F-theory Family Unification

We propose a new geometric mechanism for naturally realizing unparallel three families of flavors in string theory, using the framework of F-theory. We consider a set of coalesced local 7-branes of a particular Kodaira singularity type and allow some of the branes to bend and separate from the rest, so that they meet only at an intersection point. Such a local configuration can preserve supersymmetry. Its matter spectrum is investigated by studying string junctions near the intersection, and shown to coincide, after an orbifold projection, with that of a supersymmetric coset sigma model whose target space is a homogeneous Kahler manifold associated with a corresponding painted Dynkin diagram. In particular, if one starts from the E7 singularity, one obtains the E7/(SU(5)xU(1)^3) model yielding precisely three generations with an unparallel family structure. Possible applications to string phenomenology are also discussed.


I. INTRODUCTION
The LHC experiments finally found [1,2] the long sought-for elementary particle that was able to complete the Standard Model, the Higgs boson. They have also shown that the scale of any new physics beyond the Standard Model must be pretty much higher than the electro-weak scale. We are now more and more seriously interested in why the Standard Model is as it is: Why is the top quark so heavy? Why is the lepton-flavor mixing so large?
And in the first place, why are there three generations of quarks and leptons?
In the past 30 years after the discovery of superstring theories, an enormous amount of knowledge on elementary particles has been accumulated. Requirements, or expectations, for realistic string-phenomenology models have become more and more demanding. Indeed, the top quark was finally found [3,4] in 1995 at Tevatron, and the mass turned out to be about 10 5 times heavier than the up quark. In 1998, the zenith angle dependence of the muon atmospheric neutrino was discovered at SuperKamiokande [5], where the θ 23 lepton flavor mixing angle was revealed to be almost maximal, ∼ 45 • . Later neutrino experiments [6,7] also confirmed that another mixing angle θ 12 was large, and θ 13 was also nonzero [8][9][10][11]-all these undeniable experimental data point to a single fact: The three flavors are not on equal footing.
Superstring theory, however, has developed almost independently of these experimental discoveries. To say the least, even though it could contrive to achieve such hierarchical structures (which should be different between the quark and lepton sectors, and also between the up and down types) by more or less ad-hoc assumptions and/or fine tunings, it has never been able to explain them. Of course, it would be easy to close our eyes to all these facts and dismiss everything as an accident; this is not our attitude in this paper.
There have been numerous efforts to understand the hierarchical family structure of quarks and leptons. Particularly interesting among them is the idea of family unification.
Family unification is the idea that the quarks and leptons are the fermionic partners of the scalars of some coset supersymmetric nonlinear sigma model [12][13][14][15][16][17][18][19][20]. A remarkable observation made by Kugo and Yanagida was that [16] the supersymmetric sigma model based on E 7 /(SU (5) × SU (3) × U (1)) had precisely three sets of 10 ⊕5 of SU (5), in addition to a single 5, as its target space. What is special here is that the three generations are asymmetrically embedded into E 7 as SU (5) multiplets. Indeed, the two5's, identified as the second and third generations of the SU (5) GUT multiplets [21,22] containing the down- represents SO (10), the next to next is E 6 , and the whole square is E 7 . Exactly three sets of type quarks, come with the "symmetry breaking" from E 7 to E 6 , whereas the last5 arises when E 6 "breaks" to SO (10). In contrast, a 10 representation arises at each step when the rank of the "symmetry group" is reduced by one (FIG.1) [143]. This was called the "unparallel" family structure by Yanagida [23,24].
Therefore, a natural question to ask is: Can one get such a coset family structure in string theory? In most string phenomenological models, however, generations arise on equal footing and there is no a priori difference among them. So far, very few attempts have been made to obtain unparallel generations, and the few have met with only partial success [25,26].
In this paper, we take a modest step towards realizing this unparallel family structure in string theory. We will show that such a structure can in fact naturally arise in string theory, in the framework of local F-theory. The study of phenomenological applications of F-theory [27,28] has been of much interest in the past several years , but in this paper we will bring a slightly different perspective. We show that a certain local 7-brane system in Ftheory can realize, already at the level of six dimensions, the same quantum numbers as that of the SUSY nonlinear sigma model considered in family unification. Our key observation is that the representations of the charged matter hypermultiplets arising at the singularity are precisely (as far as the enhanced singularity is of the split type [59]) the ones consisting of a homogeneous Kähler manifold of the corresponding painted Dynkin diagram. In particular, if one starts from the E 7 singularity, one obtains a set of six-dimensional massless matter which have the same quantum numbers as those of the E 7 /(SU (5) × U (1) 3 ) Kugo-Yanagida model. To get a four-dimensional N = 1 theory, we then compactify the six dimensions on a two-torus, and project out half of the spectrum by taking an orbifold. Note that the requisite quantum numbers are already obtained in six dimensions, and there is no need for contrived assumptions to get the desired spectrum here.
Although these rules themselves must have been known for some time, the relation to homogeneous Kähler spaces or nonlinear sigma models seems to have never been discussed in the literature [144]. Indeed, as of writing this article, there is only one paper [62] in the INSPIRE database that cites both Katz-Vafa [63] and Kugo-Yanagida [16], and in [62] such a connection was not mentioned.
We are interested in some local geometric structure that can realize precisely three unparallel families. This is because if the realization of the SM were a consequence of the global details of the entire compactification space, it would be very hard, if not impossible, to find any reason or explanation for what we observe now.
The plan of the rest of this paper is as follows: In section II, we give a brief review of the basic idea of coset space family unification. In section III, we first recall the known results of F-theory/heterotic duality in six dimensions, and then explain Tani's argument of how the chiral matter at the extra zeroes of the discriminant can be understood in terms of string junctions. We are naturally led to the coset structure of the chiral matter, and propose the 7-brane configuration for the E 7 /(SU (5) × U (1) 3 ) model. In section IV, we prove that the 7-brane configuration can preserve SUSY. Section V is devoted to a brief discussion on how we can derive a four-dimensional model from the six-dimensional one obtained from the configuration. It is also pointed out that in our setup there is a possibility for the anomalies of the original model to cancel due to the anomaly inflows. In section VI, we present the explicit local expression for the curve of the brane configuration. Finally, we conclude in section VII with a summary and discussion. Appendix A contains the explicit result of the recursion relation in section IV, whereas Appendix B is a brief explanation of how the monodromy is read off by tracing the value of the J function.

II. COSET SPACE FAMILY UNIFICATION
We begin with a review of the basic idea of coset space family unification, which is what we want to achieve in string theory. We will be brief, and for more detailed discussion we refer the reader to [20], and also [64].
As we already mentioned in Introduction, family unification is the idea that the quarks and leptons can be understood as quasi-Nambu-Goldstone fermions [13] of a supersymmetric coset nonlinear sigma model [12,[15][16][17][18]20]. This means that the target space of the sigma model is some homogeneous space G/H associated with a Lie group G and its closed subgroup H. The idea of identifying all the three families as being a part of some group is an old one [65][66][67][68][69][70][71][72][73][74][75][76], going back before the superstring theories were found, but it is important to note that being a coset is essential for the chiral nature of the spectrum, which is in contrast to the models in the earlier literature.
Originally, such a nonlinear sigma model was thought of as arising from a spontaneous supersymmetry breaking of some global symmetry caused by a strong gauge dynamics of the underlying "preon" theory [12]. Later, we will show an alternative, geometric origin of these sigma models in F-theory.
To characterize D = 4, N = 1 supersymmetric sigma models the following two facts are essential: The first fact is that, in order to have D = 4, N = 1 SUSY, the scalar manifold must be Kähler [77,78], which is well-known. The second is a classic result due to Borel [79]: Let G and H be a semi-simple Lie group and its closed subgroup, then the coset space G/H is Kähler if and only if H is the group consisting of all elements that commute with some U (1) n subgroup of G. Thus it follows that every element in G/H has a nonzero charge for some U (1) subgroup, since otherwise such an element must belong to H by construction.
Borel's theorem also states that the set of all G-invariant complex structures correspond oneto-one to that of all Weyl chambers of the Lie algebra. This statement can be translated into a useful way of distinguishing different complex structures as follows [20]: Suppose that we have chosen some U (1) generator Y which commutes with any generator of H. We can decompose the Lie algebra of G into a direct sum of eigenspaces of adY , that is, into a sum of spaces of "states" with different U (1) "Y -charges". Then G/H consists of the spaces with negative Y -charges. If the charge vector of Y is varied so that it passes across into the next Weyl chamber, then one of the signs of the Y -charges flips, and this corresponds to the change of the complex structure.  [24].
In the present E 7 /(SU (5) × U (1) 3 ) case, the Y -charge can be a linear combination of the form for any negative s, t and u.
As we said, the target space of a coset supersymmetric nonlinear sigma model is a homogeneous Kähler manifold. There is available a useful representation of homogeneous Kähler manifolds in terms of painted Dynkin diagrams [64]; their "cook-book recipe" states that [64] one first draws the Dynkin diagram for the numerator Lie group, paints a subset of the vertices (note that the roles of the black (painted) and white nodes are traded here) for the denominator U (1) subgroups so that the remaining white Dynkin diagrams correspond to the semi-simple part of the denominator group. In this way all homogeneous Kähler manifolds are classified [64]. The corresponding painted Dynkin diagrams for Basically, this kind of family unification models utilize the Froggatt-Nielsen mechanism [80] to account for the origin of the hierarchical family structure. The In this section we recall the beautiful results on F-theory/heterotic duality in six dimensions. Although this has already been well known for some time, it is useful to review the original discussion because thereby its connection to homogeneous Kähler manifolds can be uncovered.
The proposal of [92] was that the K3 compactification of the E 8 × E 8 heterotic string with instanton numbers (12 − n, 12 + n) is dual to F-theory compactified on an elliptic Calabi-Yau three-fold over the base space being the Hilzeburch surface F n . More precisely, the dual Calabi-Yau manifold was given in the Weierstrass form as [92,93] where z is the coordinate for the P 1 fiber of F n , and w is the one for the P 1 base of F n . f k (w), g k (w) are k-th order polynomials of w. Intuitively, this six-dimensional duality is understood as the fiber-wise duality in eight dimensions [27] further compactified and fibered over P 1 .
This proposed duality was examined in detail in [59]. In particular, the dimensions of the neutral hypermultiplet moduli spaces were compared between the two, and a perfect match was found in various cases of unbroken gauge symmetries. It was also found there [59] that the charged matter arose at "extra zeroes" of the discriminant of the curve (5) at particular values of w, where the singularities of the coinciding gauge 7-branes were (more) enhanced.
On the heterotic side, the number of neutral hypermultiplets is 2n + 21 [93,95], whereas on the F-theory side, it is determined by the dimensions of the complex structure moduli.
The latter is the number of coefficients of the polynomials up to an overall rescaling: (9 + n) + (13 + n) − 1, which indeed coincides with the heterotic computation.
On the other hand, charged matter in this heterotic compactification is found to be [93,95] 8 + n half-hypermultiplets in 56 of E 7 . Since 8 + n is the number of zeroes f 8+n (w), these "extra" zeroes in the discriminant implied the appearance of charged matter in Ftheory. Indeed, this was confirmed in [59] in various patterns of gauge symmetry breaking.
Their results are summarized [146] in the first three columns of TABLE II (together with the corresponding set of coalesced 7-branes and associated homogeneous Kähler manifolds, which are explained shortly).
Note that at 8 + n zero loci of f 8+n (w), the discriminant of the first term vanishes, becoming a tenth-order polynomial in z. Again, according to Tate's algorithm, this means that the singularity is enhanced from E 7 to E 8 at these "extra zeroes".
Another example is the dual curve with an E 6 unbroken gauge symmetry: The difference from (6) is that it contains a z 4 term, and also g 12+2n (w) must be in the split form [59], that is for some q 6+n (w). Then the discriminant becomes showing that there is an E 6 singularity at z = 0. The neutral moduli counting is n + 7 + n + 9 + n + 13 − 1 = 3n + 28, which again agrees with the heterotic result. Also the heterotic prediction of the charge matter is n + 6 hypermultiplets in 27, which is certainly implied by the extra zeroes of the discriminant (10).
At this point, looking at the charged matter contents in the two examples, we notice an interesting fact: They are precisely the ones found in the homogeneous Kähler manifolds (1)) and E 7 /(E 6 × U (1)), respectively. In fact, as shown in TABLE II, they all correspond to a homogeneous Kähler manifold of the relevant type. As explained in subsequent sections, the latter are classified and labeled by "painted" Dynkin diagrams [64].
The corresponding diagrams are also shown together in TABLE II.
Why does such a relationship exist? A geometric explanation has been given [63] on how the charged matter arises at the extra singularities by utilizing the Cartan deformation of the singularity. There is, however, another alternative argument using string junctions [60,61] that is more convenient to establish the connection between the charged matter spectrum at an extra zero and a homogeneous Kähler manifold. This is the main theme of the next section. one of E 8 's on the heterotic side, and r specifies how they are distributed when the commutant group is a direct product [59]. In addition to the data shown in [59], the corresponding 7-brane configurations as well as the homogeneous Kähler manifolds are also displayed.

B. Matter from string junctions-Tani's argument
We will now explain how the chiral matter spectrum is determined by investigating string junctions near the enhanced singularity, following [60] (see also [61] for a more recent discussion) . In F-theory, gauge symmetry enhancement occurs associated with a singularity of an elliptic manifold on which F-theory is compactified [27]. Singularities of elliptic fibrations were classified according to their types investigated by Kodaira [96]. Technology was developed [97] to describe these Kodaira singularities in terms of coalesced [p, q] 7-branes and string junctions stretched between them (see [98][99][100][101][102][103][104][105][106][107] for more works on string junctions in F theory). We will first briefly summarize the salient features of their description, referring to [97] for more detail.
One of the characteristic features of F-theory is that 7-branes are allowed to change their types (=(p, q) SL(2, Z) charges) as they wander about among themselves. More precisely, a 7-brane background is only single-valued up to SL(2, Z) transformations [108]. Such a path-dependent transformation is called monodromy. For instance, as we will see in section IV, in the single D7-brane solution the type IIB scalar τ behaves like ∼ 1 2πi log z near the brane locus on the complex z plane. So if one traces the value of τ as one moves around the locus, τ gets transformed to τ + 1. Thus the monodromy in this case is a fractional linear transformation specified by More generally, taking the convention that a [1, 0]-brane means a D-brane, the monodromy matrix for a [p, q] brane is given by a similarity transformation of T as p r q s where p, q, r, s are all integers satisfying ps − qr = 1. X [p,q] does not depend on the choice of r or s.
What has been shown in [97] is that Kodaira's classification of singularities of elliptic fibrations can be expressed by the joining/parting of several 7-branes, each of which is of the simplest (I 1 ) singularity type with a (relative) monodromy of either [147] The correspondence is summarized in  1, 0)) string circles around a C-brane and a B-brane, then so it turns into a (−1, 0) string. This can still end on another A brane but its sign of the charge is inverted. A pair of B and C branes are necessary elements to constitute a D-type singularity, identified as an orientifold plane [28]. String junctions are conveniently represented in the "divisor-like" form [97]; if n (p, q) strings come into a [p, q] 7-brane for a given multi-pronged string, then one associates them with the monomial nx [p,q] . Summing up these monomials over all the prongs, one obtains the expression for the string junction as a formal sum of monomials with integer coefficients.

For the example of FIG. 3, this is
where a = x [1,0] , b = x [1,−1] and c = x [1,1] . On the other hand, an ordinary fundamental string directly connecting the two D(=A)-branes is represented as In [97], it was proved that the 7-brane configuration for the E N algebra for N ≥ 2 is equivalent tõ since they are made identical by the use of monodromy transformations and an SL(2, Z) conjugation. In fact, the string junctions representing the E 8 roots are most conveniently described in terms of Freudenthal's realization of E 8 [109][110][111]; the exceptional Lie algebra where 123456789 = 123456789 = +1. The string junctions for the E 8 roots corresponding to these generators are summarized in TABLE IV.
The fact that the Kodaira singularities are described by coinciding 7-branes, and the existence of varieties of string junctions which correspond to the roots of the exceptional

Generator
String junction group, offer a natural explanation [98] for the origin of the exceptional group gauge symmetry in F-theory. When N D-branes come on top of each other, one gets U (N ) gauge symmetry [112]. D-brane systems, where the explicit quantization is possible, this is a familiar phenomenon [113]. So if one assumes that this is also true in the general case when the singularity contains not only D-branes but other [p, q] branes, then one can get a perfectly consistent picture of the chiral matter generation [60]. how to do that we would like to consider the more fundamental question: Can the 7-brane configuration as described above preserve SUSY? This is because, if not, our proposal of how to achieve supersymmetric family unification in F-theory would break down. Fortunately, however, we can prove that it indeed can, as we show in the next section.

IV. HOLOMORPHIC DEFORMATION OF 7-BRANES
A. The stringy-cosmic-string solution Let us first recall the 7-brane solution of type IIB supergravity on P 1 obtained long time ago by Greene et.al. [108]. Although this is also a well-known material, we will take a different route to it, examining the SUSY conditions; our approach is more convenient and can be directly extended to the deformed case discussed in the next section.
The metric ansatz is The type IIB complex scalar field τ = C 0 + ie −φ , where C 0 and φ are the RR scalar and the dilaton, respectively, is assumed to be a holomorphic function depending only on z: The other supergravity fields are set to zero. In this case the supersymmetry variations of the gravitino and the dilatino are [114,115] where P µ and Q µ are well-known SU (1, 1)-invariant connections given by Since we have assumed that τ is holomorphic, we have Let z,z be complex linear combinations of real coordinate x˙8 and x˙9 :[148] and take the two-dimensional gamma matrices where each component is understood to act on spinors in the eight-dimensional space-time. Then so δλ vanishes for of the form * 0 .
On the other hand, the gravitino variation δψ µ depends on the spin connections and Q µ .
The only nontrivial components of the spin connection are while since τ is assumed to be holomorphic. Let then Therefore, if ϕ = log τ −τ 2i , a Killing spinor (which is constant in this case) exists and half the supersymmetries are preserved. More generally, if for some holomorphic function F (z), then with is a Killing spinor. F (z) is chosen to be [108] F (z) = 2 log η(τ (z)) + f (z).
The first term is for the modular (or S-duality) invariance, while f (z) is some function of z to compensate the zeroes at the 7-brane loci. For instance, for a single D7-brane at z = 0 we have locally [108] τ

B. A supersymmetric deformation of 7-branes
In the previous section we have reviewed the 7-brane solutions in eight dimensions-in modern terminology this is a codimension-one singularity. We now turn to a codimension-two singularity, that is, we deform τ so that it also varies over another holomorphic coordinate w = x˙6 + ix˙7: Note that we do not specify any particular global geometry. We will show that, for any such holomorphic deformation [149] of the modulus function τ , there exists, at least locally, some Kähler metric such that it preserves a quarter of supersymmetries.
We focus on the four-dimensional part of the ten-dimensional metric, which we assume to be hermitian: Any hermitian metric can be written in this form with two real functions Φ, Ψ and a complex function ξ. The vierbein of this subspace is block diagonal: where µ = i,ī; i = z, w;ī =z,w; α = a,ā; a = 1, 2;ā =1,2 with In our convention the flat metric is so that We take so that Due to the holomorphic assumption (40), we have, again, The dilatino variation thus reads Since the leftmost columns of γ a (a = 1, 2) are zero as displayed in (47), δλ vanishes for a SUSY variation parameter of the form We will now examine under what conditions the gravitino variation δψ µ also vanishes for (50). Since the nonzero component is only the first one, we are only concerned with the first columns of ω γαβ γ αβ : Since the "Bismut-like" connection (23) It is easy to see that they are equivalent to ∂ w (e Ψ ξξ + e Φ ) = ∂ z (e Ψξ ) and (54) or which are satisfied if the metric is Kähler. Conversely, if the conditions (52), (53) are satisfied, then the spin connection turns out to be a U (2) connection and hence the metric is Kähler.
Suppose that we have a solution to the conditions (52), (53). Such a solution exists, at least locally, as we will show in a moment. Then using them in (51), we find On the other hand, Q µ 's are given by Therefore, similarly to the previous section, we have a Killing spinor if for some holomorphic function F of z i = z, w. Again, we can set F (z) similarly to (38) for some f (z) which compensates the zeroes of the brane loci, then the metric is positive and Now what remains to be done is to show the existence of Φ, Ψ and ξ that satisfy (52), (53) with the constraint (59) for a given τ (z i ). First we note that, if τ is only a function of z and does not depend on w, then the problem reduces to that discussed in the previous section with which obviously satisfy (52), (53) and (59). So for given we determine with To do this it is more convenient to write A is a real, while B is a complex function of z, w,z andw. Then where we have used (59) in the last line. Let us also set where the real function h(z, w,z,w) is determined by the given holomorphic functions τ and F . The system of equations is now which can be solved by iteration. We expand w nwn A (n,n) (z,z), w nwnB (n,n) (z,z).
Since A is real, we have whereas B (n,n) =B (n,n) .
Using (75) and (76) with the initial conditions and arbitrary functions B (0,n) (n = 1, 2, . . .), A (n,n) and B (n,n) (and henceB (n,n) ) can be determined iteratively. Thus we have shown that the equations (52) and (53)  The explicit forms of A (n,n) obtained as a result of the iteration are given in Appendix A up to n +n ≤ 3.
Finally, since the J function is also holomorphic, holomorphic deformations of the coefficient functions in the Weierstrass form lead to a local supersymmetric solution of type IIB supergravity.

V. ORBIFOLDS AND ANOMALIES
Thus the 7-branes described in III C (the ones like a bunch of raw spaghetti) preserve SUSY. So if we compactify two more dimensions to reduce the SUSY to N = 1 and drop one half of the chiral supermultiplets with a definite chirality, then we end up with precisely the E 7 /(SU (5) × U (1) 3 ) supersymmetric family unification model. This could be done in a variety of ways. The most modern way to do this is to turn on (after compactifying to four dimensions) some appropriate vortex Higgs field [29]. Then depending on the sign of the charge, similarly to the Y -charge explained in section II, a half of the solutions of the Dirac equation would become nonnormalizable and one is left with a chiral spectrum.
Alternatively, one could simply compactify two of the six dimensions on a two torus T 2 , wrap the 7-branes around it and take an orbifold (see e.g. [116,117] and references therein) to reduce the SUSY, thereby imposing a boundary condition such that the massless fields with, say, positive Y -charges are projected out. This would lead to the same effect as turning on a Higgs field, and hence is an effective and efficient way. It will be a bit ad hoc, but we emphasize that the necessary set of fields with correct gauge quantum numbers are already fixed before the projection; there is no need to adjust anything but is only need to eliminate a half to get them. More concrete discussion on this part of the construction will be given elsewhere.
There is one more thing to be discussed at this point: The E 7 /(SU (5) × H) models are anomalous, in the senses of both the gauge anomaly and the sigma-model anomaly. This problem has been for a long time [118]: The E 7 /(SU (5) × H) model includes a single 5 representation, and there is nothing else in the sigma model itself to cancel its anomaly.
In six dimensions there is no problem; the six-dimensional heterotic spectrum on K3 is of course anomaly-free, and so is it for F-theory. If one focuses on a particular extra zero point, then the anomaly balance will be lost, but upon compactification to four dimensions, they become non-chiral and hence have no anomaly. Therefore, this is the issue only after the chiral projection.
There are at least two ways out of this problem. One is the idea that there arises some extra matter from the orbifold. This idea has already been pointed out by several authors.
Another, more interesting possibility is that extra anomalous contribution to the effective action on the brane might come in from the bulk, known as the anomaly inflow mechanism [119][120][121], [25,61,122,123]. This is an interesting possibility, but if this is true, then it would lead to a non-trivial prediction in the Higgs sector, since it is not the quantum effect of some5 field, but some other effective contribution induced from the bulk, that cancels the anomaly of the 5 field of the model. We postpone a more concrete analysis to future work. Therefore, since the J function is holomorphic, we have only to consider the Weierstrass form for holomorphic functions f (z, w), g(z, w) such that they develop an E 7 singularity at z = w = 0, and vary over w according to Tate's algorithm so that the singularity is relaxed to g(z, w) = 2z 6 + a 2 12 where a = a(w), b = b(w) and = (w) are smooth functions only of w such that If one resorts to the general argument [97,98] on the 7-brane realization of the Kodaira singularities, it may easily be guessed what types of 7-branes are separating from the rest of coalesced branes. We can, however, directly see this by tracing the value of the J function (see Appendix B). This technology was developed by Tani [60]. For the purpose of illustration let us consider some special cases: Case I : a(w) = b(w) = (w) = 0, unbroken E 7 In this case (80) and (81) become simply g(z, w) = 2z 6 .
The discriminant is indicating that the curve has an E 7 singularity at z = 0 for arbitrary w. This is a coalesced  around the origin with the angle varying from 0 to 2π − π 6 (and not to full 2π, so that we may distinguish clockwise or anti-clockwise). This is anti-clockwise.
which agrees with the monodromy read off from the behavior of the J function.
In this case the equations (80), (81) read which gives an E 6 singularity at z = 0 for generic w but it is enhanced to E 7 at w = 0 since a(0) = 0 as we assumed. The discriminant in this case is ∆ = 108z 11 + 9 a(w) 2 − 4 z 10 + 4z 9 + 3a(w) 4 z 8 16 , whose roots are z = 0(multiplicity eight), The second to last complex conjugate pair is the loci of the 7-branes which were already The monodromies around the origin and the locus of the separated brane are shown in the right plots. From these we can see that the former is T −1 S as an SL(2, Z) conjugacy class, and the latter is T . Thus the separating brane is an A brane, and the monodromy around the origin agrees with the fact that If we also allow b to take nonzero value, then we have g(z, w) = 2z 6 + a 2 12 The discriminant takes the form ∆ = 108u 11 + 9 a 2 − 36b 2 + 12b − 4 u 10 + −216b 3 + 216b 2 + 4 u 9 The new discriminant locus is which is equal to T −1 S −1 T −1 . Multiplying the monodromy around the origin T , we have which is the same thing as T −1 S in P SL(2, Z) and hence is consistent with Case II. On the other hand, which is equal to T in P SL(2, Z), whereas so the separating brane is indeed identified as the C brane.
Case IV : a(w)b(w) (w) = 0, The final case is when any of a(w), b(w) or (w) does not vanish. The functions f and g are given by (80) and (81), respectively, and the discriminant is for the origin is the same as before, and that for the new locus is taken from −π to π − π 6 . ∆ = 108u 11 + 9 a 2 + 12 a + 4 −9b 2 + 3b + 9 2 − 1 u 10 +4 −54b 3 + 54b 2 + 27 (a + 6 )b − 27 2 − 18a + 1 u 9 + 3a 4 16 + 9 2 b − 5 2 a 2 − 12 −18b 2 + 9 2 − 1 a which is of order 5 as arranged for an A 4 singularity. There appear two new nonzero discriminant loci; their positions are 2 b for both to leading order in , but different in their higher order terms. The monodromy around the origin is T 5 as shown in FIG.9, indicating the existence of five collapsed A branes. On the other hand, the monodromy around each discriminant locus is T for both. Again, as shown in FIG.9, there is a locus of g between the origin and the closer locus, and there are two loci of f and another locus of g between the origin and the farther one. Therefore we must do the corresponding similarity transformations. It can be shown that the monodromy is S −1 for the former, and T 2 S −1 T ST −2 for the latter. Thus the monodromy matrix for the closer locus is and that for the farther locus is Changing their positions with each other in such a way that the farther brane passes through the branch cut of the closer brane, their monodromies become A −1 BA and A −1 CA, so they are a B and a C brane, respectively, as expected.

VII. SUMMARY AND DISCUSSION
Using Tani's argument to account for the chiral matter generation at extra zeroes, we have proposed a natural geometric mechanism for realizing the coset family structure in F-theory.
Tani's argument uses string junctions connecting the various gathering 7-branes meeting at that point and is a direct generalization of [113] for the intersecting D-brane systems. It offers a perfectly consistent picture of the chiral matter generation in six dimensions for the split-type singularities, and we have pointed out that their relations to homogeneous Kähler manifolds are readily explained in this picture. Note that these rules are not just a kind of "trick" as has been known in the literature, but can be deduced by the concrete entities responsible for the symmetry enhancement: the string junctions.
In particular, we have proposed a local 7-brane system as shown in FIG.5 which would yield the set of supermultiplets in six dimensions with exactly the same gauge quantum numbers as those in the three-generation E 7 /(SU (5) × U (1) 3 ) coset family unification model.
We have proved that for a given holomorphically varying type IIB scalar field configuration in six dimensions, there exists at least locally a Kähler metric such that a half of the supersymmetries are preserved. We have further discussed how this local model is compactified to four dimensions and half of the spectrum is projected out on orbifolds, and also suggested how the anomalies of the original models can be canceled. The last point is still incomplete and we leave this issue to future work.
One of the nice features of our mechanism is that such a gathering brane system is a local one and could emerge independently of the every global detail of the ambient space. Meanwhile, given the experimental data, several authors have recently pointed out the possible existence of the "desert", the absence of new physics between the electro-weak and string scales [124][125][126]. If this is true, and if string theory is really the theory beyond the Standard Model, then it must have a mechanism to realize close to what we observe now already at the string scale. Our proposal fits with this requirement.
It is interesting to speculate how this configuration might come to exist: Suppose that there were, perhaps in the very early universe, some set of 7-branes, and assume some attractive force to be somehow generated between them. Then such 7-branes might have become closer and closer until they collide with each other. This can happen only if they are a collapsable set of 7-branes which must be one of the types of the Kodaira classification.
If these branes were the ones that could constitute the E 7 singularity, then just after they made a collision and at the last minute before they were completely separated, they would have looked like FIG.5. This story is of course just a speculation at this moment, but it would be an interesting scenario to study.
h (p,q) (n,n) ≡ ∂ p z ∂ q z h (n,n) . B (0,n) 's are arbitrary functions of z andz, and B (p,q) (0,n) ≡ ∂ p z ∂ q z B (0,n) . A (n,n) is equal to the complex conjugate of A (n,n) . Once A (n,n) 's are determined, then so are B (n,n) 's by the equation (75).

Appendix B
Klein's J-function is a modular invariant holomorphic function from the upper-half plane H to the complex plane C, and maps one-to-one the fundamental region of the modular group to the complex plane. The definition in terms of theta functions is Suppose that τ ∈ H changes its value from some τ 0 in the standard fundamental region to τ 0 + 1, which belongs to another fundamental region next to it. Then the trajectory of J(τ ) circles clockwise around 1 and 0 (FIG. 10). On the other hand, if τ ∈ H changes from τ 0 to − 1 τ 0 , then J(τ ) only circles around 1, counter-clockwise (FIG. 11). Note that since J(τ ) has a triple zero at τ = e Likewise J(τ ) − 1 has a double zero at τ = i, so J(τ ) goes twice around 1 when τ does once around i.