GUTs and Exceptional Branes in F-theory - II: Experimental Predictions

GUTs and Exceptional Branes in F-theory - II: Experimental Predictions

arXiv:0806.0102

GUTs and Exceptional Branes in

[-0.25cm]F-theory - II:

[1cm]Experimental Predictions

Chris Beasley***e-mail: beasley@physics.harvard.edu, Jonathan J. Heckmane-mail: jheckman@fas.harvard.edu and Cumrun Vafae-mail: vafa@physics.harvard.edu

 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

May, 2008

Abstract

We consider realizations of GUT models in F-theory. Adopting a bottom up approach, the assumption that the dynamics of the GUT model can in principle decouple from Planck scale physics leads to a surprisingly predictive framework. An internal hypercharge flux Higgses the GUT group directly to the MSSM or to a flipped GUT model, a mechanism unavailable in heterotic models. This new ingredient automatically addresses a number of puzzles present in traditional GUT models. The internal hyperflux allows us to solve the doublet-triplet splitting problem, and explains the qualitative features of the distorted GUT mass relations for lighter generations due to the Aharanov-Bohm effect. These models typically come with nearly exact global symmetries which prevent bare terms and also forbid dangerous baryon number violating operators. Strong curvature around our brane leads to a repulsion mechanism for Landau wave functions for neutral fields. This leads to large hierarchies of the form where and are order one parameters and . This effect can simultaneously generate a viably small term as well as an acceptable Dirac neutrino mass on the order of eV. In another scenario, we find a modified seesaw mechanism which predicts that the light neutrinos have masses in the expected range while the Majorana mass term for the heavy neutrinos is GeV. Communicating supersymmetry breaking to the MSSM can be elegantly realized through gauge mediation. In one scenario, the same repulsion mechanism also leads to messenger masses which are naturally much lighter than the GUT scale.

1 Introduction

Despite many theoretical advances in our understanding of string theory, this progress has not produced a single verifiable prediction which can be tested against available experiments.  Part of the problem is that in its current formulation, string theory admits a vast landscape of consistent low energy vacua which look more or less like the real world.

Reinforcing this gloomy state of affairs is the fact that the particle content of the Standard Model is generically of the type encountered in string theory.  Indeed, the gauge group of the Standard Model is of the form and the chiral matter content corresponds to bi-fundamental fields transforming in representations such as .  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 specifically stringy principles.  For instance, we have learned that the large limit of many gauge theories causes the gauge system to ‘melt’ into a dual gravitational background [1].  Moreover, this large gauge theory can undergo a duality cascade to a small gauge theory [2].  Indeed, the Standard Model could potentially emerge at the end of such a process.  In the string theory literature, this idea has been explored in [3, 4, 5]. Interesting as this idea is, it does not incorporate the idea of grand unification 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 to incorporate the important fact that the gauge coupling constants of:

 Gstd≡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 low energies, but also suggests that the multiple gauge group factors of the Standard Model unify into a single simple group such as or .  Moreover, the fact that the matter content of the Standard Model economically organizes into representations of the groups and provides a strong hint that the basic idea of grand unified theories (GUTs) is correct. For example, it is quite intriguing that all of the chiral matter of a single generation precisely organizes into the spinor representation of .  Hence, we ask whether the principle of grand unification can narrow down the large list of candidate vacua in the landscape to a more tractable, and predictive subset.

Despite the many attractive features of the basic GUT framework, the simplest implementations of this idea in four-dimensional models suffer from some serious drawbacks.  For example, the minimal four-dimensional supersymmetric  GUT with standard Higgs content seems to be inconsistent with present bounds on proton decay [6].  In the absence of higher dimensional representations of or somewhat elaborate higher dimension operator contributions to the effective superpotential, this model also leads to mass relations and over-simplified mixing matrices which are generically too strong to be correct.  This presents an opportunity for string theory to intervene: Can string theory preserve the nice features of GUT models while avoiding their drawbacks?

Indeed, the heterotic string seems very successful in this regard because the usual GUT groups and can naturally embed in one of the factors.  See [7] for an early review on how GUT models could potentially originate from compactifying the heterotic string on a Calabi-Yau threefold.  Moreover, because no appropriate four-dimensional GUT Higgs field is typically available to break the GUT group to the Standard Model gauge group, it is necessary to employ a higher-dimensional breaking mechanism.  When the internal space has non-trivial fundamental group, the gauge group can break via a discrete Wilson line.  In this way, the gauge group in four dimensions is always the Standard Model gauge group but the matter content and gauge couplings still unify.  Moreover, such higher dimensional GUTs provide natural mechanisms to suppress proton decay and avoid unwanted mass relations.  See [8, 9, 10, 11, 12] for some recent attempts in this direction.

However, the heterotic string has its own drawbacks simply because it is rather difficult to break the gauge symmetry down to .111At a pragmatic level, the perturbative regime of the heterotic string also seems to be inconsistent with the relation between the GUT scale and the four-dimensional Planck scale .  A discussion of this discrepancy and related issues may be found in [13].  One potential way to bypass this problem requires going to the regime of strong coupling [14].  One popular method is to use internal Wilson lines to directly break the gauge symmetry to that of the MSSM.  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 is simply connected and this mechanism is unavailable.  Moreover, when the GUT group has rank five or higher, gauge group breaking by Wilson lines can also leave behind additional massless gauge bosons besides hypercharge.  Present constraints on additional long rang forces are quite stringent, and in many cases it is not 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 fundamental group, it therefore seems reasonable to look for other potential realizations of the GUT paradigm in string theory.

There are two other natural ways that GUTs can appear in string theory.  These possibilities correspond to non-perturbatively realized four-dimensional compactifications of type IIA and IIB string theory.  In the type IIA case, the GUT models originate from the compactification of M-theory on manifolds with holonomy.  For type IIB theories, the corresponding vacua are realized as compactifications of F-theory on Calabi-Yau fourfolds. In the latter case, the gauge theory degrees of freedom of the GUT localize on the worldvolume of a non-perturbative seven-brane.  The gauge group on the seven-brane corresponds to the discriminant locus of the elliptic model where the degeneration is locally of type.  Of these two possibilities, the holomorphic geometry of Calabi-Yau manifolds provides a more tractable starting point for addressing detailed model building issues.  It was with this aim that 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 distinction between the Planck scale and the GUT scale .  In the bottom up approach, the situation is completely reversed.  Indeed, insofar as effective field theory is valid at the GUT scale, it is quite important that is small and not an order one number.  For example, in the extreme situation where the only chiral matter content of a four-dimensional GUT model originates from the MSSM, the resulting theory is asymptotically free.

In geometrically engineered gauge theories in string theory, asymptotic freedom translates to the existence of a consistent decompactification limit.  It is therefore quite natural to ask if at least in principle we could have decoupled the two scales and .  This is also in accord with the bottom up approach to string phenomenology [18, 19, 20, 21].  In the present paper our main focus will therefore be to search for vacua which at least in principle admit a limit where while remains finite.  Of course, in realistic applications should also remain finite.  For completeness, we shall also present some examples of models where and cannot be decoupled.  In such cases, we note that it is not a priori clear whether the correct value of can be achieved.

Nevertheless, the mere existence of a decoupling limit turns out to endow the resulting candidate models with surprising predictive power.  It turns out that the only 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 internal geometry 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 surface because all of the other del Pezzo surfaces can be obtained from this one by blowing down various two-cycles.

At the next level of analysis, we must determine what kind of seven-brane should wrap the del Pezzo surface.  As explained in [15], realizing the primary ingredients of GUT models requires that the singularity type associated with the seven-brane should correspond to a subgroup of the exceptional group .  Because the Standard Model gauge group has rank four, this determines a lower bound on the rank of any putative GUT group. At rank four, is the only available GUT group.  Hence, the most ‘minimal’ choice is to have an seven-brane wrapping the del Pezzo 8 surface. We will indeed find that this minimal scenario is viable.  The upper bound on the rank of a candidate GUT group is six.  This bound comes about from the fact that if the rank is any higher, the model will generically contain localized light degrees of freedom at points on the del Pezzo surface which do not appear to admit a standard interpretation in gauge theory [22, 15].  This is because on complex 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 the del Pezzo with singularities of rank nine and higher which do not admit a standard gauge theoretic interpretation because is the maximal compact exceptional group.

In the minimal scenario where the seven-brane has gauge group , we find that there is an essentially unique mechanism by which the GUT group can break to a four-dimensional model with gauge group .  This breaking pattern occurs in vacua where the hypercharge flux in the internal directions of the seven-brane is non-trivial.  This mechanism is unavailable in heterotic compactifications because the hypercharge always develops a string scale mass via the Green-Schwarz mechanism [23]. As noted for example in [23], in order to preserve a massless hypercharge gauge boson, additional factors must mix non-trivially with this direction, which runs somewhat counter to the idea of grand unification. Nevertheless, for suitable values of the gauge coupling constants for these other factors, a semblance of unification can be maintained. See [24, 25, 26, 27] for further discussion on vacua of this type.

In F-theory, we show that there is no such generic obstruction.  This is a consequence of the fact that while the cohomology class of the flux on the seven-brane can be non-trivial, it can nevertheless represent a trivial class in the base of the F-theory compactification.  This topological condition is necessary and also sufficient for the corresponding four-dimensional gauge boson to remain massless.  An important consequence of this fact is that these F-theory vacua do not possess a heterotic dual.

The particular choice of internal flux which breaks the GUT group is also unique.  To see how this comes about, we first recall that the middle cohomology of the del Pezzo 8 surface splits as the span of the canonical class and the collection of two-cycles orthogonal to this one-dimensional lattice.  With respect to the intersection form on two-cycles, this orthogonal subspace corresponds to the root lattice of .  Moreover, the admissible fluxes of the hypercharge are in one to one correspondence with the roots of .  This restriction occurs because for more generic choices of flux, the low energy spectrum contains exotic matter which if present would ruin the unification of the gauge coupling constants.  In keeping with the general philosophy outlined in [15], we always specify the appropriate line bundle first and only then determine whether an appropriate Kähler class exists so that the vacuum is supersymmetric.  In this sense, there is a unique choice of flux because the Weyl group of acts transitively on the roots of . 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.222After our work appeared, this question has been studied in [28, 29].

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 the GUT model seven-brane and additional seven-branes in the full compactification.  Along these intersections, the rank of the singularity type increases by one.  This severely limits the available representation content so that the matter fields can only transform in the or along an enhancement to and the or for local enhancement to .

The internal hypercharge flux automatically distinguishes the Higgs fields from the other chiral matter content of the MSSM.  The Higgs fields localize on matter curves where the hypercharge flux is non-vanishing, and the chiral matter of the MSSM localizes on Riemann surfaces where the net flux vanishes.  In other words, the two-cycles for the Higgs curves intersect the root corresponding to this internal flux while all the other chiral matter of the MSSM localizes on two-cycles orthogonal to this choice of flux.  This internal choice of flux implies that the chiral matter content will always fill out complete representations of , while the Higgs doublets can never complete to full GUT multiplets.  Moreover, by a suitable choice of flux on the other seven-branes, the spectrum  will contain no extraneous Higgs triplets, thus solving the doublet-triplet splitting problem.  In certain cases, superheavy Higgs triplets can still cause the proton to decay too quickly.  In traditional four-dimensional GUT models the missing partner mechanism is often invoked to avoid generating dangerous dimension five operators which violate baryon number.  Here, this condition translates into the simple geometric condition that the Higgs up and down fields must localize on distinct matter curves.

In our study of Yukawa couplings, we shall occasionally encounter situations involving two fields charged under the GUT group and one neutral field (for example a interaction). In such cases, the neutral field lives on a matter curve normal to the del Pezzo which intersects this surface at a point. In order to determine the strength of the Yukawa couplings, we need to estimate the strength of the corresponding zero mode wave functions at the intersection point. It turns out that since the del Pezzo is strongly positively curved (), the normal geometry is negatively curved. Moreover, this leads to the wave function being either attracted to, or repelled away from our brane, depending on the choice of the gauge flux on the normal intersecting seven-branes. In one case the wave function is attracted to our seven-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 an exponentially small amplitude at our brane. The exponential hierarchy is given by where is a positive order one constant, is the radius of the normal geometry to the brane, and is the length associated to GUT. The estimate for depends on assumptions about how the geometry normal to our brane looks, and in particular to what extent it is tubular. We find that:

 R⊥RGUT=ε−γ (1.2)

where is a measure of the normal eccentricity and is a small parameter:

 ε∼MGUTαGUTMpl∼7.5×10−2. (1.3)

This leads to a natural hierarchy given by

 exp(−cR2⊥R2GUT)∼exp(−c1ε2γ). (1.4)

There are various vector-like pairs which can only develop a mass through a cubic Yukawa 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 problem and also obtaining a small Dirac neutrino mass leading to realistic light neutrino masses without using the seesaw mechanism.

There are two ways we can solve the problem.  Perhaps most simply, we can consider geometries where the Higgs up and down fields localize on distinct matter curves which do not intersect.  In this case, the term is identically zero. When these curves do intersect, the value of the term depends on the details of a gauge singlet wave function which localizes on a matter curve normal to the del Pezzo surface.  In the case of attraction, the term is near the GUT scale, which is untenable.  In the repulsive case, the term is suppressed to a much lower value:

 μMGUT∼exp(−c1ε2γ)%, (1.5)

so that the resulting value of can then naturally fall in a phenomenologically viable range.

In fact, a similar exponential suppression in the wave functions of the right-handed neutrinos can generate small Dirac neutrino masses of the form:

 mDν∼με−γ⟨Hu⟩×⟨Hu⟩2MGUT∼0.5×10−2±0.5 eV (1.6)

which differs by a factor of from the value predicted by the simplest type of seesaw mechanisms with Majorana masses at the GUT scale.  We note that the value we obtain is in reasonable agreement with recent experimental results on neutrino oscillations.  In this case, the Majorana mass term must identically vanish to 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 this case, the Majorana mass terms in the neutrino sector are suppressed by some overall volume factors. Although the standard seesaw mechanism again generates naturally light neutrino masses eV, we find that the Majorana mass term is naturally 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 are both in a range of values consistent with leptogenesis, as well as the observed light neutrino masses.

Non-trivial flavor structures can potentially arise in a number of ways in this class of models.  For example, one common approach in the model building literature is to use a discrete symmetry to induce additional structure in the form of the Yukawa couplings.  The Weyl group symmetries of the exceptional groups naturally act on the del Pezzo surfaces.  This symmetry can be partially broken by the choice of the Kähler classes of two-cycles. This may potentially lead to a model of flavor based on the discrete symmetry groups , or .  Indeed, these are all subgroups of the Weyl group of .

One of the main conceptual issues with the usual GUT framework is to explain why at the GUT scale while the lighter generations do not satisfy such a simple mass relation.  At a qualitative level, the behavior of the omnipresent internal hypercharge flux again plays a central role in the resolution of this issue.  Although the net hypercharge flux vanishes on curves which support full GUT multiplets, in general it will not vanish pointwise.  Hence, the hypercharge flux can still leave behind an important imprint on the wave functions of the fields in the MSSM.  Indeed, because the individual components of a GUT multiplet have different hypercharge, the Aharonov-Bohm effect will alter the distinct components of a 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 the matter curve, the amount of flux which can pierce the curve also decreases.  In this way, the most naive mass relations remain approximately intact for the heaviest generation but will in general receive corrections for the lighter generations.

In the next to minimal GUT scenario, we can consider seven-branes where the bulk gauge group has rank five.  In this case there are three choices corresponding to , and .  In this paper we mainly focus on the case because it fits most closely with our general philosophy that the exceptional groups play a distinguished role in GUT models.  It turns out that this model can only descend to the MSSM by a sequence of breaking patterns where the eight-dimensional theory first breaks to a four-dimensional flipped model with gauge group .  The model then operates as a traditional four-dimensional flipped GUT which breaks to the Standard Model gauge group when a field in the of develops a suitable vev.  Indeed, direct breaking of to the Standard Model gauge group via fluxes taking values in a subgroup always generates exotic matter which would ruin the unification of the gauge coupling constants.  Many of the more refined features of these models such as textures and our solution to the problem share a common origin to those studied in the minimal model.

Even though our main emphasis in this paper is on models which admit a decoupling limit, we also consider models where such a limit does not exist.  In such cases the problem of engineering a GUT model becomes more flexible because the local model is incomplete.  We study examples of this situation because there are well-known difficulties in heterotic models in realizing traditional four-dimensional GUT group breaking via fields in the adjoint representation.  This is due to the fact that in many cases, the requisite adjoint-valued fields do not exist.  Indeed, gauge group breaking by Wilson lines is not so much an elegant ingredient in heterotic constructions as much as it is a necessary element of any construction.333It is also possible to avoid this constraint in heterotic models which descend to a four-dimensional flipped GUT.  See [30, 31, 25] for further details on this approach.  We also note that in certain cases, chiral superfields transforming in other representations can arise from higher Kac-Moody levels of the heterotic string.  Gauge group breaking via Wilson lines can also occur in F-theory when the surface wrapped by the seven-brane has non-trivial fundamental group.  For example, a well-studied surface with is the Enriques surface which can be viewed as the quotient of a surface.

Given the large proliferation of four-dimensional GUT models which exist in the model building literature, it is also natural to ask whether there exist purely four-dimensional GUT models in F-theory with adjoint-valued GUT Higgs fields.  We find that this can be done provided the surface wrapped by the seven-brane has non-zero Hodge number .  But in contrast to the usual approach to four-dimensional effective field theories where it is common to assume that Planck scale 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 surprisingly simple to accommodate because extra messenger fields can naturally arise from additional matter curves which do not intersect any of the other curves on which the matter content of the MSSM localizes.  Supersymmetry breaking can then communicate to the MSSM via the usual gauge mediation mechanism.  We note that because the term naturally develops a value around the electroweak scale independently of any supersymmetry breaking mechanism, we can retain many of the best features of gauge mediation such as the absence of additional flavor changing neutral currents (FCNCs) while avoiding some of the problematic elements of this scenario which are related to generating appropriate values for the and terms.  Depending on the local behavior of the wave functions which propagate in directions normal to the del Pezzo surface, the messenger scale can quite flexibly range from values slightly below the GUT scale to much lower but still phenomenologically viable mass scales.

The organization of this paper is as follows. In Section 2, we formulate what we wish to achieve in our GUT constructions.  In Section 3 we review and slightly extend our previous work on realizing GUT models in F-theory.  To this end, we describe many of the necessary ingredients for an analysis of the matter content and interaction terms of any potential model.  Before proceeding to any particular class of models, in Section 4 we discuss the various mass scales which will generically appear throughout this paper.  In Section 5, we give a general overview of the class of GUT models in F-theory we shall study. These models intrinsically divide based on how the GUT breaks to the MSSM.  We first study models where the GUT scale cannot be decoupled from the Planck scale.  In Section 6 we discuss models where GUT breaking proceeds just as in four-dimensional models.  Next, we discuss GUT breaking via discrete Wilson lines in Section 7.  In the remainder of the 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 is followed in Section 9 by a study of GUT breaking to the MSSM via an internal hypercharge flux. In Section 10 we determine which bulk gauge groups can break directly to the Standard Model gauge group via internal fluxes.  We also explain in 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 Section 12 we study the interrelation between proton decay and doublet triplet splitting in our models. After giving a simple criterion for avoiding the simplest dimension five operators responsible for proton decay, in Section 13 we explain how extra global symmetries in the low energy effective theory are encoded geometrically in F-theory, and in particular, how these symmetries can forbid potentially dangerous higher dimension operators. In Section 14 we discuss some coarse properties of Yukawa couplings and also speculate on how further details of flavor physics could in principle be incorporated.  In this same Section we also provide a qualitative explanation for why the usual mass relations of GUT models become increasingly distorted as the mass of a generation decreases.  In Section 15 we show that interaction terms involving matter fields which localize on Riemann surfaces outside of the surface can generate hierarchically small values for both the term as well as Dirac neutrino masses.  We also study a variant on the usual seesaw mechanism which generates the expected mass scale for the light neutrinos.  Intriguingly, the Majorana mass of the right-handed neutrinos is somewhat lower than the value expected in typical GUT models.  In Section 16 we propose how supersymmetry breaking could communicate to the MSSM, and in Section 17 we present an model which incorporates some (but not all!) of the mechanisms developed in previous sections.  Our expectation is that further refinements are possible which are potentially more realistic.  In a similar vein, in Section 18 we present a flipped model.  Section 19 collects various numerical estimates obtained throughout the paper, and Section 20 presents our conclusions.  The Appendices contain further background material used in the main body of the paper and which may also be of use in future model building efforts.

2 Constraints From Low Energy Physics

In this Section we define the criteria by which we shall evaluate how successfully our models reproduce features of low energy physics obtained by a minimal extrapolation of experimental data to the MSSM.  There are a number of open questions in both phenomenology and string theory which must ultimately be addressed in any approach.  See [32, 33] for an expanded discussion of some of the issues we briefly address here.

At the crudest level, we require that any viable model contain precisely three generations of chiral matter.  It is an experimental fact that the chiral matter content of the Standard Model organizes into and GUT multiplets.  Coupled with the fact that the gauge couplings of the MSSM appear to unify at an energy scale GeV, we shall aim to reproduce these features in all of the models we shall consider.  For all of these reasons, we require that the low energy content of all of our models must match to the matter content of the MSSM.  By this we mean that in addition to achieving the correct chiral matter content and Higgs content of the MSSM, all additional matter charged under the gauge groups must at the very least fit into vector-like pairs of complete GUT multiplets in order to retain gauge coupling unification.444While it is in principle possible to consider models where vector-like exotics preserve gauge coupling unification, we believe this runs contrary to the spirit of GUT models.  Although we shall not entertain this possibility here, see [34, 35] for further discussion of this possibility.  In the minimal incarnation of GUT models considered here, we shall further require that the low energy spectrum of particles charged under the Standard Model gauge group must exactly match to the matter content of the MSSM.  We note that historically, even this qualitative requirement has been difficult to achieve in Calabi-Yau compactifications of the perturbative heterotic string.

Although the correct particle content is a necessary step in achieving a realistic model, it is certainly not sufficient because we must also reproduce the superpotential of the MSSM:

 W=μHuHd+λuijQiUjHu+λdijQiDjHd+λlijLiEjHd+λνijLiNjRHu+... (2.1)

where the indices and label the three generations.  While the precise form of the Yukawa matrices labeled by the ’s will lead to masses and mixing terms between the generations, a necessary first step is that there are in principle non-zero contributions to the above superpotential!  As a first approximation, we require that the tree level superpotential of the theory at high energy scales generate a non-trivial interaction term for the third generation so that there is a rough hierarchy in mass scales.  In the context of GUT models, it is well-known that because the particle content of the Standard Model organizes into complete GUT multiplets, the Yukawa couplings couple universally to fields organized in such multiplets.  One attractive feature of the tree level superpotential in most GUT models is that the third generation obeys a simple mass relation of the form at the GUT scale.  Evolving this relation under the renormalization group to the weak scale yields the relation which is roughly in agreement with experiment.  Unfortunately, this relation is violated for the lighter generations.  Ideally, it would be of interest to find models which naturally preserve the mass relations of the third generation while modifying the relations of the first two generations.

At the next level of approximation, any model should be consistent with current experimental bounds on the lifetime of the proton ( yrs [36]).  This requires that certain operators must be absent or sufficiently suppressed in the low energy superpotential.  Indeed, note that in equation (2.1), we have implicitly only included renormalizable R-parity invariant couplings because if present, the interaction terms and will cause the proton to decay too rapidly. We shall consider models with and without R-parity.  In the latter case, we therefore must present alternative reasons to expect renormalizable operators responsible for R-parity to vanish.

Proton decay is a hallmark of GUT models.  Aside from renormalizable interaction terms, the dominant contribution to proton decay in the simplest GUT models comes from the dimension five operator [37, 38]:555There is an additional contribution to the superpotential given by .  At the level of discussion in this paper, it is sufficient to only deal with the term .

 O5=c5MGUT∫d2θQQQL (2.2)

and the dimension six operator:

 O6=c6M2GUT∫d4θQQU†E†. (2.3)

The operator can originate from the exchange of heavy Higgs triplets and can cause the decay .  The operator can originate from the exchange of heavy off-diagonal GUT group gauge bosons and can cause the decay .  To remain in accord with current bounds on nucleon decay, can typically be an order one coefficient whereas must be suppressed at least to the order of .  See [39] for further discussion on proton decay in GUT models.

In four-dimensional GUT models, this issue is closely related to the mechanism responsible for removing the Higgs triplets from the low energy spectrum.  One common approach is to invoke some continuous or discrete symmetry to sufficiently suppress this operator.  The use of discrete symmetries in compactifications of M-theory on manifolds with holonomy has been studied in [40].  Note that while the Higgs triplet must develop a sufficiently large mass in order to reproduce the particle content of the MSSM, we must also require that the supersymmetric Higgs mass should be on the order of the weak scale.

While the above problems are necessary requirements for any potentially viable model, there are many additional phenomenological constraints which must be satisfied in a fully realistic compactification. In principle, a complete model should also naturally accommodate hierarchical masses for the quarks and leptons.  For example, in conventional GUT models, the seesaw mechanism allows the neutrino masses in the Standard Model to be much lighter than the electroweak scale.  At a more refined level, a full model should explain why the CKM matrix is nearly equal to the identity matrix whereas the MNS matrix contains nearly maximal mixing between the neutrinos.

A fully realistic model must of course specify how supersymmetry is broken and provide a mechanism for communicating this breaking to the MSSM.  Our expectation is that this issue can be treated independently from the supersymmetric models which shall be our primary focus here.  We note that for general string compactifications, supersymmetry breaking is closely entangled with moduli stabilization.  While we will not specify a method for stabilizing moduli, we note that F-theory provides a natural arena for further study of this issue.  See [41] for a particular example of moduli stabilization in F-theory and [42] for a review of this active area of research.

3 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 as the effective superpotential of the four-dimensional theory.  Further details may be found in [15].

F-theory compactified on an elliptically fibered Calabi-Yau fourfold preserves supersymmetry in the four uncompactified spacetime dimensions.  Letting denote the base of the Calabi-Yau fourfold, the discriminant locus of the elliptic fibration determines a subvariety of complex codimension one in the base . Denoting by the Kähler surface defined by an irreducible component of , when this degeneration locus is a singularity of type, the resulting eight-dimensional theory defines the worldvolume of an exceptional seven-brane with gauge group of type.  This singularity type can enhance along complex codimension one curves in to a singularity of type and can further enhance at complex codimension two points in to a singularity of type .  Such points correspond to the triple intersection of three matter curves.  Because the Cartan subalgebra of each singularity type is visible to the geometry [43, 44], these enhancements satisfy the containment relations:

 GS×U(1)×U(1)⊂GΣ×U(1)⊂Gp. (3.1)

As argued in [15], many necessary features of even semi-realistic GUT models require that .  In particular, this implies that the rank of the bulk gauge group is at most six.  This significantly limits the available bulk gauge groups because the rank of must be at least four in order to contain the Standard Model gauge group.

In this paper we shall assume that given a choice of matter curves, there exists a Calabi-Yau fourfold which contains the corresponding local enhancement in singularity 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 engineer a local model where the bulk gauge group enhances along a matter curve in to an singularity.  A local elliptic model of this type is:

 y2=x3+fxz3+q2z4. (3.2)

In the above, is a section of , is a section of and the coordinates transform as a section of [15]:

 L2⊕L3⊕L⊗KS (3.3)

where denotes the canonical bundle on and is a line bundle which can be expressed in terms of and .  The essential point of this example is that in a local model, there always exists a line bundle such that the resulting local model is well-defined.  For example, in this case we have:

 L=OS(Σ)⊗K2S. (3.4)

Further, we shall make the additional assumption that there is no mathematical obstruction to various twofold enhancements in the rank of the singularity at points of the surface .  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 superfields, the matter content of the theory consists of an vector multiplet which transforms as a scalar on , a collection of chiral superfields which transform as a form on (the bulk gauge bosons) and a collection of chiral superfields which transform as a holomorphic form on .  The bulk modes couple through the superpotential term:

 WS=∫STr[(¯¯¯∂A+A∧A)∧Φ]. (3.5)

When two irreducible components and of intersect on a Riemann surface , the singularity type enhances further.  In this case, additional six-dimensional hypermultiplets localize along .  As in [44], the representation content of these fields is given by decomposing the adjoint representation of the enhanced singularity to the product associated with the gauge groups on and .  In terms of four-dimensional superfields, the matter content localized on a curve consists of chiral superfields and which transform as spinors on .  The bulk modes couple to matter fields localized on the curve via the superpotential term:

 WΣ=∫Σ⟨Λc,(¯¯¯∂+A+A′)Λ⟩ (3.6)

where denotes the natural pairing which is independent of any metric data.

Finally, when three irreducible components of intersect at a point , the singularity type can enhance even further.  Evaluating the overlap of three ’s for three matter curves yields a further contribution to the four-dimensional effective superpotential:

 Wp=Λ1Λ2Λ3|p. (3.7)

An analysis similar to that given below equation (3.2) shows that given three matter curves which form a triple intersection, so long as the resulting interaction term is consistent with group theoretic considerations, there exists a local Calabi-Yau fourfold with the desired twofold enhancement in singularity type.

Having specified the individual contributions to the quasi-topological eight-dimensional theory, the superpotential is:

 W[Φ,A,Λ]=WS1+...+WSl+WΣ1+...+WΣm+Wp1+...+Wpn+Wflux+Wnp. (3.8)

In the above, the corresponding fields entering the above expression are to be viewed as a large collection of four-dimensional chiral superfields labeled by points of the complex surfaces and the Riemann surfaces .  We have also included the contribution from the flux-induced superpotential which couples to the various forms of the seven-branes and indirectly to matter fields localized on curves.  As explained in [15], the vevs for the form and fields localized on matter curves correspond to complex deformations of the Calabi-Yau fourfold.  Because the flux-induced superpotential couples to the complex structure moduli of the Calabi-Yau fourfold, such terms will generically be present.  In equation (3.8), we have also included the term which denotes all non-perturbative contributions from wrapped Euclidean three-branes.  These terms are proportional to where is an order one positive constant.  In a GUT model where the gauge coupling constants unify perturbatively, such contributions are negligible.

The fields of the four-dimensional effective theory correspond to zero mode solutions in the presence of a background field configuration.  As in [15], we shall confine our analysis of the matter spectrum to backgrounds where all fields other than the bulk gauge field are expanded about zero.  In the presence of a non-trivial background gauge field configuration, the chiral matter content of the four-dimensional effective theory descends from bulk modes on and Riemann surfaces which we denote by the generic label .  An instanton taking values in a subgroup will break to the commutant subgroup.  Decomposing the adjoint representation of to the maximal subgroup of the form , the chiral matter transforming in a representation of descends from the bundle-valued cohomology groups:

 τ∈ H0¯¯¯∂(S,T∗)∗⊕H1¯¯¯∂(S,T)⊕H2¯¯¯∂(S,T∗)∗ (3.9)

where denotes a bundle transforming in the representation of obtained by the decomposition of the adjoint representation of the associated principle bundle on .  When is a del Pezzo surface, the cohomology groups and vanish for supersymmetric gauge field configurations so that the number of zero modes transforming in the representation is given by an index:

 nτ=χ(S,T)=−(1+12c1(S)⋅c1(T)+12c1(T)⋅c1(T)). (3.10)

An analogous computation holds for the zero mode content localized on a Riemann surface transforming in a representation of :

 ν×ν′∈H0¯¯¯∂(Σ,K1/2Σ⊗V⊗V′) (3.11)

so that the net number of zero modes is given by the index:

 nν×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 always be done when the curve corresponds to the vanishing locus of the holomorphic form in the eight-dimensional theory.

When , it is also possible to consider vacua with non-trivial Wilson lines.  In order to avoid complications from the reduction of additional supergravity modes, we shall always assume that is a finite group.  The discussion closely parallels a similar analysis in heterotic compactifications (see for example [45]).  Recall that admissible Wilson lines are specified by a choice of element .  In order to maintain continuity with the discussion reviewed above, we shall require that the non-trivial portion of the discrete Wilson line takes values in the subgroup defined above.  More generally, this restriction can be lifted and may allow additional possibilities for projecting out phenomenologically unviable representations from the low energy spectrum.  Under these restrictions, the unbroken four-dimensional gauge group is given by the commutant subgroup of in .

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 heterotic string, our strategy will be to lift all computations to a covering theory.  Because is finite, the universal cover of denoted by is a compact Kähler surface.  Letting denote the covering map, the bundle on now lifts to a bundle on .  Under the present restrictions, the Wilson line corresponds to a flat -bundle induced from the covering map from to .  The deck transformation defined by the action of on also determines a group action of on the cohomology groups .  Treating as a complex vector space, the eigenspace decomposition of is of the form:

 Hi¯¯¯∂(˜S,˜T)≃⊕λCλ (3.13)

in the obvious notation.  The irreducible representation of defined by decomposes into irreducible representations of the maximal subgroup as:

 τ≃⊕iτi⊗Ri. (3.14)

The zero modes transforming in the representation are therefore specified by the invariant subspaces:

 τi:[H0¯¯¯∂(˜S,˜T∗)∗⊗Ri]ρS⊕[H1¯¯¯∂(˜S,˜T∗)∗⊗Ri]ρS⊕[H2¯¯¯∂(˜S,˜T∗)∗⊗Ri]ρS. (3.15)

Having specified the zero mode content of the theory, we can now in principle determine the full superpotential of the low energy effective theory by integrating out all Kaluza-Klein modes from equation (3.8).  This is similar to the treatment of Chern-Simons gauge theory as a string theory [46].  For quiver gauge theories defined by D-brane probes of Calabi-Yau threefolds, the higher order terms of the effective superpotential are given by integrating out all higher Kaluza-Klein modes from the associated holomorphic Chern-Simons theory for B-branes [47].

In the present context, we can follow the procedure outlined in [48] to determine the full expression for the effective superpotential.  This is given by a bosonic partition function with action given by the superpotential of equation (3.8).   Viewing the higher-dimensional fields as a collection of four-dimensional chiral superfields labeled by points of the internal space, the effective superpotential is now given by the bosonic path integral:

 exp(−Weff[Φ0,A0,Λ0])=∫1PI[dΦ][dA][dΛ]exp(−W[Φ+Φ0,A+A0,Λ+Λ0]) (3.16)

where the zero subscript denotes the zero mode, and the path integral is over all one particle irreducible Feynman diagrams.  In this expression, should be viewed as a bosonic action with functional dependence identical to that of equation (3.8).  The complete four-dimensional effective superpotential for the zero modes is then determined by the partition function of the quasi-topological theory.  We emphasize that this partition function is well-defined without any reference to metric data.  A very similar procedure for extracting the superpotential by integrating out Kaluza-Klein modes in heterotic compactifications has been given in [23].  Some examples of similar computations for quiver gauge theories can be found in [49].  To conclude this Section, we note that any symmetry of the full eight-dimensional theory descends to the four-dimensional effective superpotential for the zero modes.  Neglecting the contribution due to non-perturbative effects in equation (3.8), the extra factors which are always present when the singularity type enhances will provide additional global symmetries in the effective theory which will typically forbid some higher dimension operators from being generated.  Although non-perturbative effects can violate these symmetries, the corresponding contribution to will typically be small enough that we may safely neglect such contributions.

These general considerations already constrain the matter content of any candidate theory.  Modes propagating in the bulk of the surface must transform in the adjoint representation of the bulk gauge group.  Moreover, although matter fields can localize on a curve inside of , these fields must descend from the adjoint representation of .  For example, for gauge group factors which do not embed in , the only available local enhancements are to higher or type singularities.  In such cases, the decomposition of the adjoint representation only contains two index representations.  Similar restrictions apply for gauge group factors which do not embed in .  In particular, the spinor representation never appears in such cases.  In a sense, this is to be expected because these are precisely the types of configurations which can be realized within perturbative type IIB vacua.

For gauge groups, the available representations are the vector, spinor or adjoint representations, and for gauge groups, the only available representations are the one, two or three index anti-symmetric and the adjoint representations.666Strictly speaking there are additional possibilities if the rank of the bulk singularity enhances by more than one rank.  If one allows more general breaking patterns involving higher and type enhancements, it is also possible to achieve two index symmetric representations of theories.  For example, letting denote the two index anti-symmetric representation of , decomposes to as .  Higgsing this to the diagonal subgroup, we note that the product contains two index symmetric representations. This is a rather exotic possibility and we shall therefore not consider it further in this paper. For example, when , this implies that all of the matter fields transform in the or , while for , the only available representations are the , , , or .  in the specific case of del Pezzo models, this matter content is even more constrained.  Indeed, as explained in [15], the bulk zero mode content for del Pezzo models never contains chiral superfields which transform in the adjoint representation of the unbroken gauge group in four dimensions.

In fact, the type of twofold enhancement strongly determines the qualitative behavior of the associated triple intersection of matter curves.  For example, the possible rank two enhancements of are , , and .  In the case of and , the associated curves which form a triple intersection all live inside of .  Indeed, by group theory considerations, the matter fields localized on each curve transform in non-trivial representations of [15].  On the other hand, this is qualitatively different from a local enhancement to .  In this case, two of the curves of the triple intersection support matter in the fundamental and anti-fundamental of and therefore live in , while the third curve of the intersection supports matter in the singlet representation.

More generally, we note that as opposed to a generic field theory, in F-theory, vector-like pairs of the bulk gauge group can only interact through cubic superpotential terms involving a field localized on a curve which intersects at a point.  While the vev of this gauge singlet can induce a mass term for the vector-like pair, the dynamics of this field in the threefold base is qualitatively different from fields which localize inside of .

4 Mass Scales and Decoupling Limits

Before proceeding to specific models, we first present a general analysis of the relevant mass scales in the local models we treat in this paper.  Rather than specify one particular profile for the threefold base , we consider both geometries where is roughly tubular so that it decomposes as the product of  with two non-compact directions orthogonal to in , as well as more homogeneous profiles. To parameterize our ignorance of the details of the geometry, we define the length scales:

 RS ≡Vol(S)1/4 (4.1) RB ≡Vol(B3)1/6 (4.2)

as well as a cutoff length scale which measures the radius normal to :

 R⊥≡RB×(RBRS)ν (4.3)

so that the exponent ranges from when is homogeneous, to the value when is the product of with two non-compact directions.  Indeed, the approximations we consider in this paper are valid in the regime .  Note that under the assumption , the three length scales are related by:

 R⊥>RB>RS. (4.4)

See figure 2 for a comparison of the local behavior of for and .  To clarify, although the directions normal to 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 is quite 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 , the ten-dimensional Einstein-Hilbert action is:

 SEH∼M8∗∫R3,1×B3R√−gd10x (4.5)

where is a particular mass scale associated with the supergravity limit of the F-theory compactification.  In perturbative type IIB string theory, the parameter is given in string frame by the relation .  Upon reduction to four dimensions, the four-dimensional Planck scale satisfies the relation:

 M2pl∼M8∗Vol(B3). (4.6)

The tension of a seven-brane wrapping a Kähler surface in determines the gauge 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:777The 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 algebra generators in order to conform to conventions typically used in topological gauge theory.  Because our emphasis here is on the four-dimensional effective field theory, in this paper we have reverted back to the standard sign convention in the physics literature so that all Lie group generators are hermitian.

 Skin∼−M4∗∫R3,1×STr(F∧∗8F). (4.7)

The value of the gauge coupling constant at the scale of unification is therefore:

 α−1GUT∼M4∗Vol(S). (4.8)

Equations (4.6) and (4.8) now imply:

 Vol(B3)∼(αGUTMplVol(S))2 (4.9)

or:

 R6B∼(αGUTMplR4S)2. (4.10)

We now convert these geometric scales into mass scales in the low energy effective theory.  To this end, we next relate to the GUT scale .  In most of the cases we consider, non-zero flux in the internal directions of will partially break the bulk gauge group of the seven-brane.  Letting denote the mass scale of the internal flux, we therefore require:

 M2GUT∼⟨FS⟩. (4.11)

Because the flux is measured in units of length on the surface , this implies:

 Vol(S)∼M−4GUT. (4.12)

Equation (4.9) therefore yields:

 Vol(B3)∼(αGUTMplM−4GUT)2. (4.13)

The radii and are therefore given by:

 1RS ∼MGUT=3×1016 GeV (4.14) 1RB ∼MGUT×ε1/3∼1016 GeV (4.15)

where we have introduced the small parameter:

 ε≡MGUTαGUTMpl∼7.5×10−2%. (4.16)

Collecting equations (4.9) and (4.12), the parameter now takes the form:

 1R⊥=MGUT×εγ∼5×1015±0.5 GeV (4.17)

where .  We note that these numerical values for the radii satisfy the inequality of line (4.4).

We conclude this Section by discussing the normalization of Yukawa couplings in models where the superpotential originates from the triple intersection of matter curves.  In a holomorphic basis of wave functions, the F- and D-terms are:

 LholF =∑pψi(p)ψj(p)ψk(p)∫d2θ˜ϕi˜ϕj˜ϕk (4.18) ≡λholijk∫d2θ˜ϕi˜ϕj˜ϕk (4.19) LholD =M2∗∫Σd4θK(˜ϕ,˜ϕ†) (4.20)

where in the above, denotes the internal value of the wave function associated with the four-dimensional chiral superfield evaluated at a point in , and the holomorphic Yukawa couplings are defined as:

 λholijk=∑pψi(p)ψj(p)ψk(p). (4.21)

The behavior of the wave functions near these points can generate hierarchically small values near nodal points, and order one values away from such nodal points.

We eventually wish to extract numerical estimates for the physical Yukawa couplings, defined in a basis of four-dimensional chiral superfields with canonically-normalized kinetic terms. However, if we reduce the -term in (4.20) over , we find that the kinetic term for is multiplied by the -norm on of the corresponding zero-mode wave function .

In general, transforms on as a holomorphic section of , where is a line bundle on determined by the gauge field on . Both and carry natural hermitian metrics inherited from the bulk metric and gauge field on . Fixing the holomorphic wave function , we are interested in how the -norm of scales with the metric on , since the volume of effectively determines . For concreteness, let us write the metric on in local holomorphic coordinates as , where is a local holomorphic coordinate along and is a holomorphic coordinate normal to . Under an overall scaling , the hermitian metric on is unchanged, so the norm of behaves as

 ⟨ψ|ψ⟩ =∫Σd2zgz¯z(gz¯z)1/2ψ¯¯¯¯ψ, (4.22) ⟼ℓ1/2⟨ψ|ψ⟩. (4.23)

Since the volume of scales with , we see from (4.22) that scales with .

At first glance, the dependence of on might appear to be the only source of -dependence in the respective - and -terms in (4.18) and (4.20), since the -term is determined by the overlap of fixed holomorphic wavefunctions. However, in making precise sense of this overlap, an additional -dependence also enters.

To explain this -dependence, let us consider a slightly simplified situation, for which the holomorphic curves , , and meet transversely at a point inside a Calabi-Yau threefold . The role of the line bundle is inessential, so on each curve we take the wavefunction to transform as a holomorphic section of . In local holomorphic coordinates around the point of intersection, the wavefunction overlap is defined by

 ψ1(p)ψ2(p)ψ3(p)√dz√dw√dv√Ω(p). (4.24)

Here is a holomorphic three-form on which we must introduce so that the overlap in (4.24) does not depend on the particular holomorphic coordinates chosen at .

Of course, is unique up to scale — but it is precisely the scale of the overlap that we are trying to fix! Given that carries a metric, we fix the norm of by the requirement that , where is the Kähler form associated to the metric on . Once we impose this condition, scales as under an overall scaling of the metric on . Hence the wavefunction overlap in (4.24) and thus the holomorphic Yukawa coupling actually scales as .

After canonically normalizing all kinetic terms, the physical Yukawa couplings are given by

 λphysijk=λholijk√M2∗⟨ψi|ψi⟩M2∗⟨ψj|ψj⟩M2∗⟨ψk|ψk⟩. (4.25)

By the preceding discussion, under an overall scaling of the metric on , the physical Yukawa coupling scales as . Restoring the dependence on the volumes of each curve, we find the result which one would naively guess,

 λphysijk=λ0ijk√M2∗Vol(Σi)M2∗Vol(Σj)M2∗Vol(Σk). (4.26)

Here denotes the fiducial, order one Yukawa coupling defined by (4.24) when has unit volume.

Although we have phrased the preceding discussion in the very special case that , , and are holomorphic curves intersecting transversely in a Calabi-Yau threefold, the result (4.26) holds quite generally in F-theory. According to the discussion in § of [15], when , , and are matter curves intersecting at a point inside , one must choose a trivialization of to evaluate the wavefunction overlap. This choice, analogous to the choice of in (4.24), introduces the same scaling with .

Once we introduce four-dimensional chiral superfields with canonical kinetic terms, the -terms become

 LF=λ0ijk∫d2θϕiϕjϕk√M2∗Vol(Σi)M2∗Vol(Σj)M2∗Vol(Σk). (4.27)

We note that when all matter curves have comparable volumes set by the overall size of , .  In this case, (4.8) implies:

 LF=α3/4GUTλ0ijk∫d2θϕiϕjϕk. (4.28)

In rescaling each field by an appropriate power of the volume factor, we shall typically use the classical value of .  Strictly speaking, this approximation is only valid in the supergravity limit.  Due to the fact that in F-theory there is at present no perturbative treatment of quantum corrections, most of the numerical results obtained throughout this paper can only be reliably treated as order of magnitude estimates.

5 General Overview of the Models

In this Section we provide a guide to the class of models we study.  The choice of Kähler surface already determines many properties of the low energy effective theory.  In keeping with our general philosophy, we require that the spectrum at low energies must not contain any exotics.  When , we expect the low energy spectrum to contain additional states obtained by reduction of the bulk supergravity modes of the compactification.  For this reason we shall always require that is a finite group.  There are two further possible refinements depending on whether or not the model in question admits a limit in which remains finite while .  In order to fully decouple gravity, the extension of the local metric on to a local Calabi-Yau fourfold must possess a limit in which the surface can shrink to zero size.  In particular, this imposes the condition that must be ample.  This is equivalent to the condition that is a del Pezzo surface, in which case .  We note that the degree Hirzebruch surfaces satisfy but do not define fully consistent decoupled models.

In fact, even the way in which the gauge group of the GUT breaks to that of the MSSM strongly depends on whether or not such a decoupling limit exists.  For surfaces with , the zero mode content will contain contributions from the bulk holomorphic form.  Because the form determines the position of the exceptional brane inside of the threefold base , a non-zero vev for the associated zero modes corresponds to the usual breaking of the GUT group via an adjoint-valued chiral superfield.888The potential application of this GUT breaking mechanism was noted in a footnote of [52] and has also been discussed in [15, 16].  Along these lines, we present some examples of four-dimensional GUT models which can originate from surfaces of general type.  An important corollary of this condition is that the usual four-dimensional field theory GUT models cannot be fully decoupled from gravity!  We believe this is important because it runs counter to the usual effective field theory philosophy that issues pertaining to the Planck scale can always be decoupled.  This is in accord with the existence of a swampland of effective field theories which may not admit a consistent UV completion which includes gravity [53].  Moreover, as we explain in greater detail later, it is also possible that a generic surface of general type may not support sufficiently many matter curves of the type needed to engineer a fully realistic four-dimensional GUT model.

When available, discrete Wilson lines in higher-dimensional theories provide another way to break the GUT group to .  Indeed, most models based on compactifications of the heterotic string on Calabi-Yau threefolds require discrete Wilson lines to break the gauge group and project out exotics from the low energy spectrum.  When , a similar mechanism for gauge group breaking is available for exceptional seven-brane theories.  As an example, we present a toy model where is an Enriques surface and .  In our specific example, we find that the zero mode content contains additional vector-like pairs of fields in exotic representations of .

We next turn to the primary case of interest for bottom up string phenomenology where is a del Pezzo surface.  Because and for del Pezzo surfaces, the two mechanisms for gauge group breaking mentioned above are now unavailable.  In this case, the GUT group breaks to a smaller subgroup due to non-trivial internal fluxes.  For example, the group can break to when the internal flux takes values in the factor.  In heterotic compactifications this mechanism is unavailable because a non-zero internal field strength would generate a string scale mass for the hypercharge gauge boson in four dimensions [23].  We find that in F-theory compactifications without a heterotic dual, there is a natural topological condition for the four-dimensional gauge boson to remain massless.  Our expectation is that this condition is satisfied for many choices of compact threefolds .  In the remainder of this Section we discuss further properties of del Pezzo models.

Along these lines, we present models based on where the gauge group of the eight-dimensional theory breaks directly to in four dimensions, as well as a hybrid scenario where breaks to in four dimensions and then subsequently descends from a flipped GUT model to the MSSM.  In fact, we also present a general no go theorem showing that direct breaking of to via abelian fluxes always generates extraneous matter in the low energy spectrum.  In both the regular and flipped scenarios, we find that in order to achieve the exact spectrum of the MSSM, all of the matter fields must localize on Riemann surfaces.  In the models, the matter fields organize into the and of .  In the models, a complete multiplet in the of localizes on the matter curves.  In both cases, all matter localizes on curves so that all of the tree level superpotential terms descend from the triple intersection of matter curves.  When some of the matter localizes on different curves, this leads to texture zeroes in the Yukawa matrices.

In addition to presenting some examples of minimal del Pezzo models, one of the primary purposes of this paper is to develop a number of ingredients which can be of use in further more refined model building efforts.  A general overview of these ingredients has already been given in the Introduction, so rather than repeat this here, we simply summarize the primary themes of the minimal model which recur throughout this paper.  The most prominent ingredient is the internal hypercharge flux which facilitates GUT breaking. This hyperflux also provides a natural solution to the doublet-triplet splitting problem and generates distorted GUT mass relations for the lighter generations.  More generally, the presence of additional global symmetries in the low energy theory forbids a number of potentially problematic interaction terms from appearing in the superpotential.  Topologically, the absence of dangerous operators translates into conditions on how the matter curves intersect inside of .  For example, proton decay is automatically suppressed when the Higgs up and down fields localize on different matter curves.  When these curves do not intersect, the term is zero.  When the Higgs matter curves do intersect, the resulting term can be naturally suppressed.  Indeed, an important feature of all the models we consider is that while expectations from effective field theory 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 the gauge 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 different choice of sign (when the normal wave function is repelled from our brane) we find the opposite situation, where is highly suppressed.  The ostensibly large mass term corresponding to the vev of a gauge singlet is in fact exponentially suppressed since its wave function is very small near our brane.  Here, the principle of decoupling is especially important because the large positive curvature of the del Pezzo surface can lead to a natural suppression of the normal wave functions.  This provides an explanation for why the term is far below the GUT scale, as well as why the neutrino masses are so far below the electroweak scale.  While we discuss many of these mechanisms in the specific context of the minimal model, these same features carry over to the flipped GUT models as well.  In such cases, additional well-established field theoretic mechanisms are also available.  For example, four-dimensional flipped models already contain an elegant mechanism for doublet-triplet splitting which also naturally suppresses dangerous dimension five operators responsible for proton decay.  In this case, we can also utilize a conventional seesaw mechanism to generate hierarchically light neutrino masses.

6 Surfaces of General Type

In this Section we present some examples of models where Planck scale physics cannot be decoupled from local GUT models.  Recall that in a traditional four-dimensional GUT, the GUT group breaks to when an adjoint-valued chiral superfield develops a suitable vev.  In F-theory, this requires that the seven-brane wraps a surface with .  Before proceeding to a discussion of GUT models based on such surfaces, we first discuss some important constraints on matter curves and supersymmetric gauge field configurations for such surfaces.

In many cases, some of the chiral fields of the low energy theory will localize on matter curves in .  When , the number of available matter curves will typically be much smaller than the dimension of would suggest.  To see this, suppose that an element of  corresponds to a holomorphic curve in .  We shall also refer to the class as an “effective” divisor.  Given a form on , note that:

 ∫ΣΩ=∫SΩ∧PD(Σ)=0 (6.1)

where denotes the element of which is Poincaré dual to .  This last equality follows from the fact that corresponds to the first Chern class of an appropriate line bundle and therefore is of type .999This last correspondence follows from the link between divisors and line bundles.  We thus see that although the condition is satisfied by a large class of vacua, at generic points in the complex structure moduli space each element of imposes an additional constraint of the form given by equation (6.1).  At the level of cohomology, the divisor classes are parameterized by the Picard lattice of :

 Pic(S)=H1,1(S,C)∩H2(S,Z). (6.2)

For example, we note that for a generic algebraic surface, has rank one.  Indeed, this lattice is generated by the hyperplane class inherited from the projective embedding of a general quartic in .  It is only at special points in the complex structure moduli space that additional holomorphic curves are present. An example of a surface of this type occurs when the quartic is of Fermat type.  In this case, the rank of is instead .  Because there is a one to one correspondence between line bundles and divisors on , we conclude that a similar condition holds for the available line bundles on a generic surface.

Having stated these caveats on what we expect for generic surfaces of general type, we now construct an GUT model with semi-realistic Yukawa matrices.  In order to have a sufficient number of matter curves, we consider a seven-brane with worldvolume gauge group wrapping a surface defined by the blowup at points of a degree hypersurface in with odd.  Some properties of hypersurfaces in are reviewed in Appendix B.  We have introduced these blown up curves in order to simplify several properties of our example.  Indeed, as explained 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 and of the degree hypersurface remain invariant under these blowups.  Thus, for many purposes we will be able to perform many of our calculations of the zero mode content as if the surface were a degree hypersurface in .

For , we expect to find a large number of additional adjoint-valued chiral superfields.  Geometrically, the vevs of these fields correspond to complex structure moduli in the Calabi-Yau fourfold which can develop a mass in the presence of a suitable background flux.  We show that in the present context, a suitable profile of vevs can simultaneously break the GUT group and lift all excess fields from the low energy spectrum.

As explained in Section 3, in the context of a local model, we are free to specify the enhancement type along codimension one matter curves inside of .  We first introduce four curves where the singularity type enhances to so that a half-hypermultiplet in the of localizes on each curve.  With notation as in Appendix B, the homology class of each curve is:

 [Σ1] =E2 (6.3) [Σ2] =E4 (6.4) [Σ3] =E6 (6.5) [ΣB] =−a1l1−E8−E9. (6.6)

where we have written for some generators of such that for .  Using the genus formula , we conclude that the genera of