four-dimensional effective superpotential for the zero modes. Neglecting thecontribution due to non-perturbative effects in equation (3.8), the extra U (1) factorswhich are always present when the singularity type enhances will provide additionalglobal symmetries in the effective theory which will typically forbid some higherdimension operators from being generated. Although non-perturbative effects canviolate these symmetries, the corresponding contribution to W eff [Φ , A , Λ ] willtypically be small enough that we may safely neglect such contributions.These general considerations already constrain the matter content of any candi-date theory. Modes propagating in the bulk of the surface S must transform in the24djoint representation of the bulk gauge group. Moreover, although matter fieldscan localize on a curve Σ inside of S , these fields must descend from the adjointrepresentation of G Σ . For example, for SU ( N ) gauge group factors which do notembed in E , the only available local enhancements are to higher A or D type singu-larities. In such cases, the decomposition of the adjoint representation only containstwo index representations. Similar restrictions apply for SO ( N ) gauge group factorswhich do not embed in E . In particular, the spinor representation never appears insuch cases. In a sense, this is to be expected because these are precisely the typesof configurations which can be realized within perturbative type IIB vacua.For SO ( N ) ⊂ E gauge groups, the available representations are the vector,spinor or adjoint representations, and for SU ( N ) ⊂ E gauge groups, the only avail-able representations are the one, two or three index anti-symmetric and the adjointrepresentations. For example, when G S = SO (10), this implies that all of the mat-ter fields transform in the 10 , ,
16 or 45, while for G S = SU (5), the only availablerepresentations are the 5, 5, 10, 10 or 24. in the specific case of del Pezzo models, thismatter content is even more constrained. Indeed, as explained in [15], the bulk zeromode content for del Pezzo models never contains chiral superfields which transformin the adjoint representation of the unbroken gauge group in four dimensions.In fact, the type of twofold enhancement strongly determines the qualitativebehavior of the associated triple intersection of matter curves. For example, thepossible rank two enhancements of SU (5) are E , SO (12), and SU (7). In the caseof E and SO (12), the associated curves which form a triple intersection all liveinside of S . Indeed, by group theory considerations, the matter fields localized oneach curve transform in non-trivial representations of SU (5) [15]. On the otherhand, this is qualitatively different from a local enhancement to SU (7). In thiscase, two of the curves of the triple intersection support matter in the fundamentaland anti-fundamental of SU (5) and therefore live in S , while the third curve of theintersection supports matter in the singlet representation. Strictly speaking there are additional possibilities if the rank of the bulk singularity enhancesby more than one rank. If one allows more general breaking patterns involving higher SU ( N ) and SO (2 N ) type enhancements, it is also possible to achieve two index symmetric representations of SU ( N ) theories. For example, letting A N denote the two index anti-symmetric representation of SU (2 N ), A N decomposes to SU ( N ) × SU ( N ) as A N → A N ⊗ ⊗ A N + F N ⊗ F N . Higgsing thisto the diagonal SU ( N ) subgroup, we note that the product F N ⊗ F N contains two index symmetricrepresentations. This is a rather exotic possibility and we shall therefore not consider it further inthis paper. S at a point. Whilethe vev of this gauge singlet can induce a mass term for the vector-like pair, thedynamics of this field in the threefold base B is qualitatively different from fieldswhich localize inside of S . Before proceeding to specific models, we first present a general analysis of the rele-vant mass scales in the local models we treat in this paper. Rather than specify oneparticular profile for the threefold base B , we consider both geometries where B is roughly tubular so that it decomposes as the product of S with two non-compactdirections orthogonal to S in B , as well as more homogeneous profiles. To parame-terize our ignorance of the details of the geometry, we define the length scales: R S ≡ V ol ( S ) / (4.1) R B ≡ V ol ( B ) / (4.2)as well as a cutoff length scale R ⊥ which measures the radius normal to S : R ⊥ ≡ R B × (cid:18) R B R S (cid:19) ν (4.3)so that the exponent ν ranges from ν = 0 when B is homogeneous, to the value ν = 2 when B is the product of S with two non-compact directions. Indeed, theapproximations we consider in this paper are valid in the regime 0 . ν .
2. Notethat under the assumption R B > R S , the three length scales are related by: R ⊥ > R B > R S . (4.4)See figure 2 for a comparison of the local behavior of B for ν ∼ ν ∼
2. Toclarify, although the directions normal to S are “non-compact” in our local model,in a globally consistent compactification of F-theory they will still be quite small,and all on the order of the GUT scale, as will be discussed below. Indeed, this is26uite different from models based on large extra dimensions which can be either flat,but still compact [50], or potentially highly warped and of infinite extent [51].Compactifying on a threefold base B , the ten-dimensional Einstein-Hilbert ac-tion is: S EH ∼ M ∗ Z R , × B R √− gd x (4.5)where M ∗ is a particular mass scale associated with the supergravity limit of theF-theory compactification. In perturbative type IIB string theory, the parameter M ∗ is given in string frame by the relation M ∗ = M s /g s . Upon reduction to fourdimensions, the four-dimensional Planck scale M pl satisfies the relation: M pl ∼ M ∗ V ol ( B ). (4.6)The tension of a seven-brane wrapping a K¨ahler surface S in B determines thegauge coupling constant of the four-dimensional effective theory. More precisely,the coefficient of the kinetic term for the gauge field strength is of the form: S kin ∼ − M ∗ Z R , × S T r ( F ∧ ∗ F ) . (4.7)The value of the gauge coupling constant at the scale of unification is therefore: α − GUT ∼ M ∗ V ol ( S ). (4.8)Equations (4.6) and (4.8) now imply: V ol ( B ) ∼ ( α GUT M pl V ol ( S )) (4.9)or: R B ∼ (cid:0) α GUT M pl R S (cid:1) . (4.10) The astute reader will notice a difference in sign between the gauge kinetic term used here,and the convention adopted in [15]. In [15], we adopted an anti-hermitian basis of Lie algebragenerators in order to conform to conventions typically used in topological gauge theory. Becauseour emphasis here is on the four-dimensional effective field theory, in this paper we have revertedback to the standard sign convention in the physics literature so that all Lie group generators arehermitian. 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ΣΣ Figure 2: Depiction of F-theory compactified on a local model of a Calabi-Yaufourfold with non-compact base threefold B . The diagram shows the behaviorof the geometry in the neighborhood of a compact K¨ahler surface S on which thegauge degrees of freedom of the GUT model can localize in the cases where B isgiven by a roughly tubular geometry, as in case a), as well as geometries where B is more homogeneous, as in case b). In both cases, the directions orthogonal to S in B are large compared to S , but not warped. To regulate the geometry of thelocal model it is necessary to introduce a cutoff length scale which we denote by R ⊥ .The intersection locus between the compact surface S and a non-compact surface S ′ appears as a curve Σ in S . When seven-branes wrap both surfaces, additional lightstates will localize on this matter curve. 28e now convert these geometric scales into mass scales in the low energy effectivetheory. To this end, we next relate V ol ( S ) to the GUT scale M GUT . In most of thecases we consider, non-zero flux in the internal directions of S will partially breakthe bulk gauge group of the seven-brane. Letting p h F S i denote the mass scale ofthe internal flux, we therefore require: M GUT ∼ h F S i . (4.11)Because the flux is measured in units of length − on the surface S , this implies: V ol ( S ) ∼ M − GUT . (4.12)Equation (4.9) therefore yields:
V ol ( B ) ∼ (cid:0) α GUT M pl M − GUT (cid:1) . (4.13)The radii R B and R S are therefore given by:1 R S ∼ M GUT = 3 × GeV (4.14)1 R B ∼ M GUT × ε / ∼ GeV (4.15)where we have introduced the small parameter: ε ≡ M GUT α GUT M pl ∼ . × − . (4.16)Collecting equations (4.9) and (4.12), the parameter R ⊥ now takes the form:1 R ⊥ = M GUT × ε γ ∼ × ± . GeV (4.17)where 1 / ≤ γ ≤
1. We note that these numerical values for the radii satisfy theinequality of line (4.4).We conclude this Section by discussing the normalization of Yukawa couplingsin models where the superpotential originates from the triple intersection of matter29urves. In a holomorphic basis of wave functions, the F- and D-terms are: L holF = X p ψ i ( p ) ψ j ( p ) ψ k ( p ) Z d θ e φ i e φ j e φ k (4.18) ≡ λ holijk Z d θ e φ i e φ j e φ k (4.19) L holD = M ∗ Z Σ d θK ( e φ, e φ † ) (4.20)where in the above, ψ i ( p ) denotes the internal value of the wave function associatedwith the four-dimensional chiral superfield e φ i evaluated at a point p in S , and theholomorphic Yukawa couplings are defined as: λ holijk = X p ψ i ( p ) ψ j ( p ) ψ k ( p ). (4.21)The behavior of the wave functions near these points can generate hierarchicallysmall values near nodal points, and order one values away from such nodal points.We eventually wish to extract numerical estimates for the physical Yukawa cou-plings, defined in a basis of four-dimensional chiral superfields with canonically-normalized kinetic terms. However, if we reduce the D -term in (4.20) over Σ, wefind that the kinetic term for e φ is multiplied by the L -norm on Σ of the correspond-ing zero-mode wave function ψ .In general, ψ transforms on Σ as a holomorphic section of K / ⊗ L , where L is aline bundle on Σ determined by the gauge field on S . Both K / and L carry naturalhermitian metrics inherited from the bulk metric and gauge field on S . Fixing theholomorphic wave function ψ , we are interested in how the L -norm of ψ scaleswith the metric on S , since the volume of S effectively determines M GUT . Forconcreteness, let us write the metric on S in local holomorphic coordinates ( z, w ) as ds = g zz dzdz + g ww dwdw , where z is a local holomorphic coordinate along Σ and w is a holomorphic coordinate normal to Σ. Under an overall scaling g ℓg , the30ermitian metric on L is unchanged, so the norm of ψ behaves as h ψ | ψ i = Z Σ d z g zz (cid:0) g zz (cid:1) / ψψ , ℓ / h ψ | ψ i . (4.22)Since the volume of Σ scales with ℓ , we see from (4.22) that h ψ | ψ i scales withVol(Σ) / .At first glance, the dependence of h ψ | ψ i on ℓ might appear to be the only sourceof ℓ -dependence in the respective F - and D -terms in (4.18) and (4.20), since the F -term is determined by the overlap of fixed holomorphic wavefunctions. However,in making precise sense of this overlap, an additional ℓ -dependence also enters.To explain this ℓ -dependence, let us consider a slightly simplified situation, forwhich the holomorphic curves Σ , Σ , and Σ meet transversely at a point insidea Calabi-Yau threefold B . The role of the line bundle L is inessential, so on eachcurve we take the wavefunction ψ i to transform as a holomorphic section of K / i .In local holomorphic coordinates ( z, w, v ) around the point p of intersection, thewavefunction overlap is defined by ψ ( p ) ψ ( p ) ψ ( p ) √ dz √ dw √ dv p Ω( p ) . (4.23)Here Ω is a holomorphic three-form on B which we must introduce so that theoverlap in (4.23) does not depend on the particular holomorphic coordinates ( z, w, v )chosen at p .Of course, Ω is unique up to scale — but it is precisely the scale of the overlapthat we are trying to fix! Given that B carries a metric, we fix the norm of Ω bythe requirement that − i Ω ∧ Ω = ω ∧ ω ∧ ω , where ω is the K¨ahler form associated tothe metric on B . Once we impose this condition, Ω scales as Ω ℓ / Ω under anoverall scaling of the metric on B . Hence the wavefunction overlap in (4.23) andthus the holomorphic Yukawa coupling λ holijk actually scales as ℓ − / .After canonically normalizing all kinetic terms, the physical Yukawa couplingsare given by λ physijk = λ holijk p M ∗ h ψ i | ψ i i M ∗ h ψ j | ψ j i M ∗ h ψ k | ψ k i . (4.24)31y the preceding discussion, under an overall scaling g ℓ g of the metric on B , thephysical Yukawa coupling scales as λ physijk ℓ − / λ physijk . Restoring the dependence onthe volumes of each curve, we find the result which one would naively guess, λ physijk = λ ijk p M ∗ Vol(Σ i ) M ∗ Vol(Σ j ) M ∗ Vol(Σ k ) . (4.25)Here λ ijk denotes the fiducial, order one Yukawa coupling defined by (4.23) when B has unit volume.Although we have phrased the preceding discussion in the very special case thatΣ , Σ , and Σ are holomorphic curves intersecting transversely in a Calabi-Yauthreefold, the result (4.25) holds quite generally in F-theory. According to the dis-cussion in § . , Σ , and Σ are matter curves intersecting at a point p inside S , one must choose a trivialization of (cid:16) K / ⊗ K / ⊗ K / (cid:17)(cid:12)(cid:12)(cid:12) p to evaluate thewavefunction overlap. This choice, analogous to the choice of Ω in (4.23), introducesthe same scaling with ℓ .Once we introduce four-dimensional chiral superfields { φ i } with canonical kineticterms, the F -terms become L F = λ ijk Z d θ φ i φ j φ k p M ∗ Vol(Σ i ) M ∗ Vol(Σ j ) M ∗ Vol(Σ k ) . (4.26)We note that when all matter curves have comparable volumes set by the overall sizeof Vol( S ), Vol(Σ) ∼ Vol( S ) . In this case, (4.8) implies: L F = α / GUT λ ijk Z d θ φ i φ j φ k . (4.27)In rescaling each field by an appropriate power of the volume factor, we shalltypically use the classical value of Vol(Σ i ). Strictly speaking, this approximationis only valid in the supergravity limit. Due to the fact that in F-theory there isat present no perturbative treatment of quantum corrections, most of the numeri-cal results obtained throughout this paper can only be reliably treated as order ofmagnitude estimates. 32 General Overview of the Models
In this Section we provide a guide to the class of models we study. The choiceof K¨ahler surface S already determines many properties of the low energy effectivetheory. In keeping with our general philosophy, we require that the spectrum at lowenergies must not contain any exotics. When h , ( S ) = 0, we expect the low energyspectrum to contain additional states obtained by reduction of the bulk supergravitymodes of the compactification. For this reason we shall always require that π ( S )is a finite group. There are two further possible refinements depending on whetheror not the model in question admits a limit in which M GUT remains finite while M pl → ∞ . In order to fully decouple gravity, the extension of the local metric on S to a local Calabi-Yau fourfold must possess a limit in which the surface S can shrinkto zero size. In particular, this imposes the condition that K − S must be ample. Thisis equivalent to the condition that S is a del Pezzo surface, in which case h , ( S ) = 0.We note that the degree n ≥ h , ( S ) = 0 but do notdefine fully consistent decoupled models.In fact, even the way in which the gauge group of the GUT breaks to that ofthe MSSM strongly depends on whether or not such a decoupling limit exists. Forsurfaces with h , ( S ) = 0, the zero mode content will contain contributions fromthe bulk holomorphic (2 ,
0) form. Because the (2 ,
0) form determines the positionof the exceptional brane inside of the threefold base B , a non-zero vev for theassociated zero modes corresponds to the usual breaking of the GUT group via anadjoint-valued chiral superfield. Along these lines, we present some examples offour-dimensional GUT models which can originate from surfaces of general type.
Animportant corollary of this condition is that the usual four-dimensional field theoryGUT models cannot be fully decoupled from gravity!
We believe this is importantbecause it runs counter to the usual effective field theory philosophy that issuespertaining to the Planck scale can always be decoupled. This is in accord with theexistence of a swampland of effective field theories which may not admit a consistentUV completion which includes gravity [53]. Moreover, as we explain in greaterdetail later, it is also possible that a generic surface of general type may not supportsufficiently many matter curves of the type needed to engineer a fully realistic four- The potential application of this GUT breaking mechanism was noted in a footnote of [52] andhas also been discussed in [15, 16]. Σ ΣΣ Σ Σ
S z
Σ Σψ
Figure 3: The bulk group on the K¨ahler surface S corresponds to a singularity oftype G S . Over complex codimension one matter curves in S which we denote byΣ, this singularity type can further enhance so that six-dimensional matter fieldslocalize on these curves. Over complex codimension two points in S the singularitytype can enhance further. On the left of the figure we depict a triple intersectionof matter curves in S . It is also possible for one of the matter curves to intersect S at a point. Depending on the background gauge fluxes and local curvatures, wavefunctions localized on curves normal to the GUT brane are either exponentiallysuppressed or of order one near the point of contact with the GUT brane.34imensional GUT model.When available, discrete Wilson lines in higher-dimensional theories provide an-other way to break the GUT group to G std . Indeed, most models based on com-pactifications of the heterotic string on Calabi-Yau threefolds require discrete Wilsonlines to break the gauge group and project out exotics from the low energy spectrum.When π ( S ) = 0, a similar mechanism for gauge group breaking is available for ex-ceptional seven-brane theories. As an example, we present a toy model where S isan Enriques surface and G S = SU (5). In our specific example, we find that the zeromode content contains additional vector-like pairs of fields in exotic representationsof G std .We next turn to the primary case of interest for bottom up string phenomenologywhere S is a del Pezzo surface. Because h , ( S ) = 0 and π ( S ) = 0 for del Pezzosurfaces, the two mechanisms for gauge group breaking mentioned above are nowunavailable. In this case, the GUT group breaks to a smaller subgroup due tonon-trivial internal fluxes. For example, the group SU (5) can break to SU (3) × SU (2) × U (1) Y when the internal flux takes values in the U (1) Y factor. In heteroticcompactifications this mechanism is unavailable because a non-zero internal fieldstrength would generate a string scale mass for the U (1) hypercharge gauge boson infour dimensions [23]. We find that in F-theory compactifications without a heteroticdual , there is a natural topological condition for the four-dimensional gauge bosonto remain massless. Our expectation is that this condition is satisfied for manychoices of compact threefolds B . In the remainder of this Section we discuss furtherproperties of del Pezzo models.Along these lines, we present models based on G S = SU (5) where the gaugegroup of the eight-dimensional theory breaks directly to G std in four dimensions,as well as a hybrid scenario where G S = SO (10) breaks to SU (5) × U (1) in fourdimensions and then subsequently descends from a flipped SU (5) GUT model tothe MSSM. In fact, we also present a general no go theorem showing that directbreaking of SO (10) to G std via abelian fluxes always generates extraneous matter inthe low energy spectrum. In both the regular SU (5) and flipped SU (5) scenarios,we find that in order to achieve the exact spectrum of the MSSM, all of the matterfields must localize on Riemann surfaces. In the G S = SU (5) models, the matterfields organize into the 5 and 10 of SU (5). In the G S = SO (10) models, a completemultiplet in the 16 of SO (10) localizes on the matter curves. In both cases, all35atter localizes on curves so that all of the tree level superpotential terms descendfrom the triple intersection of matter curves. When some of the matter localizes ondifferent curves, this leads to texture zeroes in the Yukawa matrices.In addition to presenting some examples of minimal del Pezzo models, one ofthe primary purposes of this paper is to develop a number of ingredients which canbe of use in further more refined model building efforts. A general overview ofthese ingredients has already been given in the Introduction, so rather than repeatthis here, we simply summarize the primary themes of the minimal SU (5) modelwhich recur throughout this paper. The most prominent ingredient is the internalhypercharge flux which facilitates GUT breaking. This hyperflux also provides anatural solution to the doublet-triplet splitting problem and generates distorted GUTmass relations for the lighter generations. More generally, the presence of additionalglobal U (1) symmetries in the low energy theory forbids a number of potentiallyproblematic interaction terms from appearing in the superpotential. Topologically,the absence of dangerous operators translates into conditions on how the mattercurves intersect inside of S . For example, proton decay is automatically suppressedwhen the Higgs up and down fields localize on different matter curves. When thesecurves do not intersect, the µ term is zero. When the Higgs matter curves dointersect, the resulting µ term can be naturally suppressed. Indeed, an importantfeature of all the models we consider is that while expectations from effective fieldtheory would suggest that vector-like pairs will always develop a suitably large mass,here we find two distinct possibilities depending on the choice of the sign for thegauge fluxes: In one case (when the normal wave function is attracted to our brane)we essentially recover the field theory intuition. On the other hand, with a differentchoice of sign (when the normal wave function is repelled from our brane) we find theopposite situation, where µ is highly suppressed. The ostensibly large mass termcorresponding to the vev of a gauge singlet is in fact exponentially suppressed sinceits wave function is very small near our brane. Here, the principle of decouplingis especially important because the large positive curvature of the del Pezzo surfacecan lead to a natural suppression of the normal wave functions. This provides anexplanation for why the µ term is far below the GUT scale, as well as why the neutrinomasses are so far below the electroweak scale. While we discuss many of thesemechanisms in the specific context of the minimal SU (5) model, these same featurescarry over to the flipped SU (5) GUT models as well. In such cases, additional36ell-established field theoretic mechanisms are also available. For example, four-dimensional flipped SU (5) models already contain an elegant mechanism for doublet-triplet splitting which also naturally suppresses dangerous dimension five operatorsresponsible for proton decay. In this case, we can also utilize a conventional seesawmechanism to generate hierarchically light neutrino masses. In this Section we present some examples of models where Planck scale physicscannot be decoupled from local GUT models. Recall that in a traditional four-dimensional GUT, the GUT group breaks to G std when an adjoint-valued chiralsuperfield develops a suitable vev. In F-theory, this requires that the seven-branewraps a surface with h , ( S ) = 0. Before proceeding to a discussion of GUT modelsbased on such surfaces, we first discuss some important constraints on matter curvesand supersymmetric gauge field configurations for such surfaces.In many cases, some of the chiral fields of the low energy theory will localize onmatter curves in S . When h , ( S ) = 0, the number of available matter curves willtypically be much smaller than the dimension of H ( S, Z ) would suggest. To seethis, suppose that an element of H ( S, Z ) corresponds to a holomorphic curve Σ in S . We shall also refer to the class [Σ] as an “effective” divisor. Given a (2 ,
0) formΩ on S , note that: Z Σ Ω = Z S Ω ∧ P D (Σ) = 0 (6.1)where
P D (Σ) denotes the element of H ( S, Z ) which is Poincar´e dual to Σ. Thislast equality follows from the fact that P D (Σ) corresponds to the first Chern classof an appropriate line bundle and therefore is of type (1 , We thus see thatalthough the condition h , ( S ) = 0 is satisfied by a large class of vacua, at genericpoints in the complex structure moduli space each element of H , ( S, C ) imposes anadditional constraint of the form given by equation (6.1). At the level of cohomology,the divisor classes are parameterized by the Picard lattice of S : P ic ( S ) = H , ( S, C ) ∩ H ( S, Z ). (6.2) This last correspondence follows from the link between divisors and line bundles. K P ic ( S ) has rank one.Indeed, this lattice is generated by the hyperplane class inherited from the projectiveembedding of a general quartic in P . It is only at special points in the complexstructure moduli space that additional holomorphic curves are present. An exampleof a K P ic ( S ) is instead 20. Because there is a one to one correspondencebetween line bundles and divisors on S , we conclude that a similar condition holdsfor the available line bundles on a generic surface.Having stated these caveats on what we expect for generic surfaces of generaltype, we now construct an SO (10) GUT model with semi-realistic Yukawa matrices.In order to have a sufficient number of matter curves, we consider a seven-branewith worldvolume gauge group SO (12) wrapping a surface S defined by the blowupat k points of a degree n ≥ P with n odd. Some propertiesof hypersurfaces in P are reviewed in Appendix B. We have introduced theseblown up curves in order to simplify several properties of our example. Indeed, asexplained around equation (6.2), the Picard lattice of a surface may have low rank.An important point is that some of the numerical invariants such as h , ( S ) and χ ( S, O S ) of the degree n hypersurface remain invariant under these blowups. Thus,for many purposes we will be able to perform many of our calculations of the zeromode content as if the surface were a degree n hypersurface in P .For n ≥
5, we expect to find a large number of additional adjoint-valued chiralsuperfields. Geometrically, the vevs of these fields correspond to complex structuremoduli in the Calabi-Yau fourfold which can develop a mass in the presence of asuitable background flux. We show that in the present context, a suitable profile ofvevs can simultaneously break the GUT group and lift all excess fields from the lowenergy spectrum.As explained in Section 3, in the context of a local model, we are free to specifythe enhancement type along codimension one matter curves inside of S . We firstintroduce four curves Σ , Σ , Σ , Σ B where the singularity type enhances to E so thata half-hypermultiplet in the 32 of SO (12) localizes on each curve. With notation as38n Appendix B, the homology class of each curve is:[Σ ] = E (6.3)[Σ ] = E (6.4)[Σ ] = E (6.5)[Σ B ] = − a l − E − E . (6.6)where we have written K H n = a l + a l + ... for some generators l i of H ( H n , Z )such that l i · l j = 0 for i = j . Using the genus formula C · ( C + K S ) = 2 g − , Σ , Σ are all zero while Σ B has genus one. Wenote that in order for Σ B to represent a holomorphic curve, it may be necessary togo to some special points in the moduli space of the surface S . In the presence of asuitable internal flux, a single generation in the 16 of SO (10) will localize on eachof the Σ i ’s. The fields localized on Σ B will instead develop a suitable vev to liftextraneous matter from the low energy spectrum.We next introduce the curve Σ R where the singularity type enhances to SO (14)so that a hypermultiplet transforming in the 12 of SO (12) localizes on this curve.The homology class of Σ R is:[Σ R ] = − a l − E − E (6.7)so that Σ R has genus one.A supersymmetric U (1) gauge field configuration can simultaneously break SO (12)to SO (10) × U (1) P Q and also induce a net chiral matter content in the four-dimensionaleffective theory. Representations of SO (12) decompose under the subgroup SO (10) × U (1) P Q as: SO (12) ⊃ SO (10) × U (1) P Q (6.8)66 → + 1 + 10 + 10 − (6.9)32 → + 16 − (6.10)12 → + 1 − + 10 . (6.11)All candidate Higgs fields in the 10 − are equally charged under the group U (1) P Q and we shall therefore loosely refer to it as a Peccei-Quinn symmetry. We consider39onfigurations such that one generation in the 16 of SO (10) localizes along each Σ i for i = 1,2 ,
3. In addition to the matter content of the MSSM, we shall also requirethat there is extra vector-like matter in the 16 and 16 − localized along Σ B and a10 and 1 localized along Σ R . When the extra vector-like 16’s develop a vev atsuitably large energy scales, they will remove an additional U (1) B − L gauge bosonfrom the low energy spectrum. Further, interaction terms between the 10 and 1 − can also serve to remove extraneous matter from the spectrum.The above requirements are satisfied by a large class of supersymmetric linebundles. For concreteness, we consider the line bundle: L = O S ( E − E + E − E + E − E − E + E + N ( E − E )) (6.12)where to simplify some cohomology calculations, we shall sometimes take N to be alarge integer. By inspection, there exists a parametric family of K¨ahler classes suchthat the condition: ω ∧ c ( L ) = 0 (6.13)holds. In the above, ω denotes a particular choice of K¨ahler form on S . While all of the chiral matter of the MSSM localizes on the matter curves Σ , Σ and Σ , the internal U (1) flux specified by the line bundle of equation (6.12) willalso induce additional bulk zero modes. The bulk matter content all descends fromthe adjoint representation of SO (12). First consider the number of chiral superfieldstransforming in the representation 45 + 1 . These fields are neutral under U (1) P Q so that the total number of chiral superfields transforming in this representation is h ( S, O S ) + h ( S, O S ). In the present case, h ( S, O S ) = 0 so that it is enough tocompute h ( S, O S ) = h , ( S ). The Hodge numbers of S are computed in AppendixB with the end result: (cid:18)
16 ( n − n + 11 n ) − (cid:19) × (45 + 1 ) ∈ H ∂ ( S, O S ). (6.14)When these fields develop a suitable vev, the GUT group will break to G std .The chiral superfields transforming in the 10 ± are classified by the bundle-valued40ohomology groups:10 ± ∈ H ∂ ( S, L ∓ ) ∗ ⊕ H ∂ ( S, L ± ) ⊕ H ∂ ( S, L ∓ ) ∗ . (6.15)Now, when the integer N of equation (6.12) is sufficiently large, both H ∂ ( S, L ∓ ) ∗ and H ∂ ( S, L ∓ ) ∗ will indeed vanish. The resulting dimension of H ∂ ( S, L ± ) can thenbe computed via an index formula: h ( S, L ± ) = − (cid:18) χ ( O S ) + 12 c ( S ) · c ( L ± ) + 12 c ( L ± ) (cid:19) (6.16)= −
16 ( n − n + 11 n ) + (16 + 4 N ) (6.17)so that there are an equal number of 10 +2 and 10 − ’s. Based on their coupling to thefields localized along the matter curve, we shall tentatively identify these as Higgsfields. We now study the chiral matter content localized on matter curves. By construction, L restricts to a degree one line bundle on the genus zero matter curves Σ , Σ , Σ so that a single generation transforming in the 16 localizes on each matter curve.Further, L restricts to a trivial line bundle on Σ B so that a single vector-like pairof 16 and 16 − localizes along Σ B . Finally, L restricts to a degree − O Σ R ( − p ) on the genus one matter curve Σ R where p denotes a degree one divisor ofΣ R . In order to achieve one copy of the 10 , we also include a contribution to theflux from the other seven-brane intersecting the GUT model seven-brane along Σ R so that L ′ Σ R = O Σ R ( p ′ ), where p ′ is another degree one divisor of Σ R . The total fieldcontent on Σ R is therefore given by one 10 , , three 1 − , ’s and one 1 − , − , wherethe two subscripts indicate the U (1) charge with respect to the two U (1) factors. The representation content and type of matter curve are summarized in the following As we explain later in Section 10, the overall normalization of the U (1) charges is somewhatinconsequential so long as the fields transform in mathematically well-defined line bundles. SO (10) Model Curve Class g Σ L Σ L ′ n Σ × Σ E O Σ (1) O Σ × Σ E O Σ (1) O Σ × Σ E O Σ (1) O Σ × (cid:0) + 16 − (cid:1) Σ B − a l − E − E O Σ B (0) O Σ B × , + 3 × − , + 1 × − , − Σ R − a l − E − E O Σ R ( − p ) O Σ R ( p ′ ) .(6.18)As will be clear when we discuss the high energy superpotential, although the 1 − , − couples non-trivially with the 10 , to bulk modes on S , the 1 − , ’s do not contributeto the cubic superpotential, and we shall therefore neglect their contribution to thelow energy theory. To simplify notation, we shall therefore refer to the 10 , as the10 and the 1 − , − as the 1 − . In the present model, the Yukawa couplings of the MSSM originate from purely bulkcouplings and couplings between bulk gauge fields and matter fields localized alongmatter curves. In addition, a background flux configuration in the Calabi-Yau four-fold will also couple to the complex structure moduli of the compactification. Indeed,as shown in [15], the vevs of the bulk (2 ,
0) form and fields localized along mattercurves all determine complex deformations of the background compactification. Inthe case of fields localized along the matter curve, this corresponds to the “mesonic”branch of moduli space. We therefore conclude that fluxes can induce a non-trivialmass and vev for the corresponding fields. At energy scales close to M GUT but belowthe energy scale where the first Kaluza-Klein mode can contribute an appreciableamount, the high energy superpotential is: W high = W S + W S ΣΣ + W flux + W np (6.19)42here: W S = f iIJ ( I )+2 × ( J ) − × (45 ( i )0 + 1 ( i )0 ) (6.20) W S ΣΣ = λ aJ ( a )1 × ( a )1 × ( J ) − + α a ( a )2 × × − (6.21)+ (cid:16) β J × × ( J ) − + γ J − × − × ( J )2 (cid:17) (6.22) W flux = Z CY Ω ∧ G (6.23) W np = µ ( IJ ) − ( I )+2 × ( J )+2 + µ ( IJ )+4 ( I ) − × ( J ) − (6.24)In the above, terms proportional to the coefficients λ aJ descend from the three mattercurves Σ , Σ , Σ , while terms proportional to β J and γ J descend from the mattercurve Σ B . Here, we have also included the effects of non-perturbatively gener-ated mass terms for the 10’s which explicitly violate the U (1) P Q global symmetry.Such terms can originate from exponentially suppressed higher-dimensional opera-tors which couple the fields of the GUT model to additional GUT group singlets.When these singlets develop a suitable vev, they can generate terms of the type givenby W np . In this case, the resulting µ term will naturally be exponentially suppressed.A similar mechanism has been analyzed in the context of type II intersecting D-branemodels as a potential solution to the µ problem [54].While stabilizing the moduli in a realistic compactification is certainly a non-trivial task, in a local model, the vevs of the complex structure moduli can effectivelybe tuned to an arbitrary value. Letting Ω (0) denote the value of the holomorphicfour form of the Calabi-Yau fourfold with the desired values of the complex structuremoduli, we note that the critical points of W flux with G = λ (Ω (0) + Ω (0) ) will indeedyield such a configuration. For compact models, this must be appropriately adjustedbecause the potential for the overall volume of the Calabi-Yau fourfold will developa non-supersymmetric minimum. We now show that an appropriate choice of vevs in W high given by equation (6.19)can yield a low energy effective theory with precisely the matter content of the MSSMand a semi-realistic low energy superpotential. We first demonstrate that the above43odel can indeed remove all excess matter at sufficiently high energies. To this end,first note that when a 45 ( i )0 develops the vev: h i = iσ y ⊗ diag ( a, a, a, b, b ) (6.25)the resulting gauge group will break to SU (3) C × SU (2) L × U (1) Y × U (1) B − L . Byinspection of equation (6.20), when a ∼ M GUT , this vev will also remove the Higgstriplets of 10 ( J ) − (and the 10 ( I )+2 ’s) from the low energy spectrum. When the zeromode content contains at least two 45’s which have distinct couplings to the product10 ( I )+2 × ( J ) − , a suitable choice of b for each 45 can be arranged so that at mostone pair of SU (2) L doublets from one linear combination of the 10 − ’s will remainmassless. We note that this is simply a variant on the well-known Dimopoulos-Wilczek mechanism for achieving doublet-triplet splitting in four-dimensional SO (10)GUT models [55, 56].In the absence of other field vevs, the resulting spectrum would contain two SU (2) L doublets from a bulk 10 − as well as its counterpart 10 . In fact, we nowdemonstrate that when the flux induces a suitably large mass term for the 10 aswell as a vev for the 1 − , the resulting low energy spectrum will not contain anyfields transforming in the representation 10 . With the above choice of fluxes, themass matrix for the 10 and remaining 10 is schematically of the form: W eff ⊃ h i " h − ih − i h M flux i (6.26)so that all extraneous 10 +2 ’s can indeed lift from the low energy spectrum.The resulting spectrum is almost that of the MSSM at low energies. The onlyadditional matter content is an additional U (1) B − L gauge boson and a vector-likepair of matter fields 16 and 16 − localized on Σ B . In fact, when the 16 and 16 − develop a suitable vev, they will break U (1) B − L .Maximally utilizing conventional four-dimensional field theoretic mechanisms toachieve the correct matter spectrum, this model yields the spectrum of the MSSMat low energies. Moreover, by placing the three generations on three distinct mattercurves, a large hierarchy in scales can be generated by a suitable choice of K¨ahlerclass. 44he effective superpotential is now schematically of the form: W eff = µH u H d + λ uij Q i U j H u + λ dij Q i D j H d + λ lij L i E j H d + λ νij L i N jR H u + ... (6.27)where the λ ij ’s are all diagonal.While it is of course possible to further refine the above model, we believe thisprovides a fruitful starting point for analyzing how traditional four-dimensional GUTmodels can embed in F-theory. Again, we emphasize that strictly speaking, a purelyfour-dimensional effective field theory approach breaks down in this case because nodecoupling limit between M GUT and M pl is available. In the previous Section we presented an example of a four-dimensional GUT modelwhich breaks to the MSSM when a collection of adjoint-valued chiral superfieldsdevelop appropriate vevs. This requires that the surface S wrapped by the seven-brane satisfies h , ( S ) = 0. When π ( S ) = 0, it is also possible for the GUT groupto spontaneously break to the gauge group of the Standard Model via an appropriatechoice of Wilson lines. In this Section we describe some features of models based onthe case where S is an Enriques surface. After reviewing some basic properties ofsuch surfaces, we present a toy model with bulk gauge group G S = SU (5). Althoughthe correct matter content of the MSSM can localize on matter curves, we find thatthe discrete Wilson lines also generically produce additional vector-like pairs of zeromodes transforming in exotic representations of G std . This can be traced back tothe fact that the universal cover of an Enriques surface is a K S is defined by the conditions: K S = O S but K S = O S (7.1)and that the “irregularity” h , ( S ) = q ( S ) = 0. The non-vanishing Hodge numbersof an Enriques surface are h , ( S ) = 10 and h , ( S ) = h , ( S ) = 1. The fundamentalgroup of S is π ( S ) = Z . Moreover, the universal cover of S is a K h , ( K
3) = 1 does not survive in the quotient space.Nevertheless, we shall see that in the presence of discrete Wilson lines, the zeromode content retains some imprint from the underlying K K H ( K , Z ) is isomorphicto: H ( K , Z ) = ( − E ) ⊕ ( − E ) ⊕ U ⊕ U ⊕ U (7.2)where − E denotes minus the intersection form for the Lie algebra E and the “hy-perbolic element” U is the intersection form with entries given by the Pauli matrix σ x . The intersection form on S is instead given by: H ( S, Z ) /T or = ( − E ) ⊕ U (7.3)where in the above we have modded out by possible torsional elements. As anintegral lattice, H ( S, Z ) is isomorphic to: H ( S, Z ) ≃ Z ⊕ Z . (7.4)We label the generators of H ( S, Z ) as α , ..., α in correspondence with the roots of E and d and d for the generators associated with U such that d i · d j = 1 − δ ij .Finally, we label the torsion element as t . An important feature of Enriques surfacesis that the Poincar´e dual homology classes for d and d both represent holomorphicelliptic curves in S .We now present a toy model with S an Enriques surface with bulk gauge group G S = SU (5) which spontaneously breaks to G std due to a discrete Wilson line takingvalues in the U (1) Y factor. The example we shall now present cannot be consideredeven semi-realistic because in addition to containing exotic matter, the tree levelsuperpotential contains too many texture zeroes. Nevertheless, it illustrates someof the elements which are necessary in more realistic constrictions. To simplify our46iscussion, we shall emphasize elements unique to having non-trivial discrete Wilsonmodels.Because bulk modes descend from the adjoint representation of SU (5) and allof the matter of the Standard Model descends from other representations of SU (5),the chiral superfields of the MSSM must localize on matter curves. The generic G S = SU (5) singularity enhances to SU (6) along the Higgs and 5 M matter curvesand enhances to SO (10) along the 10 M matter curve. The matter curves and choiceof line bundle assignment are given in the following table:Enriques Model Curve K g Σ L Σ L ′ n Σ × (cid:0) H + 5 H (cid:1) Σ H e Σ H d Z ⊗ O Σ H O Σ H × M Σ (1) M e Σ (1) M ∐ e Σ ′ (1) M d O Σ (1) M O Σ (1) M ( − p )3 × M Σ (2) M e Σ (2) M ∐ e Σ ′ (2) M d O Σ (2) M O Σ (2) M (3 p ) .(7.5)In the above, we have also indicated how each curve lifts to K
3. In this case bothmatter curves Σ ( i ) M lift to the disjoint union of two curves in K H lifts to a curve which is fixed by the Z involution in K
3. As an explicitexample, we can consider the case where the covering space of S is a real K Z involution corresponds to complex conjugation. In this case, the curveΣ ( i ) M lifts to a generic holomorphic curve and its image under complex conjugationwhile Σ H lifts to a real algebraic curve in K
3. We now show that in this case thediscrete Wilson line projects out the Higgs triplet from the low energy spectrum.Because the Higgs curve is fixed by the Z involution, the fields localized on thiscurve will transform non-trivially in the presence of a Z Wilson line. The analysisbelow equation (3.12) applies equally well to fields localized on matter curves. Underthe breaking pattern SU (5) ⊃ SU (3) × SU (2) × U (1), the 5 of SU (5) decomposesto (1 , + (3 , − . In this case, the relevant cohomology group lifts to the Z oddeigenspace: 5 H ∈ H ∂ ( e Σ H , O e Σ H ) ≃ C ( − ) . (7.6)Hence, we conclude that the total wave function for the components of the 5 H and47 H take values in the invariant subspaces:(1 , − , (1 , ∈ h C ( − ) ⊗ H ∂ ( e Σ H , O e Σ H ) i Z ≃ C (7.7)(3 , − , (3 , ∈ h C (+) ⊗ H ∂ ( e Σ H , O e Σ H ) i Z = 0. (7.8)Hence, the Higgs triplet is absent from the low energy spectrum while the Higgs upand down doublets remain.The matter content of this example is not fully realistic because it also containscontributions from the bulk zero modes which appear as vector-like pairs transform-ing in exotic representations of G std . To compute the bulk zero mode content inthe presence of the discrete Wilson line, we again apply the analysis below equation(3.12) in the special case where the bundle T is trivial. Decomposing the adjointrepresentation of SU (5) to SU (3) × SU (2) × U (1), the only irreducible representa-tions which transform non-trivially under the U (1) factor are the (3 , − and (3 , .We now compute the number of bulk zero modes transforming in the (3 , − . Inthe covering K ,
0) is given by the Z invariant subspace:(3 , − ∈ (cid:2) C ( − ) ⊗ H ∂ ( K , O K ) (cid:3) Z (7.9)where the C ( − ) factors indicates the charge of the representation (3 , − under the Z subgroup of U (1) Y . Next recall that the Z group action on the holomorphic(2 ,
0) form sends ϕ
7→ − ϕ . In particular, this implies that the cohomology group H ∂ ( K , O K ) ≃ C ( − ) . A similar analysis also holds for zero modes transforming inthe representation (3 , − . Because C ( − ) ⊗ C ( − ) is Z invariant, we conclude thatthe low energy spectrum contains exotic vector-like pairs.There are potentially several ways to avoid the presence of these exotics. For ex-ample, when G S = SO (10), a combination of U (1) flux breaking and discrete Wilsonline breaking might avoid any contributions from bulk zero modes. Moreover, evenif additional exotic particles are present in the low energy spectrum, it is conceivablethat an appropriately engineered superpotential could cause these exotics to developa large mass.It is also possible to consider a more general class of surfaces with non-trivialdiscrete Wilson lines. In the present context the maximal case of interest would be48urfaces with h , ( S ) = h , ( S ) = 0 and π ( S ) a finite group. Some examples ofsurfaces such as the classical Godeaux and Campadelli surfaces may be found in [57].As a technical aside, we note that one particularly interesting class of surfaces canbe obtained by choosing n distinct points of a del Pezzo 9 surface and performingan order a i logarithmic transformation at the i th point. The resulting surface hasthe same Hodge numbers, Euler character and signature as the del Pezzo 9 surfaceand is called a Dolgachev surface, D ( a , ..., a n ). For example, when n = 2 and a and a have a common divisor, the fundamental group is π ( D ( a , a )) ≃ Z m where m = gcd( a , a ). See [58, 59] and references therein for further discussion ofDolgachev surfaces defined by two logarithmic transformations. We note that thecase a = 2, a = 2 corresponds to the Enriques surface. It is also common inthe mathematics literature to treat the more general case as well. When the a i arepairwise co-prime integers, the resulting fundamental group is [60]: π ( D ( a , ..., a n )) = h t , ..., t n | t a i i = 1 , t · · · t n = 1 i . (7.10)Given the prominent role that the del Pezzo 9 surface has played in recent heteroticmodels such as [8,9], it would be interesting to study models based on such Dolgachevsurfaces. In the remainder of this paper we focus on the case of primary interest where S isa del Pezzo surface. In this case, it is at least in principle possible to consistentlydecouple the Planck scale from the GUT scale. Because much of the analysis tofollow relies on properties of del Pezzo surfaces, in this Section we collect variousrelevant facts about the geometry of such surfaces. After giving the definition of delPezzo surfaces, we catalogue the moduli of such surfaces which must be stabilizedin a globally consistent model. Next, we review the beautiful connection betweenthe homology groups of del Pezzo surfaces and the root lattices of exceptional Liealgebras. In particular, we show that the line bundles L on S such that both L and L − have trivial cohomology are in one to one correspondence with the roots ofthe corresponding exceptional Lie algebra. This classification will prove important See [57] for the definition and further properties of logarithmic transformations of surfaces. P × P and P . There areeight additional surfaces defined as the blowup of P at up to eight points in generalposition. We shall refer to such surfaces as del Pezzo N ( dP N ) surfaces for the caseof N blown up points.We now describe the K¨ahler and complex structure moduli spaces of these sur-faces. First consider the K¨ahler moduli of del Pezzo surfaces. P × P has twoK¨ahler moduli corresponding to the volume of the two P factors. There is a singleK¨ahler modulus which fixes the overall size of P . In addition to the overall size ofthe P , for the del Pezzo N surfaces, there are N further moduli corresponding tothe volume of each blown up cycle. Further properties of the K¨ahler cone for eachdel Pezzo surface are reviewed in Appendix A of [15].In addition to the K¨ahler moduli of each del Pezzo surface, these surfaces mayalso possess a moduli space of complex structures. For P × P and P there is aunique choice of complex structure. When S = dP N , the overall P GL (3) symmetryof P implies that the number of complex structure moduli is 2 N − ≤ N ≤ P GL (3) action on P may not properlyextend to the compact threefold base.We next describe the homology groups of the del Pezzo surfaces. The homologygroup H ( P × P , Z ) is two dimensional and has generators σ and σ correspondingto the two P factors. These generators have intersection product: σ i · σ j = 1 − δ ij (8.1)where δ ij is the Kronecker delta. The canonical class for P × P is: K P × P = − c ( P × P ) = − σ − σ . (8.2)In particular, − K P × P defines a K¨ahler class on P × P where both P factors havevolume two in an appropriate normalization.The homology group H ( dP N , Z ) is N + 1 dimensional and has generators H , E , ..., E N where H denotes the hyperplane class inherited from P and the E i de-50ote the exceptional divisors associated with the blowup. These generators haveintersection product: H · H = 1, H · E i = 0, E i · E j = − δ ij (8.3)so that the signature of H ( dP N , Z ) is (+ , − N ). The canonical class for dP N is: K dP N = − c ( dP N ) = − H + E + ... + E N . (8.4)There is a beautiful connection between del Pezzo N ≥ H ( dP N , Z )orthogonal to K dP N is identified with the root space of the corresponding Lie algebra E N . Because dP admits a different treatment, first consider the dP N surfaces with N ≥
3. The generators of the lattice h K dP N i ⊥ are: α = E − E , ..., α N − = E N − − E N , α N = H − E − E − E . (8.5)The intersection product of the α i ’s is identical to minus the Cartan matrix for thedot product of the simple roots for the corresponding Lie algebra E N . For dP , thesingle generator of the lattice h K dP N i ⊥ is given by E − E , which we identify as aroot of su (2).This correspondence further extends to include the Weyl group of the exceptionalLie algebras. In the following we shall adopt a “geometric” convention so that thesignature of the root space is negative definite. The Weyl group for a simplyconnected Lie algebra with simple roots α , ..., α N is generated by the Weyl reflections w α i . Given an element α of the root lattice, the Weyl reflected vector w α i ( α ) is: w α i ( α ) = α + ( α · α i ) α i . (8.6)This is precisely the action of the large group of diffeomorphisms for the del Pezzo N surfaces on the corresponding generators orthogonal to K dP N . Indeed, note thatthe canonical class is invariant under the action of the Weyl group.Anticipating future applications, we now show that when S is a del Pezzo N ≥ With this sign convention, a root α satisfies α · α = − L such that: H i∂ ( S, L ± ) = 0 (8.7)for all i are in one to one correspondence with the roots of the Lie algebra E N .Because the indices defined by L and L − must separately vanish, the differencein the two indices also vanishes:0 = χ ( dP N , L ) − χ ( dP N , L − ) = c ( dP N ) · c ( L ) = − K dP N · c ( L ). (8.8)Treating c ( L ) as an element of H ( dP N , Z ), c ( L ) is therefore a vector in the orthog-onal complement of the canonical class. Hence, c ( L ) corresponds to an element ofthe root lattice of E N . Utilizing equation (8.8), the index χ ( dP N , L ) now takes theform: χ ( dP N , L ) = 1 + 12 c ( L ) · ( c ( L ) + c ( dP N )) = 1 + 12 c ( L ) · c ( L ) (8.9)which vanishes provided: c ( L ) · c ( L ) = − c ( L ) to correspond to a root of E N . Conversely, we notethat given a root α of h K dP N i ⊥ , the line bundle L = O dP N ( α ) defines a supersymmet-ric gauge field configuration. The vanishing theorem of [15, 63] and the vanishing ofthe corresponding index now imply that all cohomology groups are trivial.A similar analysis holds for the remaining del Pezzo surfaces P , P × P and dP .When S = P , we note that because H ( P , Z ) has a single generator given by thehyperplane class of P , all non-trivial line bundles L have c ( L ) · c ( P ) = 0 so thatequation (8.8) is never satisfied.To treat the cases S = P × P , dP and in order to partially widen the scope ofour discussion, we note that these del Pezzo surfaces are also Hirzebruch surfaces.More generally, recall that the middle homology of the degree n Hirzebruch surface F n has generators σ and f which have intersection pairing: f · f = 0, f · σ = 1, σ · σ = − n . (8.11)52he canonical class for F n is: K F n = − c ( F n ) = − ( n + 2) f − σ . (8.12)We now show that F is the only Hirzebruch surface which admits line bundlessatisfying equation (8.7). To this end, consider a line bundle L = O F n ( af + bσ ). Inorder to satisfy equation (8.7), we must have:0 = χ ( F n , L ) − χ ( F n , L − ) = c ( F n ) · c ( L ) = b ( n + 2) + 2 a − bn . (8.13)When this condition is satisfied, the index χ ( F n , L ) vanishes provided:0 = χ ( F n , L ) = 1 + 12 c ( F n ) · c ( L ) + 12 c ( L ) · c ( L ) (8.14)= 1 + 12 (2 ab − b n ) = 1 + 12 ( b n − b ( n + 2)) (8.15)or, − b ( n − ( n + 2)). (8.16)In order for this equation to possess a solution over the integers, b = ± n − n = 0so that n = 0 or n = 1. First consider the case where n = 1. Returning to equation(8.13), when n = 1 and b = ±
1, we find that a = ± /
2, which is not an integer.We therefore conclude that the only remaining case is n = 0. For F , the only linebundles satisfying equation (8.7) are L = O F ( ± f ∓ σ ) = O P × P ( ± σ ∓ σ ) where inthe final equality we have reverted to the notation of equation (8.1). U (1) Fluxes
When S is a del Pezzo surface, the zero mode content does not contain any adjoint-valued chiral superfields which could potentially play the role of a four-dimensionalGUT Higgs fields. In this Section we present an alternative mechanism where theGUT group breaks due to non-trivial hypercharge flux in the internal directions.Experience with other string compactifications suggests that a non-trivial internalfield strength would cause the photon to develop a string scale mass because thisgauge boson couples non-trivially to the p-form gauge potentials of the closed stringsector. In this Section we present a topological criterion for this U (1) gauge boson53o remain massless. This then provides a novel mechanism for GUT group breakingin F-theory.To analyze whether the coupling to closed string modes will generate a massfor the U (1) gauge boson, first recall that the ten-dimensional supergravity actioncontains the terms (neglecting the overall normalization of individual terms by orderone constants): S (10 d ) ⊃ M ∗ Z R , × B C △ C − M ∗ Z R , × S T r ( F ∧ ∗ F ) + M ∗ Z R , × S C (4) ∧ T r ( F ∧ F ) (9.1)where C (4) denotes the RR four-form gauge potential and F denotes the eight-dimensional field strength of the seven-brane. Letting h F S i denote the non-vanishingfield strength in the internal directions, integrating out C (4) yields a term in the ef-fective action of the form: S (10 d ) eff ⊃ Z R , × B δ R , × S ∧ h F S i ∧ F △ δ R , × S ∧ h F S i ∧ F (9.2)where δ R , × S denotes the delta function for the seven-brane and we have droppedthe overall trace because our primary interest is in abelian instanton configurations.Next, expand δ R , × S ∧ h F S i in a basis of eigenmodes so that: δ R , × S ∧ h F S i = X α f α ψ α (9.3)where △ ψ α = λ α ψ α denote eigenmodes of the Laplacian on B and f α denote theassociated Fourier coefficients. We thus arrive at a non-local term in the four- For D-branes, the relative normalizations between these terms contains factors of g s . In thepresent class of models, this distinction is ambiguous because these vacua exist in a regime of strongcoupling. L (4 d ) eff ⊃ X α Z S F ∧ k f α ψ α | S k △ + λ α F (9.4)= X α =0 Z S k f α ψ α | S k A + X α =0 Z S F ∧ k f α ψ α | S k △ + λ α F . (9.5)so that the contribution from zero modes of ∆ induces a mass term for the four-dimensional gauge boson. The remaining modes induce a non-local operator whichtends to zero in the decompactification limit.The zero modes of ∆ which can potentially couple to the internal field on S correspond to harmonic representatives of the cohomology group H ( B , R ) whichare Poincar´e dual to elements of H ( B , Z ). For concreteness, we let Γ denote such afour-cycle. In the same spirit as [64], we therefore conclude that the four-dimensional U (1) gauge boson will remain massless provided the class in H ( S, Z ) correspondingto h F S i integrates trivially when wedged with any element of H ( B , Z ). In otherwords, given any four-cycle Γ in B , Γ must intersect trivially with the Poincar´e dualof h F S i which we denote as [ F S ] for some element of H ( S, Z ). This implies that thecycle [ F S ] must be trivial in B . We note that just as in [64], this entire discussioncan be phrased in terms of the relative cohomology between S and B , and we referthe reader there for more details on this type of argument.Our expectation is that this condition can be met in a large number of cases.Indeed, in backgrounds where the (2 ,
0) form vanishes, a line bundle L correspondsto a supersymmetric gauge field configuration when [15]: ω ∧ c ( L ) = 0 (9.6)where ω denotes the K¨ahler form on S . In particular, if this ω descends from theK¨ahler form in the threefold base B , this is a necessary condition for the Poincar´edual of h F S i to lift to a trivial class in H ( B , Z ). Note that when dim H ( B , Z ) = 1,this condition is in fact sufficient.For illustrative purposes, we now show that there exist compactifications of F-theory where this condition can be met. To this end, we consider an elliptically This same observation has been made independently by M. Wijnholt. B = P . In this case, the homology ring H ∗ ( P , Z ) is generated by the hyperplane class H P . Introducing homogeneous co-ordinates x , x , x , x , we recall that the vanishing locus of a generic degree twopolynomial in the x i defines a P × P in B , and the vanishing locus of a generic de-gree three polynomial defines a del Pezzo 6 surface in B . As reviewed in AppendixB, a multiple of the generator H P restricts to the anti-canonical class of a degree n hypersurface in P .Letting σ and σ denote the generators of H ( P × P , Z ) corresponding to thetwo P factors, the class σ − σ lifts to a trivial class in P due to the fact that K P × P · ( σ − σ ) = 0. Similar considerations apply for the del Pezzo 6 surfacebecause all of the two-cycles corresponding to elements in the root lattice of E areorthogonal to K dP . Given the usual heterotic/F-theory duality, it is natural to ask whether GUT groupbreaking via internal fluxes can also occur in the heterotic string. A general obstruc-tion to using U (1) fluxes in heterotic models was already noted in [23]. In fact, inall F-theory models which admit a heterotic dual, the mechanism described above isunavailable! To establish this, first recall that the basic heterotic/F-theory dualityrelates compactifications of the heterotic string on an elliptic curve to compactifica-tions of F-theory on an elliptically fibered K K P fibration over a K¨ahlersurface S .We now establish that in this case, an internal hypercharge flux will always causethe corresponding four-dimensional gauge boson to lift from the low energy theory.As explained previously, it is enough to determine whether this internal flux wedgesnon-trivially with any two forms in H ( B , R ). To see why this occurs, first considerthe case where the fibration is trivial so that the threefold base is of the form S × P = B . In this case, we note that: H ( B , R ) = H ( S × P , R ) ≃ H ( S, R ) ⊕ H ( P , R ). (9.7)56his implies that all non-zero elements of H ( S, R ) wedge non-trivially with someelement of H ( B , R ). Next consider the case of a non-trivial fibration. Theonly consequence of the non-trivial fibration structure is that the cohomology group H ( B , R ) could potentially contain additional contributions on top of those alreadypresent in the product formula of equation (9.7). In particular, all of the elementsof the cohomology group of H ( S, R ) again wedge non-trivially with some elementof H ( B , R ).
10 Avoiding Exotica
As argued in the previous Section, abelian fluxes provide a potentially generic mech-anism for breaking the GUT group to G std . As shown in [15], such fluxes alsodetermine the zero mode content of the low energy effective theory. It thus fol-lows that the zero mode content of the theory may not match to the MSSM. Inkeeping with our general philosophy, we require that all of the zero modes otherthan the Higgs fields must organize into complete GUT multiplets. Indeed, if thesezero modes do not fill out complete GUT multiplets, they can potentially spoil theunification of the gauge couplings.It is in principle possible that these restrictions can be relaxed. If all exoticscome in vector-like pairs, effective field theory arguments would appear to suggestthat such pairs will develop a large mass and lift from the low energy spectrum. Wenote that in the present case, all mass terms descend from cubic or higher ordersuperpotential terms. Large mass terms will only result when a singlet develops asufficiently large vev. As will be clear in all of the models considered here, suchsinglets are charged under additional gauged symmetries. In this case, such massterms may not be sufficiently large to avoid spoiling gauge coupling unification. Forthese reasons, we shall always require that the zero mode content of the low energytheory contains no vector-like pairs of fields in exotic representations of G std .This constraint imposes important restrictions on admissible gauge bundle con-figurations which can break the bulk gauge group G S to G std . In particular, when G S = SU (5), we show that the gauge bundle configurations with no exotica are inone to one correspondence with the roots of an exceptional Lie algebra corresponding At a more formal level, this is a direct consequence of the Leray-Serre spectral sequence.
57o the del Pezzo surface in question. Moreover, when G S = SO (10), we present a nogo theorem which shows that direct breaking of G S to G std via internal fluxes alwaysproduces exotica in the low energy theory. In this Section we determine which internal fluxes can break the GUT group and si-multaneously do not generate any extraneous zero modes in the low energy spectrum.In fact, a cursory analysis would incorrectly suggest that such states are unavoidable.For example, the decomposition of the adjoint representation of SU (5) decomposesunder G std as: SU (5) ⊃ SU (3) × SU (2) × U (1) (10.1)24 → (1 , + (8 , + (1 , + (3 , − + (3 , . (10.2)We note that no fields of the MSSM transform in the representation (3 , − or (3 , .Letting L denote the supersymmetric line bundle associated with this breaking pat-tern, the bulk zero mode content therefore descends to:(3 , − ∈ H ∂ ( S, L ) ∗ ⊕ H ∂ ( S, L − ) ⊕ H ∂ ( S, L ) ∗ (10.3)(3 , ∈ H ∂ ( S, L − ) ∗ ⊕ H ∂ ( S, L ) ⊕ H ∂ ( S, L − ) ∗ . (10.4)Mathematically, the collection of admissible line bundles are those which have van-ishing cohomology group. As explained in Section 8, when S is a del Pezzo N surface, such line bundles are in one to one correspondence with the roots of the Liealgebra E N , with a similar result for P × P . By definition, a root α satisfies thecondition that nα is also a root only when n = ±
1. It now follows that if L is a linebundle, L cannot correspond to a root of the Lie algebra E N . Said differently, theintegral quantization of fluxes in the bulk theory would appear to present a generalobstruction towards realizing the spectrum of the MSSM without any additional bulkmatter with exotic U (1) Y charges.We now argue that so long as all fields transform in mathematically well-definedline bundles, fractional powers of line bundles also define consistent gauge field con-figurations for the bulk theory. To establish this, first recall that when all fieldsof a theory with gauge group SU ( N ) transform in the adjoint representation, all58bservables are invariant under SU ( N ) modulo the center. Hence, the actual gaugegroup of the theory is SU ( N ) / Z N so that the flux quantization condition allowsgauge field configurations with 1 /N fractional flux units [66]. In the presence ofquark fields charged in the fundamental of SU ( N ), we note that the gauge group isindeed SU ( N ) rather than SU ( N ) / Z N .In the present class of models, a similar fractional quantization condition holdsbecause all of the resulting gauge groups descend from an E gauge group. Indeed,recall that the E N groups canonically embed in E as: E N × SU ( K ) Z K ⊂ E (10.5)where N + K = 9. This result can be established as follows. Decomposing theadjoint representations of E N and SU ( K ) to E N − × U (1) and SU ( K − × U (1), wefind that the resulting representations all have charge 0 or ± K . As two examples,consider the decomposition of the adjoint representations of the algebras E and E = SO (10): E ⊃ SO (10) × U (1) (10.6)78 → + 45 + 16 − + 16 (10.7) E ⊃ SU (5) × U (1) (10.8)45 → + 24 + 10 + 10 − . (10.9)Returning to the weight space decomposition of the charged representations, it fol-lows that the relative normalization of the matrices which generate the Cartan sub-algebras of E N and E differ by 1 /K . Exponentiating these matrices, we arrive atthe desired condition in the corresponding subgroups.This fractional quantization condition demonstrates that in the above example,we may treat L as a line bundle, with L a “fractional power” of a line bundle.Moreover, fields localized on a matter curve Σ transform as sections of K / ⊗ L a Σ ⊗ L ′ b Σ for integers a and b , where L Σ and L ′ Σ respectively denote the restriction of potentiallyfractional line bundles on S and S ′ . Indeed, the common identification of the centersof the gauge groups in (10.5) illustrates that although the individual restrictions of L and L ′ to Σ may correspond to ill-defined line bundles, their tensor product may stilldetermine a mathematically well-defined line bundle. We therefore conclude that so59ong as the resulting fields all transform in well-defined bundles, the correspondingfractional line bundles are physically well-defined. G S = SO (10) The analysis of the previous subsection establishes that when G S = SU (5), there areno exotic bulk zero modes if and only if the gauge bundle corresponds to a fractionalline bundle of the form O S ( α ) / where α corresponds to a root associated withan element of H ( S, Z ). In this Section we show that when G S = SO (10), directbreaking to G std via fluxes always results in exotica in the low energy spectrum.To establish this result, we note that the classification of Appendix C shows thatthe only instanton configurations which break SO (10) to G std take values in thesubgroup U (1) × U (1) so that the commutant subgroup in SO (10) is SU (3) × SU (2) × U (1) × U (1) . With respect to this decomposition, the adjoint, spinor andvector representations of SO (10) decompose as: SO (10) ⊃ SU (5) × U (1) ⊃ SU (3) × SU (2) × U (1) × U (1) (10.10)45 → (1 , , + (1 , , + (8 , , + (1 , , (10.11)+ (3 , − , + (3 , , + (3 , , + (3 , − , − (10.12)+ (3 , − , + (3 , , − + (1 , , + (1 , − , − (10.13)16 → (1 , , − + (3 , , + (1 , − , (10.14)+ (1 , , − + (3 , , − + (3 , − , − (10.15)10 → (3 , − , + (1 , , + (3 , , − + (1 , − , − . (10.16)In the MSSM, fields charged under the subgroup SU (3) × SU (2) transform inthe representations (3 , ,
2) and (3 , , , , (3 , − , − or (3 , , − .In F-theory, all of the matter content of the MSSM descend from the 45, 16,16 or 10 of SO (10). As reviewed in Appendix C, there are precisely two linearcombinations of U (1) and U (1) which can correspond to U (1) Y in the Standard60odel: U (1) Y = U (1) (10.17) U (1) Y = − U (1) − U (1) . (10.18)While the first case corresponds to embedding hypercharge in the usual way insideof the SU (5) factor, the second possibility corresponds to a “flipped” embedding ofhypercharge [67].First suppose that U (1) Y is given by equation (10.17). Letting A ≡ L and B ≡ L − ⊗ L − , the condition that the zero mode content must contain no exoticmatter requires that the following cohomology groups must vanish:(3 , , ∈ H ∂ ( S, A ) = 0 (10.19)(3 , − , ∈ H ∂ ( S, A − ) = 0 (10.20)(3 , − , − ∈ H ∂ ( S, B ) = 0 (10.21)(3 , , − ∈ H ∂ ( S, A ⊗ B ) = 0 (10.22)(1 , − , − ∈ H ∂ ( S, A − ⊗ B ) = 0. (10.23)For a supersymmetric configuration, it follows from the vanishing theorem of [15]that the cohomology groups H ∂ and H ∂ vanish for all of the above line bundles. Thecohomology group H ∂ therefore vanishes when the index of each line bundle vanishes.Equations (10.19) and (10.20) imply:0 = χ ( S, A ) + χ ( S, A − ) = 2 + c ( A ) · c ( A ). (10.24)On the other hand, equations (10.21)-(10.23) imply:0 = χ ( S, A ⊗ B ) + χ ( S, A − ⊗ B ) − χ ( S, B ) = c ( A ) · c ( A ) (10.25)which contradicts equation (10.24). The resulting low energy spectrum will thereforealways contain some exotic matter.Next consider the flipped embedding of U (1) Y given by equation (10.18). Withnotation as above, the condition that the zero mode content must contain no exotic61atter now requires that the following cohomology groups vanish:(3 , − , − ∈ H ∂ ( S, B ) = 0 (10.26)(3 , , ∈ H ∂ ( S, B − ) = 0 (10.27)(3 , , ∈ H ∂ ( S, A ) = 0 (10.28)(3 , , − ∈ H ∂ ( S, A ⊗ B ) = 0 (10.29)(1 , , ∈ H ∂ ( S, A ⊗ B − ) = 0. (10.30)These conditions are the same as those of equations (10.19-10.23) with the roles of A and B interchanged. We therefore conclude that in all cases, the resulting spectrumwill contain exotic matter.More generally, we note that the classification of possible breaking patterns pro-vided in Appendix C requires at least one U (1) factor. When G S has rank five ormore, direct breaking to G std therefore requires the instanton configuration to takevalues in a subgroup of G S with rank at least two. We note that while only abelianinstanton configurations are available for rank four and five bulk gauge groups, it isin principle possible that an SU (2) valued instanton could partially break the bulkgauge group when G S = E . However, decomposing the adjoint representation to G std , the number of different exotic representations appears to always be greaterthan the rank of the subgroup in which the instanton takes values. The requirementthat so many different cohomology groups must simultaneously vanish is then anover-constrained problem so that in such cases exotics are unavoidable. In this Section we explain how to obtain the exact spectrum of the MSSM when S is a del Pezzo surface. As explained in subsection 10.2, direct breaking via internalfluxes will generate exotics when the bulk gauge group is not SU (5). Restrictingto the case G S = SU (5), the only candidate bundles which will not generate exoticbulk zero modes are in one to correspondence with the roots of an exceptional Liealgebra. In this case, all of the matter content of the MSSM must localize on mattercurves.Individual components of a GUT multiplet will interact differently with the in-ternal hypercharge flux. In keeping with our general philosophy, we require that62 F ] S [Σ ] M [Σ ] H H (S,Z) Figure 4: Letting [ F S ] denote the two-cycle in H ( S, Z ) which is Poincar´e dual to thebackground hypercharge flux h F S i , there is a natural distinction between the class ofthe Higgs curve [Σ H ] and the class of the chiral matter curves [Σ M ]. Indeed, whilethe net flux on Σ M must vanish to preserve a full GUT multiplet, the gauge fieldconfiguration must restrict non-trivially on the Higgs curves in order to solve thedoublet triplet splitting problem. When the net flux on the Higgs curve is not zero,this corresponds to the condition that [Σ M ] and [ F S ] are orthogonal while [Σ H ] and[ F S ] are not.a complete GUT multiplet must localize on a given matter curve so that on suchcurves, the net hypercharge flux must vanish. Otherwise, a different index will de-termine the number of zero modes coming from each component of a complete GUTmultiplet. On the other hand, the gauge field must restrict non-trivially on theHiggs curves in order to solve the doublet-triplet splitting problem. See figure 4 fora depiction of how the corresponding elements in H ( S, Z ) intersect.In order to achieve a chiral matter spectrum in four dimensions, the net fluxon the matter curve cannot vanish. As an example, consider a six-dimensionalhypermultiplet in the 5 of G S × G S ′ = SU (5) × U (1) which localizes on an exceptionalcurve Σ with homology class E . The overall normalization of the U (1) charge is notparticularly important because we shall consider vacua with fractional line bundles.When L = O S ( E − E ) / the restriction of L to Σ is trivial. Letting L ′ denotethe supersymmetric line bundle on the seven-brane which intersects the GUT modelseven-brane along Σ, the restriction of L ′ to Σ must be non-trivial in order to achievea chiral matter spectrum. For example, when L ′ Σ = O Σ ( − × ∈ H ∂ (Σ , K / ⊗ O Σ ( − × ∈ H ∂ (Σ , K / ⊗ O Σ (3)) = H ∂ (Σ , O Σ (2)) (10.32)where we have also indicated the multiplicity of the zero modes. Similar considera-tions apply for other GUT multiplets.On the other hand, the Higgs fields of the MSSM do not fill out complete GUTmultiplets at low energies. In this case, the net hypercharge flux piercing this mattercurve must be non-zero. More precisely, recall that for minimal supersymmetric SU (5) GUT models, the Higgs up and down fields respectively descend from the 5and 5 of SU (5), where the 5 decomposes to (3 , − + (1 , . Letting L Σ denote therestriction of the bulk gauge bundle L to the matter curve Σ with similar notationfor L ′ Σ , we note that the zero mode content is determined by the cohomology groups:(1 , ∈ H ∂ (Σ , K / ⊗ L ⊗ L ′ n Σ ) (10.33)(3 , − ∈ H ∂ (Σ , K / ⊗ L − ⊗ L ′ n Σ ) (10.34)where n is an integer associated with the U (1) charge associated with the branewrapping S ′ . Mathematically, we wish to find line bundles such that K / ⊗ L ⊗ L ′ n Σ has non-vanishing cohomology whereas K / ⊗ L − ⊗ L ′ n Σ has trivial cohomology. Anecessary condition for K / ⊗ L − ⊗ L ′ n Σ to have trivial cohomology is that the degreeof the line bundle L − ⊗ L ′ n Σ must vanish. As a brief aside, we recall the well-knownfact that degree zero line bundles are in one to one correspondence with points onthe Jacobian of the curve.As an example, consider a genus one matter curve where the line bundles L Σ and L ′ Σ are given by: L Σ = O Σ ( − np + np ) (10.35) L ′ Σ = O Σ (3 p − p ) (10.36)where p and p denote distinct degree one divisors on Σ which are not linearlyequivalent. Because these divisors are not linearly equivalent, the divisor p − p is64ot effective. Assuming that K / is trivial, we have:(1 , ∈ H ∂ (Σ , O Σ (0)) ≃ C (10.37)(3 , − ∈ H ∂ (Σ , O Σ (5 n ( p − p )) = 0 (10.38)since the divisor p − p is not effective. In this case, we achieve a vector-like pairof Higgs up/down fields on the curve Σ.Now, there is no reason that the Higgs up and down fields must localize on thesame matter curve. In a certain sense, the above implementation of doublet tripletsplitting is somewhat artificial precisely because the distinguishing feature of theHiggs curve is that a non-trivial flux is present. With this in mind, it seems far morenatural to consider line bundles which have non-trivial degree on the Higgs curves.In this case, a given Higgs curve will automatically contain more Higgs up than Higgsdown fields.To give an explicit example of this type, consider a six-dimensional hypermultipletin the 5 of SU (5) localized on a genus zero curve Σ. In this case, the zero modecontent is determined by the cohomology groups:(1 , ∈ H ∂ (Σ , K / ⊗ L ⊗ L ′ Σ ) (10.39)(1 , − ∈ H ∂ (Σ , K / ⊗ L − ⊗ L ′− ) (10.40)(3 , − ∈ H ∂ (Σ , K / ⊗ L − ⊗ L ′ Σ ) (10.41)(3 , ∈ H ∂ (Σ , K / ⊗ L ⊗ L ′− ). (10.42)The zero mode content on Σ yields precisely one Higgs up field for fractional linebundle assignments: L Σ = O Σ (1) / and L ′ Σ = O Σ (1) / . (10.43)Similarly, a single Higgs down field can also localize on another matter curve.It is also possible to localize a single Higgs up field on a higher genus mattercurve. For example, with notation as above, when Σ is a genus one curve, the More generally, recall that on a general genus g Riemann surface, a divisor D with degree ≥ g is linearly equivalent to an effective divisor [68]. This imposes a non-trivial constraint on the waysin which doublet-triplet splitting can arise for a general matter curve. L Σ = O Σ ( p ) / ⊗ O Σ ( p − p ) / and L ′ Σ = O Σ ( p ) / ⊗ O Σ ( p − p ) − / (10.44)will again yield a single H u field localized on Σ.In fact, in Section 12 we will show that in order to remain in accord with currentbounds on the lifetime of the proton, the Higgs fields must localize on different mattercurves. These matter curves may or may not intersect inside of S . When thesecurves do not intersect, these fields do not couple in the superpotential and the µ term is automatically zero. Moreover, when these curves do intersect, they mustinteract with a third gauge singlet which localizes on a curve that only intersects S ata point. In Section 15 we estimate the behavior of this gauge singlet wave functionnear the surface S and show that this naturally yields an exponentially suppressed µ term. In the MSSM with R-parity, the lightest supersymmetric partner (LSP) could bea viable dark matter candidate. In fact, in the context of a local model, it isnatural to expect a large number of additional gauge degrees of freedom which onlyinteract gravitationally with the MSSM. This appears to be an automatic feature ofmany consistent string compactifications which will typically contain several hiddensectors. For example, in the perturbative heterotic string, this role can be playedby the hidden E factor. A rough comparison of the two E factors would thensuggest that half of the matter content in such a model could be visible, and theother half could be dark matter. In F-theory, the analogue of the hidden E factorcould be the additional seven-branes which are required for the compactification tobe globally consistent. For example, F-theory compactified on K B is given by12 c ( B ). Integrating this Chern class over an appropriate two-cycle would thenyield a rough estimate on the amount of dark matter from seven-branes. It is alsoin principle possible that the total number of three-branes in the compactificationcould also contribute to the dark matter content of the model. In the absence of66uxes, the total number of three-branes is given by χ ( CY ) /
24. We note that inorder for the Calabi-Yau fourfold to be elliptically fibered, the threefold base B must be a Fano variety. For example, B = P , gives 48 seven-branes. Note thatthe GUT group involves a bound state of O (10) such seven-branes. We find it quiteamusing that this is in rough agreement with the observed ratio between visible anddark matter in our Universe! Of course, this depends on the relative masses for thevarious visible and hidden fields. There is a finite list of such manifolds [69], and itwould therefore be of interest to compare the relative number of three-branes andseven-branes in such compactifications.
11 Geometry and Matter Parity
From a phenomenological viewpoint, matter parity provides a simple way to forbidrenormalizable terms in the four-dimensional superpotential which can potentiallyinduce proton decay. It also naturally leads to an LSP which could potentially be adark matter candidate. In a Lorentz invariant theory, this is equivalent to assigningan appropriate R-parity to the individual components of a superfield. Indeed, theessential point is that this discrete symmetry distinguishes the Higgs superfields fromall of the other chiral superfields of the MSSM. In this Section we argue that thepresence of such a Z symmetry is quite natural from the perspective of F-theory.As explained in subsection 10.3, the Higgs fields localize on matter curves piercedby a net amount of internal hypercharge flux while the chiral matter localizes oncurves where the net hypercharge flux is trivial. This is a discrete choice whichnaturally distinguishes the Higgs superfields from the rest of the chiral superfields ofthe MSSM.From a more global perspective, these fluxes correspond to the localization offour-form G -flux in the compactification. If the Calabi-Yau fourfold admits a geo-metric Z symmetry, then these fluxes will decompose into even and odd elements of H ( CY , Z ) which we denote by H ( CY , Z ) + and H ( CY , Z ) − . If this symmetry iswell-defined, it follows that on a given seven-brane, the corresponding line bundlesmust have a definite parity under this choice of sign. For example, the parity of theline bundle on the S brane can be even while the parity of the line bundles on theother branes may have other parities. 67t now follows that the net flux on a matter curve can only be non-zero when theflux and matter curve have the same parity. Indeed, letting F ± denote a flux withparity ± ± , the unbroken Z symmetryimplies: Z Σ − F +Σ − = Z S F + ∧ P D (Σ − ) = − Z S F + ∧ P D (Σ − ) = 0 (11.1) Z Σ + F − Σ + = Z S F − ∧ P D (Σ + ) = − Z S F − ∧ P D (Σ + ) = 0 (11.2)where P D (Σ ± ) ∈ H ( S, Z ) denotes the Poincar´e dual element of [Σ ± ] ∈ H ( S, Z ).In other words, when the integral of the flux over a curve does not vanish, the fluxand curve have the same parity.In order for this group action to remain well-defined on the matter curves, theinternal wave functions which are sections of appropriate bundles must also have adefinite sign under the group action. First consider the parity of the Higgs fields.These wave functions are defined as sections of line bundles which depend non-trivially on the restriction of line bundles from both S as well as other transverselyintersecting seven-branes. We therefore conclude that both fluxes must have thesame parity. In particular, we conclude if the parity of the bulk gauge field is even,then the Higgs fields will also have even parity.Next consider the parity of the remaining matter fields. Here it is essential thatthe net flux contribution from S is trivial on all such matter curves. In particular, ifthe gauge bundle from the transversely intersecting seven-brane is odd under matterparity, then the corresponding sections on each matter curve will also be odd underthe Z action on the Calabi-Yau fourfold. Hence, we obtain on rather generalgrounds a geometric version of matter parity.
12 Proton Decay and Doublet-Triplet Splitting
As argued in subsection 10.3, there exist vacua which yield the exact spectrumof the MSSM for an appropriate choice of flux in a local intersecting seven-braneconfiguration. In particular, we found that the Higgs triplets can typically be68emoved from the low energy spectrum. While this mechanism provides a naturalway to achieve the correct zero mode spectrum in the Higgs sector, when the Higgsup and down fields localize on the same matter curve, the higher Kaluza-Klein modesof the corresponding six-dimensional fields will generate higher order superpotentialterms of the form
QQQL/M KK with order one coefficients. While here we havepresented the operator in terms of the Kaluza-Klein mass scale M KK , for minimal SU (5) GUT models, we can reliably approximate M KK by M GUT . If present, suchoperators can significantly shorten the lifetime of the proton.We now explain how such terms could potentially be generated in our class ofmodels. When all Yukawa couplings to the Higgs triplets are order one parameters,the superpotential terms: W GUT = QQT u + QLT d + M KK T u T d (12.1)will give a large mass to the Higgs triplets T u T d of order M KK . Integrating out T u and T d , the coefficient of the operator QQQL/M KK would then be too large tosatisfy present constraints. In fact, the geometry of the matter curves indicatesprecisely when we can expect such terms to be generated. The tree level diagramwhich generates the offending operator is given by drawing the intersection locus ofthe matter curves and interpreting each matter curve as a leg of the correspondingFeynman diagram. See figure 5 for a depiction of how the geometry of the mattercurves quite literally translates into a statement about diagrams in the low energytheory.While it is in principle possible to suppress the value of this coefficient by incor-porating flavor symmetries, in the context of four dimensional supersymmetric GUTmodels, this problem can be avoided by having T u and T d develop masses by pairingwith additional heavy triplet states T ′ u and T ′ d so that the superpotential insteadtakes the form: W GUT = QQT u + QLT d + M T u T ′ d + M T d T ′ u (12.2) When we present some examples of four-dimensional flipped SU (5) models which descend froman eight-dimensional SO (10) model, there can be a small discrepancy between the four-dimensionalGUT scale M GUT and M KK . We thank S. Raby for emphasizing this point to us. TQQ QL T TQQ QL M M M H HM M H
5 + 5
H M M M Figure 5: Depiction of how the geometry of matter curves directly translates intoamplitudes in the low energy theory. In case a), the Higgs up and down fields localizeon the same matter curve. The resulting field theory diagram which generates theoperator
QQQL is given by interpreting each matter curve as the leg of a Feynmandiagram. In case b), the Higgs up and down fields localize on distinct matter curves.In this case, the Feynman diagram involving the exchange of massive Higgs tripletsis unavailable. 70hich does not generate the offending dimension five operator from integrating outmassive fields at tree level. Note that this occurs automatically when the Higgs upand down fields localize on distinct matter curves.In compactifications of the heterotic string on Calabi-Yau threefolds, the Higgstriplet is typically projected out of the low energy spectrum by discrete Wilson lines.In general, it is not clear to us whether this sufficiently suppresses proton decay.Indeed, while the Higgs triplet zero mode may be absent from the spectrum, thereis an entire tower of Kaluza Klein modes which must also be considered. If any ofthese modes contribute an interaction term of the form given by equation (12.1), thecoefficient of the offending dimension five operator may still be too large to remainin accord with observation.To summarize, we have seen that the proton decays too rapidly when the Higgs upand down fields localize on the same matter curve. As a necessary first step, we haveshown that when these Higgs fields localize on distinct matter curves, integrating outthe higher Kaluza-Klein modes for the Higgs fields does not generate the offendingbaryon number violating term
QQQL . Even so, it is still in principle possible thatsome exotic process could generate the operator
QQQL . In fact, placing the Higgsfields on different matter curves automatically equips them with additional globalsymmetries in the low energy effective theory. As we now explain, these symmetriessignificantly extend the lifetime of the proton.
13 Extra U (1) ’s and Higher Dimension Operators In Section 12 we have shown that the dimension five operators responsible for protondecay are naturally suppressed when the Higgs up and Higgs down fields localize ondifferent matter curves. In this Section we explain from a different perspective whythis suppression occurs and also discuss on more general grounds when we expectother higher dimensional operators to suffer a similar fate.Imposing additional global symmetries provides one common way to suppressundesirable interaction terms in field theory. Indeed, so long as the global symmetryremains unbroken, all of the higher order terms of the effective superpotential will alsorespect this symmetry. In F-theory, these U (1) factors occur automatically becausethe breaking direction in the Cartan subalgebra of a given singularity determines71he location of the matter curves in the geometry. Matter localizes on the curveprecisely when it is charged under the appropriate subgroup. While this genericallyallows local triple intersections of matter curves to take place, all of the fields of theMSSM will therefore be charged under additional U (1) factors. These extra U (1)’scan therefore naturally suppress higher dimension operators. When two curvesdo not intersect inside of S , fields localized on each curve will be charged underdistinct U (1) groups. This can forbid cubic interaction terms as well as many higherorder contributions to the effective superpotential. It would be very interesting todetermine the precise mapping between topological properties of intersecting curvesand the associated U (1) fields.From a bottom up perspective, the fields of the MSSM contain various accidentalsymmetries. Assuming generic values of the Yukawa couplings and that the µ termoriginates from the vev of a gauge singlet, the classical action is invariant underfour U (1) symmetries. These can be identified with U (1) Y hypercharge, U (1) B baryon number, U (1) B − L baryon minus lepton number and a U (1) P Q
Peccei-Quinnsymmetry. Of these four possibilities, only U (1) Y and U (1) B − L are potentiallynon-anomalous.In a quantum theory of gravity, any global symmetry must be promoted to a gaugesymmetry. One potential worry is that because the fields of the MSSM are naturallycharged under these U (1)’s, the presence of these gauge bosons could lead to conflictwith experiment. While these U (1)’s will typically be anomalous and therefore liftfrom the low energy spectrum, it is interesting to ask whether a massless U (1) ofthis type is already ruled out by experiment. This is not very promising becausecurrent constraints from fifth force experiments have set a strong limit on the gaugecoupling of extra massless U (1) gauge bosons: g extra . m n M pl ∼ − (13.1)where m n denotes the mass of the neutron. In the absence of a natural explanationfor why the gauge coupling would be so weak for such couplings, this appears quitefine-tuned. In fact, such a small value is already in conflict with the conjecture thatgravity is the weakest force [70]. See [71, 72] for further discussion on extra massless U (1) gauge bosons. 72he analogue of equation (4.8) for α extra = g extra / π is of the form: α − extra = M ∗ V ol ( S extra ) ∼ M ∗ R ⊥ R S (13.2)where as before, R ⊥ denotes the length scale associated with the direction normal tothe surface S . In tandem with equation (4.8) this implies: α extra ∼ α GUT R S R ⊥ = α GUT × ε γ ∼ × − ± . . (13.3)Based on the above estimate, we conclude that all additional U (1) gauge bosonsmust develop a sufficiently large mass in order to lift from the low energy spectrum.In fact, our expectation is that this only imposes a mild constraint on the compactifi-cation. When the U (1) symmetry is anomalous, the Green-Schwarz mechanism willgenerate a string scale mass for the gauge boson. Even when the U (1) symmetry isnon-anomalous, the gauge boson can still develop a large mass. Indeed, althoughthe analysis of Section 9 shows that four-dimensional U (1) gauge bosons can remainmassless in the presence of internal fluxes, it also establishes sufficient conditions forsuch bosons to develop a large mass on the order of R − ⊥ . In either case, we thereforeexpect that it is always possible for all extraneous U (1) gauge bosons to develop asuitably large mass. In the low energy effective theory, some imprint of the gaugesymmetry will remain as an approximate global symmetry in the low energy effectivetheory. These global symmetries can be violated by non-perturbative contributionsto the superpotential from Euclidean branes wrapping the various K¨ahler surfacesof the compactification. Such contributions are naturally suppressed by an expo-nential factor of the form exp( − c/α extra ) where c is an order one positive number.Similar instanton effects have been proposed as a possible solution to the cosmolog-ical constant problem [73]. Such exponentials could also provide a novel methodof generating contributions to the flavor sector of the theory. We present one briefspeculation along these lines in Section 14. As a brief aside, recall that in Section6 we presented an example of a four-dimensional GUT model where an appropriateoperator generated by non-perturbative contributions could produce an effective µ term. Indeed, when a strict decoupling limit does not exist, it is likely that non-perturbative contributions to the superpotential could play a more prominent rolein the effective theory. 73 Finding vacua with the correct matter spectrum of the MSSM is only the first stepin constructing a semi-realistic model. In models where all chiral matter localizes onmatter curves, the leading order contribution to the four-dimensional effective super-potential originates from the triple intersection of matter curves. After presentinga general analysis of how matter curves can form triple intersections in S , we showthat in order to achieve one generation with mass which is hierarchically larger thanthe two lighter generations, some of the matter curves must self-intersect or “pinch”inside of S . See figure 6 for a depiction of a pinched curve. While a completetheory of flavor is beyond the scope of this paper, we can nevertheless provide aqualitative explanation for why the heaviest generation obeys an approximate GUTmass relation which is violated by the lighter generations. In fact, the effect we dis-cover is generically realized in vacua with non-zero internal hypercharge flux becausethe Aharanov-Bohm effect distorts the wave functions of individual components ofa GUT multiplet by different amounts. Moreover, this distortion becomes morepronounced as the mass of the generation decreases. We conclude by presentingsome speculations on how more detailed properties of flavor physics could originatefrom a local del Pezzo model. As reviewed in Section 3, cubic contributions to the superpotential of an exceptionalseven-brane can originate from three sources. These correspond to interactionsamongst three bulk zero modes, interactions between a single bulk zero mode andtwo zero modes localized on a matter curve, and interaction terms between threezero modes on matter curves. As explained in subsection 10.3, in a minimal SU (5)GUT all of the field content of the MSSM localizes on curves. Thus, the leadingorder contribution to the effective superpotential comes from the triple intersectionof matter curves.Locally, the triple intersection of matter curves in S occurs when the bulk singu-larity type G S undergoes an at least twofold enhancement to a singularity of type G p ⊃ G S × U (1) × U (1) . Following the general philosophy of [44], we note thatmatter localized along curves in S is charged under the corresponding U (1) × U (1) t and t denote the local deformation parameters associ-ated with the two U (1) factors, this curve is locally described by an equation of theform: at + bt = 0. (14.1)In the above, the constants a and b are determined by the decomposition of theadjoint representation of G p to G S × U (1) × U (1) so that the appropriate irreduciblerepresentation of G S has U (1) × U (1) charge ( a, b ). This is simply the statementthat because the Cartan subgroup is visible to the geometry, this local enhancementin singularity type has been Higgsed in the bulk to G S .The triple intersection of three curves Σ , Σ and Σ requires that the intersectionproduct of the corresponding homology classes satisfies [Σ i ] · [Σ j ] > i = j .Even so, generic curves representing each class which all intersect pairwise will notform a triple intersection in S . However, in certain cases there exist representativeholomorphic curves of each homology class which can form a triple intersection insideof S . For this to occur, it must be possible to deform the point of intersection ofone pair of curves to coincide with the point of intersection of another pair. Inother words, the normal bundle N Σ /S of one of the curves must possess at least oneglobal section. Although from the perspective of the surface S this may appear tobe a somewhat non-generic situation, we note that in F-theory such points of tripleintersection occur automatically. Indeed, as explained in [15], this follows from thefact that in F-theory, rank two enhancements in the singularity type will genericallyoccur at points in S . The claim now follows from group theoretic considerations.At a pragmatic level, given curves Σ i = ( f i = 0), it is possible to engineer a tripleintersection by requiring that one of the f i is a linear combination of the other two f i ’s in the ring of sections on S . Assuming without loss of generality that f is givenby a linear combination of f and f , this can be written as: f = α f + α f (14.2)where the α i correspond to holomorphic sections of some line bundles on S . Forexample, this condition is satisfied when both [Σ ] − [Σ ] and [Σ ] − [Σ ] are “effective”divisors, namely divisors which correspond to holomorphic curves.This geometric condition can be used to narrow the search for vacua which arephenomenologically viable. For example, to forbid cubic matter parity violating75ontributions to the superpotential, it is enough to require that the curves supportingthe chiral matter of the MSSM must not form a triple intersection. On the otherhand, in order to have at non-trivial interaction terms, some of the matter contentof the MSSM must localize on a curve which is not exceptional . Indeed, threeexceptional curves cannot triple intersect in S . This follows from the fact that thenormal bundle of a curve in S has degree [Σ] · [Σ] which equals − H ∂ ( P , O ( − At zeroth order, it is most important to obtain a naturally heavy third generationin the quark sector. Indeed, the mass of the top quark is roughly 170 GeV, whichis significantly higher than the next heaviest up type quark. This requires that thecorresponding Yukawa coupling must be sufficiently large. In a suitable basis, wetherefore require that the up-type Yukawa couplings are of the form: λ u ∼ ε ε ε ε ε ε ε ε (14.3)where the ε ’s are all parametrically smaller than 1.When all of the cubic terms of the superpotential originate from the triple inter-section of matter curves in S , there is additional structure in the form of the Yukawacouplings. First consider the Yukawa couplings for fields charged in the 10 of SU (5).In this case, the interaction terms: W ⊃ λ uij H × ( i ) M × ( j ) M (14.4)are non-zero whenever the curves defined by Σ H , Σ i and Σ j form a triple intersection.When none of the Σ i self-intersect, or “pinch”, it follows that the general form of λ uij λ uij = A BA CB C (14.5)where A , B and C are constants given by evaluating wave function overlaps. Wenow argue that this matrix cannot yield one generation which is hierarchically heavierthan the first two generations. In order for such a hierarchy to exist, we requirethat there exists a limit in the parameters A , B and C where two of the massesdetermined by λ uij tend to zero while the third mass remains large.In the limit in which one of the generations has zero mass, the determinant ofthe matrix λ uij vanishes: 2 ABC = 0 (14.6)so that without loss of generality, we may assume that the strictly massless limitcorresponds to A = 0. Since the trace of λ u is zero, we conclude that when A = 0,two of the eigenvalues of λ u are equal in magnitude and have opposite sign. Thisimplies that there does not exist a limit in which two of the generations are para-metrically lighter than the third. On the contrary, this would suggest that two ofthe generations are significantly heavier than the lightest generation. We emphasizethat this result holds independent of how the kinetic terms are normalized. This isbecause it is always possible to switch to a basis of fields where the kinetic terms arecanonically normalized. This alters the form of λ u by a similarity transformationand an overall rescaling. In this new basis, the determinant and trace will still vanishso that the above argument proceeds as before. Rather than appeal to non-perturbative effects, we note that such a hierarchy caneasily be achieved provided the Yukawa matrix possesses at least one non-zero di-agonal element. Geometrically, this requires that one of the matter curves mustpinch off so that globally, the curve intersects itself inside of S . We caution thatthis notion of self-intersection is somewhat stronger than what is usual meant byself-intersection at the level of homology. At the level of homology, a class is typi-cally said to self intersect when two distinct representatives of a given homology class77 ? Figure 6: Depiction of how a local enhancement in the singularity type can enhanceto the intersection of two distinct curves (left), or a single curve which self-intersects(right).intersect inside of S . See figure 6 for a depiction of how a curve can self-intersectby pinching off inside of S .We now extend the analysis of [15] for smooth matter curves to the present caseof interest where the curve may pinch off, or self-intersect. Before describing the caseof self intersection, let us recall what happens when two distinct curves intersect. Inthis case near the generic intersection point the two curves can be modeled by theequation: z z = 0 (14.7)where z = 0 describes one curve and z = 0 denotes the other so that the intersectionpoint is located at z = z = 0. By group theory considerations explained in [15],it is clear that a third matter curve will also pass through this point, with a localdefining equation z = z + z = 0. This gives rise to a Yukawa interaction of theform: W ⊃ φ φ φ (14.8)where φ i denotes a field associated with the local vanishing locus z i = 0.From a global perspective, this description does not specify whether φ and φ localize on distinct matter curves or whether they localize on the same curve. Inthe case where these fields localize on the same curve, the locus z = 0 curve mustconnect to the z = 0 curve in a more global description inside of S . In other words,these two loci must form a single Riemann surface. Hence, a self-intersecting curvecorresponds to a genus g + 1 curve which pinches to a genus g curve in such a way78hat this pinching process does not lead to two disconnected surfaces. Conversely,when this pinching process produces two disconnected curves, this describes the casewhere the matter curves are distinct.To analyze the matter content localized on a self-intersecting curve, we note thatthe overlap of wave functions at the pinching point determines a single linear relationamongst the various zero modes of the form: α i φ ( i ) = 0 (14.9)where the φ ( i ) label the zero modes of the genus g curve obtained by pinching theassociated genus g + 1 curve. This identification reduces the value of the associatedindex by one.The number of self-intersection points as well as their proximity will clearly havean impact on the properties of the Yukawa couplings in the low energy theory. Toillustrate this point, it is enough to consider the up type Yukawa couplings of theminimal GUT model which descend from the cubic interaction term: W ⊃ λ uij H × iM × jM . (14.10)Suppose that three generations in the 10 of SU (5) all localize on the same self-intersecting matter curve. If there is only one point of self-intersection which wedenote by 0, the Yukawa matrix is given by the outer product of the wave functionfor the three generations: λ uij = ψ H (0) ψ i (0) ψ j (0) (14.11)so that it automatically has rank one. By a suitable change of basis, the leading orderbehavior of the up-type Yukawa couplings is given by equation (14.3) as requiredin a semi-realistic model. Additional points of self-intersection will increase therank of the up-type Yukawa coupling matrix. In this case, the relative proximitybetween these points of intersection as well the analogous expressions for the down-type Yukawa couplings will control the masses and mixing angles in the quark sector.It would be interesting to determine whether a hierarchical pattern of masses andmixing angles could emerge from such a treatment.79 In this subsection we show that the usual GUT mass relations present in the simplestfour-dimensional GUT models can be significantly distorted in the presence of aninternal hypercharge flux. In the simplest four-dimensional GUT models, the massesof the up and down type quarks are determined by the superpotential terms: W ⊃ λ uij H × iM × jM + λ dij H × iM × jM . (14.12)Assuming that the individual components of a GUT multiplet have the same wavefunction normalization, this would imply that m q = m l for the quarks and leptonswhich unify in a 5 M of SU (5). Evolving the values of the masses observed at lowenergies up to the GUT scale, it is well-known that only the third generation obeysa relation of the form m b ∼ m τ . At the level of precision we can perform here, theoriginal analysis of mass relations in the non-supersymmetric SU (5) GUT analyzedin [74] is certainly sufficient. In this case, the actual mass relations at the GUTscale are: m b ∼ m τ , m s ∼ m µ / m d ∼ m e . (14.13)See [75] for an updated analysis of the various mass relations obtained by extrapo-lating the observed values of the masses to the GUT scale. This problem is evenmore pronounced for the simplest SO (10) GUTs where all interaction terms descendfrom the coupling 16 M × M × H . Letting i = 1 , ,
3, we can parameterize theviolation of the expected mass relation for each generation: δ i = m ( i ) q − m ( i ) l m ( i ) q + m ( i ) l . (14.14)Returning to equation (14.13), the violation of the simplest mass relation for eachgeneration is: δ = 0, δ ∼ δ ∼ − SU (5) GUT model of the type treated here, one extremesolution would be to invoke the mechanism of doublet triplet splitting via fluxesdescribed in Section 12 so that individual components of a full GUT multiplet couldlocalize on distinct matter curves.While this provides one possible way to avoid incorrect mass relations amongstmembers of the lighter generations, we find it somewhat anti-thetical to the wholeidea of grand unification that the matter content of the Standard Model neatlyfits into GUT multiplets. Indeed, it would seem unfortunate to sacrifice such anaesthetic motivation for grand unification. Moreover, the usual GUT mass relationdoes work relatively well for the third generation. We now argue that even whena complete GUT multiplet localizes on a matter curve, the relative normalization ofthe kinetic terms between different components of the GUT multiplet will in generalbe different. Moreover, we give a qualitative explanation for why the mass relationsbecome increasingly distorted for the lighter generations.Recall that in the minimal SU (5) GUT, the net hypercharge flux vanishes oncurves supporting complete GUT multiplets. Indeed, the converse of this conditionfor the Higgs curves provides a qualitative explanation for why these fields do notfill out full GUT multiplets. Although the average hypercharge flux vanishes onchiral matter curves, the field strength will in general not vanish pointwise . Becausethe individual components of a GUT multiplet have different hypercharge, the cor-responding wave functions will couple differently to this background flux leading todistinct zero mode wave functions. The fact that the zero mode wave functions are81ot the same, and may in particular have different magnitudes, can be interpretedas Aharanov-Bohm interferences in a varying B-field background.In a minimal SU (5) GUT, all of the interaction terms originate from evaluatingthe wave functions at points of triple intersection and now there is no reason whythe magnitude of different matter fields within a GUT multiplet are the same. Thisleads to different Yukawa couplings and thus to different mass relations. In particu-lar, assuming for simplicity no mixing between generations, we have modified massrelations of the form: m q = m l (cid:12)(cid:12)(cid:12)(cid:12) ψ q (0) ψ l (0) (cid:12)(cid:12)(cid:12)(cid:12) . (14.16)It would be interesting to examine whether modified GUT mass relations for thelighter generations of the general type proposed in [74] admit a geometric interpre-tation.We now estimate the expected distortion in the usual GUT mass relations dueto the Aharanov-Bohm effect with a varying B-field. To this end, let F Σ denotethe internal U (1) hypercharge field strength on the matter curve Σ. The overallscaling dependence of the mass relation violation δ can be determined by rescalingthe overall volume of Σ by ε . Because the reduction of the instanton to Σ scales as | F Σ | /ε , it follows that F Σ rescales by a factor of √ ε . This reduction is explained infurther detail in [77]. It now follows that the violation of the mass relation will beproportional to: δ ∼ √ ε . (14.17)Note that as the volume of Σ tends to zero, the amount of violation in the massrelation also vanishes. Equation (4.25) implies that the masses of fields localized onΣ scale as: M ∼ /V ol (Σ) ∼ /ε , (14.18)because in a canonical normalization of all fields, each wave function contributes afactor of ψ (0) / p M ∗ V ol (Σ) to the Yukawa couplings. Hence, the violation of themass relation obeys the scaling law: δ ∼ / √ M . (14.19)While a mass relation will still hold for each generation, the particular numericalcoefficient relating the masses will depend on the generation in question.82o conclude this Section, we note that a common theme running throughoutmuch of this paper is the central role of the internal hypercharge flux. Indeed, anintra-generational distortion in the usual GUT mass relations requires the presenceof an internal hypercharge flux. In a sense, we can view the violation of the GUTmass relation as the first experimental evidence for the existence of extra dimensions! In this subsection we speculate on one possible way to achieve semi-realistic masshierarchies and mixing angles in the context of our compactification. To frame thediscussion to follow, we first review the field theory Froggatt-Nielsen Mechanism forgenerating a hierarchical structure in both the masses and mixing angles of the quarksector. As observed in [78], this naturally occurs when the up and down Yukawacouplings assume the form: λ uij = g uij ε a i + b j , λ dij = g dij ε a i + c j , (14.20)where the g ’s are order one 3 × ε is a small parameter which isrelated to the Cabbibo angle θ c ∼ .
2. With this ansatz, the quark sector exhibitshierarchical masses and mixing angles determined by appropriate powers of ε [78].From a field theory perspective, this type of power law suppression naturallyoccurs in theories with additional global U (1) symmetries. For example, if thesuperfields Q i , U i and D i have charges a i , b i and c i under a global U (1) symmetry,then the corresponding fields interact by coupling to an appropriate power of a gaugesinglet charged under this global U (1). For example, letting φ denote a gauge singletsuperfield with charge +1 under this global symmetry, the lowest order coupling inthe superpotential is given by: W ⊃ g uij (cid:18) φM pl (cid:19) − a i − b j Q i U j H u + g dij (cid:18) φM pl (cid:19) − a i − c j Q i D j H d (14.21)where for the purposes of this discussion we assume that H u and H d are neutralunder the global U (1) symmetry. When φ develops a vev less than M pl , we obtainthe expected hierarchy in the Yukawa couplings of equation (14.20).83e now speculate as to how such a hierarchy could potentially occur in compact-ifications of F-theory. Given a sufficiently generic configuration of matter curveswhich form triple intersections, in a holomorphic basis of wave functions the result-ing holomorphic Yukawa couplings introduced in Section 4 will be given by orderone complex numbers. To extract the values of the physical up and down typeYukawa couplings, all of these fields must be rescaled to a canonical normalization ofall kinetic terms. In the large volume limit, this simply rescales each wave functionby an appropriate power of the overall volume factor so that the up and down typeYukawa couplings are: λ uij = g uij Z (10) i Z (10) j Z H u , λ dij = g dij Z (10) i Z (5) j Z H d (14.22)where we have introduced the notation Z = ( M ∗ V ol (Σ)) − / . In the above, thesuperscript on each Z denotes the representation and as usual, the indices i and j label the generations. In the extreme case where the volumes of the mattercurves are hierarchical, this would provide a crude analogue of the Froggatt-Nielsenmechanism. It is not clear to us, however, that such a hierarchy is always availablefor self-intersecting curves. Indeed, it is likely that the Z ’s differ by order one factors.While this is typically enough to sufficiently distort the usual GUT mass relations,it may prove insufficient to produce the large hierarchy in mass scales between thetop quark and the charm quark, for example.Implicit in the above discussion is the assumption that the Z ’s of equation (14.22)only depend on the classical volumes of the matter curves. Indeed, as explainedin Section 4, the overall normalization of each wave function will receive quantumcorrections away from the large volume limit. While we do not have a systematicmethod for computing these corrections, experience in perturbative string theorystrongly suggests that these corrections are exponentially suppressed as functionsof the K¨ahler moduli. Moreover, these corrections may induce small off-diagonalterms in the K¨ahler metric for the fields of the required type to generate a hierarchicalstructure in the physical Yukawa couplings.In a similar vein, it is also tempting to speculate that non-perturbative con-tributions to the superpotential from Euclidean 3-branes wrapping divisors in theCalabi-Yau fourfold base could also contribute to a viable model of flavor physics.Indeed, because such corrections will typically violate global U (1) symmetries present84n the low energy effective theory, the corresponding exponential factor can in prin-ciple have a form compatible with the Froggatt-Nielsen mechanism. While theseremarks are admittedly speculative, it would be interesting to see whether thereexist calculable examples of the desired type. Discrete symmetries provide another possible way to achieve semi-realistic Yukawacouplings and interaction terms because such models can mimic the primary fea-tures of the Froggatt-Nielsen mechanism, but with the global continuous symmetryreplaced by a discrete symmetry. In this approach, it is common to search for fi-nite groups which admit two- and three-dimensional irreducible representations. Forexample, the two lightest generations could transform in a two-dimensional represen-tation while the heaviest generation could transform as a singlet. As one application,these symmetries are typically enough to alleviate potential problems with FCNCsin gravity mediation scenarios. A list of candidate discrete flavor groups withorder at most thirty one which are of phenomenological interest has been tabulatedin [79]. Some common choices in the model building literature are the symmetricgroup on three or four letters denoted by S and S as well as A , the alternatingsubgroup of S . See [80, 81] for a recent review of some possibilities along theselines. In the present context, the group of large diffeomorphisms of a del Pezzosurface provide a potentially attractive starting point for a theory of flavor basedon discrete symmetries. We note that some version of this gauged symmetry willsurvive even away from the large volume regime. It is therefore possible that suchsymmetries could undergird a theory of flavor.The group of large diffeomorphisms for the del Pezzo surfaces has a natural actionon the matter curves of the del Pezzo which automatically lifts to a group action onthe matter fields of the MSSM. For example, the del Pezzo 3 surface corresponds tothe exceptional group E = SU (3) × SU (2) which has Weyl group S × S . The S factor could potentially play the role of the desired flavor group.One potential caveat to the above proposal is that the action of the Weyl groupon the matter curves corresponds to an integral representation. In other words, the We thank K.S. Babu for emphasizing this point to us. A and S are given by various powers of a thirdroot of unity, this direct application of discrete symmetries may be too trivial.We note that no similar obstruction is present in the case of the discrete group S . Indeed, consider as a toy model the case where the three generations havelocalized on the exceptional curves E , E and E of the del Pezzo 3 surface. In thiscase, the S Weyl group permutes the exceptional curves. The three dimensionalrepresentation spanned by the three curves also determines how S acts on the threegenerations. This three dimensional representation decomposes to the sum of a twodimensional representation and singlet which are respectively spanned by: h E , E , E i ≃ h E − E , E − E i doublet ⊕ h E + E + E i singlet . (14.23)This suggests that the wave function for the heavy generation transforms as thesinglet, while the two light generations transform as the doublet. It would beinteresting to develop such a theory of flavor in more detail.
15 Suppression Factors From Singlet Wave Func-tions
So far we have only considered contributions to the superpotential from matter fieldswhich all transform as non-trivial representations of G std . A fully realistic modelwill most likely contain contributions to the effective superpotential from chiral su-perfields which transform as gauge singlets under G std . For example, the µ termcould originate from a cubic interaction term between the Higgs fields and a gaugesinglet. The vev of this singlet would then set the size of µ . As another example, wenote that because neutrino oscillations are now well-established, the superpotentialmust contain terms of the form LN R H u where N R denotes the right-handed neutrinosuperfields which transform as gauge singlets.Generating appropriately small neutrino masses as well as a value for the µ termnear the scale of electroweak symmetry breaking has historically been a challenge in86tring-based models. Some discussion on neutrino masses in string theory may befound for example in [82]. In type II D-brane constructions, contributions to thesuperpotential from wrapped Euclidean branes can produce an appropriately largeMajorana mass term for right-handed neutrinos [54, 83]. Similar effects may alsogenerate exponentially suppressed µ terms [54]. More recently, it has also been shownthat D-brane instantons can also potentially generate suppressed Dirac neutrinomasses [84]. In this Section, we show that the Yukawa couplings which involve asinglet of G S can in suitable circumstances be exponentially suppressed relative tothe Yukawa couplings which only involve fields charged under G S .The rest of this Section is organized as follows. In subsection 15.1, we studythe behavior of gauge singlet wave functions which contribute to the low energysuperpotential. After performing this analysis, in subsection 15.2, we estimate theoverall normalization of the Yukawa couplings for such gauge singlet wave functions.For interaction terms involving three singlets, there is a natural volume suppressioneffect. For gauge singlets which are attracted to the GUT model seven-brane, thewave function behaves as if it had localized on a matter curve inside of S . For gaugesinglets which are repelled away from the GUT model seven-brane, we find thatthe Yukawa couplings are naturally suppressed. In the remaining subsections weshow that these effects can naturally generate both hierarchically small µ terms andneutrino masses. In both cases, we find that order one parameters in the high energytheory naturally can yield values which are in rough agreement with observation. To setup notation, we consider three seven-branes which wrap surfaces S , S ′ , and S ′′ inside the compactification threefold B and which carry respective gauge groups G S , G S ′ , and G S ′′ . By assumption, S , S ′ , and S ′′ intersect transversely along smoothcurves Σ X = S ∩ S ′ , Σ Y = S ∩ S ′′ , Σ ⊥ = S ′ ∩ S ′′ , (15.1)which give rise to corresponding chiral superfields X , Y , and Φ in four dimen-sions. Each superfield transforms as a bifundamental under the respective products See § . S × G S ′ , G S × G S ′′ , and G S ′ × G S ′′ . Finally, if the curves Σ X , Σ Y , and Σ ⊥ them-selves intersect transversely at a single point, the low-energy effective superpotentialcontains a cubic coupling of the form W ⊥ = λ Φ XY , (15.2)invariant under G S × G S ′ × G S ′′ .By assumption, the kinetic terms for X , Y , and Φ have the canonical normal-ization in four dimensions, so the dimensionless coupling λ in W ⊥ depends upon the L -norms of the associated zero-mode wavefunctions on the curves in (15.1). Sinceboth Σ X and Σ Y are compact curves inside S , the norms of wavefunctions for X and Y merely scale with the volumes of the curves in S . However, unlike Σ X and Σ Y ,the curve Σ ⊥ is not embedded in S but rather intersects S transversely at a point in B . From the perspective of the four-dimensional effective theory, this distinction ingeometry is reflected by the fact that Φ transforms as a singlet under G S , whereas X and Y form a vector-like pair. We are interested in the limit that S contracts inside B , or equivalently, in the limit that the volume of Σ ⊥ goes to infinity. In the limitthat Σ ⊥ becomes non-compact, we clearly need to be careful in our estimate for thenorm of the wavefunction ψ associated to the singlet Φ.We are ultimately interested in the behavior of ψ near the point where Σ ⊥ inter-sects S , so let us introduce local holomorphic and anti-holomorphic coordinates ( z, z )on Σ ⊥ such that z = 0 is the location of the intersection with S . As we reviewedin Section 3, ψ generally transforms on Σ ⊥ as a holomorphic section of the bundle K / ⊥ ⊗ L , ψ ∈ H ∂ (cid:0) Σ ⊥ , K / ⊥ ⊗ L (cid:1) , L = L ′ (cid:12)(cid:12) Σ ⊥ ⊗ L ′′ (cid:12)(cid:12) Σ ⊥ , (15.3)where L ′ and L ′′ are line bundles on S ′ and S ′′ . Because ψ is holomorphic, ψ satisfies ∂ † ∂ψ = 0 , (15.4)where ∂ is the Dolbeault operator acting on K / ⊥ ⊗ L , and ∂ † is the adjoint operatordefined with respect to the induced metric on Σ ⊥ and the hermitian metric on L inherited from L ′ and L ′′ .Besides the Dolbeault operator ∂ , the bundle K / ⊥ ⊗ L also carries a unitary con-nection which defines a covariant derivative ∇ and an associated Laplacian △ = ∇ † ∇ .88y a standard Hodge identity reviewed in Appendix E of [15], the Laplacian △ isrelated to the operator ∂ † ∂ via △ = 2 ∂ † ∂ − R + F . (15.5)Here R is the scalar curvature of the metric on Σ ⊥ , and F is the scalar curvature ofthe unitary connection on L .The positive constants in (15.5) will not be important for the following analysis,but the signs will be essential. First, the relative sign between R and F in (15.5)arises because R is the scalar curvature of the induced metric on Σ ⊥ and hence isthe curvature of a connection on the holomorphic tangent bundle T Σ ⊥ ∼ = K − ⊥ , asopposed to a connection on the spin bundle K / ⊥ . To fix the overall sign multiplying R , we note that the Laplacian △ is a positive-definite hermitian operator. On theother hand, because ψ is holomorphic, △ ψ = (cid:18) − R + F (cid:19) ψ . (15.6)According to (15.6), if F = 0 and R > ψ must vanish. Sucha vanishing is consistent with the fact that K / ⊥ = O ( −
1) admits no holomorphicsections on Σ ⊥ = P , and this observation fixes the sign of R in the Hodge identity(15.5).In a local unitary frame, the Laplacian △ takes the standard Euclidean form △ = − ∂ /∂z ∂z , and (15.6) reduces to the wave equation4 ∂ ψ∂z∂z + (cid:18) F − R (cid:19) ψ = 0 . (15.7)Thus if ψ is normalized so that ψ (0) = 1, then ψ behaves near z = 0 as ψ ( z, z ) = exp (cid:18) − m | z | (cid:19) + · · · ,m = (cid:20) F − R (cid:21) z =0 , (15.8) The factor ‘1 /
2’ multiplying R in (15.5) arises from the square-root in the spin bundle K / ⊥ . · · · ’ indicate terms in ψ that vanish at z = 0, and the curvatures whichdefine m are evaluated at that point. In general, m can be either negative orpositive, and the sign of m determines whether ψ exponentially grows or decaysaway from the origin.At first glance, one might be perplexed as to how such exponential behavior in ψ can arise, since nothing so far really distinguishes the point z = 0. In fact, given that ψ is written in a unitary frame, the behavior in (15.8) merely reflects the behaviorof the metric on K / ⊥ ⊗ L .As a very concrete example, let us take Σ ⊥ to be P , with a metric which weparameterize in Liouville form as ds = e φ ( z,z ) dz dz . (15.9)For instance, if the metric on P is round with constant curvature Λ , then φ ( z, z ) = − ln (cid:18) | z | (cid:19) . (15.10)The role of the particular line bundle K / ⊥ ⊗ L is inessential, so for simplicity wejust take ψ to transform in the holomorphic tangent bundle T P . As is well-known,holomorphic tangent vectors on P take the global form u ( z ) ∂∂z , u ( z ) = a + a z + a z , (15.11)where ( a , a , a ) are complex parameters. However, if φ ( z, z ) in (15.9) varies non-trivially over P , the holomorphic vector ∂/∂z does not have constant length. Todescribe ψ in a unitary frame, we instead introduce a new basis vector ˆ e for T P ,ˆ e = 12 e − φ ( z,z ) ∂∂z . (15.12)Though ˆ e is not holomorphic, ˆ e does have constant, unit length in the metric (15.9). Holomorphy of ψ implies that the total curvature satisfies R Σ ⊥ ⋆ ( F − R ) ≥
0, but the sign of
F − R may vary from point to point on Σ ⊥ .
90n the frame described by ˆ e , a holomorphic tangent vector ψ therefore takes the form ψ = e φ ( z,z ) u ( z ) ˆ e. (15.13)Because the scalar curvature of the metric in (15.9) is given in terms of φ as R = − − φ ∂ φ∂z ∂z , (15.14)the behavior near z = 0 of ψ in (15.13) is controlled by the local curvature. To make use of (15.8), we must still estimate m at the point where Σ ⊥ intersectsthe surface S . Since m receives contributions from both R and F , we consider eachcontribution in turn.To estimate R , we recall that S is a del Pezzo surface shrinking to zero sizeinside the elliptic Calabi-Yau fourfold X . As a result, the scalar curvature on S islarge and positive, of order M GUT . On the other hand, since X is Calabi-Yau, thetotal scalar curvature on X vanishes. Because the elliptic fiber of X is genericallynon-degenerate, with negligible curvature, the large positive curvature of S near itspoint of intersection with Σ ⊥ must be locally cancelled by a corresponding negativecurvature on Σ ⊥ itself. The scalar curvature R on Σ ⊥ near z = 0 is thus negativeand of order R ∼ − M GUT . (15.15)We note that if Σ ⊥ has genus zero or one, then R must become positive elsewhereon Σ ⊥ as dictated by the Euler characteristic.We apply a similar argument to estimate the curvature F on L near z = 0. Bydefinition, the line bundle L is a tensor product L ′ (cid:12)(cid:12) Σ ⊥ ⊗ L ′′ (cid:12)(cid:12) Σ ⊥ of line bundles L ′ and L ′′ on respective surfaces S ′ and S ′′ , and both L ′ and L ′′ carry anti-self-dualconnections. The following observations are symmetric between L ′ and L ′′ , but forconcreteness let us focus on the bundle L ′ over S ′ .The surface S ′ contains two curves Σ X = S ∩ S ′ and Σ ⊥ = S ′′ ∩ S ′ which intersecttransversely at the point z = 0 on Σ ⊥ . Since S is shrinking inside X , the curveΣ X is similarly shrinking inside the surface S ′ . In this situation, an anti-self-dualconnection on L ′ over S ′ must restrict to a solution of the two-dimensional Yang- Because of the conventions adopted, R in (15.14) plays the role of F in (15.8). X . Hence the curvature of L ′ on Σ X mustbe constant and uniform, of order d Vol(Σ X ) − ∼ d M GUT , where d is the degree of L ′ on Σ X .Without loss, we assume that the metric on S ′ at the intersection of Σ X andΣ ⊥ takes the diagonal form ds = dzdz + dwdw , where w is a local holomorphiccoordinate on Σ X and z is a local holomorphic coordinate on Σ ⊥ . Because thecurvature of the connection on L ′ is anti-self-dual, the curvature at z = 0 along Σ ⊥ must be opposite to the curvature along Σ X . Hence the curvature of L ′ on Σ ⊥ is oforder − d M GUT .Including a similar contribution from L ′′ , we find F ∼ − " deg( L ′ (cid:12)(cid:12) Σ X )Vol(Σ X ) + deg( L ′′ (cid:12)(cid:12) Σ Y )Vol(Σ Y ) ∼ ± M GUT . (15.16)Both R and F are of roughly the same magnitude, but whereas the sign of R is fixed,the sign of F generally depends upon the degrees of L ′ and L ′′ as well as the relativevolumes of the matter curves Σ X and Σ Y in S . We see no particular reason why thecontributions to F from Σ X and Σ Y should be correlated in either sign or absolutevalue. So depending upon the choices for L ′ and L ′′ , the parameter m = F − R can be either positive or negative, of order M GUT .We are left to estimate the norm of the singlet wavefunction ψ . Now, the greatvirtue of writing ψ in a unitary frame is that the L -norm of ψ is given directly by || ψ || = M ∗ Z Σ ⊥ ω | ψ | , ω = i φ ( z,z ) dz ∧ dz , | ψ | ≡ ψψ . (15.17)Here ω is the K¨ahler form for the induced metric on Σ ⊥ , which for concretenesswe parameterize in the Liouville form (15.9). According to (15.8) and (15.14), theintegrand of (15.17) then behaves to leading order near z = 0 ase φ ( z,z ) | ψ | ≈ exp (cid:20) −
12 ( m + R ) | z | (cid:21) , = exp (cid:20) − (cid:18) F + 12 R (cid:19) | z | (cid:21) . (15.18)If the combination F + R is positive at z = 0, the integral over Σ ⊥ in (15.17)92as rapid Gaussian decay at the scale M GUT , so immediately || ψ || ∼ M ∗ M GUT , (cid:20) F + 12 R (cid:21) z =0 > . (15.19)In this case the normal wave function is attracted to our brane.Conversely, if F + R is negative at z = 0, the expression in (15.18) rapidlyblows up away from the origin. In this case the normal wave function is repelled fromour brane. To make sense of || ψ || , we impose a cutoff in the integral over Σ ⊥ at ascale | z | ∼ R ⊥ . As we discuss briefly below, we expect the Gaussian approximationin (15.18) to be valid up to the cutoff, so we estimate || ψ || as h ψ | ψ i = || ψ || ∼ M ∗ M GUT exp (cid:0) c M GUT R ⊥ (cid:1) , (cid:20) F + 12 R (cid:21) z =0 < . (15.20)In this estimate, c > R ⊥ , which roughly encodes the behavior of themetric on B away from S . We recall that R ⊥ is parameterized as R ⊥ = M − GUT ε − γ , ε = M GUT α GUT M pl , (15.21)where γ typically lies in the range 1 / < γ < || ψ || , we assume that the curvature of theCalabi-Yau metric on X (and similarly the connection on L ) is slowly varying andof order M GUT in a region of size R ⊥ away from S . This behavior of the Calabi-Yaumetric on X is suggested by similar behavior of the local Calabi-Yau metric on thecotangent bundle T ∗ CP , as exhibited for instance in § T ∗ CP ,the scalar curvature R along the cotangent fiber experiences only a slow, power-lawdecay away from CP , and we roughly expect the same behavior normal to S in X .However, a more precise estimate of || ψ || clearly demands a more detailed analysisof the local Calabi-Yau metric on X . 93 Having estimated the local behavior of gauge singlet wave functions near the delPezzo surface, we now determine the corresponding values of the Yukawa couplingsin the low energy theory. With notation as above, to estimate the size of the Yukawacoupling in equation (15.2), we introduce the wave function x (resp. y ) for the chiralsuperfield X (resp. Y ) which localizes on the matter curve Σ X (resp. Σ Y ) in S . Thesuperpotential term of equation (15.2) due to a triple overlap between Σ X , Σ X , Σ ⊥ at a point p is: W ⊥ = λ Φ XY (15.22)= x ( p ) p M ∗ V ol (Σ X ) y ( p ) p M ∗ V ol (Σ Y ) ψ ( p ) p h ψ | ψ i Φ XY (15.23)where in the above, we have adopted the physical normalization of Yukawa couplingsdetailed in Section 4. The value of the Yukawa coupling strongly depends on whetherthe del Pezzo surface attracts or repels the gauge singlet wave function from the pointof intersection. By contrast, we note that because X and Y localize on matter curvesinside of S , the values of x ( p ) and y ( p ) are order one numbers. Making the roughapproximation M ∗ V ol (Σ) ∼ α − / GUT , the resulting Yukawa coupling is: λ = α / GUT ψ ( p ) p h ψ | ψ i . (15.24)We now estimate the value of the Yukawa coupling depending on whether theGUT model seven-brane attracts or repels the gauge singlet wave function. Tothis end, we shall frequently refer back to the estimates of the various length scalesobtained in Section 4. In the repulsive case, equation (15.20) now implies: λ repel ∼ α / GUT × (cid:18) α / GUT R S R ⊥ exp (cid:16) − cε γ (cid:17)(cid:19) (15.25)= α / GUT × ε γ exp (cid:16) − cε γ (cid:17) (15.26)where the second equality follows from equation (4.17) and as in the previous sub-section, c is a positive order one number.94y contrast, in the undamped case described by equation (15.19), the associatedYukawa coupling is: λ attract ∼ α / GUT M GUT M ∗ ∼ α / GUT . (15.27)Physically, the value of λ attract agrees with the intuition that in the attractive case, alldetails of the compactification decouple because the gauge singlet behaves as thoughit localizes on a curve in S . In general, we see that: | λ attract | ≫ | λ repel | . (15.28)In addition to interaction terms between matter fields inside of S and a singlegauge singlet, it is also possible for three gauge singlet wave functions to interactoutside of S . When one such gauge singlet develops a non-zero vev, the resultinginteraction term will determine the mass of the remaining gauge singlets. Letting ψ i denote gauge singlet wave functions for i = 1 , ,
3, the value of the physical Yukawacoupling from wave function overlap at a point b outside of S is now given by: λ singlet ∼ ψ ( b ) p M ∗ V ol (Σ ) ψ ( b ) p M ∗ V ol (Σ ) ψ ( b ) p M ∗ V ol (Σ ) ∼ M ∗ R ⊥ ) (15.29) ∼ α / GUT (cid:18) R S R ⊥ (cid:19) = α / GUT × ε γ . (15.30)We note that in comparison to Yukawa couplings on S which are on the order of α / GUT , this naturally yields an overall suppression factor by a non-trivial power of ε . µ Term
We now discuss a natural mechanism for obtaining small supersymmetric µ terms.For concreteness, suppose that the bulk gauge group G S = SU (5) and that the H u and H d fields localize on distinct matter curves where the singularity type enhances to SU (6). In the case where these curves do not intersect, the µ term is automaticallyzero. In the case where they do intersect, the matter fields will interact with a gaugesinglet which localizes on a curve normal to S . Letting Φ denote the chiral superfield95or this gauge singlet, the superpotential now contains the interaction term: W µ ⊃ λ Φ H u H d ∼ α / GUT ψ ( p ) p h ψ | ψ i Φ H u H d (15.31)with notation as in equation (15.24). When Φ develops a vev, the superpotentialwill contain a µ term for the Higgs up and Higgs down fields. The value of this vevis controlled by the dynamics orthogonal to S and therefore scales as: h Φ i ∼ R ⊥ ∼ M GUT × ε γ . (15.32)Returning to equations (15.25) and (15.27), it thus follows that in the attractive case,the resulting value of µ is far above the electroweak scale, and would lift the Higgsdoublets from the low energy spectrum. On the other hand, in the exponentiallydamped case, the value of the µ term is: µ = λ repel h Φ i ∼ α / GUT × ε γ exp (cid:16) − cε γ (cid:17) . (15.33)This leads to a large hierarchy between the µ term and the GUT scale. For example,with γ = 1 and c = 1 / µ ∼
140 GeV. In Section 19 we present someadditional estimates of µ . At a conceptual level, the µ term and Dirac mass terms for the neutrinos bothoriginate from interactions between two fields on curves in S and a third field whichlocalizes on a curve normal to S . Indeed, in the previous subsection we found thatwhen the gauge singlet wave function is exponentially suppressed near S , the µ termis hierarchically suppressed below the GUT scale. We now estimate the values ofthe light neutrino masses of the MSSM depending on the profile of the right-handedneutrino wave function near the surface S . When the gauge singlet is attractedto S , a variant on the usual seesaw mechanism yields neutrino masses which areapproximately correct. On the other hand, when the gauge singlet is repelled awayfrom S , the value of the Dirac masses is already quite low, and the seesaw mechanismwould yield unviable neutrino masses. In fact, the Dirac mass terms are already in96 viable range so that in this case the neutrinos are purely of Dirac type.For simplicity, we perform our estimates for a single neutrino species, because asexplained in Section 14, a detailed model of flavor is currently beyond our reach. Inthis case, the neutrino sector of the superpotential is: W ν = λ D LN R H u + λ singlet Θ N R N R (15.34)where N R denotes the right-handed neutrino chiral superfield, and Θ is another gaugesinglet. In certain cases, the second interaction term may not be present. In thefollowing we analyze the interplay between the behavior of the right-handed neutrinowave functions near S and this second interaction term. We now consider the case where the second interaction term Θ N R N R does not vanishand show that a phenomenologically viable scenario requires that the right-handedneutrino wave function is attracted to S . When Θ develops a vev, it induces aMajorana mass term for the right-handed neutrinos. Using the value of λ singlet givenby equation (15.30), this yields the Majorana mass: m M ≡ λ singlet h Θ i = λ singlet R ⊥ = α / GUT M GUT × ε γ ∼ × ± . GeV. (15.35)The value of the Dirac masses strongly depends on the profile of the gauge singletwave function near S . By inspection of equations (15.25) and (15.27), the value of λ attract will induce a Dirac mass term for neutrinos which is around the electroweakscale, while the value of λ repel will induce a far smaller Dirac mass term. The massmatrix for the neutrinos is: M ν = " m D m D m M ∼ α / GUT " h H u ih H u i M GUT × ε γ . (15.36)Because the Majorana mass term is non-zero, it is much larger than the Diracmass terms so that the smaller eigenvalue of M ν is given by the usual seesaw mech-97nism: m light ∼ m D m M . (15.37)Due to the fact that the Majorana mass term is in the usual range expected for aseesaw mechanism, m D must be on the order of the electroweak scale in order toyield a viable light neutrino mass. Restricting to this case, m light is now given by: m light ∼ (cid:16) α / GUT × ε − γ (cid:17) × h H u i M GUT ∼ × − ± . eV. (15.38)We note that in this case, we automatically find an enhancement over the naiveseesaw value h H u i /M GUT ! Indeed, in the GUT literature it is often necessary tolower the Majorana mass term below M GUT to obtain more realistic neutrino masses.
Next consider the possibility that the interaction term between Θ and N R in equation(15.34) does not exist so that the neutrinos are purely of Dirac type. In the previoussubsection we found that a variant of the standard seesaw mechanism requires thatthe right-handed neutrino wave function is attracted towards S . Indeed, the Diracmass terms for the undamped wave functions were automatically on the order ofthe electroweak scale. In the absence of a seesaw mechanism, this profile for thewave functions would yield an unacceptably large value for the neutrino masses. Onthe other hand, the wave functions which are repelled away from S will naturallygenerate much smaller Dirac neutrino mass terms.Restricting to the repulsive case, the Dirac mass term is: m Dirac = λ repel h H u i ∼ h H u i × h α / GUT × ε γ exp (cid:16) − cε γ (cid:17)i . (15.39)The essential point of the above formula is that the Dirac mass can be quite light, andfor an appropriate order one value of c , yields a phenomenologically viable mass forthe light neutrinos. For example, setting c = 5 and γ = 1 / m Dirac ∼ × − eV. Before closing this subsection, we note that while large Majorana mass termswhich violate lepton number are typically invoked as a primary cause of leptogenesisin early universe cosmology, there do exist viable alternative scenarios which onlyrequire Dirac neutrino masses. See [86] and references therein for a recent account98f Dirac leptogenesis. µ and ν In the previous subsection we presented a general formula which naturally generatesan exponentially suppressed value for the masses of purely Dirac type neutrinos.Indeed, the exponential damping terms for both the µ term of equation (15.33) andthe Dirac mass term of equation (15.39) are both sensitive to an order one parameterwhich we denote by c . We now present a relation between µ and m Dirac in which theoverall dependence on this exponential factor cancels out. This expression is modelindependent in the sense that it does not depend as strongly on the details of theexponential suppression factor.The exponential suppression factors of the µ term and the purely Dirac massterm both originate from a gauge singlet wave function which is repelled away fromthe surface S so that: m Dirac = λ repel ( c ) h H u i (15.40) µ = λ repel ( c ′ ) h Φ i (15.41)where h Φ i denotes the vev of a gauge singlet which localizes on a matter curve normalto S . In the above, we have allowed two potentially different suppression factorssuch that c and c ′ may differ by some small amount.Making the simplifying assumption c = c ′ , all exponential effects cancel, and weobtain the rough estimate: m Dirac = µ h H u ih Φ i = µε − γ h H u i × h H u i M GUT ∼ × − ± . eV (15.42)for µ ∼
100 GeV. Of course, for small mismatches between the parameters c and c ′ ,slightly higher (or lower) values are also in principle possible.99 Up to now, our analysis has assumed that the four-dimensional effective theory pre-serves N = 1 supersymmetry. See [87, 88] for recent discussions of supersymmetrybreaking in F-theory and [89] for an explicit realization of gauge mediated super-symmetry breaking in an intersecting D-brane model. In this Section we brieflysketch how supersymmetry breaking can be communicated to the MSSM in a gaugemediation scenario. Further details will appear in [90]. A more general frameworkwhich interpolates between gauge mediation and gravity mediation is given in [52].In that context, supersymmetry breaking takes place on a seven-brane distinct froma GUT model seven-brane. When these branes intersect, supersymmetry breakingis communicated via gauge mediation. As the separation between the seven-branesincreases, this interpolates to a gravity mediation scenario. In the present case,most of our seven-branes form non-trivial topological intersections which cannot dis-appear. While we shall present some brief speculations on generating hierarchicallysmall values for the scale of supersymmetry breaking, a complete analysis wouldentail a broader discussion which is beyond the scope of this paper.To frame the discussion to follow, we now briefly sketch the basic features of gaugemediated supersymmetry breaking. See [91] for a review of gauge mediation. Ingeneral, most mediation mechanisms consist of three sectors. These are given by thesector of the theory which breaks supersymmetry, the sector of communication, andthe MSSM itself. Although we do not specify how supersymmetry can be broken,we can still parameterize this breaking in terms of at least one chiral superfield X which develops a supersymmetry breaking vev: h X i = x + θ F . (16.1)To specify the messenger sector, we introduce vector-like pairs of GUT multipletswhich will communicate supersymmetry breaking to the MSSM. As an explicitexample, we take Y to transform in the fundamental of SU (5) and Y ′ in the anti-fundamental. These fields can then localize on matter curves inside of S . Themessengers couple to X via an interaction term of the form: W d ⊃ W mess = λXY Y ′ . (16.2)100nce X develops a vev of the type given by equation (16.1), the messengers will geta mass: M mess = λx . (16.3)Supersymmetry breaking then communicates to the MSSM because the messengerfields interact with the gauge bosons of the MSSM. In this setup, the soft massesfor the gauginos are generated at one loop order while the soft scalar masses aregenerated at two loop order. One attractive feature of the gauge mediation scenariois that FCNCs are automatically suppressed.Although precise numerical estimates are beyond the scope of the present paper,to simply get a sense of the mass scales involved, recall that in gauge mediation, themasses of the gauginos are: m i ∼ α i ( M weak )4 π Fx . (16.4)We note that this estimate does not require any knowledge of the overall normal-ization factors appearing in equation (16.2). The lightest gaugino in this caseis the Bino which in viable models has a mass of ∼
100 GeV. Plugging in theproperly normalized value of the hypercharge coupling at the weak scale given by α ( M weak ) ∼ (5 / × (1 / ∼ − , we see that the scale of supersymmetry break-ing √ F and the messenger scale x are related via: √ F ∼
300 GeV / √ x . (16.5)Depending on the origin of the X field in the F-theory GUT model, the resultingmessenger mass scales can potentially be quite different. In the following subsectionswe discuss three natural candidates for X in the present class of compactifications.The field X can correspond to a bulk gauge boson on a transversely intersectingseven-brane, or a field which localizes on a matter curve orthogonal to S . In thelatter case, there are two further refinements depending on whether the GUT modelseven-brane attracts or repels the corresponding gauge singlet wave function. When the matter fields Y and Y ′ localize on the same curve, these fields will automat-ically couple to the bulk gauge fields of a seven-brane which transversely intersects101he GUT model seven-brane. In this case, we can interpret x as the supersymmetricvev of the bulk gauge field. The value of x depends on the volume of the mattercurve containing the messenger fields as well as the remaining bulk worldvolume ofthe other seven-brane. Using the basic scaling relations obtained in Section 4, weestimate h X i ∼ /R ⊥ so that the resulting messenger mass is: M mess = α GUT M GUT ε γ (16.6) ∼ × ± . GeV. (16.7)
It is also possible that X could correspond to a gauge singlet which localizes ona matter curve which intersects S at a point. In this case, much of the analysisperformed in Section 15 carries over. For example when the gauge singlet wavefunction for X is attracted towards the seven-brane, it couples to the messengerfields with the same strength as a field inside of S . In this case, the messenger massis on the order of: M mess = λ attract R ⊥ = α / GUT M GUT × ε γ ∼ × ± . GeV. (16.8)On the other hand, the seven-brane can also repel the gauge singlet wave func-tion. In this case, the messenger mass scale can be hierarchically much lighter thanthe GUT scale due to the exponential suppression factor present at the point of in-tersection with the seven-brane. In this case, the resulting messenger mass is givenby a similar expression to that derived for the µ term in equation (15.33): M mess = λ repel R ⊥ ∼ M GUT × α / GUT ε γ exp (cid:16) − cε γ (cid:17) . (16.9)In this case, the messenger mass scale can potentially range over many candidatevalues. For example, we obtain a value of ∼ GeV when c = 1 and γ = 1 /
3, anda value of ∼
300 TeV when c = 1 /
10 and γ = 1.102 A well-known difficulty with the gauge mediation scenario is that it is typicallydifficult to simultaneously generate the correct values for the µ and Bµ terms. Inthe present context, we note that the µ term is naturally light and on the orderof the electroweak scale. Indeed, this setup decouples the issue of supersymmetrybreaking from the µ problem. In fact, at the GUT scale, the Bµ term is zero at highenergies, and is instead radiatively generated. Phenomenological fits to this rangeof parameter space favor larger values for tan β = h H u i / h H d i [92].We also expect that higher order terms in the superpotential of the form: W quart = c ijk M KK X Λ i Λ j Λ k (16.10)where Λ i denotes a generic field of the MSSM cannot be generated by integratingout Kaluza-Klein modes. As explained in Section 13, this is due to the fact thatsuch terms will typically violate a global U (1) symmetry in the low energy theory.Indeed, matter fields in F-theory are always charged under additional U (1) factors ofprecisely this type. Letting σ i denote the bosonic component of the chiral superfieldΛ i , this suggests that the values of the soft breaking A -terms in the effective potential: V eff = A ijk σ i σ j σ k (16.11)will automatically vanish at the scale set by x . Because both the Bµ and A termsvanish, there is a common rephasing symmetry of the fields which naturally avoidsadditional CP violating phases. To conclude this Section, we now briefly speculate on ways in which supersymmetrybreaking can take place in the various scenarios outlined above. First consider thecase where X is identified with a bulk gauge field on a seven-brane which inter-sects the GUT seven-brane. Returning to the equations of motion for fields on thetransversely intersecting seven-brane S ′ derived in [15], the value of F is: F ∗ = ∂ ′ φ ′ + δ Σ hh Y c , Y ii ad ( P ) + ... (16.12)103here φ ′ denotes the holomorphic (2 ,
0) form for this brane and the ... denotes con-tributions to the F-term localized on other matter curves in the surface S ′ . Whenthe righthand side of the above equation is non-zero, this will break supersymmetry.This can easily occur when the background value of the G-flux in the Calabi-Yaufourfold is incompatible with the complex structure on S ′ . Because this differencecan be quite small in principle, we can obtain small values for F in this case.Next consider scenarios where the X field corresponds to a gauge singlet localizedon a matter curve intersecting S at a point. While we have primarily focussed onthe behavior of this wave function in supersymmetric backgrounds, presumably asimilar analysis will also carry through in a non-supersymmetric background. Inthis vein, it may be possible to extend the discussion of Section 15 to this moregeneral case. It would be interesting to see whether a suitable hierarchy in the scaleof supersymmetry could be arranged in this way. SU (5) Model
Having presented a number of potential model building ingredients in the previousSections, we now proceed to some semi-realistic examples of models based on a delPezzo 8 surface which incorporates at least some of these ideas. Our expectationis that significant refinements are possible in the actual examples we present. Asexplained in previous sections, the GUT group directly breaks to G std via an internalhypercharge flux. Moreover, to avoid exotic matter representations, the availableinternal fluxes are in one to one correspondence with the roots of an exceptional Liealgebra. In this case, all of the matter content of the MSSM must localize on curvesin S . The fields in the 5 or 5 localize on curves where the bulk SU (5) singularityenhances to SU (6), while fields in the 10 and 10 localize on curves where SU (5)enhances to SO (10).As explained in [15], the interaction terms 5 H × M × M originate from pointswhere the bulk singularity G S = SU (5) undergoes a twofold enhancement in rank toan SO (12) singularity. Similarly, the interaction terms 5 M × M × M originatefrom a twofold enhancement in rank to an E singularity. As in [15], we deducethe local behavior of the matter curves near such points by decomposing the adjoint104epresentations of SO (12) and E to the product SU (5) × U (1) × U (1): SO (12) ⊃ SU (5) × U (1) × U (1) (17.1)66 → , + 1 , + 24 , (17.2)+ 5 , + 5 − , + 5 − , + 5 − , − + 10 , + 10 − , (17.3) E ⊃ SU (5) × U (1) a × U (1) b (17.4)78 → , + 1 , + 1 − , − + 1 , + 24 , (17.5)+ 5 − , + 5 , − + 10 − , − + 10 , + 10 , + 10 − , . (17.6)Consider first the fields associated with the Cartan of SO (12). Labeling the localCartan generators as t , t , we conclude that a six-dimensional field in the 5 localizeson the matter curve ( t + t = 0) and another field in the 5 localizes along ( t − t = 0),while a 10 localizes on the matter curve ( t = 0). Similar considerations apply for E ,from which we conclude that a six-dimensional field in the 5 localizes on the mattercurve ( − t a + t b = 0), while distinct six-dimensional 10’s localize on the matter curves( t a + 3 t b = 0) and ( t a = 0). The gauge singlets of SU (5) localize on curves whichonly intersect S at a discrete set of points. To generate naturally suppressed µ termsand light Dirac masses for the neutrinos, we also consider local enhancements to SU (7).For illustrative purposes, we first present an example which we shall refer to as“Model I” which exhibits the correct matter spectrum of the MSSM at low energies,but which also contains unrealistic interaction terms. Indeed, in this model the thirdgeneration is not hierarchically heavier than the two lighter generations. Moreover,the neutrinos of the Standard Model are exactly massless. Finally, the model con-tains superpotential terms which lead to rapid proton decay. After explaining theprimary features of this model, we next present a more refined example of admissiblematter curves which rectifies all of the above issues.As a first example, consider a model with fractional line bundle L = O S ( E − ) / and matter content localized on the following choice of matter curves:Model I Curve Class g Σ L Σ L ′ n Σ × (cid:0) H + 5 H (cid:1) Σ H − K S O Σ H ( p − p ) / O Σ H ( p − p ) − / × M Σ (1) M E O Σ (1) M O Σ (1) M ( − × M Σ (2) M H − E − E O Σ (2) M O Σ (2) M (2)1 × M Σ (3) M E O Σ (3) M O Σ (3) M (1) (17.7)where p and p denote two divisors on Σ H which are not linearly equivalent andwe have indicated how L restricts on each matter curve as well as the gauge bundlecontent of each GUT multiplet due to the restriction of the line bundle L ′ on S ′ tothe various matter curves. By construction, we find that a vector-like pair of Higgsdoublets localizes on Σ H . The degree of the line bundles on each of the chiral mattercurves has been chosen to reproduce the correct multiplicity in the MSSM.In terms of SU (5) GUT multiplets, the schematic form of the superpotential is: W SU (5) = λ dij · H × ( i ) M × ( j ) M + λ uj · H × M × ( j ) M (17.8)where i = 1 , , M all localized on a single mattercurve and j = 1 , M localized on the matter curveΣ (2) M . More generally, the superpotential may also contain interactions which involvegauge singlets which take the schematic form 1 × ×
5. Such interactions can thenlead to a µ term for the Higgs and a Dirac mass term for the neutrinos.As the above example demonstrates, there are potentially many admissible localmodels of this type which can all yield the matter content of the MSSM. Althoughthis model possesses non-trivial interaction terms, it is unclear whether these termsare consistent with constraints from low energy physics. As argued in subsection14.2, when no curves self-intersect or pinch inside of S , the corresponding Yukawacouplings do not produce the correct hierarchy in quark masses. Moreover, asexplained in Section 12, because H u and H d localize on the same matter curve, liftingthe Higgs triplets via fluxes can still induce quartic terms in the superpotential of theform QQQL/M KK with order one coefficients. Finally, in addition to an incorrecthierarchy for the quarks, the neutrinos are exactly massless in this model.106e now present a different configuration of matter curves which resolves all ofthe problems mentioned above. To this end, we require that at least one generationlocalize on a self-intersecting curve. For simplicity, we place all three generationsof 10 M ’s on a self-intersecting P and all three generations of 5 M ’s on a smooth P which does not self-intersect. With the same choice of L = O S ( E − E ) / as in theprevious example, the matter content, line bundle assignments and effective class ofeach matter curve are:Model II Curve Class g Σ L Σ L ′ n Σ × H Σ ( u ) H H − E − E O Σ ( u ) H (1) / O Σ ( u ) H (1) / × H Σ ( d ) H H − E − E O Σ ( d ) H ( − / O Σ ( d ) H ( − / × M Σ (1) M (pinched) 2 H − E − E O Σ (1) M O Σ (1) M (3)3 × M Σ (2) M H O Σ (2) M O Σ (2) M (3) .(17.9)See figure 7 for a depiction of the various matter curves in this model. In computingthe multiplicities on the self-intersecting curve we have neglected all subtleties whichcould occur based on viewing this curve as a pinched genus one curve because theflux data from the non-compact brane is a free discrete parameter which we canalways tune to give the correct number of generations. The superpotential now takesthe form: W SU (5) = λ dij · H × ( i ) M × ( j ) M + λ uij · H × ( i ) M × ( j ) M (17.10)+ ρ ia repel · H × ( i ) M × N ( a ) R + λ repel · Φ × H × H (17.11)where in the above, the intersection between Σ ( u ) H and Σ (2) M leads to a two-fold en-hancement in rank to an SU (7) singularity so that the singlet N ( a ) R may be identifiedwith the right-handed neutrinos and the vev of Φ determines the supersymmetric µ term. In this model, the neutrino masses are purely of Dirac type. As explainedin Section 15, these gauge singlet wave functions can generate an exponential sup-pression of the expected type. Finally, as explained in greater detail in Section 12,because the H u and H d fields localize on distinct matter curves, the operator QQQL is automatically suppressed by a phenomenologically acceptable amount.107
U(5)H d H uM
3 10 5 5 N
H M R
5 5 10
H M M
H H du µ
5 10 10
M MH
3 5 M Figure 7: Depiction of the various matter curves in the SU (5) model referred to as“Model II”. In this case, all three generations in the 10 of SU (5) localize on onecurve and three generations in the 5 localize on another curve. The Higgs up anddown curves localize on distinct matter curves and intersect at a point in S . Thecontributions to the superpotential from the intersection of various matter curves isalso indicated. 108 In the previous sections we have presented many potential ingredients for buildingmodels based on G S = SU (5). This is partially due to the analysis of subsection10.2 which shows that for G S = SO (10), direct breaking to G std via internal fluxeswill always as a byproduct generate exotic matter fields. For surfaces of generaltype, a partial breaking to SU (5) × U (1) would not present a serious obstructionbecause after breaking to a four-dimensional GUT group, the remaining breakingcan proceed when an adjoint-valued field develops a suitable vev. For del Pezzomodels, a similar mechanism exists for flipped GUT models.We now recall the primary features of four-dimensional flipped SU (5) GUT mod-els [30, 67, 93]. The gauge group of flipped SU (5) is SU (5) × U (1), which naturallyembeds in SO (10). Indeed, the chiral matter content of the Standard Model is givenby the flipped SU (5) multiplets:Matter : 3 × (1 − + 5 + 10 − ) (18.1)MSSM Higgs : 1 × (5 + 5 − ) (18.2)GUT Higgs : 1 × (10 − + 10 ) (18.3)where U (1) Y of the MSSM corresponds to a linear combination of the U (1) generatorin SU (5) and the overall U (1) factor. Due to the fact that the U (1) hypercharge isgiven by a flipped embedding, the 5 contains the Higgs down of the MSSM, while the5 − contains the Higgs up. In addition to interaction terms which descend from the16 M × M × H in an SO (10) GUT, a flipped SU (5) model includes the interactionterms 5 × − × − and 5 − × × between the MSSM and GUT Higgs fields.These interaction terms descend from 16 h × h × H and 16 h × h × H in an SO (10) GUT. As explained in [30] there is a unique F- and D-flat direction alongwhich the GUT Higgs 10 − and 10 develop a vev. This vev simultaneously breaks SU (5) × U (1) to SU (3) × SU (2) × U (1) while also giving a large mass to the Higgstriplets of the 5 and 5 − . In order to emphasize the embedding in SO (10), we shallorganize all of the matter content in terms of representations of SO (10). Explicitly,109e have: SO (10) ⊃ SU (5) × U (1) (18.4)16 M = 1 − + 5 + 10 − (18.5)10 H = 5 + 5 − (18.6)Because the GUT Higgs fields 10 − and 10 do not fill out a complete SO (10) mul-tiplet, we shall refer to these fields as Π and Π, respectively.We now explain how in F-theory a higher dimensional SO (10) GUT can naturallybreak to a four-dimensional flipped SU (5) GUT. For concreteness, we considermodels based on the del Pezzo 8 surface. The adjoint representation of SO (10)decomposes into representations of SU (5) × U (1) as: SO (10) ⊃ SU (5) × U (1) (18.7)45 → + 24 + 10 + 10 − . (18.8)By inspection, the U (1) charge assignment of the 10 does not correspond to therepresentation content of any field in a flipped SU (5) model. We therefore requirethat the zero mode content of the theory must not contain any 10 ’s or 10 − ’s. Inthis case, the only gauge bundle configurations which do not contain any such exoticsare all of the form O S ( α ) / where α corresponds to a simple root of H ( S, Z ).So long as the instanton configuration breaks G S to a four-dimensional flippedGUT group with all matter fields in well-defined flipped GUT multiplets, we canavoid additional exotica in the low energy spectrum. For example, in breaking E to SO (10) × U (1), the adjoint decomposes as: E ⊃ SO (10) × U (1) (18.9)78 → + 45 + 16 − + 16 . (18.10)Further breaking SO (10) to SU (5) × U (1), if we again require that no zero modesdescend from the 45 of SO (10) × U (1), we will generically produce zero modes whichdescend from the 16 − and 16 . We note that in this case, the zero modes can stillorganize into complete flipped multiplets.110 SU (5) Model
We now present a hybrid model which partially unifies to a flipped SU (5) GUT asa four-dimensional model and then further unifies to a higher dimensional SO (10)GUT model. Because none of the matter fields of the flipped model descend fromthe adjoint representation of SO (10), all of the chiral matter content of the flipped SU (5) model must localize on matter curves. Hence, the SO (10) interaction term16 M × M × H must originate from the triple intersection of matter curves. Tothis end, we consider a geometry where the generic SO (10) singularity undergoes atwofold enhancement in rank to E and SO (14) singularities.Decomposing the adjoint representation of E with respect to the subgroup SO (10) × U (1) × U (1) yields: E ⊃ SO (10) × U (1) × U (1) (18.11)133 → , + 1 , + 1 , − + 1 , + 45 , (18.12)+ 10 , + 10 − , + 16 − , + 16 − , − + 16 , + 16 , − (18.13)so that six-dimensional hypermultiplets in the 16 localize on the two matter curves( − t + t = 0) and ( − t − t = 0) and a six-dimensional hypermultiplet in the 10localizes on the matter curve ( t = 0). By inspection, we see that a local enhancementto E can accommodate interaction terms of the form 16 × ×
10 and 16 × × SO (14) can accommodatean interaction term of the form 1 × × We now present a toy hybrid scenario which we refer to as the “Hybrid I” model.Some deficiencies with this example will be rectified in the “Hybrid II” model. The SO (10) GUT group breaks to SU (5) × U (1) with no bulk exotics when the gaugebundle configuration corresponds to the fractional line bundle L = O S ( E − E ) / .In the Hybrid I model, the matter curves and gauge bundle assignments for each In fact, in a previous version of this paper, these local U (1) charge assignments for the explicitflipped models considered were not properly taken into account. We thank J. Marsano, N. Saulinaand S. Sch¨afer-Nameki for bringing this error to our attention. g Σ L Σ L ′ n Σ × M Σ (1) M E O Σ (1) M O Σ (1) M (1)2 × M Σ (2) M H − E − E O Σ (2) M O Σ (2) M (2)1 × ( d ) H Σ ( d ) H H − E − E O Σ ( d ) H (1) / O Σ ( d ) H (1) / × ( u ) H Σ ( u ) H H − E − E O Σ ( u ) H ( − / O Σ ( u ) H ( − / × (Π + Π) Σ h (pinched) 3 H − E − E O Σ h ( p − p ) / O Σ h ( p − p ) / (18.14)with notation as in (17.7).By construction, we find one chiral generation of the MSSM localized on Σ (1) M with two generations localized on Σ (2) M . The matter curve Σ ( d ) H supports a zero modetransforming in the representation 5 ( d )2 which contains the Higgs down of a flippedGUT model, and Σ ( u ) H supports a single zero mode in the 5 ( u ) − . Finally, in additionto the matter content of the MSSM, we have also included a single vector-like pairof GUT Higgs fields Π and Π.Including terms up to quartic order, the resulting superpotential of the four-dimensional flipped SU (5) model is therefore: W SU (5) × U (1) = W Matter + W Higgs + W Quartic (18.15)where the interaction terms for the chiral matter are: W Matter = λ ui (5 ( u ) − × ( i )3 × (3) − + 5 ( u ) − × ( i ) − × ( i )3 ) (18.16)+ λ di (5 ( d )2 × ( i ) − × (3)3 + 5 ( d )2 × ( i ) − × (3) − ) (18.17)and i = 1 , W Higgs = λ repel · Φ × ( d )2 × ( u ) − + λ Π · ( d )2 × Π × Π + λ Π · ( u ) − × Π × Π. (18.18)The final term W Quartic originates from integrating out the heavy Kaluza-Klein modes112ssociated with the Higgs fields: W Quartic = c i M KK ( i ) − × Π × (3) − × Π. (18.19)In the above, the mass scale M KK is the overall Kaluza-Klein mass scale. In general,this can be slightly higher than the vev of the GUT Higgs fields. We note that whenΠ develops a vev which also lifts the Higgs triplets from the low energy spectrum, italso generates a large Majorana mass term for the right-handed neutrinos.Because the matter curves Σ (1) M and Σ (2) M do not self-intersect, the resulting modelhas two heavy generations. In contrast to the minimal SU (5) models considered pre-viously, the field-theoretic missing partner mechanism already lifts the Higgs tripletsand prevents the higher dimension QQQL operator from being generated. Moreover,the model already incorporates a natural seesaw mechanism.Before proceeding to a slightly more realistic model, we note that although itwould at first appear to be more economical to place the Higgs up and Higgs downon the same matter curve, this leads to certain undesirable consequences in the lowenergy theory. The reason is that the Higgs up and down fields would then beequally or oppositely charged under a common U (1) symmetry. This would eitherforbid the coupling 16 × ×
10 or 16 × ×
10 in the low energy theory. Theformer interaction is necessary for generating semi-realistic Yukawa couplings, whilethe latter is necessary for implementing doublet-triplet splitting using the missingpartner mechanism. In order to achieve both couplings, it appears necessary tolocalize these fields on different matter curves.A more realistic hierarchy in quark masses can be achieved when the chiral mattercurves self-intersect. As a small refinement on the above model, we take L = O S ( E − E ) / as before, while the matter curves and gauge bundle assignments for113ach curve are now:Hybrid II Curve Class g Σ L Σ L ′ n Σ × ( d ) H Σ ( d ) H H − E − E O Σ ( d ) H (1) / O Σ ( d ) H (1) / × ( u ) H Σ ( u ) H H − E − E O Σ ( u ) H ( − / O Σ ( u ) H ( − / × M Σ M (pinched) 3 H O Σ M O Σ M (3 p ′ )1 × (cid:0) Π + Π (cid:1) Σ h (pinched) 3 H − E − E O Σ h ( p − p ) / O Σ h ( p − p ) / (18.20)so that all three generations localize on the matter curve Σ M . See figure 8 for adepiction of the Hybrid II model. While the zero mode content of this case is thesame as the Hybrid I model, the self-intersection of the matter curves allows themodel to have one generation which is hierarchically heavier than the lighter twogenerations, much as in the second minimal SU (5) example of Section 17. Asidefrom this difference, the structure of the superpotential is quite similar to that givenby equation (18.15). Indeed, just as in the Hybrid I model, there exist higherdimension operators which can generate large Majorana mass terms for the right-handed neutrinos.
19 Numerology
Throughout this paper we have given numerical estimates of various quantities whichappear to be in rough agreement with observation. In this Section we demonstratethat for an appropriate choice of order one constants, many of the relations obtainedthroughout are in agreement with experimental observation. Our point here is notso much to show that we can match to the precise numerical values, but rather thatthe numbers we have obtained are not wildly different from the expected ranges.Indeed, although we shall typically evaluate all quantities at the GUT scale, in amore accurate analysis these quantities would of course have to be evolved underrenormalization group flow to low energies. In this regard, our order of magnitudeestimates will be somewhat naive, although we believe it still gives a reliable guidefor the ranges of energy scales involved in our models. Moreover, for concreteness,in this Section we focus on the case of the minimal SU (5) model.At the level of precision with which we can reliably estimate parameters, all of114 O(10) M
3 16
Π + Π
H H du µ M M H(u)
16 16 10 16 16 10
MM H(d) H(d)
Π ΠΠ Π H(u) H (u) H (d) Figure 8: Depiction of the various matter curves in the flipped SU (5) model referredto as “Hybrid II” in the text. The background instanton configuration breaks thebulk gauge group SO (10) to SU (5) × U (1). In this case, all three generationstransform in the 16 of SO (10) and localize on a single self-intersecting matter curve.The MSSM Higgs fields descend from two different 10’s of SO (10). The model alsocontains a single vector-like pair transforming in the 10 − and 10 +1 of SU (5) × U (1)which facilitates GUT group breaking and doublet-triplet splitting. These GUTHiggs fields descend from a six-dimensional hypermultiplet transforming in the 16 of SO (10). 115ur estimates depend on order one coefficients, the Planck mass M pl , the GUT scale M GUT , the Higgs up vev, and the value of the gauge coupling constants at the GUTscale, α GUT . Throughout, we use the following approximate values: M pl ∼ × GeV (19.1) M GUT ∼ × GeV (19.2) h H u i ∼
246 GeV (19.3) α GUT = g Y M ( M GUT )4 π ∼
125 . (19.4)In general, factors of 2 and π are typically beyond the level of precision which wecan reliably estimate.The above parameters appear geometrically as the length scale R S associatedwith the size of the del Pezzo, R B which is associated with the size of the threefoldbase, and R ⊥ which may be viewed as a local cutoff on the behavior of wave functionsin the model. These length scales are related by appropriate powers of the smallparameter: ε = M GUT α GUT M pl ∼ . × − . (19.5)The various length scales are then given by:1 R S = M GUT ∼ × GeV (19.6)1 R B = M GUT × ε / ∼ × GeV (19.7)1 R ⊥ = M GUT × ε γ ∼ × ± . GeV (19.8)where the parameter 1 / . γ . / B is homogeneous, to 1when B is given by a tubular geometry.We now collect and slightly expand on the estimates obtained throughout thispaper. We begin by discussing the mass scales associated with quarks. In this case,the masses of the quarks at the GUT scale are very roughly given by: m q ∼ α / GUT h H u i ∼
20 GeV. (19.9)116ote that the top quark mass is about a factor of 3 higher than this (taking intoaccount the RG flow), which suggests that perhaps the corresponding curves aresmaller by that factor to give the correct wave function normalization.We have also seen that matter fields which localize on curves normal to S inthe threefold base B can provide a natural mechanism for generating light neutrinomasses as well an exponentially suppressed µ term. As an intermediate case, we haveshown that right-handed neutrino wave functions which are attracted to the seven-brane can potentially realize a viable seesaw mechanism. Reproducing equation(15.38) for the convenience of the reader, the light neutrino mass in the seesawscenario is: m light ∼ α / GUT h H u i M GUT ε − γ ∼ × − ± . eV. (19.10)Gauge singlet wave functions can also exhibit more extreme behavior. Indeed,when the Higgs up and down fields localized on different matter curves which in-tersect, they interact with a gauge singlet wave function outside of S . When thiswave function is exponentially suppressed, the induced µ term is given by equation(15.33): µ ( c, γ ) ∼ M GUT × α / GUT ε γ exp (cid:16) − cε γ (cid:17) . (19.11)We find that when c and γ are order one numbers, this value can naturally fall nearthe electroweak scale. For example, we have: µ ( c = 1 / , γ = 1) ∼
140 GeV (19.12) µ ( c = 1 , γ = 0 . ∼
107 GeV. (19.13)In a scenario where the neutrinos are purely of Dirac type, an exponentially smallvalue can also be achieved when the gauge singlet wave function is exponentiallydamped near S . The Dirac mass is given by equation (15.39): m Dirac ( c, γ ) ∼ h H u i × α / GUT ε γ exp (cid:16) − cε γ (cid:17) . (19.14)As for the µ term, order one values of c and γ yield reasonable values for the masses.Indeed, as explained in subsection 15.5, when the exponential suppression factors are117dentical for the Dirac neutrino mass and µ term, we obtain the estimate: m Dirac ( c, γ ) ∼ µ ( c, γ ) h H u i M GUT × ε − γ ∼ . × − ± . eV (19.15)when µ ( c, γ ) ∼
100 GeV. We have also observed that a similar analysis of Yukawacouplings also applies in estimates of the messenger mass scales for gauge mediatedsupersymmetry breaking scenarios.
20 Conclusions
F-theory provides a natural setup for studying GUT models in string theory. Inthis paper we have adopted a bottom up approach to string phenomenology andhave found that it provides a surprisingly powerful constraint on low energy physics.One’s natural expectation is that there should be a great deal of flexibility in localmodels where issues pertaining to a globally consistent compactification can alwaysbe deferred to a later stage of analysis. This is indeed the case in models where asufficiently loose definition of “local data” is adopted so that gravity need not decou-ple, and we have given some examples along these lines. Strictly speaking, though,a local model is well-defined by local data when the model admits a limit where itis in principle possible to decouple the GUT scale from the Planck scale. Perhapssurprisingly, this qualitative condition endows these GUT models with considerablepredictive power.The main lesson we have learned is that the mere existence of a decoupling limitconstrains both the local geometry of the compactification as well as the type ofseven-brane which can wrap a compact surface in the local model. To realize aGUT model with no low energy exotics, the bulk gauge group of the seven-branecan only have rank four, five or six, and in order for a decoupling limit to even existin principle, the seven-brane must wrap a del Pezzo surface. Moreover, all of thevacua which descend at low energies to the MSSM in four dimensions all possess aninternal U (1) hypercharge flux on the del Pezzo which at least partially breaks theGUT group. For concreteness, in this paper we have primarily focused on the caseswhere the bulk gauge group in eight dimensions is SU (5) or SO (10).In the minimal SU (5) model, all of the matter content at low energies derives118rom the intersection of the GUT model seven-brane with additional non-compactseven-branes. We have explained how the fields which localize at such intersectionscan only transform in the 5, 10 or complex conjugate representations. Moreover,the interaction terms are all cubic in the matter fields because the superpotentialderives from the triple intersection of matter curves. Matter fields which are neutralunder the GUT group localize on matter curves which are orthogonal to the brane.When the gauge singlet is attracted to the seven-brane, the corresponding Yukawacouplings behave as though the gauge singlet had localized inside of S . On theother hand, when the gauge singlet wave function is repelled away from the seven-brane, this can yield a significant exponential suppression in the value of the Yukawacouplings on the order of exp( − c/ε γ ) where c and γ are order one positive numbersand ε ∼ α − GUT M GUT /M pl . In particular, vector-like pairs in such compactificationsdo not always develop masses on the order of M GUT . This runs counter to a coarseeffective field theory analysis which would otherwise suggest that such pairs shouldalways develop large masses. In fact, we have seen that this is consistent witha more refined effective field theory analysis because there are typically additionalglobal symmetries present in the low energy theory.The exponential suppression of such Yukawa couplings naturally solves the µ problem and also provides a natural mechanism for generating acceptably light neu-trino masses. The wave function for the right-handed neutrino is either attracted orrepelled away from the del Pezzo surface. In the repulsive case, the neutrino massterm is purely of Dirac type and is on the order of 0 . × − ± . eV. In the attractivecase, we find a natural implementation of a modified seesaw mechanism so that thelight neutrinos masses are 2 × − ± . eV and the Majorana mass is ∼ × ± . GeV, which is naturally smaller than the simplest GUT seesaw models.The combination of non-trivial hypercharge flux in the internal dimensions andthe existence of additional fluxes derived from the transversally intersecting seven-branes alleviates a number of problems which plague four-dimensional supersym-metric GUT models. The doublet-triplet splitting problem reduces to the conditionthat the hypercharge flux and flux from the other seven-branes both pierce the Higgsmatter curves, while the net hypercharge flux vanishes on curves which support fullGUT multiplets.The internal U (1) hypercharge flux also provides a qualitative explanation forwhy the b − τ GUT mass relation approximately matches with observation while119he lighter two generations at best obey distorted versions of this relation. Thisis in a sense the remnant of the mechanism that solves the doublet-triplet splittingproblem. Even though the net hypercharge flux vanishes on a matter curve whichsupports a complete GUT multiplet, the field strength is not identically zero. In thisway, the GUT multiplet wave functions experience an Aharanov-Bohm effect whichincreasingly distorts the GUT mass relations as the mass of the GUT multipletdecreases. In fact, this mechanism requires that the internal hypercharge gaugefield be non-trivial.This flux will also typically generate a threshold correction to the unification ofthe gauge couplings. While there are potentially many other such threshold correc-tions due to Kaluza-Klein modes, it would clearly be of interest to see whether atleast some of these corrections can be reliably estimated in our setup.The geometry of the compactification can also prevent the proton from decayingtoo rapidly. Cubic terms in the superpotential are typically excluded in a bottomup approach by requiring that the theory is invariant under R-parity. We havefound two ways that the geometry can forbid the same interaction terms which R-parity removes. In one case, R-parity corresponds to a suitable Z symmetry in thegeometry of the Calabi-Yau fourfold. At a topological level, the absence of R-parityviolating cubic interaction terms corresponds to a technically natural restriction onwhich matter curves intersect. In the scenario where R-parity descends from a Z group action on the Calabi-Yau, the hypercharge flux and the Higgs matter curvesare invariant under this group action while the matter curves are odd. Due tothe Z symmetry, the net hypercharge flux must vanish on matter curves whichare odd under this group action. Hence, this automatically forces the localizedmatter to organize in complete GUT multiplets. Note that this symmetry alsopermits a non-vanishing hypercharge flux on the Higgs curves, which is consistentwith our solution to the doublet-triplet splitting problem. At higher order in theeffective superpotential, the topological condition determining which curves intersectalso forbids potentially dangerous baryon number violating quartic operators in thesuperpotential. Indeed, placing the Higgs up and down fields on distinct mattercurves equips the matter fields with additional global symmetries which can forbidsuch operators.We have also shown how the geometry of the matter curves translates in thelow energy effective theory into non-trivial structure in the Yukawa couplings. The120oarsest features of textures follow from the discrete data determining how mattercurves intersect inside the seven-brane so that texture zeroes are generically present.We have also presented some speculations on potential ways that additional structurein the Yukawa couplings could arise from a geometrical realization of the Froggatt-Nielsen mechanism, or through an interpretation of the discrete automorphism groupof a del Pezzo surface as a flavor group symmetry.Communicating supersymmetry breaking is also straightforward in this setup.Indeed, we have shown that the geometry of del Pezzo surfaces can easily accommo-date vector-like pairs of GUT multiplets localized on isolated matter curves. Thesevector-like pairs can then serve as the messenger fields in gauge mediated supersym-metry breaking. We have presented different scenarios showing the flexibility of thisapproach. Depending on the case at hand, the messenger masses can range from nearthe GUT scale, to energy scales which are significantly lower. Moreover, becausewe have an independent mechanism for naturally suppressing the µ term, this classof models preserves the best features of gauge mediation models while avoiding thenotoriously difficult issue of generating µ and Bµ at around the electroweak scale.It is perhaps surprising that a few key ideas seem to resolve many problemssimultaneously. Indeed, the overall economy in these ingredients lends substantialcredence to the basic framework. On the other hand, it is also clear that we have byno means exhausted the potential avenues of investigation. A more systematic studyof textures and choices of matter curves, as well as the geometric underpinning ofthe corresponding Calabi-Yau fourfold are all issues which deserve further attention.In addition, the communication of supersymmetry breaking is simple enough in oursetup that it could potentially lead to observable predictions at the LHC. It wouldclearly be of interest to study such a scenario further. Acknowledgements
We thank K.S. Babu, V. Bouchard, F. Denef, A.L. Fitzpatrick, J. Marsano, S. Raby,N. Saulina, S. Sch¨afer-Nameki, P. Svrˇcek, A. Tomasiello, M. Wijnholt, E. Wittenand S.-T. Yau for helpful discussions. The work of the authors is supported inpart by NSF grants PHY-0244821 and DMS-0244464. The research of JJH is alsosupported by an NSF Graduate Fellowship.121 ppendicesA Gauge Theory Anomalies and Seven-Branes
In this Appendix we further elaborate on the geometric condition for the low energyspectrum to be free of gauge theory anomalies. First recall the well-studied caseof perturbatively realized gauge theories obtained as the low energy limit of D-brane probes of non-compact Calabi-Yau singularities. The condition that all gaugetheory anomalies must cancel is equivalent to the requirement that in a consistentbound state of D3-, D5- and D7-branes, the total RR flux measured over a compactcycle must vanish [94]. Even in a non-compact Calabi-Yau threefold given by thetotal space O ( K S ) → S with S a K¨ahler surface, the theory of a stack of D7-branes wrapping S is inconsistent because the self-intersection of the divisor S inthe Calabi-Yau threefold is a compact Riemann surface. In a globally consistentmodel, additional O7-planes must be introduced to cancel the corresponding RRtadpole. Indeed, a consistent compactification of F-theory on an elliptically fiberedCalabi-Yau fourfold will automatically contain similar contributions so that the netmonodromy around all seven-branes is trivial.Next consider the potential contribution from D5-branes to a candidate boundstate. Letting [Σ D ] denote the total homology class of D5-branes wrapping compacttwo-cycles in H ( S, Z ), the resulting theory is consistent provided:[Σ D ] · K S = 0. (A.1)There is no analogous condition for D3-branes in a non-compact model because theflux lines can escape to infinity in the non-compact model.In this Appendix we consider more general intersecting seven-brane configurationswith chiral matter induced from a non-trivial field strength. Using the fact thata low energy theory must be free of non-abelian gauge anomalies, we determinethe geometric analogue of equation (A.1) for intersecting A × A and D × A braneconfigurations in a broader class of F-theory compactifications. We also present anexample of anomaly cancelation for an E exceptional brane theory.122 .1 A × A Anomalies
We now consider seven-branes wrapping two K¨ahler surfaces S and S ′ such that thegauge group of the respective seven-branes is G S = SU ( N ) and G S ′ = SU ( N ′ ) witha six-dimensional bifundamental localized along a matter curve Σ = S ∩ S ′ . Becauseonly instanton configurations with an overall U (1) factor can induce chirality in thebulk and on matter curves, it is enough to consider instanton configurations in S and S ′ taking values in U (1) n and U (1) n ′ for some n ≤ N − n ′ ≤ N ′ − SU ( N ) decomposes into non-abeliansubgroup factors SU ( N ) , ..., SU ( N n ). Similar conventions will also hold for thedecomposition of the gauge group SU ( N ′ ). Letting −→ q denote the charge of a repre-sentation under the U (1) n − subgroup, the fundamental and adjoint representationdecompose as: SU ( N + ... + N n ) ⊃ SU ( N ) × ... × SU ( N n ) × U (1) n − (A.2) N → ( N ) −→ q ⊕ .... ⊕ ( N n ) −→ q n (A.3) A N → ( A N ) −→ q ⊕ .... ⊕ ( A N n ) −→ q n (A.4) ⊕ (cid:20) ⊕ i We now consider seven-branes wrapping two K¨ahler surfaces S and S ′ such that thegauge group of the respective seven-branes is G S = SU ( N ) and G S ′ = SO (2 R + 2 M )with six-dimensional matter fields localized along the curve Σ = S ∩ S ′ . Decomposing SO (2 N + 2 R + 2 M ) ⊃ SU ( N ) × SO (2 R + 2 M ) × U (1), the six-dimensional fieldslocalized on Σ now transform in the representation ( A N , ⊕ ( N, R ) of SU ( N ) × SO (2 R +2 M ). As before, it is enough to treat instanton configurations taking valuesin the subgroups U (1) n ⊂ SU ( N ) and U (1) t ⊂ SO (2 R ). In order to simplify thecombinatorics associated with breaking patterns of the SO gauge group factor, weconfine our analysis to the breaking pattern SO (2 R + 2 M ) ⊃ SO (2 R ) × SU ( M ) × U (1). The fundamental and adjoint representations of SO (2 R + 2 M ) decomposeinto the commutant subgroup of U (1) as: SO (2 R + 2 M ) ⊃ SO (2 R ) × SU ( M ) × U (1) (A.16)2 R → (2 R ) ⊕ (cid:0) ( M ) p ⊕ ( M i ) − p (cid:1) (A.17) ad ( SO (2 R )) → ⊕ ad ( SO (2 R i )) ⊕ ad ( SU ( M i )) (A.18) ⊕ ( A M ) p ⊕ ( A M ) − p ⊕ (2 R, M ) p ⊕ (2 R, M ) − p . (A.19)Consider first non-abelian anomalies associated to the gauge group factor SU ( N i ).In this case, we recall that in a normalization of group generators where the fun-damental has anomaly coefficient +1, the two index anti-symmetric representationhas anomaly coefficient N i − 4. Repeating a similar analysis to that given in theprevious Section, the total anomaly coefficient for the non-abelian group SU ( N i ) is: a i = − N n − X k =1 π k ( −→ q i ) d k + (2 R + 2 M ) n − X k =1 π k ( −→ q i ) d k | Σ (A.20)+ ( N i − n − X k =1 π k (2 −→ q i ) d k | Σ + X j = i N j n − X k =1 π k ( −→ q i + −→ q j ) d k | Σ ! (A.21)= − N n − X k =1 π k ( −→ q i ) d k + (2 R + 2 M ) n − X k =1 π k ( −→ q i ) d k | Σ − n − X k =1 π k ( −→ q i ) d k | Σ (A.22)= 2 Z S c ( L π ( −→ q i )1 ⊗ ... ⊗ L π n − ( −→ q i ) n − ) c (cid:0) O S ( K S ) N ⊗ O S (Σ) R + M − (cid:1) . (A.23)125omparing equations (A.15) and (A.23), the shift R + M → R + M − O SU ( M ) factor. In this case, thetotal anomaly coefficient for the non-abelian group SU ( M ) is: b i = − p ( M − d ′ − Rpd ′ + 2 pN d ′ Σ (A.24)= 2 Z S ′ c ( L ′ p ) c (cid:0) O S ′ (Σ) N ⊗ O S ′ ( K S ′ ) R + M − (cid:1) . (A.25)Proceeding by induction, it now follows that a similar result also holds for the moregeneral breaking pattern where each SU ( M ) and SO (2 R ) factor decomposes further. A.3 E Anomalies The analysis of the previous subsections demonstrates that for A - and D - type seven-branes, the geometric condition for anomaly cancelation in the four-dimensional ef-fective theory relates the total matter content in the bulk with that localized onmatter curves. We now determine the analogous condition for a seven-brane withgauge group G S = E and M copies of the 56 localized on a curve Σ. We consider a U (1) gauge field configuration which breaks E to SU (7) × U (1). The representationcontent of E decomposes as: E ⊃ SU (8) ⊃ SU (7) × U (1) (A.26)56 → − + 7 + 21 + 21 − (A.27)133 → + 7 + 7 − + 48 + 35 + 35 − (A.28)where the 21, 35 and 35 denote the two, three and four index anti-symmetric repre-sentations of SU (7). It now follows that the chiral matter content derived from S and Σ is: = − c ( S ) · c ( L ) (A.29) − c ( S ) · c ( L − ) (A.30) − = M deg L − | Σ (A.31) M deg L | Σ . (A.32)126o compute the anomaly of the SU (7) theory, we first recall that the anomalycoefficient for the i -index anti-symmetric representation A ( i ) k of SU ( n ) in 2( k − A (2) k = n − k − (A.33) A (3) k = 12 n − n (2 k + 1) + 3 k − (A.34) A (4) k = 112 (2 n − n (2 k + 2) + n (4 × k + 3 × k + 4) − × k ) (A.35)so that in four dimensions, the anomaly coefficients of the SU (7) theory are A (2) k = 3 ,A (3) k = 2, A (4) k = − 2. Returning to equations (A.29)-(A.32), we note that thecontribution to the total anomaly from S and Σ separately cancel in this particularcase so that we do not deduce an analogue of equation (A.1). B Hypersurfaces in P n hypersurfaces H n in P . Furtherdetails can be found for example in [96]. Letting H denote the hyperplane class of P , the total Chern class of H n is given by the adjunction formula: c ( H n ) = c ( P ) c ( N H n / P ) = 1 + (4 − n ) H + (6 − n + n ) H . (B.1)It thus follows that the Euler character e ( H n ), holomorphic Euler characteristic χ ( O H n ) and signature τ ( H n ) are: e ( H n ) = Z H n c ( H n ) = Z P n (6 − n + n ) H = n − n + 6 n (B.2) χ ( O H n ) = Z H n c ( H n ) + c ( H n )12 = 16 ( n − n + 11 n ) (B.3) τ ( H n ) = Z H n c ( H n ) − c ( H n )3 = − 13 ( n − n ). (B.4)127e next determine the Hodge numbers of H n . Using the Lefschetz hyperplanetheorem, h , ( P ) = 0 implies h , ( H n ) = 0. Moreover, because e ( H n ) = 2 + 2 h , + h , and χ ( O H n ) = 1 − h , + h , = 1 + h , , equations (B.2)-(B.4) imply: h , ( H n ) = 13 (2 n − n + 7 n ) (B.5) h , ( H n ) = 16 ( n − n + 11 n ) − b ( H n ) = n − n + 6 n − 2. (B.7)The last expression determines the dimension of H ( H n , Z ) as a lattice over theintegers. It follows from Poincar´e duality that when equipped with the intersectionpairing of the geometry, this lattice is self-dual. Moreover, returning to equation(B.1), reduction of c ( H n ) mod 2 implies that H n is spin when n is even. This inturn implies that the lattice H ( H n , Z ) is even (resp. odd) for n even (resp. odd).Because the signature and dimension uniquely determine a lattice with indefinitesignature, we conclude that the lattice is of the general form: H ( H n , Z ) ≃ (+1) ⊕ ( b + τ ) / ⊕ ( − ⊕ ( b − τ ) / ( n odd) (B.8) H ( H n , Z ) ≃ ( − E ) τ/ ⊕ U ⊕ ( b − τ ) / ( n even) (B.9)where − E is minus the Cartan matrix for E and U is the “hyperbolic element”with entries specified by the Pauli matrix σ x . The canonical class has self intersectionnumber: K H n · K H n = Z H n c ( H n ) = n ( n − . (B.10)For many purposes, it is of practical use to have a large number of contractiblerational curves inside of a given surface which can serve as matter curves for a givenmodel. We note, however, that general results from the mathematics literature[97,98] demonstrate that for a generic hypersurface of degree at least five, the minimalgenus of a curve is at least two. Indeed, typically a given homology class onlycorresponds to a holomorphic curve for a specific choice of complex structure. Toavoid such subtleties, we consider the blowup of a degree n hypersurface at k points, B k H n . While the value of h , remains invariant under this process, the canonical This follows from Wu’s theorem and the fact that H n is simply connected. K B k H n = K H n + E + ... + E k . (B.11)where the E i denote the effective classes associated with blown up rational curves. C Classification of Breaking Patterns In this Appendix we classify all possible breaking patterns via instantons for a the-ory defined by a seven-brane filling R , × S with bulk gauge group G S such thatthe resulting spectrum can in principle contain the matter content of the StandardModel. While breaking patterns for GUT groups is certainly a well-studied topicin the phenomenology literature, as far as we are aware, this question has not beenstudied from the perspective of F-theory. Indeed, although much of our analysis inthis paper has focussed on the cases where the bulk gauge group is SU (5) or SO (10),it seems of use for future potential efforts in this direction to catalogue a broaderclass of candidate breaking patterns which could in principle arise from compacti-fications of F-theory. We note that by appealing to gauge invariance and certainbasic phenomenological requirements, a partial classification of candidate breakingpatterns which can appear in string theory has been given in [99].Throughout our analysis, we shall assume that our model is generic in the sensethat along complex codimension one and two subspaces, the rank of the singularitytype can enhance by one or two. While in this paper we have focussed on a minimalclass of models where the bulk gauge group is G S = SU (5) or SO (10), there areadditional possibilities at higher rank. For example, in higher rank cases it maybe possible to allow some of the matter fields of the MSSM to originate from bulkzero modes. We now proceed to an analysis of all possible breaking patterns viainstantons which can accommodate the matter content of the MSSM. The relevantgroup theory material on the decomposition of various irreducible representationsmay be found in [100, 101].In keeping with our general philosophy, we shall also assume that the group cor-responding to the rank two enhancement in singularity type is a subgroup of E . Forthis reason, the rank of the singularity type can be at most six. Moreover, becausethe Standard Model gauge group has rank four, it is enough to classify breaking pat-129erns associated with singularities of rank four, five and six. The relevant ADE -typeof the singularities are therefore:Rank 4: A , D (C.1)Rank 5: A , D (C.2)Rank 6: A , D , E . (C.3)The singularity type does not fully determine the gauge group G S . When thecollapsed cycles of the singularity type are permuted under a monodromy in thefiber direction, the resulting gauge group is given by the quotient of the originalsimply laced group by an outer automorphism. In this way, we can also obtain allnon-simply laced groups such as SO (2 n + 1), U Sp (2 n ), F and G . In what followswe adopt the convention U Sp (2) ≃ SU (2). It therefore follows that we must analyzethe breaking patterns for the following possibilities:Rank 4: SU (5), SO (8), SO (9), F (C.4)Rank 5: SU (6), SO (10), SO (11) (C.5)Rank 6: SU (7), SO (12), E . (C.6)Note in particular that the bulk gauge group is never of U Sp type. There are ingeneral many possible ways in which the Standard Model gauge group can embed inthe above gauge groups. To classify admissible breaking patterns to the StandardModel gauge group, we shall require that all of the matter content of the StandardModel must be present. While much of our analysis will hold for non-supersymmetrictheories as well, we shall typically focus on the field content and interactions of theMSSM. In terms of the gauge group SU (3) × SU (2) × U (1), the representationcontent of the fields of the MSSM are: Q U D H d , L E H u (3 , (3 , − (3 , (1 , − (1 , (1 , . (C.7)In addition, any realistic model must allow the three superpotential terms: W ⊃ QU H u + QDH d + ELH d . (C.8)130tarting from representations which descend from the decomposition of the ad-joint representation of E , our strategy will be to rule out as many possible breakingpatterns as possible because the representation content is incorrect, or because gaugeinvariance in the parent theory forbids a required superpotential term. For SO gaugegroups, we assume the matter organizes into the fundamental, spinor or adjoint rep-resentations. For SU gauge groups, we assume that in addition to the adjointrepresentation, the matter organizes into one, two or three index anti-symmetricrepresentations.To classify the possible breaking patterns of a given bulk gauge group G S , wefirst list all maximal subgroups. Next, we determine all maximal subgroups ofeach such subgroup and proceed iteratively until we arrive at the Standard Modelgauge group. We note that even for a unique nested sequence of subgroups, theremay be several distinct subgroups whose commutant contains the Standard Modelgauge group. The classification of these possible subgroups is aided by the factthat the gauge group of the Standard Model has rank four so that the correspondinginstanton configuration can only take values in a rank one or two subgroup of a givenbulk gauge group G S .Although they cannot serve as a bulk gauge group, it is also convenient to listall maximal subgroups of some common lower rank groups which appear frequently.131he maximal subgroups of SO (7), SU (4), U Sp (6), U Sp (4) and G are: SO (7) ⊃ SU (4) (C.9) SO (7) ⊃ SU (2) × SU (2) × SU (2) (C.10) SO (7) ⊃ U Sp (4) × U (1) (C.11) SO (7) ⊃ G (C.12) SU (4) ⊃ SU (3) × U (1) (C.13) SU (4) ⊃ SU (2) × SU (2) × U (1) (C.14) SU (4) ⊃ U Sp (4) (C.15) SU (4) ⊃ SU (2) × SU (2) (C.16) U Sp (6) ⊃ SU (3) × U (1) (C.17) U Sp (6) ⊃ SU (2) × U Sp (4) (C.18) U Sp (6) ⊃ SU (2) (C.19) U Sp (6) ⊃ SU (2) × SU (2) (C.20) U Sp (4) ⊃ SU (2) × SU (2) (C.21) U Sp (4) ⊃ SU (2) × U (1) (C.22) U Sp (4) ⊃ SU (2) (C.23) G ⊃ SU (3) (C.24) G ⊃ SU (2) × SU (2) (C.25) G ⊃ SU (2). (C.26)In the remainder of this Appendix, we classify possible breaking patterns viainstantons of the bulk gauge group. To further specify the order of breaking in anested sequence of subgroups, we shall sometimes enclose separate subgroup factorsin square brackets. C.1 Rank Four We now classify all breaking patterns of rank four groups. Although SU (5) isthe only group of line (C.4) which contains complex representations, for our higherdimensional theories, it is a priori possible that a suitable U (1) field strength ineither the compact or non-compact directions of an intersecting seven-brane theory132an induce a net chirality in the resulting gauge group. In the rank four case welist all maximal subgroups even if they do not contain the Standard Model gaugegroup. This is done because for the higher rank cases, such breaking patterns maybecome available. In the rank four case, we find that only G S = SU (5) is a viablepossibility. C.1.1 SU (5)There is a single maximal subgroup of SU (5) which contains the Standard Modelgauge group. Indeed, the representation content is given by the Georgi-Glashowmodel: SU (5) ⊃ SU (3) C × SU (2) L × U (1) Y ≡ G std (C.27)5 → (1 , + (3 , − (C.28)10 → (1 , + (3 , − + (3 , (C.29)24 → (1 , + (1 , + (3 , − + (3 , + (8 , . (C.30)By turning on an instanton in U (1) Y , we break to the desired gauge group. C.1.2 SO (8)We now proceed to the case of SO (8). The maximal subgroups of SO (8) are [100]: SO (8) ⊃ SU (2) × SU (2) × SU (2) × SU (2) (C.31) SO (8) ⊃ SU (4) × U (1) (C.32) SO (8) ⊃ SU (3) (C.33) SO (8) ⊃ SO (7) (C.34) SO (8) ⊃ SU (2) × U Sp (4). (C.35)Returning to lines (C.9-C.26), it follows that there does not exist a breaking patternwhich yields G std . 133 .1.3 SO (9)The maximal subgroups of SO (9) are: SO (9) ⊃ SO (8) (C.36) SO (9) ⊃ SU (2) × SU (2) × U Sp (4) (C.37) SO (9) ⊃ SU (2) × SU (4) (C.38) SO (9) ⊃ SU (2) (C.39) SO (9) ⊃ SU (2) × SU (2). (C.40)Of the above possibilities, only line (C.38) contains G std . Breaking to G std via a U (1) instanton yields: SO (9) ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × [ U (1)]] (C.41)9 → (3 , + (1 , + (1 , − (C.42)16 → (2 , + (2 , − + (2 , − + (2 , +1 (C.43)36 → (3 , + (1 , + (1 , − + (1 , + (1 , + (3 , + (3 , − . (C.44)By inspection, all singlets of SU (2) × SU (3) are also neutral under the U (1) factor.It thus follows that SO (9) is ruled out as a candidate. C.1.4 F The maximal subgroups of F are: F ⊃ SO (9) (C.45) F ⊃ SU (3) × SU (3) (C.46) F ⊃ SU (2) × U Sp (6) (C.47) F ⊃ SU (2) (C.48) F ⊃ SU (2) × G (C.49)the first case is excluded by the previous analysis of SO (9), leaving only lines (C.46)and (C.47). 134irst consider the breaking pattern of (C.46): F ⊃ SU (3) × SU (3) (C.50)26 → (8 , 1) + (3 , 3) + (3 , 3) (C.51)52 → (8 , 1) + (1 , 8) + (6 , 3) + (6 , SU (3) ⊃ SU (2) × U (1) via a U (1) instanton, we note thatall resulting SU (3) × SU (2) singlets are also neutral under U (1). We thereforeconclude that the breaking pattern of line (C.46) is also excluded.Next consider the remaining breaking pattern of (C.47) which can descend to theStandard Model gauge group: F ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × U (1)] (C.53)26 → (2 , + (2 , − + (2 , − + (2 , + (2 , (C.54)52 → (3 , + (1 , + (1 , + (1 , − + (1 , (C.55)+ (2 , + (2 , − + (2 , − + (2 , . (C.56)As before, the resulting singlets of the non-abelian factor are also neutral under the U (1) factor. Summarizing, we find that the only available rank four bulk gaugegroup which can contain the Standard Model is SU (5). C.2 Rank Five We now proceed to rank five bulk gauge groups. While it is in principle possiblethat an SU (2) instanton configuration could produce a consistent breaking patternto the particle content of the Standard Model, we find that in all cases, the relevantbreaking pattern is again always an instanton configuration with structure group U (1) or U (1) × U (1). C.2.1 SU (6)We assume that the matter content organizes into the representations 6, 15, 20 and35 of SU (6), as well as their dual representations. The maximal subgroups of SU (6)135re: SU (6) ⊃ SU (5) × U (1) (C.57) SU (6) ⊃ SU (2) × SU (4) × U (1) (C.58) SU (6) ⊃ SU (3) × SU (3) × U (1) (C.59) SU (6) ⊃ SU (3) (C.60) SU (6) ⊃ SU (4) (C.61) SU (6) ⊃ U Sp (6) (C.62) SU (6) ⊃ SU (2) × SU (3) (C.63)of which only the first three contain G std . By inspection, it now follows thatfor n ≥ 2, an SU ( n ) instanton will break too much of the gauge group to pre-serve G std . Moreover, it follows from lines (C.57)-(C.59) that up to linear com-binations of the U (1) charge for the other breaking patterns, it is enough to ana-lyze the U (1) instanton configuration which breaks SU (6) via the nested sequence SU (6) ⊃ SU (5) × U (1) ⊃ G std × U (1). Restricting to U (1) valued instanton con-figurations, the decomposition of the one two and three index anti-symmetric andadjoint representations of SU (6) are: SU (6) ⊃ SU (5) × [ U (1)] ⊃ [ SU (3) × SU (2) × [ U (1)]] × [ U (1)] (C.64)6 → (1 , , + (3 , − , − + (1 , , − (C.65)15 → (1 , , − + (3 , − , − + (1 , , (C.66)+ (3 , − , − + (3 , , (C.67)20 → (1 , , − + (3 , − , − + (3 , , − (C.68)+ (1 , − , + (3 , , + (3 , − , (C.69)35 → (1 , + (1 , , + (3 , − , (C.70)+ (1 , − , − + (3 , , − + (1 , , + (1 , , (C.71)+ (3 , − , + (3 , , + (8 , , . (C.72)The above decomposition illustrates the fact that there are a priori different ways inwhich the representation content of the MSSM can be packaged into higher dimen-sional representations.We now determine all possible choices consistent with obtaining the correct spec-136rum and interaction terms. We first require that at least one linear combinationof the U (1) charges may be identified with U (1) Y of the Standard Model. Labelingthe U (1) charges as a and b , this implies that the charges of the MSSM fields mustsatisfy the relations: E : 5 b = ± a + 2 b = ± a − b = ± Q : a + 2 b = ± a − b = ± − a = ± U : 2 a + b = − a + 4 b = − − a − b = − a − b = − D : 2 a + b = 2 or 2 a + 4 b = 2 or − a − b = 2 or 2 a − b = 2 (C.76) H d , L : 3 a − b = ± a + 6 b = ± H u : 3 a − b = ± a + 6 b = ± 3. (C.78)First suppose that the E -relation 5 b = ± a are: Q = ⇒ ± a = 175 or 75 or 235 or 135 or 15 (C.79) L = ⇒ ± a = 75 or 35 or 175 (C.80) D = ⇒ a = 85 or 25 or − 75 or 175 or − 75 or 45 or 235 or − 135 (C.81) U = ⇒ a = − 135 or − 75 or − 225 or 25 or 110 or 1910 or 85 or − 285 (C.82)so that the only common solution to all of the above conditions requires a = − / U con-dition requires a = − / b = − / 5, the Q condition requires a = − / b = +6 / E -relation 6 a + 2 b = ± b is nowdetermined by the relations: Q = ⇒ b = ± 65 or ± 35 or ± 185 or ± 125 or 0 (C.83) U = ⇒ b = − 18 or − − 95 or − 35 or 245 or 910 or 310 or 0. (C.84)It thus follows that in this case that the only consistent choice of U (1) Y requires137 = 0. Note that in this case the U (1) charge assignments match to those of the SU (5) GUT.Finally, suppose that the E -relation 6 a − b = ± b is nowdetermined by the possible Q -relations to be: Q = ⇒ b = ± 45 or ± 125 or ± 85 or 0 (C.85) U = ⇒ b = − − ± 65 or ± 25 or 85 or 0 (C.86) D = ⇒ b = 2 or ± 45 or − 65 or 0 (C.87)so that the only consistent solution requires b = 0, as before. C.2.2 SO (10)We assume that the matter content organizes into the representations 10, 16, 16 and45 of SO (10). The maximal subgroups of SO (10) which contain G std are: SO (10) ⊃ SU (5) × U (1) (C.88) SO (10) ⊃ SU (2) × SU (2) × SU (4) (C.89) SO (10) ⊃ SO (9) (C.90) SO (10) ⊃ SU (2) × SO (7) (C.91) SO (10) ⊃ SO (8) × U (1) (C.92) SO (10) ⊃ U Sp (4) (C.93) SO (10) ⊃ U Sp (4) × U Sp (4). (C.94)Of the above maximal subgroups, only the first four contain SU (3) × SU (2) as asubgroup. Whereas lines (C.88) and (C.89) lead to well-known GUTs, the maximalsubgroups of lines (C.90) and (C.91) are typically not treated in the GUT literature.We now demonstrate that no breaking pattern of the latter two cases can yieldthe MSSM spectrum. In the case SO (10) ⊃ SO (9), the 10, 16, 16 and 45 of SO (10)descend to the 9, 16, and 36 of SO (9). It now follows from the analysis of subsectionC.1.3 that no breaking pattern will yield the matter content of the Standard Model.Next consider the maximal subgroup SU (2) × SO (7). Because there is only one138aximal subgroup of SO (7) which contains SU (3), the unique candidate breakingpattern in this case is: SO (10) ⊃ SU (2) × SO (7) ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × [ U (1)]] (C.95)10 → (3 , + (1 , + (1 , + (1 , − (C.96)16 → (2 , + (2 , − + (2 , − + (2 , (C.97)45 → (3 , + (1 , + (1 , − + (1 , + (1 , − + (1 , . (C.98)By inspection, we note that all singlets of SU (3) × SU (2) are also neutral under the U (1) factor. We therefore conclude that such a breaking pattern cannot include E -fields.We now analyze breaking patterns of the two remaining cases of lines (C.88) and(C.89) which are both well-known in the GUT literature. In the present context,we wish to determine whether a non-standard embedding of the fields in an SO (10)representation could also be consistent with the field content of the MSSM. SO (10) ⊃ SU (5) × U (1) Consider first the maximal subgroup SU (5) × U (1). Inthis case, the unique nested sequence of maximal subgroups which contains the gaugegroup G std is: SO (10) ⊃ SU (5) × [ U (1)] ⊃ SU (3) × SU (2) × [ U (1) a ] × [ U (1) b ] (C.99)10 → (1 , , + (3 , − , + (1 , − , − + (3 , , − (C.100)16 → (1 , , − + (1 , − , + (3 , , + (1 , , − (C.101)+ (3 , − , − + (3 , , − (C.102)45 → (1 , + (1 , , + (3 , − , + (3 , , (C.103)+ (1 , − , − + (3 , , − + (3 , − , − + (1 , , (C.104)+ (1 , , + (8 , , + (3 , − , + (3 , , . (C.105)As usual, we require that at least one linear combination of the U (1) charges may beidentified with U (1) Y of the Standard Model and that all of the necessary interactionterms of the MSSM are present. We begin by classifying all possible combinations139f Q -, U - and D -fields which can yield the gauge invariant combination QU H u : Q U H u ( a, b )1 (3 , , − (3 , , − (2 , − , OU T , , − (3 , , (2 , − , − ( − / , − / , , − (3 , − , − (2 , , (1 , , , − (3 , − , (2 , , − (1 , , , (3 , , − (2 , − , − ( − / , / , , (3 , , OU T OU T , , (3 , − , − (2 , , − (1 , , , (3 , − , OU T OU T , − , (3 , , − (2 , , ( − / , / , − , (3 , , (2 , , − ( − / , − / , − , (3 , − , − OU T OU T 12 (3 , − , (3 , − , OU T OU T (C.106)In the above list, entries in the H u column listed by “ OU T ” indicate that of theavailable representations, no choice yields a gauge invariant quantity in the parenttheory. Similarly, an “ OU T ” entry in the ( a, b ) column indicates that no consistentsolution of U (1) Y exists in this case. We next require that a consistent choice ofrepresentation for D and H d to admit the interaction QDH d also exists amongst the140emaining possibilities: Q D H d ( a, b )2 (3 , , − (3 , − , − (2 , , ( − / , − / , , − (3 , , (2 , − , − (1 , , , − (3 , , − (2 , − , (1 , , , OU T (2 , , ( − / , / , , OU T (2 , − , (1 , , − , OU T (2 , − , − ( − / , / , − , OU T (2 , − , ( − / , − / 5) . (C.107)Of the three remaining possibilities, we next require that the interaction term ELH d be present: E L H d ( a, b )2 a (1 , , − (1 , − , (2 , , ( − / , − / b (1 , − , − (1 , , (2 , , ( − / , − / a (1 , , − (1 , − , − (2 , − , − (1 , b (1 , , (1 , − , (2 , − , − (1 , a (1 , , − (1 , − , − (2 , − , (1 , b (1 , , OU T (2 , − , (1 , 0) . (C.108)We therefore conclude that there are in fact five distinct ways in which the fieldcontent of the MSSM can be packaged in representations of SO (10). We note inparticular that in some cases, the chiral matter of the MSSM does not descend fromeither of the spinor representations of SO (10). The above classification can alsobe obtained without imposing the condition that non-trivial interaction terms bepresent in the superpotential. Indeed, by listing all possible consistent choices of U (1) charge assignments, we arrive at the same list of admissible configurations.Finally, we note that the choice b = 0 corresponds to the breaking pattern where U (1) Y embeds in SU (5) and the other consistent choice corresponds to the flippedembedding of hypercharge [67]. 141 O (10) ⊃ SU (2) × SU (2) × SU (4) We next analyze the other nested sequence ofmaximal subgroups given by decomposing SO (10) as: SO (10) ⊃ SU (2) × SU (2) × SU (4) ⊃ SU (2) × SU (2) × [ SU (3) × U (1)] (C.109)10 → (2 , , + (1 , , + (1 , , − (C.110)16 → (2 , , + (2 , , − + (2 , , − + (2 , , (C.111)45 → (3 , , + (1 , , + (1 , , + (1 , , − (C.112)+ (1 , , + (1 , , + (2 , , + (2 , , − . (C.113)While an SU (2) instanton configuration can indeed yield the gauge group G std , wenote that the putative U (1) Y would then be incorrect. It thus follows that it isenough to consider U (1) × U (1) instanton configurations. Because the representationcontent of this decomposition is identical to that of the previous case, we concludethat there are again two possible ways to package the MSSM fields into SO (10)representations. C.2.3 SO (11)We assume that the matter content organizes into the representations 11, 32 and 55of SO (11). The maximal subgroups of SO (11) are: SO (11) ⊃ SO (10) (C.114) SO (11) ⊃ SU (2) × SO (8) (C.115) SO (11) ⊃ U Sp (4) × SU (4) (C.116) SO (11) ⊃ SU (2) × SU (2) × SO (7) (C.117) SO (11) ⊃ SO (9) × U (1) (C.118) SO (11) ⊃ SU (2) (C.119)so that only the first five maximal subgroups contain G std .142 O (11) ⊃ SO (10) In the case SO (11) ⊃ SO (10), the representations of SO (11)decompose as: SO (11) ⊃ SO (10) (C.120)11 → → 16 + 16 (C.122)55 → 10 + 45 (C.123)so that all of the analysis of breaking patterns performed for SO (10) carries over tothis case as well. In this case, it less clear whether the resulting matter spectrumcan be chiral, but all matter fields of the MSSM can indeed be present. SO (11) ⊃ SU (2) × SO (8) In the case SO (11) ⊃ SU (2) × SO (8), the representationcontent of SO (11) decomposes as: SO (11) ⊃ SU (2) × SO (8) (C.124)11 → (3 , 1) + (1 , v ) (C.125)32 → (2 , s ) + (2 , c ) (C.126)55 → (3 , 1) + (1 , 28) + (3 , v ). (C.127)The two maximal subgroups of SO (8) which contain an SU (3) factor are SU (4) × U (1) and SO (7) ⊃ SU (4). SO (11) ⊃ SU (2) × SO (8) ⊃ SU (2) × [ SU (4) × [ U (1)]] The decomposition to SU (2) × [ SU (4) × [ U (1)]] is: SO (11) ⊃ SU (2) × SO (8) ⊃ SU (2) × [ SU (4) × [ U (1)]] (C.128)11 → (3 , + (1 , + (1 , − + (1 , (C.129)32 → (2 , s ) + (2 , c ) → (2 , + (2 , − + (2 , − (C.130)+ (2 , (C.131)55 → (3 , + (1 , + (1 , + (1 , − + (1 , (C.132)+ (3 , + (3 , − + (3 , (C.133)143o that the decomposition to G std × U (1) along this path is: SO (11) ⊃ SU (2) × [ SU (3) × [ U (1)] a × [ U (1)] b ] (C.134)11 → (3 , , + (1 , , + (1 , , − + (1 , , + (1 , − , (C.135)32 → (2 , , + (2 , − , + (2 , − , − + (2 , , − (C.136)+ (2 , , − + (2 , − , − + (2 , − , + (2 , , (C.137)55 → (3 , , + (1 , , + (1 , , + (1 , − , (C.138)+ (1 , , − + (1 , − , − + (1 , , + (1 , − , (C.139)+ (1 , , + (1 , , + (3 , , + (3 , , − (C.140)+ (3 , , + (3 , − , . (C.141)In order to achieve the correct U (1) Y charge assignment for the E -fields and Q -fields,we require: 2 b = ± − a ± b = 1 (C.143)so that: b = ± a = − U (1) Y charge assignment for the L -fields, we mustalso require: ± a ± b = ± a = 2 and without loss of generality, we may choose a sign convention for b so that b = 3. In this case, the candidate representations for Q , D and H d are: Q D H d (2 , − , (1 , − , (2 , − , − (C.147)so that the product QDH d is not neutral under U (1) a . We therefore conclude thatthis breaking pattern cannot yield the spectrum of the Standard Model.144 O (11) ⊃ SU (2) × SO (7) ⊃ SU (2) × SU (4) ⊃ SU (2) × SU (3) × U (1) In thiscase, breaking to G std proceeds via the nested sequence: SO (11) ⊃ SU (2) × SO (8) ⊃ SU (2) × SO (7) (C.148) ⊃ SU (2) × SU (4) ⊃ SU (2) × SU (3) × U (1) (C.149)11 → (3 , + (1 , + (1 , + (1 , + (1 , − (C.150)32 → (2 , + (2 , − + (2 , − + (2 , + (2 , (C.151)+ (2 , − + (2 , − + (2 , (C.152)55 → (3 , + (1 , + (1 , + (1 , − + (1 , (C.153)+ (1 , − + (1 , + (1 , − + (1 , + (1 , (C.154)+ (3 , + (3 , + (3 , + (3 , − . (C.155)By inspection, the above decomposition does not contain any E -fields. We thereforeconclude that in all cases, breaking patterns of SO (11) with maximal subgroup SU (2) × SO (8) cannot contain G std . SO (11) ⊃ U Sp (4) × SU (4) Because U Sp (4) does not contain SU (3) as a subgroup,it follows that in this case, SU (4) must decompose to SU (3) × U (1). The decom-position must therefore proceed via the path: SO (11) ⊃ U Sp (4) × SU (4) ⊃ U Sp (4) × SU (3) × U (1) (C.156)11 → (5 , 1) + (1 , → (5 , + (1 , + (1 , − (C.157)32 → (4 , 4) + (4 , → (4 , + (4 , − + (4 , − (C.158)55 → (10 , 1) + (1 , 15) + (5 , → (10 , + (1 , (C.159)+ (1 , − + (1 , + (1 , . (C.160)To proceed further, we specify a maximal subgroup of U Sp (4) among the ones listedin lines (C.21)-(C.23). Because a given instanton configuration must preserve thenon-abelian factor SU (3) × SU (2) of the G std , we conclude that only the first twoare viable breaking patterns. SO (11) ⊃ U Sp (4) × SU (4) ⊃ U Sp (4) × SU (3) × U (1) ⊃ [ SU (2) × SU (2)] × [ SU (3) × U (1)] In this case, the decomposition of the matter content contains the145epresentation content of the breaking pattern SO (10) ⊃ SU (2) × SU (2) × SU (4).Explicitly: SO (11) ⊃ U Sp (4) × SU (4) ⊃ [ SU (2) × SU (2)] × [ SU (3) × [ U (1)]] (C.161)11 → (1 , , + (2 , , + (1 , , + (1 , , − (C.162)32 → (2 , , + (2 , , − + (1 , , + (2 , , − (C.163)+ (2 , , − + (2 , , + (1 , , − + (2 , , (C.164)55 → (3 , , + (1 , , + (1 , , − + (1 , , (C.165)+ (1 , , + (1 , , + (1 , , − + (2 , , (C.166)+ (2 , , − + (1 , , + (2 , , . (C.167)It follows that the analysis of breaking patterns for SO (10) directly carries over tothis case as well. SO (11) ⊃ U Sp (4) × SU (3) × U (1) ⊃ [ SU (2) × U (1)] × [ SU (3) × U (1)]While this is seemingly quite similar to the breaking pattern described previously,we now show that the embedding of the U (1) factor in U Sp (4) does not admit anembedding of the matter content of the Standard Model. To this end, we firstdecompose SO (11) via: SO (11) ⊃ U Sp (4) × SU (4) ⊃ [ SU (2) × [ U (1)] a ] × [ SU (3) × [ U (1)] b ] (C.168)11 → (1 , , + (1 , − , + (3 , , + (1 , , + (1 , , − (C.169)32 → (2 , , + (2 , , − + (2 , − , + (2 , − , − + (2 , − , − (C.170)+ (2 , − , + (2 , , − + (2 , , (C.171)55 → (1 , , + (3 , , + (3 , , + (3 , − , + (1 , , (C.172)+ (1 , , − + (1 , , + (1 , , + (1 , , + (1 , , − (C.173)+ (1 , − , − + (1 , − , + (3 , , + (3 , , − . (C.174)It follows from the above decomposition that the E -fields correspond to the represen-tation (1 , ± , of the above decomposition. It thus follows that a = ± 3. Becausethe Q -fields correspond to the representation (2 , ± , − and the L fields correspondto the representation (2 , ± , ± , we conclude that without loss of generality, fixingthe sign of a to be positive so that a = +3, there is a unique linear combination of U (1) charges so that a = 3 and b = 2. The field content of the MSSM thus descends146rom the above representations as: E Q U D L H u H d (1 , , (2 , , − (1 , , − (1 , , − (2 , , − (2 , − , (2 , , − (C.175)By inspection, we note that whereas the product QU H u is indeed invariant under allgauge group factors, QDH d violates U (1) b . We therefore conclude that the abovebreaking pattern cannot yield the MSSM. SO (11) ⊃ SU (2) × SU (2) × SO (7) Because there is a single maximal subgroup of SO (7) which contains SU (3), we find that the unique breaking pattern which canreproduce G std proceeds as: SO (11) ⊃ SU (2) × SU (2) × SO (7) ⊃ SU (2) × SU (2) × SU (4) (C.176)11 → (2 , , 1) + (1 , , 1) + (1 , , 6) (C.177)32 → (1 , , 4) + (1 , , 4) + (2 , , 4) + (2 , , 4) (C.178)55 → (3 , , 1) + (1 , , 1) + (2 , , 1) + (2 , , 6) + (1 , , 6) + (1 , , 15) (C.179)By inspection, this decomposition again contains all of the matter content of the SO (10) breaking pattern which proceeds via SO (10) ⊃ SU (2) × SU (2) × SU (4).We therefore conclude that the analysis of the breaking patterns via instantons isidentical to this case. SO (11) ⊃ SO (9) × U (1) The final maximal subgroup which contains G std is givenby SO (9) × U (1). In this case, SU (2) × SU (4) is the only maximal subgroup of SO (9) which contains the product SU (3) × SU (2). Decomposing with respect to147his path yields: SO (11) ⊃ SO (9) × [ U (1)] b ⊃ [ SU (2) × SU (4)] × [ U (1)] b (C.180) ⊃ [ SU (2) × SU (3) × [ U (1)] a ] × [ U (1)] b (C.181)11 → (1 , , − + (1 , , + (3 , , + (1 , , + (1 , − , (C.182)32 → (2 , , + (2 , − , + (2 , − , + (2 , , (C.183)+ (2 , − , − + (2 , , − + (2 , , − + (2 , − , − (C.184)55 → (1 , , + (3 , , + (1 , , + (1 , − , + (3 , , − (C.185)+ (1 , , − + (1 , − , − + (3 , , + (3 , , + (3 , − , (C.186)+ (1 , , + (1 , − , + (1 , , + (1 , , . (C.187)In this case, the E -fields must correspond to the representation (1 , , ± . Thisimplies the relation b = ± 3. Moreover, because the Q and L -fields respectivelycorrespond to the representations (2 , − , ± and (2 , ± , ± , it follows that withoutloss of generality a = 2 and b = +3 is the unique choice of U (1) charges which canyield the correct value of U (1) Y for all fields. In this case, the representation contentof the Q , D and H d fields is uniquely determined to be: Q D H d (2 , − , (1 , − , (2 , − , − . (C.188)Because the product QDH d violates U (1) b , we conclude that the correspondingbreaking pattern cannot lead to the MSSM. C.3 Rank Six We now proceed to the classification of all breaking patterns of rank six groups.Because it is the case of primary phenomenological interest in many cases, we beginour analysis with breaking patterns of E . We next determine all possible breakingpatterns of SU (7) and conclude with an analysis of breaking patterns of SO (12).148 .3.1 E The non-trivial representations of E which can descend from the adjoint represen-tation of E are the 27, 27 and 78 of E . The maximal subgroups of E are: E ⊃ SO (10) × U (1) (C.189) E ⊃ SU (2) × SU (6) (C.190) E ⊃ SU (3) × SU (3) × SU (3) (C.191) E ⊃ U Sp (8) (C.192) E ⊃ F (C.193) E ⊃ SU (3) × G (C.194) E ⊃ G (C.195) E ⊃ SU (3). (C.196)Of the above configurations, only the maximal subgroups of lines (C.189)-(C.194)contain G std . In particular, the first three breaking patterns can descend to moreconventional GUT theories. We begin our analysis by demonstrating that none ofthe remaining possibilities can produce a consistent embedding of the MSSM. E ⊃ U Sp (8) The maximal subgroups of U Sp (8) are: U Sp (8) ⊃ SU (4) × U (1) (C.197) U Sp (8) ⊃ SU (2) × U Sp (6) (C.198) U Sp (8) ⊃ U Sp (4) × U Sp (4) (C.199) U Sp (8) ⊃ SU (2) (C.200) U Sp (8) ⊃ SU (2) × SU (2) × SU (2). (C.201)Of these possibilities, only line (C.198) contains SU (3) × SU (2). Further, by inspec-tion of lines (C.17)-(C.20), the only maximal subgroup of U Sp (6) which contains SU (3) is: U Sp (6) ⊃ SU (3) × U (1). (C.202)149n this case, the unique candidate breaking pattern is: E ⊃ U Sp (8) ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × [ U (1)]] (C.203)which is obtained by a non-trivial U (1) instanton in the U Sp (6) factor. In this case,the representations of E decompose as: E ⊃ U Sp (8) ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × [ U (1)]] (C.204)27 → (2 , + (2 , − + (1 , − + (1 , + (1 , + (1 , (C.205)78 → (3 , + (1 , + (1 , + (1 , − + (1 , + (2 , (C.206)+ (2 , − + (1 , − + (1 , + (1 , + (2 , + (2 , − (C.207)+ (2 , − + (2 , . (C.208)By inspection, all singlets of SU (3) × SU (2) are neutral under the only U (1) factorso that the resulting model cannot contain any E -fields. E ⊃ F The representation content of E decomposes under F as: E ⊃ F (C.209)27 → 26 + 1 (C.210)78 → 26 + 52. (C.211)Returning to our previous analysis of breaking patterns for F , we therefore concludethat this breaking pattern cannot produce the correct matter content of the MSSM. E ⊃ SU (3) × G Although G contains SU (3) as a maximal subgroup, it is notpossible to arrange for an instanton configuration to break G to SU (3). For thisreason, we conclude that the SU (3) factor of G std must be identified with the SU (3)factor of the maximal subgroup SU (3) × G of E . In this case, it now follows thatthe factor SU (2) × U (1) must descend from G . Returning to lines (C.24)-(C.26), itfollows that the maximal subgroups SU (3) and SU (2) × SU (2) contain SU (2) × U (1).First consider the decomposition of representations of E via the nested sequence150f maximal subgroups: E ⊃ SU (3) × G ⊃ SU (3) × [ SU (3)] ⊃ SU (3) × [ SU (2) × [ U (1)]] (C.212)27 → (6 , + (3 , + (3 , + (3 , − + (3 , − + (3 , (C.213)78 → (8 , + (1 , − + (1 , + (1 , + (1 , − (C.214)+ (1 , + (1 , + (1 , − + (1 , . (C.215)Because the ratio of the U (1) charge for the candidate E - and Q -fields does not equalsix, we conclude that this is not a viable breaking pattern.Next consider the decomposition associated with the nested sequence of maximalsubgroups: E ⊃ SU (3) × G ⊃ SU (3) × [ SU (2) × SU (2)] (C.216)27 → (6 , , 1) + (3 , , 3) + (3 , , 2) (C.217)78 → (8 , , 1) + (1 , , 3) + (1 , , 1) + (1 , , 4) + (8 , , 3) + (8 , , U (1) subgroup of either SU (2) factor, we find that the ratio of U (1) charges for the candidate E - and Q -fields again does not equal six. Hence, neither nested sequence of maximal subgroupsyields the correct spectrum of the MSSM. E ⊃ SU (3) × SU (3) × SU (3) In order to make the Z outer automorphism of E more manifest, we assume that the decomposition of E to the maximal subgroup SU (3) × SU (3) × SU (3) is given by: E ⊃ SU (3) × SU (3) × SU (3) (C.219)27 → (3 , , 1) + (3 , , 3) + (1 , , 3) (C.220)27 → (3 , , 1) + (3 , , 3) + (1 , , 3) (C.221)78 → (8 , , 1) + (1 , , 1) + (1 , , 8) + (3 , , 3) + (3 , , SU (3)factor, this is a choice of convention. Indeed, because of the Z outer automorphism,without loss of generality we require that the first SU (3) factor is common to G std aswell. First note that while an SU (3) × U (1) instanton can break E to G std , we note151hat the resulting U (1) factor of G std must descend from one of the remaining SU (3)factors. By inspection of the above decomposition of line (C.219), the purported U (1) Y is incorrect.To proceed further, we next consider the maximal subgroups of the last two SU (3)factors. The maximal subgroups of SU (3) are: a ) : SU (3) ⊃ SU (2) × U (1) (C.223) b ) : SU (3) ⊃ SU (2). (C.224)We therefore conclude that there are four distinct maximal subgroups of SU (3) × SU (3) × SU (3) which can potentially yield G std . Moreover, in order to achievethe subgroup SU (2) × U (1) of G std , we must assume that at least one SU (3) factordescends to a maximal subgroup via line (C.223). E ⊃ SU (3) × SU (3) × SU (3) ⊃ SU (3) × [ SU (2)] × [ SU (2) × U (1)]We first treat the nested sequence of maximal subgroups where the second SU (3)factor descends to SU (2) as in line (C.224) while the third descends to SU (2) × U (1)as in line (C.223). Because interchanging the last two SU (3) factors of E ⊃ SU (3) × SU (3) × SU (3) complex conjugates all representations, a similar analysiswill hold in that case as well. The representation content of E decomposes as: E ⊃ SU (3) × SU (3) × SU (3) ⊃ SU (3) × [ SU (2)] × [ SU (2) × U (1)] (C.225)27 → (3 , , 1) + (3 , , ) + (3 , , − ) + (1 , , − ) + (1 , , ) (C.226)78 → (8 , , 1) + (1 , , 1) + (1 , , 1) + (1 , , ) + (1 , , ) + (1 , , − ) (C.227)+ (1 , , ) + (3 , , ) + (3 , , − ) + (3 , , − ) + (3 , , ). (C.228)There are several ways in which an instanton configuration can yield the gaugegroup G std . First consider configurations obtained via a non-trivial SU (2) instantonconfiguration. Because the SU (2) factor of SU (3) either breaks completely or to a U (1) subgroup of SU (2), we conclude that only SU (2) instantons with values in thefactor SU (3) of line (C.219) can preserve the gauge group G std . In this case, the U (1) charge assignments for the Q - and E -fields are incompatible with the U (1) Y assignments of the Standard Model.Next consider abelian instanton configurations which break one of the SU (2)factors. Decomposing the factor SU (2) with respect to a U (1) subgroup, the152esulting representation content is: E ⊃ SU (3) × SU (3) × SU (3) ⊃ SU (3) × [ SU (2)] × [ SU (2) × U (1)] (C.229) ⊃ SU (3) × [ U (1) a ] × [ SU (2) × U (1) b ] (C.230)27 → (3 , , ) + (3 , − , ) + (3 , , ) + (3 , , ) + (3 , , − ) (C.231)+ (1 , , − ) + (1 , − , − ) + (1 , , − ) + (1 , , ) + (1 , − , ) (C.232)+ (1 , , ) (C.233)78 → (8 , , ) + (1 , , ) + (1 , − , ) + (1 , , ) + (1 , , ) (C.234)+ (1 , , ) + (1 , , ) + (1 , − , ) + (1 , − , ) + (1 , , ) (C.235)+ (1 , , ) + (1 , , − ) + (1 , , ) + (3 , , ) + (3 , − , ) (C.236)+ (3 , , ) + (3 , , − ) + (3 , − , − ) + (3 , , − ) + (3 , − , − ) (C.237)+ (3 , , − ) + (3 , , − ) + (3 , − , ) + (3 , , ) + (3 , , ). (C.238)The representation content of each MSSM field therefore descends from the followingrepresentations: E : (1 , ± , ± ) (C.239) Q : (3 , , − ) or (3 , ± , − ) (C.240) H d , L : (1 , ± , ± ) or (1 , , ± ) (C.241) U : (3 , ± , ) or (3 , , − ) or (3 , ± , − ) (C.242) D : (3 , ± , ) or (3 , , − ) or (3 , ± , − ) (C.243) H u : (1 , ± , ± ) or (1 , , ± ). (C.244)There are four possible assignments for the Q, U, H u fields which can yield a non-trivial QU H u term: Q U H u (3 , , − ) (3 , ± , ) (1 , ∓ , +1 )(3 , , − ) (3 , , − ) (1 , , +3 )(3 , ± , − ) (3 , ∓ , − ) (1 , , +3 ) (C.245)so that in the first three cases, the U (1) Y charge of Q requires b = − U (1) Y charge of H u requires b = +1. In particular, this impliesthat the second choice of charge assignments in line (C.245) is inconsistent. Nextconsider the first choice of charge assignments. In order to obtain the correct U (1) Y charge assignment for the U -field, we must therefore require a = ∓ 2. Finally, thefourth choice of charge assignments requires a = ± 1. Of these possible chargeassignments, only the first yields a choice consistent with the U (1) Y charge of the E -field in line (C.239). We therefore find that a = − b = − a . It now follows that the only candidatecharge assignments for the fields are: E Q , U D , L H u H d (1 , − , − ) (3 , , − ) (3 , , ) (3 , , − ) (1 , , − ) (1 , − , +1 ) (1 , , )(C.246)where we have also indicated the E representation content. The interaction term QU H u therefore descends from a 27 term so that in particular, Q descends fromthe 27 of E . In order to obtain a non-trivial QDH d term, this in turn requires D to descend from the 78 of E so that we finally obtain the representation content: E Q U D L H u H d (1 , − , − ) (3 , , − ) (3 , , ) (3 , , − ) (1 , , − ) (1 , − , +1 ) (1 , , )(C.247)we therefore conclude that a U (1) × U (1) of the above type can indeed yield aspectrum consistent with the MSSM. E ⊃ SU (3) × SU (3) × SU (3) ⊃ SU (3) × [ SU (2) × U (1)] × [ SU (2) × U (1)]We next treat the nested sequence of maximal subgroups where the second andthird SU (3) factors of the decomposition E ⊃ SU (3) × SU (3) × SU (3) descend to SU (2) × U (1) as in line (C.223). Under this decomposition, the resulting represen-154ation content is: E = SU (3) × SU (3) × SU (3) (C.248) ⊃ SU (3) × [ SU (2) × U (1) a ] × [ SU (2) × U (1) b ] (C.249)27 → (3 , − , ) + (3 , , ) + (3 , , ) + (3 , , − ) + (1 , , − ) (C.250)+ (1 , − , − ) + (1 , , ) + (1 , − , ) (C.251)78 → (8 , , ) + (1 , , ) + (1 , , ) + (1 , − , ) + (1 , , ) (C.252)+ (1 , , ) + (1 , , ) + (1 , , − ) + (1 , , ) + (3 , − , − ) (C.253)+ (3 , − , ) + (3 , , − ) + (3 , , ) + (3 , , ) + (3 , , − ) (C.254)+ (3 , − , ) + (3 , − , − ). (C.255)As opposed to previous examples, we now show that a non-abelian instanton canindeed yield the spectrum of the MSSM. To this end, we first show that the repre-sentation content under the subgroup SU (3) × [ U (1) a ] × [ SU (2) × U (1) b ] can yieldthe desired spectrum. We note that this will then establish the same result for a U (1) instanton which breaks this SU (2) factor to U (1).The representation content of the candidate fields is given by ignoring the first SU (2) factor: E : (1 , ε , − ε ) or (1 , ε , ε ) or (1 , ε , ) (C.256) Q : (3 , , − ) or (3 , − , ) or (3 , , ) (C.257) U : (3 , − , ) or (3 , , ) or (3 , , − ) (C.258)or (3 , , ) or (3 , − , ) (C.259) D : (3 , − , ) or (3 , , ) or (3 , , − ) (C.260)or (3 , , ) or (3 , − , ) (C.261) H d , H u , L : (1 , ε , ε ) or (1 , − ε , ε ) or (1 , , ε ) (C.262)where ε = ± 1. We begin by listing all possible distinct combinations of fields which155an potentially descend to the MSSM interaction term QU H u : Q U H u ( a, b )1 (3 , , − ) (3 , − , ) (1 , , ) (2 , − , , − ) (3 , , ) (1 , − , ) ( − , − , , − ) (3 , , − ) (1 , , ) OU T , , − ) (3 , , ) (1 , − , − ) ( − , − , , − ) (3 , − , ) (1 , , − ) (2 , − , − , ) (3 , − , ) OU T OU T , − , ) (3 , , ) (1 , , − ) ( − , − , − , ) (3 , , − ) (1 , , ) OU T , − , ) (3 , , ) (1 , , − ) ( − , − , − , ) (3 , − , ) OU T OU T 11 (3 , , ) (3 , − , ) (1 , , − ) (2 , − , , ) (3 , , ) (1 , − , − ) ( − , , , ) (3 , , − ) (1 , − , ) ( − , , , ) (3 , , ) OU T OU T 15 (3 , , ) (3 , − , ) (1 , , − ) (2 , − 1) (C.263)where we have also solved for the linear combination of U (1) a and U (1) b consistentwith U (1) Y charge assignments in the MSSM. Next, we list all possible combinationsof fields consistent with the above classification which also allow the interaction term156 DH d . Q U D , , − ) (3 , − , ) (3 , , ) or (3 , , − ) or (3 , , )2 (3 , , − ) (3 , , ) (3 , , − ) or (3 , − , )4 (3 , , − ) (3 , , ) (3 , − , ) or (3 , , − )5 (3 , , − ) (3 , − , ) (3 , , ) or (3 , , − ) or (3 , , )7 (3 , − , ) (3 , , ) OU T , − , ) (3 , , ) (3 , , − )11 (3 , , ) (3 , − , ) (3 , , ) or (3 , , − )12 (3 , , ) (3 , , ) (3 , , )13 (3 , , ) (3 , , − ) (3 , − , )15 (3 , , ) (3 , − , ) (3 , , ) or (3 , , − ) (C.264) H u H d ( a, b )1 (1 , , ) (1 , − , ) or (1 , , ) or (1 , − , − ) (2 , − , − , ) (1 , , ) or (1 , , − ) ( − , − , − , − ) (1 , , ) or (1 , , ) ( − , − , , − ) (1 , − , ) or (1 , , ) or (1 , − , − ) (2 , − , , − ) OU T ( − , − , , − ) (1 , , ) ( − , − , , − ) (1 , − , − ) or (1 , − , ) (2 , − , − , − ) OU T ( − , , − , ) (1 , , − ) ( − , , , − ) (1 , − , − ) or (1 , − , ) (2 , − 1) . (C.265)To further narrow the possible combinations of fields, we next require that the inter-actions in question properly descend from E invariant terms of the full theory. Wefind that there many ways to package the field content of the MSSM into represen-157ations of E . The complete list of possibilities is: Q U D L a (3 , , − ) ∈ 27 (3 , − , ) ∈ 27 (3 , , ) ∈ 27 (1 , − , ) ∈ b (3 , , − ) ∈ 27 (3 , − , ) ∈ 27 (3 , , − ) ∈ 27 (1 , − , − ) ∈ c (3 , , − ) ∈ 27 (3 , − , ) ∈ 27 (3 , , − ) ∈ OU T . , , − ) ∈ 27 (3 , − , ) ∈ 27 (3 , , ) ∈ 78 (1 , − , ) ∈ a (3 , , − ) ∈ 27 (3 , , ) ∈ 27 (3 , , − ) ∈ OU T b (3 , , − ) ∈ 27 (3 , , ) ∈ 27 (3 , , − ) ∈ 27 (1 , , − ) ∈ . a (3 , , − ) ∈ 27 (3 , , ) ∈ 27 (3 , − , ) ∈ 78 (1 , , − ) ∈ . b (3 , , − ) ∈ 27 (3 , , ) ∈ 27 (3 , − , ) ∈ 78 (1 , , ) ∈ 784 (3 , , − ) ∈ 27 (3 , , ) ∈ 78 (3 , − , ) ∈ OU T . , , − ) ∈ 27 (3 , , ) ∈ 78 (3 , , − ) ∈ OU T a (3 , , − ) ∈ 27 (3 , − , ) ∈ 78 (3 , , ) ∈ OU T b (3 , , − ) ∈ 27 (3 , − , ) ∈ 78 (3 , , ) ∈ 27 (1 , − , − ) ∈ . a (3 , , − ) ∈ 27 (3 , − , ) ∈ 78 (3 , , − ) ∈ OU T . b (3 , , − ) ∈ 27 (3 , − , ) ∈ 78 (3 , , − ) ∈ OU T . a (3 , , − ) ∈ 27 (3 , − , ) ∈ 78 (3 , , ) ∈ OU T . b (3 , , − ) ∈ 27 (3 , − , ) ∈ 78 (3 , , ) ∈ 78 (1 , − , ) ∈ 279 (3 , − , ) ∈ 78 (3 , , ) ∈ 78 (3 , , − ) ∈ OU T a (3 , , ) ∈ 78 (3 , − , ) ∈ 27 (3 , , ) ∈ OU T b (3 , , ) ∈ 78 (3 , − , ) ∈ 27 (3 , , ) ∈ 27 (1 , − , ) ∈ . a (3 , , ) ∈ 78 (3 , − , ) ∈ 27 (3 , , − ) ∈ OU T . b (3 , , ) ∈ 78 (3 , − , ) ∈ 27 (3 , , − ) ∈ 27 (1 , − , ) ∈ , , ) ∈ 78 (3 , , − ) ∈ 27 (3 , − , ) ∈ 27 (1 , , − ) ∈ a (3 , , ) ∈ 78 (3 , − , ) ∈ 78 (3 , , ) ∈ 27 (1 , , ) ∈ b (3 , , ) ∈ 78 (3 , − , ) ∈ 78 (3 , , ) ∈ 27 (1 , − , ) ∈ . a (3 , , ) ∈ 78 (3 , − , ) ∈ 78 (3 , , − ) ∈ 27 (1 , − , ) ∈ . b (3 , , ) ∈ 78 (3 , − , ) ∈ 78 (3 , , − ) ∈ 27 (1 , − , − ) ∈ 27 .(C.266)158 H u H d ( a, b )1 a (1 , , − ) ∈ 27 (1 , , ) ∈ 27 (1 , − , ) ∈ 27 (2 , − b (1 , , − ) ∈ 27 (1 , , ) ∈ 27 (1 , , ) ∈ 78 (2 , − c (1 , , ) ∈ 78 (1 , , ) ∈ 27 (1 , , ) ∈ 78 (2 , − . , , ) ∈ 78 (1 , , ) ∈ 27 (1 , − , − ) ∈ 27 (2 , − a (1 , − , ) ∈ 27 (1 , − , ) ∈ 27 (1 , , ) ∈ 78 ( − , − b (1 , − , − ) ∈ 27 (1 , − , ) ∈ 27 (1 , , ) ∈ 78 ( − , − . a (1 , − , ) ∈ 27 (1 , − , ) ∈ 27 (1 , , − ) ∈ 27 ( − , − . b (1 , − , − ) ∈ 27 (1 , − , ) ∈ 27 (1 , , − ) ∈ 27 ( − , − OU T (1 , − , − ) ∈ 27 (1 , , ) ∈ 27 ( − , − . OU T (1 , − , − ) ∈ 27 (1 , , ) ∈ 78 ( − , − a (1 , , − ) ∈ 27 (1 , , − ) ∈ 27 (1 , − , ) ∈ 27 (2 , − b (1 , , ) ∈ 78 (1 , , − ) ∈ 27 (1 , − , ) ∈ 27 (2 , − . a (1 , , − ) ∈ 27 (1 , , − ) ∈ 27 (1 , , ) ∈ 78 (2 , − . b (1 , , ) ∈ 78 (1 , , − ) ∈ 27 (1 , , ) ∈ 78 (2 , − . a (1 , , − ) ∈ 27 (1 , , − ) ∈ 27 (1 , − , − ) ∈ 27 (2 , − . b (1 , , ) ∈ 78 (1 , , − ) ∈ 27 (1 , − , − ) ∈ 27 (2 , − OU T (1 , , − ) ∈ 78 (1 , , ) ∈ 27 ( − , − a (1 , , − ) ∈ 27 (1 , , − ) ∈ 27 (1 , − , − ) ∈ 27 (2 , − b (1 , , ) ∈ 78 (1 , , − ) ∈ 27 (1 , − , − ) ∈ 27 (2 , − . a (1 , , − ) ∈ 27 (1 , , − ) ∈ 27 (1 , − , ) ∈ 27 (2 , − . b (1 , , ) ∈ 78 (1 , , − ) ∈ 27 (1 , − , ) ∈ 27 (2 , − , − , ) ∈ 27 (1 , − , ) ∈ 27 (1 , , − ) ∈ 27 ( − , a (1 , , − ) ∈ 27 (1 , , − ) ∈ 78 (1 , − , − ) ∈ 27 (2 , − b (1 , , ) ∈ 78 (1 , , − ) ∈ 78 (1 , − , − ) ∈ 27 (2 , − . a (1 , , − ) ∈ 27 (1 , , − ) ∈ 78 (1 , − , ) ∈ 27 (2 , − . b (1 , , ) ∈ 78 (1 , , − ) ∈ 78 (1 , − , ) ∈ 27 (2 , − OU T denotes an entry which has been ruledout because it cannot yield the correct U (1) Y charge assignment or interaction term. E ⊃ SU (10) × U (1) We now analyze breaking patterns of E which descend fromthe maximal subgroup SO (10) × U (1) such that: E ⊃ SO (10) × [ U (1)] (C.268)27 → + 10 − + 16 (C.269)78 → + 16 − + 16 + 45 . (C.270)Of the maximal subgroups of SO (10) listed in lines (C.88)-(C.94), only the first fourcontain the non-abelian group SU (3) × SU (2) so that the unique nested sequence ofmaximal subgroups of E is uniquely determined by the paths: E ⊃ SO (10) × [ U (1)] ⊃ [ SU (5) × U (1)] × U (1) (C.271) ⊃ [ SU (3) × SU (2) × U (1)] × U (1)] × U (1) (C.272) E ⊃ SO (10) × [ U (1)] ⊃ SU (2) × SU (2) × SU (4) × [ U (1)] (C.273) ⊃ SU (2) × SU (2) × [ SU (3) × U (1)] × U (1) (C.274) E ⊃ SO (10) × [ U (1)] ⊃ SO (9) × [ U (1)] ⊃ [ SU (2) × SU (4)] × [ U (1)] (C.275) ⊃ [ SU (2) × [ SU (3) × U (1)]] × [ U (1)] (C.276) E ⊃ SO (10) × [ U (1)] ⊃ SU (2) × SO (7) ⊃ [ SU (2) × SU (4)] × [ U (1)] (C.277) ⊃ [ SU (2) × [ SU (3) × U (1)]] × [ U (1)]. (C.278)Because the previous analysis of abelian instanton configurations of SO (10) whichcan yield the MSSM spectrum carry over to this case as well, we focus on breakingpatterns which do not embed purely in SO (10). While it is in principle possible topackage the field content of the MSSM fields into representations of E in more exoticways using the additional U (1) charge, all of these configurations still correspond togeneric abelian instanton configurations. E ⊃ SO (10) × [ U (1)] ⊃ SU (2) × SO (7) × [ U (1)] ⊃ SU (2) × SU (4) × [ U (1)] ⊃ SU (2) × [ SU (3) × [ U (1)]] × [ U (1)]Decomposing the 27 and 78 with respect to this nested sequence of maximal160ubgroups, we find: E ⊃ ... ⊃ SU (2) × [ SU (3) × [ U (1) a ]] × [ U (1) b ] (C.279)27 → (1 , , + (3 , , − + (1 , , − + (1 , , − (C.280)+ (1 , − , − + (2 , , + (2 , − , + (2 , − , + (2 , , (C.281)78 → (1 , , + (2 , , − + (2 , − , − + (2 , − , − (C.282)+ (2 , , − + (2 , − , + (2 , , + (2 , , + (2 , − , (C.283)+ (3 , , + (1 , , + (1 , − , + (3 , , + (3 , − , (C.284)+ (3 , , + (1 , , + (1 , − , + (1 , , + (1 , , . (C.285)We begin by classifying all combinations of representations which can yield the non-trivial interaction term QU H u : Q U D L , − , ∈ 27 (1 , − , − ∈ 27 (1 , − , ∈ 27 (2 , − , − ∈ 272 (2 , − , ∈ 27 (1 , − , − ∈ 27 (1 , , ∈ 78 (2 , , − ∈ 783 (2 , − , − ∈ 27 (1 , − , ∈ 27 (1 , − , − ∈ 27 (2 , − , ∈ 274 (2 , − , − ∈ 27 (1 , − , ∈ 27 (1 , , ∈ 78 (2 , , ∈ 78 (C.286) E H u H d ( a, b )1 (1 , , ∈ 27 (2 , , ∈ 27 (2 , , − ∈ 78 (1 / , / , , ∈ 27 (2 , , ∈ 27 (2 , − , − ∈ 27 (1 / , / , , − ∈ 27 (2 , , − ∈ 27 (2 , , ∈ 78 (1 / , − / , , − ∈ 27 (2 , , − ∈ 27 (2 , − , ∈ 27 (1 / , − / 2) (C.287)so that in this case a non-standard embedding of a U (1) × U (1) instanton can indeedyield the spectrum of the MSSM. E ⊃ SO (10) × [ U (1)] ⊃ SO (9) × [ U (1)] ⊃ SU (2) × SU (4) × [ U (1)] ⊃ SU (2) × [ SU (3) × [ U (1)]] × [ U (1)]Decomposing the 27 and 78 with respect to this nested sequence of maximal161ubgroups, we find: E ⊃ ... ⊃ SU (2) × [ SU (3) × [ U (1) a ]] × [ U (1) b ] (C.288)27 → (1 , , + (1 , , − + (3 , , − + (1 , , − + (1 , − , − (C.289)+ (2 , , + (2 , − , + (2 , − , + (2 , , (C.290)78 → (1 , , + (2 , , − + (2 , − , − + (2 , − , − + (2 , , − (C.291)+ (2 , , + (2 , − , + (2 , − , + (2 , , (C.292)+ (3 , , + (1 , , + (1 , − , + (1 , , + (1 , , + (3 , , (C.293)+ (3 , − , + (3 , , + (1 , , + (1 , − , . (C.294)By inspection, this is precisely the same matter content as in the previous example.We therefore conclude that the abelian instanton configurations analyzed previouslyproduce an identical MSSM spectrum. E ⊃ SO (10) × [ U (1)] ⊃ SU (2) × SU (2) × SU (4) × [ U (1)] ⊃ SU (2) × SU (2) × [ SU (3) × U (1)] × U (1)The decomposition of the 27 and 78 of E in this case yields: E ⊃ SO (10) × [ U (1)] ⊃ SU (2) × SU (2) × SU (4) × [ U (1)] (C.295) ⊃ SU (2) × SU (2) × [ SU (3) × U (1) a ] × U (1) b (C.296)27 → (1 , , , + (2 , , , − + (1 , , , − + (1 , , − , − (C.297)+ (2 , , , + (2 , , − , + (1 , , − , + (1 , , , (C.298)78 → (1 , , , + (2 , , , − + (2 , , − , − + (1 , , − , − (C.299)+ (1 , , , − + (2 , , − , + (2 , , , + (1 , , , (C.300)+ (1 , , − , + (3 , , , + (1 , , , + (1 , , , (C.301)+ (1 , , − , + (1 , , , + (1 , , , + (2 , , , (C.302)+ (2 , , − , . (C.303)In fact, the representation content of this decomposition is identical to that ob-tained via the previously treated nested sequence of maximal subgroups given bylines (C.248)-(C.255): E ⊃ SU (3) × SU (3) × SU (3) ⊃ SU (3) × [ SU (2) × U (1) c ] × [ SU (2) × U (1) d ] (C.304)162nder the linear change in U (1) charges: U (1) a = 12 U (1) c + 12 U (1) d (C.305) U (1) b = 12 U (1) c − U (1) d . (C.306) E ⊃ SU (2) × SU (6) Decomposing the 27 and 78 of E with respect to SU (2) × SU (6) yields: E ⊃ SU (2) × SU (6) (C.307)27 → (2 , 6) + (1 , 15) (C.308)78 → (3 , 1) + (1 , 35) + (2 , SU (6) presented in lines (C.57)-(C.63), thelist of all possible nested sequences of maximal subgroups of E descend to G std as: E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (5) × U (1)] (C.310) ⊃ SU (2) × [ SU (3) × SU (2) × [ U (1)] × U (1)] (C.311) E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (2) × SU (4) × U (1)] (C.312) ⊃ SU (2) × [ SU (2) × [ SU (3) × U (1)] × U (1)] (C.313) E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (3) × SU (3) × U (1)] (C.314) E ⊃ SU (2) × SU (6) ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × U (1)] (C.315) E ⊃ SU (2) × SU (6) ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × U (1)] (C.316) E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (2) × SU (3)]. (C.317)In the first two nested sequences the resulting breaking pattern descends to the samerepresentation content as breaking patterns analyzed previously. For this reason,we confine our analysis to breaking patterns reached via lines (C.314)-(C.317). E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (2) × SU (3)]163n this case, the representations of E decompose as: E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (2) × SU (3)] (C.318)27 → (2 , , 3) + (1 , , 6) + (1 , , 3) (C.319)78 → (3 , , 1) + (3 , , 1) + (3 , , 8) + (2 , , 1) + (2 , , SU (2) factors with respect to a U (1) subgroup, it followsthat the ratio of U (1) charge for the Q - and E -fields is incorrect so that the MSSMcannot be obtained via this path. E ⊃ SU (2) × SU (6) ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × U (1)]The representations of E descend as: E ⊃ SU (2) × SU (6) ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × U (1)] (C.321)27 → (2 , + (2 , − + (1 , + (1 , − + (1 , + (1 , (C.322)78 → (3 , + (1 , − + (1 , + (1 , + (1 , + (1 , (C.323)+ (1 , − + (1 , + (2 , + (2 , − + (2 , + (2 , − (C.324)+ (2 , − + (2 , (C.325)Because the U (1) charge assignment is incorrect, we cannot reach the MSSM via thisnested sequence either. E ⊃ SU (2) × SU (6) ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × U (1)]Here, the representations of E descend as: E ⊃ SU (2) × SU (6) ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × U (1)] (C.326)27 → (2 , + (2 , − + (1 , + (1 , − + (1 , + (1 , (C.327)78 → (3 , + (1 , + (1 , − + (1 , + (1 , + (1 , − (C.328)+ (1 , + (1 , + (2 , + (2 , − + (2 , + (2 , − (C.329)+ (2 , + (2 , − (C.330)which does not contain any candidate E -fields. E ⊃ SU (2) × SU (6) ⊃ SU (2) × [ SU (3) × SU (3) × U (1)]All of the breaking patterns in this case have already been classified in our dis-cussion of breaking patterns for the maximal subgroup SU (3) × SU (3) × SU (3).164ndeed, this essentially follows from the fact that SU (3) contains the maximal sub-group SU (2) × U (1). We therefore proceed to the other rank six bulk gauge groupsand their breaking patterns. C.3.2 SU (7)We assume that the matter content of SU (7) descends from the adjoint representationof E . For this reason, we only treat the adjoint, 7, 21, 35 and complex conjugaterepresentations of SU (7). The maximal subgroups of SU (7) are: SU (7) ⊃ SU (6) × U (1) (C.331) SU (7) ⊃ SU (2) × SU (5) × U (1) (C.332) SU (7) ⊃ SU (3) × SU (4) × U (1) (C.333) SU (7) ⊃ SO (7) (C.334)of which only the first three contain G std . SU (7) ⊃ SU (6) × U (1) There are three maximal subgroups of SU (6) which cancontain the non-abelian factor of G std and can be reached via an instanton: SU (7) ⊃ SU (6) × U (1) ⊃ SU (5) × U (1) × U (1) (C.335) SU (7) ⊃ SU (6) × U (1) ⊃ SU (2) × SU (4) × U (1) × U (1) (C.336) SU (7) ⊃ SU (6) × U (1) ⊃ SU (3) × SU (3) × U (1) × U (1). (C.337)In this case, in order to preserve an SU (3) × SU (2) factor, the only available instantonconfiguration must generically take values in U (1) so that all nested sequences ofmaximal subgroups which can be reached by an instanton configuration all descendto the group SU (3) × SU (2) × U (1) × U (1) × U (1). It is therefore enough to consider165he breaking pattern: SU (7) ⊃ SU (6) × U (1) ⊃ SU (5) × U (1) × U (1) (C.338) ⊃ SU (3) × SU (2) × U (1) × U (1) × U (1) (C.339)7 → , , + 1 , − , − + (1 , , , − + (3 , − , , − (C.340)21 → (1 , , , − + (3 , − , , − + (3 , , , − + (3 , − , , − (C.341)+ (1 , , , − + (1 , , − , + (3 , − , , + (1 , , , (C.342)35 → (1 , , − , − + (3 , − , − , − + (3 , , − , − + (1 , − , , − (C.343)+ (3 , , , − + (3 , − , , − + (1 , , , + (3 , − , , (C.344)+ (3 , , , + (3 , − , − , + (1 , , − , (C.345)48 → , , + 1 , , + (3 , − , , + (1 , , , + (3 , , − , (C.346)+ (1 , − , − , + (1 , , , + (1 , , , + (8 , , , (C.347)+ (3 , − , , + (3 , , , + (1 , , − , − + (1 , , , (C.348)+ (3 , − , , − + (1 , , , − + (3 , , − , + (1 , − , − , . (C.349)By inspection, all of the representations of the MSSM are present in the abovedecompositions. SU (7) ⊃ SU (2) × SU (5) × U (1) The representations of SU (7) now decompose as: SU (7) ⊃ SU (2) × SU (5) × U (1) (C.350)7 → (2 , + (1 , − (C.351)21 → (1 , + (1 , − + (2 , (C.352)35 → (1 , + (2 , + (1 , − (C.353)48 → (3 , + (1 , + (2 , − + (2 , . (C.354)In order to retain an SU (3) subgroup, an instanton must take values in an appropriate U (1) or SU (2) subgroup of SU (5). As before, a generic U (1) instanton will yieldthe expected MSSM spectrum. If we instead consider an SU (2) × U (1) instanton,it is also immediate that we can again obtain the desired spectrum of the MSSM.This alternative breaking pattern has the added benefit that it contains one lessextraneous U (1) factor. 166 U (7) ⊃ SU (3) × SU (4) × U (1) The representations of SU (7) decompose as: SU (7) ⊃ SU (3) × SU (4) × U (1) (C.355)7 → (3 , + (1 , − (C.356)21 → (3 , + (3 , + (1 , − (C.357)35 → (1 , + (3 , + (3 , − + (1 , − (C.358)48 → (1 , + (8 , + (1 , + (3 , + (3 , − . (C.359)First suppose that the instanton configuration preserves the SU (3) subgroup of SU (4) ⊃ SU (3) × U (1). Such an instanton must then also preserve an SU (2)subgroup of the first SU (3) factor so that the resulting U (1) instanton reduces tothe generic situation treated previously.Alternatively, an instanton can preserve all of the first SU (3) factor and break SU (4) to a smaller subgroup. To this end, recall that the maximal subgroups of SU (4) which can contain an SU (2) subgroup are: SU (4) ⊃ SU (3) × U (1) (C.360) SU (4) ⊃ SU (2) × SU (2) × U (1) (C.361) SU (4) ⊃ U Sp (4) (C.362) SU (4) ⊃ SU (2) × SU (2). (C.363)In order to preserve an SU (2) subgroup, the first case necessarily descends to thepreviously treated case of a U (1) instanton. We therefore focus on the remainingcases. SU (7) ⊃ SU (3) × SU (4) × U (1) ⊃ SU (3) × [ SU (2) × SU (2) × U (1)] × U (1)In this case, we note that the resulting nested sequence of maximal subgroupsdescends to the same subgroup as: SU (7) ⊃ SU (2) × SU (5) × U (1) ⊃ SU (2) × [ SU (3) × SU (2) × U (1)] × U (1) (C.364)whose breaking patterns have already been analyzed. SU (7) ⊃ SU (3) × SU (4) × U (1) ⊃ SU (3) × U Sp (4) × U (1)167nder this subgroup, the representations of SU (7) decompose as: SU (7) ⊃ SU (3) × SU (4) × U (1) ⊃ SU (3) × U Sp (4) × U (1) (C.365)7 → (3 , + (1 , − (C.366)21 → (3 , + (3 , + (1 , − + (1 , − (C.367)35 → (1 , + (3 , + (3 , − + (3 , − + (1 , − (C.368)48 → (1 , + (8 , + (1 , + (1 , + (3 , + (3 , − (C.369)there are two possible maximal subgroups of U Sp (4) which can be reached by ageneral breaking pattern: a ) : U Sp (4) ⊃ SU (2) × SU (2) (C.370) b ) : U Sp (4) ⊃ SU (2) × U (1). (C.371)We first consider the decomposition with respect to case a ): SU (7) ⊃ SU (3) × U Sp (4) × U (1) ⊃ SU (3) × [ SU (2) × SU (2)] × U (1) (C.372)7 → (3 , , + (1 , , − + (1 , , − (C.373)21 → (3 , , + (3 , , + (3 , , + (1 , , − (C.374)+ (1 , , − + (1 , , − (C.375)35 → (1 , , + (3 , , + (3 , , + (3 , , − (C.376)+ (3 , , − + (3 , , − + (1 , , − + (1 , , − (C.377)48 → (1 , , + (8 , , + (1 , , + (1 , , (C.378)+ (1 , , + (1 , , + (1 , , + (3 , , (C.379)+ (3 , , + (3 , , − + (3 , , − . (C.380)Without loss of generality, we may consider an instanton which breaks the first SU (2)factor to either the trivial group, or a U (1) subgroup. Indeed, we find that evenwhen the instanton configuration contains a non-abelian factor, it is possible to reach168he MSSM spectrum: Q U D L E H u H d (3 , , (3 , , − (3 , , (1 , , − (1 , , (1 , , (1 , , − .(C.381)Note that in this case, an SU (2) × U (1) instanton will break SU (7) directly to G std with no extraneous U (1) factors.Next consider the decomposition with respect to case b ): SU (7) ⊃ SU (3) × U Sp (4) × U (1) ⊃ SU (3) × [ SU (2) × U (1)] × U (1) (C.382)7 → (3 , , + (1 , , − + (1 , − , − (C.383)21 → (3 , , + (3 , , + (3 , − , + (1 , , − (C.384)+ (1 , , − + (1 , − , − + (1 , , − (C.385)35 → (1 , , + (3 , , + (3 , − , + (3 , , − (C.386)+ (3 , , − + (3 , − , − + (3 , , − + (1 , , − + (1 , − , − (C.387)48 → (1 , , + (8 , , + (1 , , + (1 , , + (1 , − , (C.388)+ (1 , , + (1 , , + (1 , − , + (3 , , (C.389)+ (3 , − , + (3 , , − + (3 , − , − . (C.390)In fact, with respect to the corresponding U (1) × U (1) instanton, we find that thematter content again organizes into the precise analogue of line (C.381) in this case aswell. We therefore conclude that these candidate breaking patterns can in principlebe used to eliminate additional U (1) factors. SU (7) ⊃ SU (3) × SU (4) × U (1) ⊃ SU (3) × SU (2) × SU (2) × U (1)The final case of interest proceeds via a different embedding of SU (2) × SU (2)169n SU (4) such that: SU (7) ⊃ SU (3) × SU (4) × U (1) ⊃ SU (3) × [ SU (2) × SU (2)] × U (1) (C.391)7 → (3 , , + (1 , , − (C.392)21 → (3 , , + (3 , , + (1 , , − + (1 , , − (C.393)35 → (1 , , + (3 , , + (3 , , − + (3 , , − + (1 , , − (C.394)48 → (1 , , + (8 , , + (1 , , + (1 , , + (1 , , (C.395)+ (3 , , + (3 , , − . (C.396)Although this decomposition is indeed different from that presented below line (C.372),we note that an SU (2) instanton can generate a very similar breaking pattern. In-deed, under the forgetful homomorphism which trivializes all representations of thefirst SU (2) factor, we find that the two decompositions are in fact identical. Inparticular, this implies that a similar packaging of the field content of the MSSM asin line (C.381) will hold in this case as well. C.3.3 SO (12)We now proceed to the final rank six bulk gauge group which can occur in a candidateF-theory GUT model. Starting from the adjoint representation of E , the mattercontent of the bulk SO (12) theory descends from the vector 12, the spinors 32, 32 ′ and adjoint 66. The maximal subgroups of SO (12) are: SO (12) ⊃ SU (6) × U (1) (C.397) SO (12) ⊃ SU (2) × SU (2) × SO (8) (C.398) SO (12) ⊃ SU (4) × SU (4) (C.399) SO (12) ⊃ SO (10) × U (1) (C.400) SO (12) ⊃ SO (11) (C.401) SO (12) ⊃ SU (2) × SO (9) (C.402) SO (12) ⊃ SU (2) × U Sp (6) (C.403) SO (12) ⊃ U Sp (4) × SO (7). (C.404) SO (12) ⊃ SU (2) × SU (2) × SU (2) (C.405)170f which all but the last entry contain G std . As in previous examples, our expecta-tion is that many distinct nested sequences of maximal subgroups can describe thebreaking pattern of the same instanton configuration. SO (12) ⊃ U Sp (4) × SO (7) The decomposition of representations of SO (12) is: SO (12) ⊃ U Sp (4) × SO (7) (C.406)12 → (5 , 1) + (1 , 7) (C.407)32 , ′ → (4 , 8) (C.408)66 → (10 , 1) + (1 , 21) + (5 , 7) (C.409)Of the two simple group factors, only SO (7) contains an SU (3) subgroup. Further,while G and SU (4) are the two maximal subgroups of SO (7) which contain an SU (3)subgroup, an instanton can only break SO (7) to SU (3) via the SU (4) path. Furtherdecomposing with respect to the nested sequence SO (7) ⊃ SU (4) ⊃ SU (3) × U (1)therefore yields: SO (12) ⊃ U Sp (4) × SO (7) ⊃ U Sp (4) × SU (4) ⊃ U Sp (4) × SU (3) × U (1)(C.410)12 → (5 , + (1 , + (1 , + (1 , − (C.411)32 , ′ → (4 , + (4 , − + (4 , − + (4 , (C.412)66 → (10 , + (1 , + (1 , + (1 , − + (1 , (C.413)+ (1 , − + (1 , + (1 , − + (1 , + (1 , (C.414)+ (5 , + (5 , + (5 , − . (C.415)With conventions as in lines (C.370) and (C.371), we now decompose U Sp (4) withrespect to the two maximal subgroups which can break to an SU (2) factor in thepresence of an SU (2) factor. SO (12) ⊃ U Sp (4) × SO (7) ⊃ [ SU (2) × SU (2)] × [ SU (3) × U (1) b ]171irst consider the maximal subgroup U Sp (4) ⊃ SU (2) × SU (2): SO (12) ⊃ U Sp (4) × SO (7) ⊃ [ SU (2) × SU (2)] × [ SU (3) × U (1) b ] (C.416)12 → (1 , , + (2 , , + (1 , , + (1 , , + (1 , , − (C.417)32 , ′ → (2 , , + (1 , , + (2 , , − + (1 , , − + (2 , , − (C.418)+ (1 , , − + (2 , , + (1 , , (C.419)66 → (3 , , + (1 , , + (2 , , + (1 , , + (1 , , (C.420)+ (1 , , − + (1 , , + (1 , , − + (1 , , + (1 , , − (C.421)+ (1 , , + (1 , , + (1 , , + (2 , , + (1 , , (C.422)+ (1 , , − + (2 , , − . (C.423)In this case it follows that an SU (2) instanton cannot yield the correct U (1) Y as-signments for the fields of the MSSM. If we instead consider a U (1) instanton whichbreaks one of the SU (2) factors to U (1) a , the following combinations of represen-tations satisfy the requirements that all U (1) Y charge assignments are correct andfurther, that all interaction terms are consistent with gauge invariance of the parenttheory: Q U D L , , (1 − , , − (1 , , − (1 , , − , , (1 , , − (1 − , , − (1 , , − (C.424) E H u H d ( a, b )1 (1 , , (1 , , (1 − , , (3 , − , , (1 − , , (1 , , ( − , 1) . (C.425) SO (12) ⊃ U Sp (4) × SO (7) ⊃ [ SU (2) × U (1)] × [ SU (3) × U (1) b ]172ext consider the maximal subgroup U Sp (4) ⊃ SU (2) × U (1): SO (12) ⊃ U Sp (4) × SO (7) ⊃ [ SU (2) × U (1) a ] × [ SU (3) × U (1) b ] (C.426)12 → (1 , ) + (1 − , ) + (3 , ) + (1 , ) + (1 , ) + (1 , − ) (C.427)32 , ′ → (2 , ) + (2 − , ) + (2 , − ) + (2 − , − ) + (2 , − ) (C.428)+ (2 − , − ) + (2 , ) + (2 − , ) (C.429)66 → (1 , ) + (3 , ) + (3 , ) + (3 − , ) + (1 , ) + (1 , ) (C.430)+ (1 , − ) + (1 , ) + (1 , − ) + (1 , ) + (1 , − ) + (1 , ) (C.431)+ (1 , ) + (1 , ) + (1 − , ) + (3 , ) + (1 , ) + (1 − , ) (C.432)+ (3 , ) + (1 , − ) + (1 − , − ) + (3 , − ) (C.433)Listing all possible Q -, U - and H u -fields we find: Q U H u (2 ± , ) (1 , − ) or (1 , ) or (1 ± , − ) (2 ± , ± ) . (C.434)Note in particular that in this case, it is not possible to form a gauge invariant QU H u ,so this path is excluded. SO (12) ⊃ SU (2) × U Sp (6) Because there is a unique maximal subgroup of U Sp (6)which contains an SU (3) factor, we may perform the unique decomposition: SO (12) ⊃ SU (2) × U Sp (6) ⊃ SU (2) × [ SU (3) × U (1)] (C.435)12 → (2 , + (2 , − (C.436)32 → (4 , + (2 , − + (2 , + (2 , (C.437)32 ′ → (3 , + (3 , − + (1 , + (1 , − + (1 , − + (1 , (C.438)66 → (3 , + (1 , + (1 , + (1 , − + (1 , + (3 , − (C.439)+ (3 , + (3 , . (C.440)By inspection, the relative U (1) Y charge assignments for the E - and Q -fields areincorrect. We therefore conclude that this breaking pattern is not viable.173 O (12) ⊃ SU (2) × SO (9) The decomposition of SO (12) representations in thiscase yields: SO (12) ⊃ SU (2) × SO (9) (C.441)12 → (3 , 1) + (1 , 9) (C.442)32 , ′ → (2 , 16) (C.443)66 → (3 , 1) + (1 , 36) + (3 , SO (9) which contain an SU (3) factor via anested sequence of maximal subgroups: SO (9) ⊃ SU (2) × SU (4) ⊃ SU (2) × SU (3) × U (1) (C.445) SO (9) ⊃ SO (8) ⊃ SO (7) ⊃ SU (4) ⊃ SU (3) × U (1) (C.446) SO (9) ⊃ SO (8) ⊃ SU (4) × U (1) ⊃ SU (3) × U (1) × U (1) (C.447) SO (9) ⊃ SO (7) × U (1) ⊃ SU (4) × U (1) ⊃ SU (3) × U (1) × U (1). (C.448)By inspection, the U (1) × U (1) valued instanton associated with the last two nestedsequences yield identical breaking patterns. SO (12) ⊃ SU (2) × SO (9) ⊃ SU (2) × SU (2) × SU (4)Decomposing the representations of SO (12) with respect to this breaking patternyields: SO (12) ⊃ SU (2) × SO (9) ⊃ SU (2) × SU (2) × SU (4) (C.449)12 → (3 , , 1) + (1 , , 1) + (1 , , 6) (C.450)32 , ′ → (2 , , 4) + (2 , , 4) (C.451)66 → (3 , , 1) + (1 , , 1) + (1 , , 15) + (1 , , 16) + (3 , , 1) + (3 , , 6) (C.452)In this case, the analysis of breaking patterns is similar to that of the maximalsubgroup SO (10) ⊃ SU (2) × SU (2) × SU (4). We therefore conclude that theappropriate U (1) × U (1) instanton configuration can produce the spectrum of theMSSM. SO (12) ⊃ SU (2) × SO (9) ⊃ SU (2) × SO (8) ⊃ SU (2) × SO (7) ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × U (1)]174n this case, the decomposition to the appropriate subgroup does not yield aviable candidate for the E -field: SO (12) ⊃ ... ⊃ SU (2) × SU (4) ⊃ SU (2) × [ SU (3) × U (1)] (C.453)12 → (3 , + (1 , + (1 , + (1 , + (1 , + (1 , − (C.454)32 , ′ → (2 , + (2 , − + (2 , − + (2 , + (2 , + (2 , − (C.455)+ (2 , − + (2 , (C.456)66 → (3 , + (1 , + (1 , + (1 , + (1 , + (1 , − (C.457)+ (1 , + (1 , − + (1 , + (1 , − + (1 , + (1 , (C.458)+ (1 , + (1 , − + (3 , + (3 , + (3 , + (3 , (C.459)+ (3 , − (C.460) SO (12) ⊃ SU (2) × SO (9) ⊃ SU (2) × SO (8) ⊃ SU (2) × SU (4) × U (1) ⊃ SU (2) × SU (3) × U (1) × U (1)The decomposition to G std now yields: SO (12) ⊃ ... ⊃ SU (2) × SU (4) × U (1) (C.461) ⊃ SU (2) × SU (3) × U (1) a × U (1) b (C.462)12 → (3 , , + (1 , , + (1 , , − + (1 , , (C.463)+ (1 , , + (1 , − , (C.464)32 , ′ → (2 , , + (2 , − , + (2 , − , + (2 , , (C.465)+ (2 , − , − + (2 , , − + (2 , , − + (2 , − , − (C.466)66 → (3 , , + (1 , , + (1 , , + (1 , , (C.467)+ (1 , − , + (1 , , − + (1 , , − + (1 , − , − (C.468)+ (1 , , + (1 , − , + (1 , , + (1 , − , (C.469)+ (1 , , + (1 , , + (3 , , + (3 , , − (C.470)+ (3 , , + (3 , , + (3 , − , . (C.471)175n this case, the candidate E - and Q -fields yield the relations: E : ± b = 6 (C.472) Q : − a ± b = 1 (C.473)so that b = ± a = 2 or − 4. Because the candidate L -fields all descend from(2 , ± , ± , we further deduce that a = 2. Without loss of generality, we fix the signof b = +3. This in turn implies that the representation content of the remainingfields is now fixed to be: Q U D L E H u H d (2 , − , − (1 , − , (1 , − , (2 , − , (1 , , (2 , , − (2 , − , .(C.474)Because some of the necessary interaction terms of the MSSM are now forbidden bygauge invariance of the parent theory, we conclude that this does not yield a viablebreaking pattern. SO (12) ⊃ SO (11) In this case, the breaking patterns of SO (12) directly descendto the analysis of SO (11) breaking patterns previously analyzed. Indeed, the rep-resentations of SO (12) descend as: SO (12) ⊃ SO (11) (C.475)12 → , ′ → 32 (C.477)66 → 11 + 55. (C.478) SO (12) ⊃ SU (6) × U (1) First recall that the maximal subgroups of SU (6) whichcontain SU (3) × SU (2) are: SU (6) ⊃ SU (5) × U (1) (C.479) SU (6) ⊃ SU (2) × SU (4) × U (1) (C.480) SU (6) ⊃ SU (3) × SU (3) × U (1) (C.481) SU (6) ⊃ SU (2) × SU (3). (C.482)176n the first three cases we find that the resulting breaking pattern must descendto the usual breaking pattern via a U (1) instanton. Finally, by inspection of thedecomposition of SO (12) ⊃ SU (6) × U (1), we note that the resulting integral U (1)charges of each decomposition are bounded in magnitude by two. Hence, only thefirst three maximal subgroups can yield a consistent breaking pattern. While itwould be of interest to classify all possible ways of packaging the field content of theMSSM in representations of SO (12) in this case, this analysis is not necessary forthe purposes of classifying breaking patterns. SO (12) ⊃ SU (2) × SU (2) × SO (8) Decomposing all relevant representations of SO (12) with respect to this maximal subgroup yields: SO (12) ⊃ SU (2) × SU (2) × SO (8) (C.483)12 → (2 , , 1) + (1 , , v ) (C.484)32 → (1 , , s ) + (2 , , c ) (C.485)32 ′ → (1 , , c ) + (2 , , s ) (C.486)66 → (3 , , 1) + (1 , , 1) + (1 , , 28) + (2 , , v ). (C.487)There are two maximal subgroups of SO (8) which are consistent with a breakingpattern generated by an instanton configuration: SO (8) ⊃ SU (4) × U (1) (C.488) SO (8) ⊃ SO (7) ⊃ SU (4). (C.489)We now consider breaking patterns which can descend from both maximal subgroups. SO (12) ⊃ SU (2) × SU (2) × SO (8) ⊃ SU (2) × SU (2) × [ SU (4) × U (1)]Because the only simple group factor which contains an SU (3) subgroup is SU (4),177e may further decompose SU (4) ⊃ SU (3) × U (1). This yields: SO (12) ⊃ SU (2) × SU (2) × SO (8) ⊃ SU (2) × SU (2) × [ SU (4) × U (1)] (C.490) ⊃ SU (2) × SU (2) × [ SU (3) × U (1) × U (1)] (C.491)12 → (2 , , , + (1 , , , + (1 , , , − (C.492)+ (1 , , , + (1 , , − , (C.493)32 → (1 , , , + (1 , , − , + (1 , , − , − (C.494)+ (1 , , , − + (2 , , , − + (2 , , − , − (C.495)+ (2 , , − , + (2 , , , (C.496)32 ′ → (2 , , , + (2 , , − , + (2 , , − , − (C.497)+ (2 , , , − + (1 , , , − + (1 , , − , − (C.498)+ (1 , , − , + (1 , , , (C.499)66 → (3 , , , + (1 , , , + (1 , , , (C.500)+ (1 , , , + (1 , , − , + (1 , , , − (C.501)+ (1 , , − , − + (1 , , , + (1 , , − , (C.502)+ (1 , , , + (1 , , , + (2 , , , (C.503)+ (2 , , , − + (2 , , , + (2 , , − , . (C.504)If we now consider a U (1) instanton which breaks one of the SU (2) factor, weagain obtain a U (1) instanton configuration. Indeed, this case is quite similar tobreaking via the maximal subgroup SU (2) × SU (2) × SU (4) ⊂ SO (10) consideredpreviously.Next suppose without loss of generality that an instanton configuration takesvalues in the first SU (2) factor such that it breaks either to U (1) or trivial group.Because the abelian case is quite similar, we assume that the non-abelian instantonbreaks all of SU (2). In this case, the list of candidate Q -, U - and H u -fields which178an yield a gauge invariant QU H u interaction are: Q U H u ( a, b )1 (1 , , − , (1 , , − , (1 , , , − (2 , , , − , (2 , , , (2 , , , − ( − / , − / , , − , (2 , , , − OU T OU T , , − , − (1 , , − , (1 , , , (2 , − , , − , − (2 , , , OU T OU T , , − , − (2 , , , − (2 , , , ( − / , / , , , (1 , , − , OU T OU T , , , (2 , , , (1 , , − , − (1 / , − / , , , (2 , , , − (1 , , − , (1 / , / 2) . (C.505)Restricting to the six viable remaining possibilities, we now find that no candidate D -field reproduces the correct U (1) Y charge assignment. We therefore conclude thatonly abelian instanton configurations can yield the spectrum of the MSSM in thiscase. SO (12) ⊃ SU (2) × SU (2) × SO (8) ⊃ SU (2) × SU (2) × SO (7) ⊃ SU (2) × SU (2) × SU (4) ⊃ SU (2) × SU (2) × [ SU (3) × U (1)]Along this nested sequence of maximal subgroups, the decomposition of the rep-resentations of SO (12) is: SO (12) ⊃ SU (2) × SU (2) × SO (8) ⊃ SU (2) × SU (2) × SO (7) (C.506) ⊃ SU (2) × SU (2) × SU (4) ⊃ SU (2) × SU (2) × [ SU (3) × U (1) b ] (C.507)12 → (2 , , + (1 , , + (1 , , + (1 , , + (1 , , − (C.508)32 , ′ → (1 , , + (1 , , − + (1 , , − + (1 , , + (2 , , (C.509)+ (2 , , − + (2 , , − + (2 , , (C.510)66 → (3 , , + (1 , , + (1 , , + (1 , , + (1 , , − (C.511)+ (1 , , + (1 , , − + (1 , , + (1 , , − + (1 , , (C.512)+ (1 , , + (2 , , + (2 , , + (2 , , + (2 , , − . (C.513)179y inspection, the above U (1) b charge assignments do not agree with those ofthe MSSM. It thus follows that we must further break one of the SU (2) factors to U (1). Without loss of generality, we assume that the first SU (2) factor decomposesfurther to a maximal U (1) a subgroup. The list of candidate Q -, U - and H u -fieldswhich can yield a gauge invariant QU H u interaction are therefore: Q U H u ( a, b )1 (1 , , − (1 , , − (1 , , OU T , , − (1 , , (1 − , , ( − , − , , − (1 − , , (1 , , (3 , − , , − (1 , , (1 , , − ( a, − , , (1 , , − (1 − , , ( − , , , (1 , , OU T OU T , , (1 − , , (1 , , − (3 , − , , (1 , , OU T OU T − , , (1 , , − (1 , , (3 , − , , (1 , , (1 , , − ( − , − − , , (1 − , , OU T OU T 12 (1 − , , (1 , , OU T OU T . (C.514)Next, we list all candidate D - and H d -fields which can yield a gauge invariant QDH d Q D H d ( a, b )2 a (1 , , − (1 , , − (1 , , ( − , − b (1 , , − (1 − , , (1 , , ( − , − a (1 , , − (1 , , − (1 , , (3 , − b (1 , , − (1 , , (1 − , , (3 , − a (1 , , − (1 , , − (1 , , ( a, − b (1 , , − (1 , , (1 − , , (3 , − c (1 , , − (1 − , , (1 , , ( − , − , , OU T OU T ( − , a (1 , , (1 , , − (1 − , , (3 , − b (1 , , (1 , , OU T (3 , − − , , (1 , , OU T (3 , a (1 − , , (1 , , − (1 , , ( − , − b (1 − , , (1 − , , OU T ( − , − 1) . (C.515)Of these remaining possibilities, we now determine all possible candidate L - and181 -fields which can yield the gauge invariant interaction term LEH d : L E H d ( a, b )2 a (1 , , (1 − , , − (1 , , ( − , − b (1 , , (1 − , , − (1 , , ( − , − b ′ (1 , , (1 − , , (1 , , ( − , − a (1 − , , (1 , , − (1 , , (3 , − b (1 − , , (1 , , (1 − , , (3 , − b ′ (1 , , (1 , , − (1 − , , (3 , − a (1 ± , , (1 ∓ , , − (1 , , ( ∓ , − b (1 − , , (1 , , (1 − , , (3 , − b ′ (1 , , (1 , , − (1 − , , (3 , − c (1 , , (1 − , , (1 , , ( − , − c ′ (1 , , (1 − , , − (1 , , ( − , − a (1 − , , (1 , , (1 − , , (3 , − a ′ (1 , , (1 , , − (1 − , , (3 , − a (1 , , (1 − , , (1 , , ( − , − a ′ (1 , , (1 − , , − (1 , , ( − , − 1) . (C.516)Note that in this case, there are many distinct ways to package the field content ofthe MSSM such that SO (12) breaks to SU (3) × SU (2) × U (1) × U (1) via a U (1) instanton configuration. 182 O (12) ⊃ SU (4) × SU (4) Decomposing representations of SO (12) with respect tothe maximal subgroup SU (4) × SU (4) yields: SO (12) ⊃ SU (4) × SU (4) (C.517)12 → (6 , 1) + (1 , 6) (C.518)12 → (6 , 1) + (1 , 6) (C.519)32 ′ → (4 , 4) + (4 , 4) (C.520)66 → (15 , 1) + (1 , 15) + (6 , SU (4) factor further breaks to SU (3) × U (1). The remaining nested sequences of maximal subgroups which canyield the Standard Model gauge group are: SU (4) ⊃ SU (2) × SU (2) × U (1) (C.522) SU (4) ⊃ U Sp (4) ⊃ SU (2) × SU (2) (C.523) SU (4) ⊃ U Sp (4) ⊃ SU (2) × U (1) (C.524) SU (4) ⊃ SU (2) × SU (2). (C.525) SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × SU (2) × SU (2) × U (1)In this case, it follows at once from the local isomorphisms SU (4) ≃ SO (6) and SU (2) × SU (2) ≃ SO (4) that the endpoint of this breaking pattern is identical tothe endpoint of the nested sequence of maximal subgroups: SO (12) ⊃ SU (2) × SU (2) × SO (8) ⊃ SU (2) × SU (2) × SU (4) × U (1) (C.526) ⊃ SU (2) × SU (2) × [ SU (3) × U (1)] × U (1). (C.527)We therefore conclude that all breaking patterns via instantons have in this casebeen catalogued. SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × U Sp (4) ⊃ [ SU (3) × U (1) a ] × [ SU (2) × SU (2)]The decomposition of the representations of SO (12) with respected to this se-183uence of maximal subgroups is: SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × U Sp (4) (C.528) ⊃ [ SU (3) × U (1) a ] × [ SU (2) × SU (2)] (C.529)12 → (3 , , + (3 , , − + (1 , , + (1 , , + (1 , , (C.530)32 , ′ → (1 , , + (1 , , + (3 , , − + (3 , , − + (1 , , − (C.531)+ (1 , , − + (3 , , + (3 , , (C.532)66 → (1 , , + (3 , , − + (3 , , + (8 , , + (1 , , (C.533)+ (1 , , + (1 , , + (1 , , + (1 , , + (3 , , (C.534)+ (3 , , − + (3 , , + (3 , , + (3 , , − + (3 , , − . (C.535)By inspection of the above representation content, we note that while an SU (2)instanton which breaks either of the SU (2) factors could yield the correct gaugegroup, the resulting U (1) Y charge assignments of the fields would be incorrect. Itis therefore enough to consider abelian instanton configurations which break one ofthe SU (2) factors to U (1) b . Due to the symmetry between the two SU (2) factors,we assume without loss of generality that the instanton preserves the first SU (2)factor. We begin by listing the candidate representations for the Q -, U - and H u -fields which can yield the interaction term QU H u as well as the correct U (1) Y chargeassignments: Q U H u ( a, b )1 (3 , , ) − (3 , , ± ) (1 , , ∓ ) ( ∓ , − , , ) − (3 , , ) (1 , , ) − ( a, − , , ± ) (3 , , ) − (1 , , ∓ ) ( ∓ , , , ± ) (3 , , ∓ ) (1 , , ) − ( ± , − 1) (C.536)where in the above, all ± ’s of a given row are correlated. Of these four possibilities,we now list all candidate representations for the D - and H d -fields which can yield184he interaction term QDH d : Q D H d ( a, b )1 a (3 , , ) − (3 , , ∓ ) (1 , , ± ) ( ∓ , − b (3 , , ) − (3 , , ) − (1 , , ) ( ∓ , − a (3 , , ) − (3 , , ∓ ) (1 , , ± ) ( ∓ , − b (3 , , ) − (3 , , ) − (1 , , ) ( a, − , , ± ) (3 , , ) − (1 , , ∓ ) ( ± , − 1) . (C.537)Finally, we list all candidate E - and L - fields which can yield the term ELH d : E L H d ( a, b )1 a (1 , , ∓ ) − (1 , , ) (1 , , ± ) ( ∓ , − a ′ (1 , , ∓ ) (1 , , ± ) (1 , , ± ) ( ∓ , − b (1 , , ∓ ) − (1 , , ± ) (1 , , ) ( ∓ , − a (1 , , ∓ ) − (1 , , ) (1 , , ± ) ( ∓ , − a ′ (1 , , ∓ ) (1 , , ± ) (1 , , ± ) ( ∓ , − b (1 , , ∓ ) − (1 , , ± ) (1 , , ) ( ∓ , − , , ± ) − (1 , , ) (1 , , ∓ ) ( ± , − ′ (1 , , ± ) (1 , , ∓ ) (1 , , ∓ ) ( ± , − 1) . (C.538)We note that in this case, while there are only two linear combinations of the two U (1) factors which can yield U (1) Y , there are different ways to package the fields ofthe MSSM in representations of SO (12). SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × U Sp (4) ⊃ [ SU (3) × U (1) a ] × [ SU (2) × U (1) b ] 185n this case, the decomposition of representations of SO (12) yields: SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × U Sp (4) (C.539) ⊃ [ SU (3) × U (1) a ] × [ SU (2) × U (1) b ] (C.540)12 → (3 , ) + (3 − , ) + (1 , ) + (1 , ) + (1 , − ) + (1 , ) (C.541)32 , ′ → (1 , ) + (1 , − ) + (3 − , ) + (3 − , − ) (C.542)+ (1 − , ) + (1 − , − ) + (3 , ) + (3 , − ) (C.543)66 → (1 , ) + (3 − , ) + (3 , ) + (8 , ) + (1 , ) + (1 , − ) (C.544)+ (1 , ) + (1 , ) + (1 , ) + (1 , ) + (1 , − ) (C.545)+ (3 , ) + (3 − , ) + (3 , ) + (3 , − ) + (3 , ) (C.546)+ (3 − , ) + (3 − , − ) + (3 − , ). (C.547)We note in passing that this indeed yields a distinct decomposition from the previousbreaking pattern. By inspection, the only candidate E -fields are (1 , ± ) so that b = ± 3. Listing all Q -, U - and H u -fields which can yield a gauge invariant interactionterm QU H u such that b = ± Q U H u ( a, b )(3 − , ± ) (3 − , ) (1 , ∓ ) (2 , ± 3) (C.548)where all ± ’s in a given row are correlated. Listing all Q -, D - and H d - fields whichcan yield the term QDH d , we find: Q D H d ( a, b )(3 − , ± ) (3 − , ∓ ) (1 − , ± ) (2 , ± 3) . (C.549)Now, we find that in this case, the only candidate L - and H d -fields are (1 − , ± ). Inparticular, it follows that the purported ELH d interaction will violate U (1) a becausethe only candidate E -field is neutral under U (1) a so that this breaking pattern cannotyield the spectrum of the MSSM. SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × [ SU (2) × SU (2)]186n this case, the decomposition of the representations of SO (12) is given by: SO (12) ⊃ SU (4) × SU (4) ⊃ [ SU (3) × U (1) a ] × [ SU (2) × SU (2)] (C.550)12 → (3 , , 1) + (3 − , , 1) + (1 , , 1) + (1 , , 3) (C.551)32 , ′ → (1 , , 2) + (3 − , , 2) + (1 − , , 2) + (3 , , 2) (C.552)66 → (1 , , 1) + (3 − , , 1) + (3 , , 1) + (8 , , 1) + (1 , , 3) (C.553)+ (1 , , 1) + (1 , , 3) + (3 , , 3) + (3 , , 1) + (3 − , , 3) (C.554)+ (3 − , , SU (2) factors to a U (1) b subgroup. Without loss of generality, we assume thatthe instanton preserves the first SU (2) factor. In this case, the resulting candidate E -fields are all of the form (1 , , ± ) so that b = ± 3. Listing all candidate Q -, U -and H u -fields which can yield a gauge invariant term of the form QU H u , we find: Q U H u ( a, b )(3 − , , ± ) (3 − , , ) (1 , ∓ ) (2 , ± 3) (C.556)where all ± ’s are correlated in the above. This in turn implies that there is a uniquecandidate H d -field given by (1 − , ± ). This in turn requires that in order to obtaina non-zero QDH d interaction term, a candidate D -field must have representationcontent (3 , , ∓ ) which is not present in the given decomposition described above.We therefore conclude that this breaking pattern cannot yield the spectrum of theMSSM. SO (12) ⊃ SO (10) × U (1) This is the final maximal subgroup of SO (12) which canin principle contain G std . The representation content of SO (12) decomposes under187his maximal subgroup as: SO (12) ⊃ SO (10) × U (1) (C.557)12 → + 1 − + 10 (C.558)32 → + 16 − (C.559)32 ′ → + 16 − (C.560)66 → + 10 + 10 − + 45 . (C.561)Recall that the maximal subgroups of SO (10) are listed in lines (C.88)-(C.94), ofwhich only lines (C.88)-(C.91) contain an SU (3) × SU (2) subgroup. In the presentcontext, we wish to determine whether the presence of the additional U (1) factor canyield a new breaking pattern distinct from those already treated for G S = SO (10).Moreover, while it is in principle of interest to classify all ways of packaging the fieldsof the MSSM into SO (12) representations, our primary interest is in the classificationof all possible breaking patterns. For this reason, we again confine our classificationto this more narrow question. SO (12) ⊃ SO (10) × U (1) ⊃ SU (5) × U (1) × U (1)In this case, there is a unique way in which the SU (5) factor can further breakto G std . Indeed, this is the natural extension of the analogous breaking pattern of SO (10) analyzed previously. We thus conclude that in this case the abelian U (1) instanton breaks SO (12) to G std × U (1) × U (1). SO (12) ⊃ SO (10) × U (1) ⊃ [ SU (2) × SU (2) × SU (4)] × U (1)Under this nested sequence of maximal subgroups, SU (4) is the only factor whichcontains an SU (3) subgroup. The representation content of SO (12) therefore must188ecompose as: SO (12) ⊃ SO (10) × U (1) ⊃ [ SU (2) × SU (2) × SU (4)] × U (1) b (C.562) ⊃ [ SU (2) × SU (2) × [ SU (3) × U (1) a ]] × U (1) b (C.563)12 → (1 , , , + (1 , , , − + (2 , , , + (1 , , , + (1 , , − , (C.564)32 → (2 , , , + (2 , , − , + (1 , , − , + (1 , , , (C.565)+ (2 , , − , − + (2 , , , − + (1 , , , − + (1 , , − , − (C.566)32 ′ → (1 , , , + (1 , , − , + (2 , , − , + (2 , , , (C.567)+ (1 , , − , − + (1 , , , − + (2 , , , − + (2 , , − , − (C.568)66 → (1 , , , + (2 , , , + (1 , , , + (1 , , − , (C.569)+ (2 , , , − + (1 , , , − + (1 , , − , − + (3 , , , (C.570)+ (1 , , , + (1 , , , + (1 , , − , + (1 , , , (C.571)+ (1 , , , + (2 , , , + (2 , , − , . (C.572)In the present context, breaking one of the SU (2) factors to a U (1) subgroupyields a breaking pattern identical to that already studied in the context of thesequence of maximal subgroups SO (12) ⊃ SO (10) × U (1) ⊃ SU (5) × U (1) × U (1) ⊃ SU (3) × SU (2) × U (1) × U (1) × U (1). In order to classify all candidate breakingpatterns, it is therefore enough to restrict to cases where one of the SU (2) factorsis completely broken. Without loss of generality, we assume that the candidatenon-abelian instanton preserves the second SU (2) factor. Listing all candidate Q -, U - and H u -fields which can yield the gauge invariant interaction term QU H u , wefind: Q U H u ( a, b )1 (1 , , − , ± (1 , , − , (1 , , , ∓ (2 , ± , , − , ± (1 , , , (1 , , − , ∓ ( − , , , − , ± (1 , , − , ∓ (1 , , , ± (1 / , ± / , , − , ± (2 , , , ± (2 , , , ∓ ( − / , ∓ / , , , (2 , , , ± (1 , , − , ∓ (1 / , ∓ / , , , (1 , , − , ∓ (2 , , , ± (1 / , ± / 2) . (C.573)189isting all choices of representations for candidate D - and H d -fields which also admitthe gauge invariant interaction term QDH d , we find: Q D H d ( a, b )2 a (1 , , − , ± (1 , , − , (1 , , , ∓ ( − , b (1 , , − , ± (1 , , − , ∓ (1 , , , ± ( − , a (1 , , − , ± (1 , , , (1 , , − , ∓ (1 / , ± / b (1 , , − , ± (2 , , , ± (2 , , , ∓ (1 / , ± / a (2 , , , (1 , , − , ± (2 , , , ∓ (1 / , ± / b (2 , , , (2 , , , ± (1 , , − , ∓ (1 / , ± / 2) . (C.574)Because the only candidate E -fields are given by (1 , , , ± or (1 , , ± , ± , we nowobserve that all consistent choices of U (1) Y given previously cannot yield the correctvalue for the E -fields. Hence, an instanton configuration must break one of the SU (2) factors to a U (1) subgroup in order to reproduce the spectrum of the MSSM. SO (12) ⊃ SO (10) × U (1) ⊃ SO (9) × U (1)In order to obtain an SU (3) × SU (2) subgroup along this nested sequence ofmaximal subgroups, the SO (9) factor must also contain such a subgroup. Returningto lines (C.36)-(C.40), we again conclude that the only maximal subgroup of SO (9)satisfying this criterion is SU (2) × SU (4). Further decomposing the SU (4) factor tothe maximal subgroup SU (3) × U (1), the decomposition of representations of SO (12)190ow descends to: SO (12) ⊃ SO (10) × U (1) b ⊃ SO (9) × U (1) b (C.575) ⊃ [ SU (2) × SU (4)] × U (1) b (C.576) ⊃ [ SU (2) × [ SU (3) × U (1) a ]] × U (1) b (C.577)12 → (1 , , + (1 , , − + (1 , , + (3 , , (C.578)+ (1 , , + (1 , − , (C.579)32 , ′ → (2 , , + (2 , − , + (2 , − , + (2 , , (C.580)+ (2 , − , − + (2 , , − + (2 , , − + (2 , − , − (C.581)66 → (1 , , + (1 , , + (1 , , − + (3 , , (C.582)+ (3 , , − + (3 , , + (1 , , + (1 , − , (C.583)+ (1 , , − + (1 , − , − + (3 , , + (1 , , (C.584)+ (1 , , + (1 , − , + (1 , − , + (1 , , (C.585)+ (1 , , + (3 , , + (3 , − , . (C.586)Listing all Q -, U - and H u - fields which can yield the term QU H u , we find: Q U H u ( a, b )1 (2 , − , ± (1 , − , (2 , , ∓ (2 , ± , − , ± (1 , − , ∓ (2 , , ± (1 / , ± / , − , ± (1 , , (2 , − , ∓ ( − , 0) . (C.587)Because the candidate E -fields all descend from the representation (1 , , ± , it fol-lows that b = ± D -fields are therefore (1 , − , ± , where the ± sign iscorrelated with that given in the first case. In order to obtain a gauge invariant QDH d interaction term, the resulting H d -field must transform in the representation(2 , , ∓ , which does not descend from a representation of SO (12). We thereforeconclude that this breaking pattern cannot yield the spectrum of the MSSM. SO (12) ⊃ SO (10) × U (1) ⊃ [ SU (2) × SO (7)] × U (1)In this final case, SU (4) and G or the only maximal subgroups of SO (7) whichcontains an SU (3) subgroup. Of these two possibilities, an instanton can only break191he former case to SU (3). Decomposing the representations of SO (12) under thecorresponding nested sequence of maximal subgroups yields: SO (12) ⊃ SO (10) × U (1) v ⊃ [ SU (2) × SO (7)] × U (1) b ⊃ [ SU (2) × SU (4)] × U (1) b (C.588) ⊃ [ SU (2) × [ SU (3) × U (1) a ]] × U (1) b (C.589)12 → (1 , , + (1 , , − + (1 , , + (3 , , (C.590)+ (1 , , + (1 , − , (C.591)32 , ′ → (2 , , + (2 , − , + (2 , − , + (2 , , (C.592)+ (2 , − , − + (2 , , − + (2 , , − + (2 , − , − (C.593)66 → (1 , , + (1 , , + (1 , , − + (3 , , (C.594)+ (3 , , − + (3 , , + (1 , , + (1 , − , (C.595)+ (1 , , − + (1 , − , − + (3 , , + (1 , , (C.596)+ (1 , , + (1 , − , + (1 , − , + (1 , , (C.597)+ (1 , , + (3 , , + (3 , − , . (C.598)In fact, this decomposition is identical to that given for the previously considerednested sequence of maximal subgroups described by lines (C.575)-(C.586). Wetherefore conclude that just as in that case, this breaking pattern cannot yield thespectrum of the MSSM. 192 eferences [1] J. M. 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