IFT-UAM/CSIC-12-110

Non-perturbative effects and Yukawa hierarchies

in F-theory Unification

[10mm] A. Font, L. E. Ibáñez, F. Marchesano and D. Regalado

[6mm] Departamento de Física, Centro de Física Teórica y Computacional

[-0.3em] Facultad de Ciencias, Universidad Central de Venezuela

[-0.3em] A.P. 20513, Caracas 1020-A, Venezuela

[2mm] Instituto de Física Teórica UAM-CSIC, Cantoblanco, 28049 Madrid, Spain

[2mm] Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049 Madrid, Spain

[8mm] Abstract

[5mm]

Local F-theory models lead naturally to Yukawa couplings for the third generation of quarks and leptons, but inducing Yukawas for the lighter generations has proven elusive. Corrections coming from gauge fluxes fail to generate the required couplings, and naively the same applies to instanton effects. We nevertheless revisit the effect of instantons in F-theory GUT constructions and find that contributions previously ignored in the literature induce the leading non-perturbative corrections to the Yukawa couplings. We apply our results to the case of couplings in local F-theory GUTs, showing that non-perturbative effects naturally lead to hierarchical Yukawas. The hypercharge flux required to break down to the SM does not affect the holomorphic Yukawas but does modify the profile of the wavefunctions, explaining the difference between the D-quark and lepton couplings at the unification scale. The combination of non-perturbative corrections and magnetic fluxes allows to describe the measured lepton and D-quark masses of second and third generations in the SM.###### Contents

- 1 Introduction
- 2 Review of local F-theory models
- 3 The SO(12) model
- 4 Non-perturbative effects in local models
- 5 Non-perturbative zero modes at the SO(12) point
- 6 Yukawa couplings
- 7 Quark-lepton mass hierarchies and hypercharge flux
- 8 Conclusions and outlook
- A SO(12) wavefunctions at tree-level
- B Fluxes in the SO(12) model and local chirality
- C Non-perturbative superpotential and D-term
- D Non-perturbative Yukawas and residues

## 1 Introduction

One of the most difficult puzzles of the Standard Model (SM) is the structure of fermion masses and mixings. If the SM arises as a low-energy limit of an underlying string theory [1] , it should be possible to understand this structure in terms of more fundamental parameters characterizing the string vacuum. In particular the values of Yukawa couplings in string compactifications are determined by the geometric properties of extra compactified dimensions. An explicit computation of Yukawa coupling constants seems then quite difficult since we would need detailed information about the geometry of the corresponding compact space.

It has been however realized [2] that, due to the localization properties of branes, some quantities of physical interest do not depend on the full geometry of the compactification space but rather on local information around the region in which the SM fields are localized. This is particularly the case of Type IIB string compactifications with the SM fields localized on D-branes with and also on F-theory constructions. In these Type IIB compactifications the Yukawa couplings are obtained as overlap integrals involving the three wavefunctions of the quark/lepton and Higgs states involved. In particular, in the case of F-theory GUT’s the quark/lepton fields are localized on complex matter curves with quantum numbers. Yukawa couplings appear at the points of intersection of three matter curves corresponding to a right-handed fermion, a left-handed fermion and a Higgs multiplet transforming also as a 5-plet. With the wavefunctions of the three fields localized on the matter curves, the overlap integral is then dominated by the local properties of the wavefunction around the intersection point and hence no global information is required to compute the holomorphic Yukawa couplings. This opens the door to the explicit computation of Yukawa couplings in string compactifications with non-trivial curved compact spaces.

Particularly interesting from this point of view are the mentioned local F-theory GUT models
[3, 4, 5, 6], which have been recently the subject of intense study (for reviews see e.g. [7]).
These models are able to combine advantages of heterotic compactifications (gauge coupling unification) and Type IIB orientifolds
(localization of the SM fields and moduli fixing through closed string fluxes). In these F-theory models the local dynamics on the matter curves
is governed by the 8d effective action of 7-branes, and one can obtain explicit local expressions for the wavefunctions of the matter fields.
The Yukawa couplings arise at the triple intersection of matter curves [3, 4, 5, 6, 8, 9]
, and can be computed from a superpotential of the form
, where is the field strength of the 8-dimensional gauge fields and is a field parametrizing
fluctuations in the transverse dimensions to the branes, and the integral extends over the 7-brane worldvolume where the degrees of
freedom are localized. Within this simple scheme one finds that only one generation of quark/lepton fields gets a Yukawa coupling and may eventually
become massive [10, 11]. This is analogous to the result obtained in Type II toroidal orientifolds
in which the Yukawas may be computed explicitly [15, 16, 17].
This is an interesting starting point since indeed in the SM the third generation is much heavier than the rest and one may think that some
additional corrections could give masses to the first two generations.^{1}^{1}1For different approaches to the generation of hierarchies of fermion masses in F-theory unification see e.g. [12, 13, 14]. It was first thought [10] that the presence of the world-volume fluxes required
both to get chirality and break the symmetry down to the SM could be the source of these corrections.
However it was soon realized [18, 19, 20]
that open string fluxes do not modify at all the holomorphic Yukawa couplings and hence cannot
give rise to Yukawa couplings for the lighter generations.

In [21, 22] it was pointed out that non-perturbative effects from distant D3-instantons could be the source of the required corrections. The most obvious such corrections where found to be proportional to and turn out to have an alternative useful description in terms of non-commutative geometry [18]. Indeed such corrections where shown to lead to the required corrections in a simple toy model with gauge group [22]. It was however already pointed out in this reference that in realistic cases, namely GUT’s with an enhanced symmetry group or at the Yukawa triple intersection point, such corrections identically vanish, since the cubic trace is zero in orthogonal and exceptional groups. This seemed again to make problematic the generation of fermion hierarchies in F-theory GUT constructions.

In this paper we reexamine all these issues and point out that non-perturbative D3-instanton effects give rise to additional corrections, some of them previously overlooked. Such corrections to the superpotential have the form

(1.1) |

where is a small parameter and are holomorphic functions of the local coordinates. As mentioned above the contribution with vanishes for the realistic cases where the Yukawa enhancement groups are or . We hence study in detail the remaining leading corrections to the Yukawa couplings induced by the and terms, applied to the case which is relevant for the Yukawa couplings of charged leptons and D-quarks.

Describing non-perturbative corrections as in (1.1) simplifies the procedure to compute corrected Yukawas. More precisely, one may apply dimensional reduction techniques to express them in terms of a triple overlap of zero mode internal wavefunctions. In this sense, the presence of has a two-fold effect. On the one hand it modifies the zero mode internal wavefunction profile and on the other hand it induces new 8d couplings that upon dimensional reduction become new sources of Yukawa couplings. As in [22], taking both effects into account gives an interesting Yukawa pattern, in which the holomorphic Yukawas depend on but are independent of worldvolume fluxes. While in [22] this result can be guessed based on a dual non-commutative description of the 7-brane superpotential [18, 21], for the general case (1.1) such description is not available. Nevertheless, one can still generalize the results of [18] to obtain a residue formula that computes holomorphic Yukawas, and where the flux-independence of the latter is manifest.

Interestingly enough we find that a hierarchy of mass eigenvalues of the form is automatically present, explaining the observed hierarchical structure. Here is a small non-perturbative parameter measuring the size of the effects induced by the distant instantons. The holomorphic Yukawa couplings obtained are identical for both D-quarks and leptons, since they live in the same representations and the holomorphic Yukawas are flux independent. This looks problematic since running up in energies the observed D-quark and lepton masses, unification of Yukawa couplings does not hold experimentally, rather leptons of the second and third generations tend to have larger Yukawa couplings than the respective D-quarks at the unification scale. We find however that the hypercharge flux required for the symmetry breaking may explain this difference. Roughly speaking, the difference may be understood as arising from the fact that the wavefunctions for leptons are more localized than those of D-quarks, due to the fact that they have larger hypercharge quantum numbers.

The structure of this paper is as follows. In Section 2 we review the construction of local F-theory GUTs. In section 3 we construct a local GUT model with enhanced symmetry, which describes the Yukawa couplings of charged leptons and D-quarks. The spectrum of zero modes reproduces the matter content of the MSSM, but the Yukawa couplings exhibit the rank-one structure mentioned above. In section 4 we introduce the non-perturbative effects that will give rise to the superpotential (1.1), and compute the corrected zero mode equations. Such equations are solved in section 5 for the model of enhancement constructed before, while the corresponding Yukawas are computed in section 6. The discussion of these last two sections is slightly technical, and the reader not interested in such details may safely skip to section 7, where a phenomenological analysis of the final Yukawa couplings is performed. In particular, we confront our results with the measured masses of D-quarks and charged leptons, showing that a natural hierarchy of masses arises and that the effect of the hypercharge flux allows us to understand the ratios between them.

Several technical details have been relegated to the appendices. Appendix A solves the zero mode wavefunctions for the model in absence of non-perturbative effects, and compute the wavefunction normalization factors which encode the hypercharge flux dependence of the Yukawa couplings. Appendix B discusses in some detail the choice of worldvolume fluxes made for this model, motivating them via the notion of local chirality in F-theory. Appendix C derives the non-perturbative superpotential (1.1), and shows that the D-term is not corrected. Finally, in appendix D we derive a residue formula for the non-perturbative Yukawa couplings, that allows to cross-check and extend the results obtained in the main text.

## 2 Review of local F-theory models

Following the general scheme of [4, 5, 3, 6] (see also [23, 24, 25, 26, 27, 28, 29, 30]), in order to construct a local F-theory GUT model one may consider a stack of 7-branes wrapping a compact divisor of the threefold base of an elliptically-fibered Calabi-Yau fourfold. The gauge degrees of freedom that arise from are specified by the particular set of 7-branes that are wrapped on or, in geometrical terms, by the singularity type of the elliptic fiber on top of such 4-cycle. Hence, one may easily engineer local models where the GUT gauge group is given by , or even .

Besides the stack of 7-branes on , a semi-realistic F-theory model will contain further 7-branes that wrap another set of divisors , which intersect on certain curves . On top of the latter set of curves of the singularity type of the elliptic fiber is enhanced, in the sense that the Dynkin diagram that is associated to the singularity corresponds to a higher rank Lie group that contains . In practice, this implies that new degrees of freedom appear at the intersection of the 7-branes, more precisely chiral matter multiplets in a certain representation of , localized at the so-called matter curves .

Finally, two or more matter curves may meet at a point and at that point the singularity is promoted to an even higher one, such that the corresponding Lie group not only contains but also each of the involved. This time there are no new degrees of freedom arising at the point , but rather contact interactions involving the chiral multiplets from each curve . Of particular interest are those cases where three matter curves meet at , as they give rise to Yukawa couplings between chiral multiplets of the GUT matter fields.

Of course, in the process of describing a local model one must not only specify the gauge group , but also the enhanced group at each of the matter curves. This information and the intersection loci of matter curves determines the groups at each point where Yukawa couplings develop. Typically, starting with a GUT gauge group such as one may end up with enhanced groups at Yukawa points such as , , or . In the next section we will analyze a local model that describes the case where and , which corresponds to the setup describing down-type Yukawas for a local F-theory model.

While the above geometric picture is already quite illuminating, one of the most powerful results of [4, 5, 3, 6] is to provide a simple framework to compute the matter content arising at each curve and the Yukawa couplings at their triple intersections. Such framework makes use of a 8d effective action related to a stack of 7-branes which, upon dimensional reduction on a 4-cycle , provides all the dynamics of the 4d degrees of freedom [4].^{2}^{2}2See [19] for a derivation in terms of a 8d SYM Lagrangian. In particular, the Yukawa couplings between 4d chiral fields arise from the superpotential

(2.1) |

where is the F-theory characteristic scale, is the field strength of the 8d gauge vector boson arising from 7-branes, and is a (2,0)-form on the 4-cycle describing its transverse geometrical deformations. Near the Yukawa point , we can take and to transform in the adjoint of the enhanced non-Abelian group , which in our case will be given by . Further dynamics of this system is encoded in the D-term

(2.2) |

where stands for the fundamental form of . Together with the superpotential, this D-term relates the spectrum of 4d zero and massive modes to a set of internal wavefunctions along , and the couplings between these 4d modes to the overlapping integrals of such wavefunctions.

Notice that from this latter viewpoint we seem to have a single divisor with a higher gauge group . One must however take into account that both and have a non-trivial profile. On the one hand the nontrivial profile for (more precisely the fact that the rank of jumps at the curves ) takes into account the fact that we do not have a single divisor , but rather a set of intersecting divisors and . A non-vanishing then breaks the would be gauge group to the subgroup , with the gauge groups of the 7-branes wrapping the divisors , typically chosen to be .

On the other hand, the effect of is to provide a 4d chiral spectrum and to further break the GUT gauge group down to the subgroup that commutes with , as it is usual in compactifications with magnetized D-branes [16, 31, 32, 33, 34]. As a result, one may obtain a 4d MSSM spectrum from the above construction by first engineering the appropriate GUT 4d chiral spectrum via and an which commutes with , and by then turning on an extra component of along the hypercharge generator in order to break [5]. Generically, the presence of a non-vanishing field strength along the hypercharge generator is the only way to break the GUT gauge group down to the MSSM one. As a result, all the physics of the MSSM that differ from the parent GUT physics must depend on the data that describe .

Finally, in addition to the above set of divisors hosting the MSSM gauge and matter content, there will be in general other divisors also wrapped by branes which may source non-perturbative effects. Typical examples are 7-branes with a gauge hidden sector of the theory that undergoes a gaugino condensate, or Euclidean 3-branes with the appropriate structure of zero modes to contribute the the superpotential of the 4d effective theory. Such ingredients are usually not considered in the construction of F-theory local models, and indeed they will not be present in the model described in the next section. However, as we will review in section 4, they are crucial in endowing F-theory local models with more realistic Yukawa couplings. In fact, one of the main results of this work is to show this point for the class of local models that we now proceed to describe.

## 3 The SO(12) model

In this section we describe in detail the local model which we will analyze in the rest of the paper. Following the common practice in the F-theory literature, we will first specify the structure of 7-brane intersections and matter curves that breaks the symmetry down to , and then add the worldvolume flux that induces 4d chirality and breaks the GUT spectrum down to the MSSM.

While in this section we will use the language of F-theory local models, it is important to notice that the model at hand admits a more intuitive description in the framework of intersecting D7-branes in type IIB orientifolds. We will exploit such vantage point in the next section, in order to gain some insight on the non-perturbative corrections that can affect our local model.

### 3.1 Matter curves

Following the general framework described in the previous section, let us consider a local model where the symmetry group at the intersection point of three matter curves is . Away from this point, this group is broken to a subgroup because . One can then engineer a such that generically is broken to , except for some complex curves where there is an enhancement to either or . In this way, we can identify as the GUT gauge group and the enhancement curves as matter curves where chiral matter wavefunctions are localized.

In order to make the above picture more precise let us consider the generators of , in terms of which we can express the particle spectrum of our local GUT model.
These generators can be decomposed as , where the , , belong to the Cartan subalgebra of and the are step
generators.^{3}^{3}3Throughout this work we use the standard form of the generators in the fundamental representation [35].
Recall that

(3.1) |

where is the -th component of the root . The 60 non-trivial roots are given by

(3.2) |

where the underlying means all possible permutations of the vector entries.

Let us now choose the vev of the transverse position field to be

(3.3) |

where is related to the intersection slope between 7-branes as explained in section 4, and the charge operators and are the following combinations of generators of elements of the Cartan subalgebra

(3.4) |

This choice of describes a local model that is similar to the toy model analyzed in [22] in several aspects. This will allow us to apply several useful results of [22] to the more realistic case at hand.

Given (3.3) one can understand the symmetry breaking pattern described above as follows. In general the step generators satisfy

(3.5) |

with a holomorphic function of the complex coordinates , of the 4-cycle . The subgroup of not broken by the presence of this vev corresponds to those generators that commute with at any point in . This set is given by the Cartan subalgebra of and to those step generators such that for all . It is easy to see that such unbroken roots are given by

(3.6) |

together with the Cartan generators. Therefore, from the symmetry group only the subgroup remains as a gauge symmetry, and we can identify as our GUT gauge group.

On the other hand, the broken generators of , that have for generic , allow us to understand the pattern of matter curves and to classify the charged matter localized therein. Such broken roots and their charges are displayed in table 1.

root | rep. | |||||

- | ||||||

- | ||||||

- | ||||||

- |

From this table we see that there are three complex curves within where the bulk symmetry is enhanced, in the sense that there for an additional set of roots. Concretely, for there are 10 additional roots that together with those in (3.6) complete the root system. We have labeled such matter curve as , so that in the language of the previous section we would have that . These extra set of roots whose vanishes at can be split into subsets that have different away from . It is easy to convince oneself that each of these subsectors must fall into complete weight representations of , which in turn correspond to the matter localized at the curve. In the case of , there are two sectors and that correspond to the representations 5 and of , respectively, as shown in table 1.

Similarly to , at the curve there are 20 extra unbroken roots and is enhanced to , giving rise to the representations 10 and . The third matter curve is given by , where there is also an enhancement to .

Finally, let us consider a set of quantities that only depend on each root sector of the model. These are the symmetrized products^{4}^{4}4

(3.7) |

where the generators are taken in the fundamental representation of . As we will see, the equations of motion satisfied by the zero modes at the matter curves will depend on these quantities. For the broken roots we obtain

(3.8) |

where the are constants also displayed in table 1.

### 3.2 Worldvolume flux

To obtain a 4d chiral model the above pattern of matter curves is not enough, and it is necessary to add a non-trivial background worldvolume flux to our local F-theory model. Just like the position field, such flux is usually chosen along the Cartan subalgebra of , so that it commutes with and the equations of motion of our system are simplified. Moreoever, considering a component of along the hypercharge generator allows to break the GUT gauge group down to , this being in fact the only way to achieve GUT symmetry breaking for the most generic class of F-theory GUT models.

In order to construct a worldvolume flux with the desired properties we proceed in three steps. First we add a flux analog of the one introduced in the toy model of [22]. Just like in there, this flux will create chirality on the curves and , selecting the sectors and as the ones that contain the chiral matter of the model, as opposed to and . Then we add an extra piece such that the matter curve also contains a chiral spectrum: a typical requirement to achieve an acceptable Higgs sector. None of these previous fluxes further break the gauge group so, finally, we include a flux along the hypercharge generator that breaks down to .

To proceed we then consider the flux

(3.9) |

where

(3.10) |

which is the analog of the flux introduced in the toy model of [22]. To analyze the effect of this flux it is convenient to define the -charge of the roots according to

(3.11) |

The roots in (3.6) are clearly neutral under this flux component , and so the gauge symmetry is not broken further by its presence. The roots in the sectors and are however not neutral. Hence, if the integral of (3.9) over each of these curves does not vanish, they will each host a chiral sector of the theory. In the following we will assume that this is the case and that induces a net chiral spectrum of three ’s in the curve and three ’s in the curve . If this chiral spectrum can be understood in terms of local zero modes in the sense of [36], then such chiral modes should arise in the sectors and of table 1, respectively, and by the results of appendix A one should choose to describe them locally.

Notice that the roots belonging to the sector are neutral under (3.10), and so the spectrum arising from the curve is unaffected by the presence of . As the triple intersection point is where down-like Yukawa couplings arise from, we do need one in such curve, but however no so that no undesired mass terms appear. This chiral spectrum on the sector can be achieved by adding the following extra piece of worldvolume flux

(3.12) |

It is easy to check that the particles localized at the matter curve are now non-trivially charged under the flux background, and that a local chiral spectrum can be achieved if we choose . In particular, as shown in appendix B for one obtains net local chirality in the sector , yielding the desired which is the down Higgs. Notice that those particles at the curves and are also charged under (3.12). However, by construction the number of (local) families in such curves is independent of the flux , as also shown in appendix B.

Let us finally add a third piece of worldvolume flux which, unlike (3.9) and (3.12), will break the gauge group down to the MSSM. As usual, such flux should be turned along the hypercharge generator , and a rather general choice is given by

(3.13) |

where

(3.14) |

We have chosen the hypercharge flux to be a primitive (1,1)-form, so that it satisfies automatically the equations of motion for the background. Note that (3.13) has two components that are easily comparable with the previous flux components (3.9) and (3.12). The first component, proportional to the flux density , is quite similar to . Indeed, as happens for (3.12) its pullback vanishes over the matter curves and , and so it does not contribute to the (local) index that computes the number of chiral families in the sectors and . The second component, proportional to , may in principle affect the chiral index over the curves and but, following the common practice in the GUT F-theory literature, we will assume that this is not the case. Globally one requires that

(3.15) |

so that three complete families of quarks and leptons remain at the curves and after introducing the hypercharge flux. Locally, we demand that the local zero modes still arise from the sectors and , and this amounts to require flux densities such that

(3.16) |

for every possible hypercharge value in the sectors and , see table 2 below.

While innocuous for the matter spectrum at the curves , , the hypercharge flux is supposed to modify the chiral spectrum of curve , in order to avoid the doublet-triplet splitting problem of GUT models [5]. Indeed, one typically assumes that , and since (3.13) couples differently to particles with different hypercharge, this implies a different chiral index for the doublet and for the triplet of . Locally, we have that the total flux seen near the Yukawa point by the doublets on the sector is

(3.17) |

while the flux seen by the triplets is

(3.18) |

Hence, in order to have a vector-like sector of triplets in the local model we can set

(3.19) |

and then assume that such vector-like spectrum is massive. Notice that this condition still yields a chiral sector for the doublets and so forbids a -term for them. Indeed, imposing (3.19) we have that

(3.20) |

which in general will induce a net chiral spectrum of doublets in the curve . Hence, imposing (3.19) the combined effect of and is such that doublets of in the sector feel a net flux, while triplets do not. One may then choose the flux density such that it yields a single pair of MSSM down Higgses at the curve .

To summarize, the total worldvolume flux on this local model is given by

where we have defined

(3.22) | |||||

(3.23) |

and

(3.24) |

Note that the combination of flux densities corresponds to an FI-term, which will be set to vanish whenever supersymmetry is imposed.

Just like in [22], we can now express the vev of the corresponding vector potential in the holomorphic gauge defined in [11], namely as

(3.25) |

this being the quantity that will enter into the equation of motion for the zero mode wavefunctions at the curves , and . The combined effect of the background and breaks to , and as a result the sectors , and split into further subsectors compared to table 1. The content of charged particles under the surviving gauge group is shown in table 2, where we have also displayed the charges of each sector under the operators , , and . We have also included the values of and , which are defined as

(3.26) |

and which, unlike the other charges, depend on the flux densities of the model. As discussed below and in appendix A, each of these sectors obeys a different zero mode equation, and so it is described by a different wavefunction.

Sector | Root | ||||||||

X,Y | |||||||||

X,Y |

### 3.3 Perturbative zero modes

Given the above background, and ignoring for the time being non-perturbative effects, one may solve for the zero mode wavefunctions on each of the sectors of table 2. One obtains in this way the internal profile for each of the 4d chiral multiplets that arise from the matter curves , and , and in particular for the 4d chiral fermions of the MSSM.

Following [22] one may consider the 7-brane action derived in [5], more precisely the piece bilinear in fermions, and extract the equation of motion for the 7-brane fermionic zero modes. These equations can then be written in a Dirac-like form as

(3.27) |

with

(3.28) |

where the four components of represent 7-brane fermionic degrees of freedom. As pointed out in [4], these fermionic modes pair up naturally with the 7-brane bosonic modes that arise from background fluctuations

(3.29) |

More precisely we have that and form 4d chiral multiplets. In addition, form a 4d vector multiplet that should include the gauge degrees of freedom of the model. One can see that these bosonic modes feel the same zero mode equations that their fermionic partners, and so solving (3.27) gives us the wavefunction for the whole chiral multiplet.

As we started from an gauge symmetry, has gauge indices in the adjoint of . Each covariant derivatives in acts non-trivially on such indices, since they are defined as for those coordinates along the GUT 4-cycle , and as for the transverse coordinate [22]. It is then clear that each sector of table 2 will see a different Dirac equation. Hence, in order to solve for the zero mode wavefunctions of our model, we fix the roots to lie within a particular sector of table 2 and then solve sector by sector.

Within each sector , eq.(3.27) is specified in terms of the following quantities: , , and . The zero mode computation for each sector is done in detail in appendix A, and one can see that each of these solutions is of the form

(3.30) |

where we have solved for the zero modes in the holomorphic gauge of eq.(3.25). Here are holomorphic functions of the variable , with , constants that depend on the flux densities , , and the mass scale , and which are different for each sector . The index runs over the different holomorphic functions that are present on each sector, or in other words over the families of zero modes localized in the same curve. Finally, recall that is a holomorphic function of the 4-cycle coordinates . We have summarized the values of these quantities for each sector of our model in table 3,

rep. | ||||

- | ||||

- | ||||

-- | ||||

- | - + |

assuming for simplicity that (see appendix A for the general expressions). For the sectors , and , is defined as the lowest (negative) eigenvalue of the flux matrix

(3.31) |

and one can check that the three lower entries of the vector in (3.30) are the corresponding eigenvector of this matrix. The same definition applies to for the sectors , and .^{5}^{5}5Although they have a similar definition, have different values. Indeed, since we have that is the highest positive eigenvalue of . In general satisfy a complicated cubic equation discussed in appendix A, which depends on the flux densities and . Since these two quantities contain the hypercharge flux, will be different for each of the subsectors , and . Indeed, it is precisely in the value of the flux densities and that the wavefunctions within the same multiplet but with different hypercharge differ.

Although by working in the holomorphic gauge it is not easy to see which wavefunctions converge locally, by going to a real gauge we can see that if we impose condition (3.16) the locally convergent zero modes lie in the sectors , and , see appendix A. In the following we will consider this choice of signs for the fluxes, and so we will concentrate in the wavefunctions for these sectors, that in table 4 are identified with the MSSM chiral multiplets. In particular, in section 5 we will compute how these zero modes are modified in the presence of non-perturbative effects.

Sector | Chiral mult. | ||||
---|---|---|---|---|---|