Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications

The Mordell-Weil lattices (MW lattices) associated to rational elliptic surfaces are classified into 74 types. Among them, there are cases in which the MW lattice is none of the weight lattices of simple Lie algebras or direct sums thereof. We study how such"non-Cartan MW lattices"are realized in the six-dimensional heterotic/F-theory compactifications. In this paper, we focus on non-Cartan MW lattices that are torsion free and whose associated singularity lattices are sublattices of $A_7$. For the heterotic string compactification, a non-Cartan MW lattice yields an instanton gauge group $H$ with one or more $U(1)$ group(s). We give a method for computing massless spectra via the index theorem and show that the $U(1)$ instanton number is limited to be a multiple of some particular non-one integer. On the F-theory side, we examine whether we can construct the corresponding threefold geometries, i.e., rational elliptic surface fibrations over $P^1$. Except for some cases, we obtain such geometries for specific distributions of instantons. All the spectrum derived from those geometries completely match with the heterotic results.


Introduction
F-theory [1,2,3] has a unique feature in modern particle physics model building based on string theory. The SU (5) GUT, which naturally explains the hypercharges of the observed quarks and leptons, and matter in the spinor representation of SO (10), into which all the quarks and leptons of a single generation are successfully incorporated-both are readily achieved in F-theory. F-theory models have an advantage over the E 8 × E 8 heterotic models as they may evade the issue of the GUT vs. Planck scales first addressed in [4]. F-theory can also generate Yukawa couplings that are perturbatively forbidden in D-brane models [5].
Almost a decade after the first formulation of F-theory, there was considerable development in understanding the local models [6,7,8,9] in terms of twisted super Yang-Mills theories or the Higgs bundles [10,11]. One of the notable findings in the development was the mechanism of the GUT gauge symmetry breaking by the fluxes turned on the brane. A large number of studies have been carried out in local models. An incomplete list includes [12,13,14,15].
Rather soon after this development, the Higgs particle was found at LHC in 2012, and the subsequent experiments showed that there was no low-energy supersymmetry. Later, the PLANCK data became also available to reveal that the energy of the inflation can be very high, even close to the GUT scale. These two new sources of knowledge have turned the focus of F-theory model building to global models. It is also known [3] that U (1) gauge symmetries in F-theory arise when the Mordell-Weil rank is nonzero, that is, when there are nontrivial global sections. This is in sharp contrast to nonabelian gauge symmetries, which can be solely determined by the singularity in the local model. Recent works on global models include [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50].
For K3 surfaces, the Mordell-Weil rank varies depending on its Picard number. In contrast, the Mordell-Weil rank of a rational elliptic surface is always 10. Its Mordell-Weil group is known to be endowed with a lattice structure, and the possible pairs of the singularity and the corresponding Mordell-Weil lattice have been classified into 74 types [51]. Roughly speaking, the singularity lattice and the Mordell-Weil lattice are the orthogonal compliment of each other in the E 8 root lattice. In a typical case, the Mordell-Weil lattice coincides with a weight lattice of some semi-simple gauge group of the instantons in the dual heterotic string theory, and the singularity lattice is that of the unbroken gauge group. However, it is interesting to note that in the other cases the inner product matrix of the Mordell-Weil lattice is none of the (inverse of the) Cartan matrices of simple Lie algebras, nor is it their direct sum. It is these non-Cartan type Mordell-Weil lattices that we focus on in this paper. In fact, these are the cases where the gauge instanton includes some U (1) factor(s). 1 We are particularly interested in the explicit forms of the Weierstrass equations of this special class of rational elliptic surfaces with a section 2 which are fibered over P 1 to form a complex threefold. This means that the parameters of the Weierstrass equations are sections of some line bundles over P 1 [3]. 3 Each of these geometries is regarded as a part of an elliptic K3 fibration in the stable degeneration limit [61,62,63] , and hence as a "1/2 CY threefold" since two such complex manifolds can be glued together into a K3-fibered Calabi-Yau threefold (CY 3 ).
In this paper, we will specifically consider a class of rational elliptic surfaces, and their fibrations over P 1 , which satisfy the following criteria: 1. The Mordel-Weil lattice E(K) is neither a weight lattice of some semi-simple Lie algebra, nor is it a direct sum of such a weight lattice and a torsion.
2. The singularity lattice T is a sublattice of A 7 .
There are ten types of such rational elliptic surfaces in the Oguiso-Shioda classification, which are summarized in Table 1 (See section 2). To find an equation representing a "rational-elliptic-surface (RES)-fibered" complex threefold over P 1 whose fiber rational elliptic surface belongs to this class, our strategy is to realize in the geometry the singularity G of lattice T associated to a given E(K). First, we start from No.7 with singularity G = SU (2)×SU (2)×SU (2), which is not a non-Cartan type, and construct the corresponding CY 3 with K3 fibration. We then successively tune the complex structures ("unHiggsing") to achieve the necessary singularities G of the respective types given in Table 1. Next, we map the obtained K3-fibered CY 3 's to RES-fibered geometries. We carefully construct this map so that it does not change the structure of the singularity. Through this strategy, we can successfully obtain the equations for all the SU (7) series (upper rows in Table 1) and some (No. 22 and No.29) of the SU (8) series (lower rows in Table  1). We will also discuss why it is hard to find the equations for the remaining cases. We note that the equations for SU (8) series cannot be obtained by using Tate's form and we need to work in the Weierstrass form. This is because, if the coefficients a i (i = 1, 2, 3, 4, 6) of Tate's form are assumed to have the necessary factors required from each singularity of the SU (8) series, then the total degrees of some a i (including a 3 and a 6 ) would exceed i, leading additional unwanted singularities.
We also compute the massless spectra of the dual heterotic string compactifications whose vector bundles are supposed to be determined by the RES-fibered spaces above [2,3,64]. For all cases except No. 45 in Table 1, the instanton gauge group H in the dual heterotic string theory has two or more irreducible group factors. In particular, we will see that H specified by a non-Cartan Mordel-Weil lattice is typically a product of a semi-simple group and one or more U (1) group(s). Thus one can distribute the total 12 + n instanton numbers to over anything then. A recent work on non-Cartan MW lattices for rational elliptic surfaces (and not their fibrations) is [59]. 3 Although we consider in this paper rational elliptic surfaces fibered over P 1 , the Weierstrass equations we obtain can be readily converted to those for a fibration over a complex two-fold B 2 by simply replacing a degree an + b polynomial in the affine coordinate z of the P 1 , with a section of L −d ⊗ N s with d = 6a − b 2 and s = a, where L is the anti-canonical bundle of B 2 and N is the twisting line bundle determining the normal bundle of the fiber P 1 , which is the base of the rational elliptic surface. For further explanation of this correspondence, see [60]. each group factor. Although in principle there is no problem in applying the index theorem to these cases, the subtlety is that the multiplicities of some massless hypermultiplet then become fractional unless the U (1) instanton number is a multiple of some particular non-one integer. We will show why the U (1) instanton number cannot take an arbitrary integral value but must be a multiple by examining the orthogonal decomposition of the E 8 root lattice. This gives us a consistent integral number of hypermultiplets. For the cases of rank(E(K)) > 2, more than one choices of the number of U (1) direction(s) are possible. In such cases, we obtain more than one spectra for a given E(K) (and hence for a given G). We find that the spectrum is more general when the number of U (1) direction(s) is larger.
While it is possible to compute heterotic indices for the cases of distributed instantons, it is a nontrivial problem to obtain the equations for the RES-fibered spaces corresponding to such particularly distributed instantons. We will explain for the No.7 case, which is a Cartan type, how we can obtain the equations for an arbitrary distribution of instantons, and show the complete match of the six-dimensional massless spectra read off from the Weierstrass equations on the F-theory side and those obtained by the index computations on the heterotic side. On the other hand, for every case of the non-Cartan type, where we have succeeded to find an equation for the RES-fibered space, we also show that the spectrum read off from the equation agrees with that of the dual heterotic theory for a special choice of instanton distribution.
The outline of this paper is as follows: In section 2, we begin with an introduction to the non-Cartan type Mordell-Weil lattices. In section 3, we first review a general method to compute heterotic indices in six dimensions and demonstrate how it works in particular examples. We also discuss there why the instanton numbers must be a multiple of some particular non-one integer in general. In section 4 we review the basic facts on the sixdimensional F-theory/ heterotic duality, and explain our strategy to obtain the equation for a RES-fibered threefold having a non-Cartan Mordell-Weil lattice. In section 5, we start the construction by the No.7 case of the Oguiso-Shioda classification to obtain the equation for the case of a particular instanton distribution. We then deform this equation in an appropriate way to find the equations for threefolds with arbitrarily distributed instantons. The match of the spectra is also verified there. Sections 6 and 7 are devoted to the considerations of the RES-fibered threefolds for the the cases of the SU (7) series and the SU (8) series, respectively. Finally we summarize our conclusions in section 8. In appendix A and B, we present the detail of the heterotic index computations for the SU (7) and the SU (8) series, respectively. Appendix C shows the explicit forms of the functions f , g of the Weierstrass equations and the discriminant ∆ for various cases considered in the text.

Models with non-Cartan type Mordell-Weil lattices
It is known that a rational elliptic surface S possesses a lattice structure, called the Mordell-Weil lattice E(K). (K is the field over which S is defined.) When S has no singularity, E(K) is the self dual E 8 lattice. When S has a singularity of an ADE type with root lattice T , E(K) is reduced, roughly speaking, to the orthogonal complement T ⊥ of T in E 8 . More precisely, E(K) is the dual lattice of T ⊥ accompanied with a torsion part: where * denotes the dual lattice. T ⊥ satisfies In the context of the duality between F-theory and heterotic string, the root lattice T of the ADE singularity in S corresponds to the gauge symmetry G, while the orthogonal complement T ⊥ corresponds to the gauge bundle H of the heterotic string. The decomposition In other words (apart from the torsion part), The Mordell-Weil lattices E(K) are classified into 74 patterns [51]. In many cases, (T ⊥ ) * is a weight lattice of some ADE type Lie algebra or a direct sum thereof, but there are some special cases where (T ⊥ ) * is not a weight lattice of any ADE type Lie algebra. In these cases T ⊥ is also not a root lattice of any ADE type, and neither is the gauge bundle H of the heterotic string. Let us call these Mordell-Weil lattices E(K) and the corresponding gauge bundles H "non-Cartan type". In this paper, among them, we will study the cases listed in Table 1, where T is a subalgebra of A 7 and E(K) is torsion free.  [51].
In the table, Λ (N ) is a matrix representing the lattice E(K) ∼ = (T ⊥ ) * . The inverse Λ −1 representing the dual lattice T ⊥ , or equivalently, the gauge bundle H, takes one of the following non-Cartan forms: (2.5) A * n , D * n are the weight lattices of A n , D n . 1/k denotes the one-dimensional lattice with lattice spacing 1 k . Its dual lattice has a non-Cartan form unless k = 2, in which the lattice becomes the A 1 weight lattice.
In terms of heterotic string, the two series in Table 1 are obtained by Higgsing the gauge group G = SU (7) (No.25) and G = SU (8) (No.45), respectively. Their Higgsing chains are summarized as follows: 4 (the subscripts are the numbers in Table 1) (20) (2.6)

General method
We will first quickly review the general method to compute the heterotic spectrum in six dimensions.
The Dirac index for a six-dimensional compactification of heterotic string theory on a complex 2-fold is given by whereÂ(T Z) is the A-roof genus of the tangent bundle of the complex 2-fold Z on which the heterotic string is compactified, and ch(V ) is the Chern character of the vector bundle V over Z. The number of hypermultiplets is given by −1/2 of the index, where the overall minus sign is a convention and the factor of 1/2 is due to the fact that the heterotic gaugino is a Majorana-Weyl spinor in ten dimensions. We consider the cases where the instanton takes values in a subgroup H of, say, the first factor of E 8 to leave the centralizer subgroup G of H in E 8 unbroken. We are particularly interested in the cases where H contains some U (1) factors. As in [65], let the 248 of E 8 be decomposed into the representations of G × H as where L i and C i are irreducible representations of G and H, respectively. Using the fact that K3 p 1 = −48, the number of hypermultiplet n L i ,C i in a representation L i of G (and C i of H) is given by [65] where F is the instanton gauge field 2-form taking values in (the Lie algebra of) H. The trace is taken in the representation C i . The equation (3.3) is still a correct formula even for the cases where H contains some U (1) factors. Since due to the anomaly cancellation condition, where a = 1, 2 are the labels to distinguish the E 8 factors, we see that the second term of (3.3) is nothing but −30 times the contribution of each representation C i to the total instanton number (=24, taking the factor of 1/2 into account). The normalizations of the traces and U (1) charges are determined once the instanton number of each irreducible group factor of H is specified.

Example
In this case one can take SU (2) × U (1) as H. Let the instanton numbers of SU (2) and U (1) be 12 + n − r and r, respectively, so that the total instanton number of H is 12 + n. Then we have 5 1 8π 2 F A ∧ F B Tr 248 adT A adT B = 60(12 + n).
The adjoint representation of E 8 is decomposed as where the subscripts denote the U (1) charges. This yields where τ a are the 2 representation matrices of SU (2), and 14 · 60 is the sum of the U (1) charge squares. As we assumed above, the first term is equal to 60(12 + n − r), whereas the second term is 60r. Therefore we have 1 which means that (Note the factor of −1/2 in eq.(3.3)) a 2 representation of SU (2) contributes 12+n−r 2 , whereas each U (1)-charge Q component contributes Q 2 2 · r 14 , to the multiplicities of hypermultiplets transforming in the corresponding G representation.
As an illustration let us compute n 21,2 −1 . This is computed as Note that the 3rd (U (1) instanton) term is multiplied by 2 because each component of 2 −1 contributes to the index. Since n 21,2 1 is equal to n 21,2 −1 (3.9), and there is no distinction between 21 and 21 in six dimensions, the total multiplicity of 21 is in all This becomes an integer if and only if the instanton number r is a multiple of 7. So writing r = 7r , we have n 21 = n − 6r + 8.  2n − 14r + 20 The computations of the multiplicities for other representations can be worked out similarly. The result is summarized in Table 2.
Finally, we would like to comment on the computation in [66] of the heterotic string matter spectrum with an SU (7) gauge group, which is different from ours but led them to the same result as that derived here. Since the multiplicities are not integers when r is a multiple of 7, they consider U (2) instead of SU (2) with an extra U (1) and assumed a contribution of this additional U (1) to the multiplicity of the singlet (See the top row of Table 8 of [66], where the multiplicity of 1 contains the term 4c 2 (V 2 )−c 1 (L) 2 − 6 despite that the SU (7) singlets (the first two terms of (4.8) in [66]) are not charged under the U (1) ⊂ SU (7) × SU (2) × U (1)). This U (1) must commute with both the SU (7) and the SU (2) and must be different from the original U (1) factor in E 8 , but obviously, there is no room for such an extra U (1) in E 8 as the rank is already exhausted.
3.3 Orthogonal decompositions of the E 8 root lattice: Why a multiple of seven?
The fact that the U(1) instanton number must be a multiple of seven can be understood by an orthogonal decomposition of the E 8 root lattice. Let e i (i=1,. . . ,9) be a set of orthonormal vectors of nine-dimensional flat Euclidean space with inner product e i · e j = δ ij . (3.12) Then the following set of 72 + 84 + 84 = 240 vectors on an eight-dimensional hyperplane normal to 9 j=1 e j form the set of root vectors of E 8 : This fact can be most easily verified by considering Freudenthal's realization of the E 8 algebra [67,68]. The first line (3.13) is the set of root vectors of SU (9), while the second line (3.14) is the root vectors corresponding to the rank-3 tensors of SU (9). Using this presentation of E 8 roots, one can easily see where the roots of SU (7) × SU (2) are embedded and which root vector is the one corresponding to the U (1) generator. As the root vectors of SU (7) one can take , (3.15) and the roots of SU (2) ⊂ H is then ±(e 8 − e 9 ). (3.16) In the E 8 root lattice generated by the vectors (3.13) and (3.14), the orthogonal lattice normal to the SU (7) and SU (2) lattices spanned by (3.15) and (3.16) is one-dimensional, generated by  The length square of (3.17) is 14, which is 7 times as large as the simple-root length square. This explains why the instanton number of U (1) is a multiple of 7.
Note that the junction lattice [69], which is the inverse of the matrix of section parings, is given in the present case as This can be made diagonal by the change of basis which agrees with the above consideration.

Example
In this case, rank(E(K)) = 3 and we can take one or two U (1) direction(s) in H. Namely, The gauge symmetry is the same (G = SU (5) × SU (2)) for these two gauge bundles, but the resulting spectra are different. As we will explain below, the spectrum for H = SU (2) × U (1) × U (1) is more general than the one for H = SU (3) × U (1).

The case when
The junction lattice is (3.20) As H, we allow the instantons to live also in a different U (1)(≡Ũ (1)) as well as the SU (2) × U (1) for No.25 in the previous section, so that the SU (7) is further broken to SU (5)×SU (2). The instanton numbers are assumed to be 12 + n − r −r, r andr for SU (2), U (1) andŨ (1), respectively. The decomposition of 248 in representations of these subgroups is shown in Table 3.
In the present case so that The spectrum for No.17 is similarly obtained as shown in Table 4. (1, 2) n + 6 + 5 2 7 r + 1 70r (10, 2) −2 + 2 7 r + 1 70r (10, 1) n + 6 − 2 2 7 r + 1 70r For 2 7 r + 1 70r to be integer, the general solution is (r,r) = (3, 10)k + (−1, 20)l (k, l ∈ Z), (3.24) so that therefore r orr need not necessarily be a multiple of 7 or 70 in general. The orthogonal decomposition of the junction lattice is (3.26) which implies that the minimalŨ (1) charge square (measured by the E 8 root space inner product) is 35 times (and not 70 times) as large as that of the SU (2) factor. This is almost equivalent to the condition for the integrality of the multiplicities, but the latter is "twice as" severe.
Here we have defined r (17) = 10r (17) so that the multiplicities become all integers when the U (1) (17) instanton number r (17) is a multiple of 10 (which is when r (17) is an integer).

Geometries for non-Cartan Mordell-Weil lattices
Suppose that a non-Cartan Mordell-Weil lattice E(K) (non-Cartan gauge bundle H) is given. To construct the corresponding geometry, which is a rational elliptic surface fibered over P 1 , we first construct a K3-fibered CY 3 with the singularity T (gauge symmetry G) paired with the given E(K). Then, we map the CY 3 to a RES-fibered geometry.

CY 3 in six-dimensional F-theory/heterotic duality
We first review the construction of a CY 3 for six-dimensional F-theory compactification [2,3,70]. E 8 × E 8 heterotic string/K3 contains 24 instantons. Suppose 12 + n of them take values in a gauge bundle H in the first E 8 and the gauge symmetry is broken to G, while the other 12 − n instantons take values in another gauge bundle H in the second E 8 and the gauge symmetry is broken to G . It has been known [2] that the corresponding dual F-theory geometry is a K3-fibered CY 3 over P 1 and at the same time is an elliptic fibered CY 3 over the Hirzebruch surface F n with singularities G and G . The Weierstrass form is given by F n is a P 1 fibration over the base P 1 , whose coordinates are z and z , respectively. f k and g k are polynomials of z with homogeneous order k. Let us write the fiber coordinate z in a homogeneous form (u : v). Under the C * action (u : v) → (µu : µv), the other coordinates transform as (x, y, z ) → (µ 4 x, µ 6 y, z ).
The Weierstrass form (4.1) is homogeneous under these C * actions with degrees 12 and 12 + 6n. Singularities G and G are realized by demanding that f , g and the discriminant ∆, have suitable vanishing orders near z = 0 and 1/z = 0 (see Table 7).
For realizing ADE gauge symmetry in six dimensions, these Kodaira fibers should satisfy the so-called split conditions [70], which we will require in our construction since all the gauge symmetries that we would like to achieve (Table 1) are of the A type. (The explicit form of the split condition can be seen in Table 8 below.) The last row of Table 7 expresses the singularity that cannot be resolved by any blow up of the fiber. To resolve it, blowing up the base is required and an additional tensor multiplet appears in the spectrum. Such a theory does not have a heterotic dual in the perturbative regime, and hence is not the subject of our study.
The elliptic fibration over F n can also be written in Tate's form [70,71] where a i are polynomials of z and z . Because of the homogeneity under (4.3), a i are degree 2i polynomials in z: Here a ij are polynomials of z . Their degrees are determined by the homogeneity under (4.4) as deg (a ij ) = 2i + (i − j)n.

(4.8)
Discriminant ∆ is given by where b n (n = 2, 4, 6, 8) are defined in terms of a i as follows: (4.10) Singularities G and G are realized by requiring a i (i = 1, 2, 3, 4, 6) to vanish at suitable orders of z, known as Tate's algorithm (see Table 8). The vanishing orders of a i not only non-mindetermine the type of the local singularity but also control its global structure. We listed in Table 8 the fibers for which the split condition is satisfied (the subscript "s" is attatched 6 ).
The first six columns of the table represent the lowest orders of a i and ∆ in z. The next column is the coefficient of z ord(∆) , where b nj (or b n,j ) represents the coefficient of z j in b n . The "split conditions" for Kodaira fibers are the conditions that the discriminants (4.5) should be the same forms as in this column. Last two columns are the corresponding fiber degeneracy and the singularity type. 7 The last row of the table expresses the singularity that never be resolved by blowing up the fiber. The relation between the Weierstrass form (4.2) and Tate's form (4.7) is given by

Mapping CY 3 to RES-fibered geometry
The Weierstrass form of a rational elliptic surface is given by where f and g are sections of O(4) and O(6) on the base P 1 and have the form Suppose that we are given a CY 3 with the Weierstrass form (4.1), whose f and g are sections of O(8) and O(12) on P 1 as in (4.2), yielding a K3 fibration. By using their coefficients f 8+(4−i)n for i ≤ 4 and g 12+(6−j)n for j ≤ 6, one can construct a geometry with a Weierstrass form To realize the groups with dagger, one more condition should be fulfilled. There are some polynomials p and q such that (here a n,j denotes a nj ) Here we regard f and g as sections of O(4) and O(6) on P 1 (with coordinate z). Then the resulting geometry is a rational elliptic surface fibered over P 1 (with coordinate z ). This gives the map from the K3-fibered CY 3 to a RES-fibered geometry: When the rank of G is large, we have to do a slight modification to f and g in order that the map does not change the singularity G. For example, for G = SU (n) with n ≤ 5, the above procedure maps G of a CY 3 to the same G in a RES-fibered geometry, but for G = SU (6), the naïve mapping changes the singularity. The explicit form of a CY 3 with G = SU (6) is given by [66] In this CY 3 , ord(∆) = 6 and the gauge symmetry is SU (6). After mapping to f and g and calculating ∆ , one can see that ord(∆ ) = 5, i.e., the singularity is reduced by the map. The source of this reduction is f 8−n . Although f 8−n is the coefficient of the term higher than o(z 4 ) in f , it also appears in the o(z 5 ) term in g. Thus, after the map g → g , it is still contained in g . By setting this polynomial to zero in g , one recovers ord(∆ ) = 6 and the singularity remains to be SU (6). In summary, the map (4.17) is obtained by first replacing f and g to f and g , and then regarding f and g as sections of O(4) and O(6) of P 1 , and finally setting to zero the polynomials which constitute the coefficients of the terms higher than o(z 4 ) in f and o(z 6 ) in g and are still contained in f and g even after the map f, g → f , g . Hereafter, the resulting f , g and ∆ will be denoted by f res , g res and ∆ res . This last step, however, does not work for some CY 3 with sufficiently large rank G. In such cases, one cannot recover the original singularity G of CY 3 in the RES-fibered geometry and the map (4.17) does change the singularity.
There are two such cases. In the first case, additional singularities other than G are produced by the map. As an example, let us consider a CY 3 with G = SU (8) constructed by using Tate's algorithm. The orders of a i in (4.7) are set to be ord(a i ) = (0, 1, 4, 4, 8) (see Table 8). Its Weierstrass form is obtained by using (4.10) and (4.11). As a result, ord(∆) = 8 is realized. Mapping to f and g , one finds that ord(∆ ) = 5. One can see that f and g contain the following polynomials which constitute the coefficients of the terms higher than o(z 4 ) in f and o(z 6 ) in g: a 12 , a 23 , a 24 , a 34 , a 35 , a 36 , a 45 , a 46 . Setting these polynomials to zero, one recovers ord(∆ res ) = 8, but this SU (8) singularity is accompanied by additional singularities. Explicitly, ∆ res has a factorized form Since the degree of a 44 is 8 , we can write Then we obtain At z = z i , ord(∆ res ) in z is enhanced to 2. Also, one can see that ord(f res ) = ord(g res ) = 0 at these loci, yielding I 2 fibers. Thus an additional gauge symmetry SU (2) appears along each line z = z i perpendicular to z = 0 where the SU (8) singularity exists. To explain the origin of such additional singularities, let us go back to Tate's algorithm and focus on the orders of a 3 and a 6 , which are 4 and 8 for G = SU (8). It means that a 3 (z) contains a 34 , a 35 , a 36 and a 6 (z) contains a 68 , a 69 , . . . , a 6,12 . In f and g , a 6j are not contained since they only appear in f and g in the terms higher than o(z 4 ) and o(z 6 ). Also, f res and g res do not contain a 3j , since they are set to zero (4.19). It means a 3 (z) = a 6 (z) = 0 in f res and g res . Then, from (4.9) and (4.10), discriminant has a factorized form ∆ res = a 2 4 (4a 4 − b 2 2 ), leading to unwanted additional singularities (4.20). In general, when the orders of a 3 and a 6 in Tate's algorithm exceed 3 and 6 simultaneously, additional singularities appear in the resulting RES-fibered geometry.
In the second case, the map (4.17) changes the singularity type of the fiber at z = 0. This occurs when the singularity G in a CY 3 is not contained within one E 8 (z = 0) but spreads into the other E 8 (z = ∞). For example, let us consider G = SU (8) again, but take another CY 3 which is different from the one obtained from Tate's algorithm presented above and is constructed by working in the Weierstrass form directly. As seen from (4.11), a CY 3 in Tate's form is always rewritten in the Weierstrass form, but the inverse is not the case in general. This means that there may exist models that are described only in the Weierstrass form and have no Tate's counterpart (see e.g., [72]). Hence the appearance of additional singularities (4.22) in RES-fibered geometry may be an artifact in Tate's form. There may exist another SU (8) model that cannot be reached by Tate's algorithm and for that CY 3 additional singularity may not arise after the map (4.17). A candidate of such a CY 3 is the one constructed in [66]. The explicit form is given by 8 12 τ 2 f 7 z 7 + g 8 z 8 + g 9 z 9 + g 10 z 10 + g 11 z 11 + g 12 z 12 , with an irreducible polynomial The important difference from the CY 3 constructed by Tate's algorithm is that the terms higher than o(z 4 ) in f or o(z 6 ) in g contain a term written by only the polynomials that are needed to express f res and g res . It is −6λ 2 φ 4 z 7 contained in o(z 7 ) of g. It is the sign that this singularity G = SU (8) can not be realized in a rational elliptic surface fibration but we need a full-fledged K3 fibration. As a result, the map (4.17) changes the singularity as follows. Mapping f and g to f and g , one obtains ord(∆ ) = 5. The o(z 5 ) and o(z 6 ) terms of g contain the polynomials which are the coefficients of the terms higher than o(z 4 ) in f and o(z 6 ) in g. Setting them to zero, we have ord(∆ res ) = 7. Thus, the singularity is reduced by the map (4.17). 9

Geometry for
The starting point of our construction is the CY 3 for the G = SU (2)×SU (2)×SU (2) model. It is No.7 of [51] and its Mordell-Weil lattice is E(K) = D * 4 ⊕ A * 1 . As we claimed in the previous section, there are models that can be written in the Weierstrass form but cannot be written in Tate's form. Therefore, we use the Weierstrass form throughout this paper, but for the G = SU (2) × SU (2) × SU (2) model, we start from Tate's form and convert it to the Weierstrass form. 10 This is because Tate's algorithm is more convenient for realizing product gauge groups.

The heterotic spectrum
is the Cartan type, the heterotic spectrum is determined in a standard way. Let us divide the 12 + n instantons taking values in H into each factor SO(8) and SU (2) such that By using the index theorem, we obtain the spectrum as in Table 9. The number n(H) of Table 9: The spectrum for the hypermultiplets is given by while the number of vector multiplets reads n(V ) = 3 × 3 = 9. Therefore, anomaly cancellation condition is satisfied as

r = 0 case
It is necessary to put the A 1 singularities on three different lines. We first put the A 1 singularity at z = 0. From Tate's algorithm (Table 8), the orders ord(a i ) are given by (0, 0, 1, 1, 2), and hence a ij should satisfy Next we put the second A 1 on another line. This line is taken to bẽ z is shifted by an order n polynomial h n (z ), so thatz is homogeneous under the transformation (4.4). To realize the A 1 singularity atz = 0, we re-expand a i in terms ofz as Finally, we place the third A 1 atẑ ≡ z + h n = 0 (5.8) and re-expand a i as and imposeâ The resulting form of a i 's are given as follows: where (· · · ) are written by a ij , h n and h n . To translate this Tate's form into the Weierstrass form, we calculate b i (4.10). Although b i have complicated dependence of a ij , h n and h n , one can redefine a ij so that they are arranged to have simple forms: Substituting them into (4.11), we obtain f and g in terms of A ij . To arrange so that the middle polynomials f 8 and g 12 have simple form, we further redefine as Similarly, we redefine A 6j with j = 7, 8, 9, 10 by subtracting A 22 dependent terms: 8,9,10). (5.14) Moreover, we found that A 23 and A 24 can be discarded from f and g without any effect on the singularity structure. As a result, we obtain the following form: Here the dependence on h n and h n is included only through the symmetric polynomials σ 1 , σ 2 as it should be because the lines of A 1 singularity can be replaced with each other: The discriminant has the form where D 6n+24 is an irreducible polynomial written by A ij , h n and h n . The leading expansion is given by which constitute the coefficients of the terms higher than o(z 4 ) in f and o(z 6 ) in g. Setting them to zero, one obtains a RES-fibered geometry. One can see from the explicit forms of f res , g res and ∆ res that the singularity remains to be G = SU (2) × SU (2) × SU (2) (see (5.22), (5.23) and (5.24) below). From this geometry, let us extract the matter spectrum. Singlets correspond to the moduli space M(H) of the gauge bundle H, which is identified with the complex moduli of the geometry, except for the middle polynomials f 8 and g 12 belonging to the geometric moduli of the heterotic K3. The number of singlets, therefore, is the number of degrees of independent polynomials contained in f res and g res except f 8 and g 12 . Charged matters are localized at codimension two loci of singularities on the base space. Suppose the generic codimension one singularity is G and it enhances to G at a codimension two locus. Then there is a matter in a representation corresponding to the off diagonal part of G ⊂ G . 11 In the present case, f res and g res contain the following six independent polynomials (apart fromÃ 44 andÃ 66 , which are not counted because they correspond to the middle polynomials f 8 and g 12 ): Polynomial Degree h n n h n n A 20 4 + 2n A 21 4 + n A 22 4 The degree of each polynomial is given by (4.8). The number n(H 0 ) of the singlet (neutral hypermultiplet) is thus calculated as n(H 0 ) = n + n + (4 + 2n) + (4 + n) + 4 + (8 + n) + 6 − 1 = 6n + 25, (5.21) where (−1) is the overall rescaling. To see where and how the singularity enhances, we expand f res , g res and ∆ res near each of the three lines z = 0,z = 0 andẑ = 0. The results are

22)
11 It is justified in the type IIB picture by using the notion of string junctions [73].
and P 16+2n ,P 16+2n andP 16+2n are degree 16 + 2n irreducible polynomials. We represented as (· · · ) the factors irrelevant to the symmetry enhancement for simplicity. From these expansions, one can read the loci and types of enhancements, and then one finds what kind of representations of matters appear at those points. The result is given by

Zero
Degree ord(f res ) ord(g res ) ord(∆ res ) Enhancement Matter A 20 ,Ã 20 ,Â 20 4 + 2n (1, 2, 2) (5.26) This table is obtained as follows. We first pay attention on the leading term of ∆ res in (5.24). The coefficient is a product of several factors. If one of them vanishes, the order of ∆ res enhances. The list of such factors are shown in the first column and its degree is shown in the second column. When a factor vanishes, not only the order of ∆ res but also the orders of f res and g res enhance in general. To what extent they will enhance can be read from (5.22), (5.23) and (5.24) and given in the next three columns. The resulting enhancement can be read from Table 7 and is shown in the next column. The associated matter representation is given in the last column, where the three entries correspond to the three A 1 at z = 0,z = 0 andẑ = 0 in this order. As we can see, unresolvable singularity does not appear anywhere.
Let us give some comments. Consider A 20 = 0 in the first row. In this case, the fiber degeneracy is enhanced from I 2 to III, but the singularity does not change and remains to be A 1 . Therefore, there exists no matter at these points. The same is true forÃ 20 = 0 andÂ 20 = 0. When h n = 0 (the fifth row), the orders of expansions around z = 0 and z = 0 simultaneously enhance to A 3 , while the orders ofẑ do not change. This reflects the fact that the two lines z = 0 andz = 0 intersect at h n = 0, giving rise to matter in the bi-fundamental representation (2, 2, 1). A similar thing is true for h n = 0 and h n − h n = 0.

r = 0 case
We next construct a CY 3 for general distributions of instantons and then map it to a RESfibered geometry. As seen in the previous section, Tate's algorithm gives the r = 0 case only. This means that the r = 0 case is not obtained by Tate's algorithm and we have to use the Weierstrass form. So we start from the Weierstrass form (5.15) for the r = 0 case and deform the equation appropriately. In order to know how to deform it, we use one particular information from the heterotic spectrum in Table 9 as an input. The only input data is the appearance of r half-hypermultiplets in the tri-fundamental representation 1 2 (2, 2, 2). We determine the CY 3 so that they are contained. As we will show below, it turns out that this requirement uniquely determines the CY 3 . After mapping to a RES-fibered geometry, we derive the full spectrum. The resulting F-theoretic full spectrum perfectly matches with the heterotic full spectrum (not only the part we used as an input).
It is expected that tri-fundamental representation is localized at a triple intersection point of three lines of A 1 singularity. For a triple intersection point to exist, z = 0,z = 0 andẑ = 0 need to share a common solution, that is, z = 0, h n = h n = 0.
(5.28) Therefore, h n and h n have a common factor. Writing this factor as t r , we have Three lines intersect at r points satisfying z = 0 and t r (z ) = 0. The next question is how much the gauge symmetry is enhanced where a tri-fundamental appears. As we argue below, it should be In the branching of the maximal embedding G ⊃ G ×H with H = SU (2), the representation of G which is combined with 2 of H is a pseudo-real representation. In the present case (5.32), the tri-fundamental representation of G = SU (2) × SU (2) × SU (2) is pseudo-real, forming a half-hypermultiplet 1 2 (2, 2, 2). 12 In order that the singularity is enhanced to SO(8) at t r = 0, we have to tune the geometry so that (ord(f ), ord(g), ord(∆)) = (2, 3, 6) at t r = 0. Let us first notice that σ 1 = σ 2 = 0 at t r = 0. One can then easily see from (5.15) that (ord(f ), ord(g)) = (2, 3) at t r = 0 if and only if A 20 is factorized as (recall that the degree of A 20 is given by (4.8)) A 20 = p 4+2n−r t r . (5.33) Calculating ∆ explicitly, however, we find that it is over-enhanced to ord(∆) = 8 at t r = 0. We have to suppress it to ord(∆) = 6 while keeping (ord(f ), ord(g)) = (2, 3). To absorb the excessive factors of t r in ∆, we have to factorize t −1 r for at least one polynomial as Take a term in f or g. If A is contained in the coefficient of that term as a product AB, we can " interchange the factor t r " to absorb the pole as [66] On the other hand, if A is contained as A n for some n, the pole cannot be cancelled. Among the polynomials contained in the CY 3 for r = 0 case (5.15), only those listed in (5.20) can change the singularity type near z = 0. h n , h n and A 20 are already factorized. The remainings are A 21 , A 22 and A 43 . Among them, A 22 is the one that should be divided by t r . The reason is as follows. Substituting (5.29) and (5.33) into (5.15), we obtain where we defined the reduced symmetric polynomials aŝ (5.38) 12 We list below the other examples of arising half-hypermultiplets given in [70]. Why half-hypermultiplets appear in these cases has been explained from string junctions' point of view [73].

Enhancement
Maximal embedding Branching Matter Let us look at the o(z 2 ) term. It contains A 21 as − 1 3 A 2 21 . This term cannot be eliminated by any redefinitions of other polynomials in the o(z 2 ) term. Since it has the form A n , the interchange of t r (5.36) does not work. We therefore cannot remove the negative power t −2 r that would arise if we imposed A 21 = A new 21 /t r . The same argument holds for the polynomial A 43 contained in the o(z 3 ) term. As a result, we are forced to set A 22 = A new 22 /t r . In this case, the procedure of interchanging t r does work. To apply it, we have to do in advance some redefinitions of polynomials so that A 22 appears in f and g as the form A 22 × B.
One can check from (5.15) the following fact: if A 22 = A new 22 /t r was imposed, negative power terms would arise only at o(z 3 ) in f and o(z 4 ), o(z 5 ) in g. Consider the o(z 5 ) terms in g first. The negative powers of t r arise from In order to have the form A 22 × B, we should redefine Next, we rewrite the o(z 3 ) terms in f by using A 43 instead of A 43 . The negative powers of t r come from

It requires the redefinition
Now we are ready to perform the interchange of t r . It is given by (5.43) The remaining sources of negative powers of t r are contained in the o(z 4 ) terms in g. Substituting all the factorizations above into (5.15), one finds that the worst terms are proportional to t −1 r and given by The negative power is cancelled if and only if {· · · } is proportional to t r , i.e., One can show that all these factorizations and redefinitions not merely cancel the t −n r terms, 13 but indeed realize just the desired order (ord(f ), ord(g), ord(∆)) = (2, 3, 6) at t r = 0.
13 Via the replacement A 22 = q4+r tr (5.43), no pole arises in higher order terms. It is because we subtracted the A 22 dependent terms in advance in (5.14).
Also, ∆ has the form ∆ = z 2 (z + h n−r t r ) 2 (z + h n−r t r ) 2 D 6n+24 , (5.46) where D 6n+24 is the same polynomial as given in (5.17) and is still irreducible after the above factorizations and redefinitions. To summarize, we obtained a CY 3 with enhancement SU (2)×SU (2) After mapping f and g to f and g and then setting the polynomials (5.19) to zero, one obtains a RES-fibered geometry with G = SU (2) × SU (2) × SU (2). The explicit forms of f res , g res and ∆ res are given in Appendix C.1.
We have derived a RES-fibered geometry whose spectrum contains r half-hypermultiplet 1 2 (2, 2, 2). However, it is not obvious whether the geometry reproduces the other part of the spectrum in Table 9. We will show this is the case. The number of singlets is determined by the independent polynomials. They are the following 7 polynomials defined above: Apart from the −1 corresponding to the overall rescaling, one more −1 is performed, because we can choose the leading coefficient of t r as 1 when factorizing h n = h n−r t r and h n = h n−r t r . If we counted the degrees of freedom of h n−r as n − r + 1 and those of t r as r + 1, it would be an overcounting. The charged matter spectrum is derived in Appendix C.1 by using the series expansions of f res , g res and ∆ res near each of the three lines z = 0,z = 0 andẑ = 0. The resulting charged matter multiplets (C.34) together with the singlets (5.48) give the total spectrum as 6n + 25 − 4r : (1, 1, 1) 2n + 16 + 2r : (2, 1, 1), (1, 2, 1), (1, 1, 2) n − r : (2, 2, 1), (2, 1, 2), (2, 2, 1) which perfectly reproduces the heterotic result (Table 9). 14 A brief sketch for the structure of the symmetry enhancement is depicted in Figure 1. The curves express the discriminant locus ∆ res = 0 (the shapes are not accurate). Each matter is localized at each intersection point.

Looijenga's theorem
Our geometry not only reproduces the heterotic spectrum but also encodes the structure of the moduli space M(H) of the gauge bundle H. On the heterotic side, Looijenga's theorem claims that the moduli space M(H) is parameterized by the sections (we consider the sixdimensional case) Here d k is the degree of the independent Casimir of H (d 0 ≡ 0 for k = 0), and s k is the coefficient of the k-th coroot when the lowest root −θ is expanded (s 0 ≡ 1 for k = 0). L is the anti-canonical line bundle of the base P 1 of the heterotic K3 and N is the "twisting" 14 The procedure of constructing geometry for general instanton distribution described in this subsection is applicable to other cases where the gauge bundle is divided into components H = ⊗ i H i . We have studied models whose H i 's are all Cartan types. There are 7 such cases out of 74 [60]. We have examined the other cases and have constructed the CY 3 of Nos.7,10,11 and 18 with general instanton distributions. For each case, the F-theoretic spectrum precisely coincides with the heterotic spectrum. However, we have not succeeded to construct CY 3 for No.14 yet. line bundle over P 1 . Explicitly, (5.51) where p corresponds to the instanton number. Therefore, degrees of ω k with respect to the coordinate z of P 1 is given by z is identified with the coordinate of the base P 1 of F n , i.e., the variable of polynomials a ij . In the present case, H = SO(8) × SU (2) with instanton numbers (8 + n − r, 4 + r). Then we find that the sections ω k in Looijenga's theorem exactly match the 7 independent polynomials (5.47) describing the geometry on the F-theory side (see Table 10).
This correspondence is also valid for r = 0. When r = 0, the order of the polynomial t r becomes 0 and the independent polynomials are reduced to those of the r = 0 case (5.20). In other words, the geometry for the r = 0 case already captures the structure of the moduli space of the bundle H. In the following discussion, we concentrate on the r = 0 case.

Geometry for non-Cartan SU (7) series
We have constructed the CY 3 for G = SU (2) × SU (2) × SU (2) in the previous section. This is No.7 and is not contained in Table 1, but by unHiggsing the first SU (2) successively, we obtain the CY 3 for No.12 in the SU (7) series and the CY 3 for No.22 in the SU (8) series as (22) . (6.1) CY 3 's for other cases can be obtained by further unHiggsing them. Once we obtain a CY 3 for each case, we then map it to a RES-fibered geometry, extract the matter spectrum and compare it with the heterotic result given in Section 3, Appendix A and Appendix B.
Let us enhance the first SU (2) of G = SU (2) × SU (2) × SU (2) to SU (3). The discriminant for G = SU (2) × SU (2) × SU (2) with r = 0 is given by ∆ = z 2 (z + h n ) 2 (z + h n ) 2 D 6n+24 (see (5.17)). To enhance the first factor, we impose that D 6n+24 is factorized as where E 5n+24 should be an irreducible polynomial. For this, we require that the constant term of D 6n+24 vanishes. From (5.18), the condition is A solution is A 20 = 0, but it is not suitable. As seen from the explicit form of f and g (5.15), A 20 = 0 enhances the orders not only of ∆ but also of f and g near z = 0 to be (ord(f ), ord(g), ord(∆)) = (1, 2, 3). The fiber degeneracy enhances as I 2 → III, but the singularity is kept fixed as A  (3) says that the coefficient of z 3 in ∆ should be proportional to a 3 10 (see Table 8), but in this solution it is proportional to A 3 20 , as one can see by calculating the next to leading order term of D 6n+24 in (5.18). The other solution is and then A 43 = 2 p 2+n q 6 . (6.6) In this case, the coefficient of z 3 in the expansion of ∆ is proportional to p 3 2+n , and hence the split condition is fulfilled. Substituting (6.5) and (6.6) into (5.18), we obtain the explicit form of E 5n+24 (6.2) as One can check that E 5n+24 is irreducible and the gauge symmetry is in fact SU (3) × SU (2) × SU (2).
Mapping f and g to f and g , and setting the polynomials (5.19) to zero in f and g , one obtains the RES-fibered geometry with G = SU (3) × SU (2) × SU (2). The explicit forms of f res , g res and ∆ res are given in Appendix C.2. In this way, we obtain the geometry for the non-Cartan Mordell-Weil lattice E(K) = Λ 12 in Table 1.
Let us extract the matter spectrum from the resulting geometry. We first notice that f res and g res contain the following six independent polynomials: Polynomial Degree h n n h n n p 2+n 2 + n A 21 4 + n A 22 4 q 6 6 (6.8) For counting the number of singlets, it should be noted that the middle polynomialÃ 66 is written in terms of these polynomials as in (6.5). In other words, these six polynomials include the degrees of freedom ofÃ 66 , which are the geometric moduli of K3 on the heterotic side. In order to focus on the gauge bundle moduli only, we subtract 13 degrees of freedom corresponding toÃ 66 and return it to the geometric moduli of K3. As a result, we obtain n(H 0 ) = n + n + (2 + n) + (4 + n) + (4) + (6) + 6 − 1 − 13 = 4n + 8. (6.9) The derivation of the charged matter spectrum is given in Appendix C.2 and the result is summarized in (C.39). Together with the singlets (6.9), the full spectrum is given by This F-theoretic spectrum is equivalent to the heterotic spectrum (see Appendix A.2) given in Table 15 with r U (1) = 2, or the one given in Table 16 with (r,r, r (17) ) for k = 2 in (A.19). (6.2)). One expect that tuning the parameters h n , h n causes the unHigssing process as follows: h n = h n = 0 → G = SU (7). (6.11) In the first case, the tuning gives the expected enhancement. However, in the second and third cases, naïve tunings give a non-split fiber and unresolvable singularities, respectively, and hence we need some modifications of the geometry. In this subsection, we consider the G = SU (5) × SU (2) case. The CY 3 is obtained by merely setting h n = 0 (6.12) in the formulae obtained in the previous section.
We then map f and g to f and g . The resulting f and g contain polynomials which constitute the coefficients of the terms higher than o(z 4 ) in f and o(z 6 ) in g. After setting them to zero, one obtains the RES-fibered geometry with G = SU (5) × SU (2). The explicit forms of f res , g res and ∆ res are given in Appendix C.3. In this way, we obtain the geometry for the non-Cartan Mordell-Weil lattice E(K) = Λ 17 in Table 1.
The number of singlets is reduced by n + 1 (of h n ) from (6.9): n(H 0 ) = 4n + 8 − (n + 1) = 3n + 7. (6.14) The charged matter spectrum is derived in Appendix C.3 and given in (C.43). Together with the singlets (6.14), the full spectrum is given by This F-theoretic spectrum is equivalent to the heterotic spectrum (see Section 3.4) given in Table 4 with (r,r) for k = 2 in (3.24). 15
Mapping f and g to f and g , and setting the polynomials (5.19) to zero in f and g , one obtains the RES-fibered geometry with G = SU (3) × SU (4). The explicit forms of f res , g res and ∆ res are given in Appendix C.4. In this way, we obtain the geometry for the non-Cartan Mordell-Weil lattice E(K) = Λ 19 in Table 1.
In the table (6.8) of the independent polynomials, h n is eliminated and A 21 is replaced by s 2 , which reduce the degrees of freedom by (n + 1) + (5 + n − 3) = 2n + 3 from (6.9). The number of singlets is hence given by n(H 0 ) = 4n + 8 − (2n + 3) = 2n + 5. (6.23) The charged matter spectrum is derived in Appendix C.4 and given in (C.47). Together with the singlets (6.23), the full spectrum is given by 2n + 5 : (1, 1) This F-theoretic spectrum is equivalent to the heterotic spectrum (see Appendix A.1) given in Table 12 with (r,r) for k = 2 in (A.6), or the one given in Table 14 with r U (1) = 1.
To make the singularity milder, we have to do some redefinition of polynomials and "absorb" the factors of p 2+n . For this, look at the explicit form of f : where the order is enhanced to ord(f ) = 4 at p 2+n = 0. Here, one can notice that s 2 always appears as a product form p 2+n s 2 . We can hence introduce a new polynomial r 4+n and replace s 2 as The expansion of f is rewritten as whose order at p 2+n = 0 is suppressed to ord(f ) = 2. Similar calculation for g and ∆ shows that the singularity at p 2+n = 0 gets milder as (ord(f ), ord(g), ord(∆)) = (2,3,9), which is D 5 singularity and can be resolved without any problem. Therefore, unHiggsing to G = SU (7) is obtained by imposing (6.25) and (6.27) on the CY 3 for G = SU (3) × SU (4). 16 We then map f and g to f and g . The resulting f and g contain polynomials A 45 and A 46 , which constitute the coefficients of the terms higher than o(z 4 ) in f and o(z 6 ) in g. After setting them to zero, one obtains a RES-fibered geometry with G = SU (7). The explicit forms of f res , g res and ∆ res are given in Appendix C.5. In this way, we obtain the geometry for the non-Cartan Mordell-Weil lattice E(K) = Λ 25 in Table 1.
The charged matter spectrum is derived in Appendix C.5 and given in (C.50). Together with the singlets (6.29), the full spectrum is given by 2n + 6 : 1 n + 16 : 7 n + 2 : 21 This F-theoretic spectrum is equivalent to the heterotic spectrum (see section 3.2) given in Table 2 with r = 1.
7 Geometry for non-Cartan SU (8) series where F 4n+24 should be an irreducible polynomial. The explicit form of E 5n+24 is given in (6.7). One way to achieve (7.1) is to set p 2+n = 0 so that the o(1) term of (6.7) vanishes. This does not, however, lead to the enhancement to SU (4) since in this case we have (ord(f ), ord(g), ord(∆)) = (2, 2, 4), which means that the fiber type is IV , and hence the singularity does not change A 2 → A 2 . Thus we must require the other factor of the o(1) term to vanish: − A 2 22 p 2 2+n q 6 + 3A 21 p 2+n q 2 6 − 3Ã 44 p 2 2+n q 6 + A 22 A 45 p 3 2+n + 3Ã 67 p 3 2+n − 3q 3 6 h n h n = 0. (7.2) Here we used σ 2 = h n h n . There are two solutions to (7.2). One solution is given by Let us call this CY 3 , "CY 3 ". In this solution, however,Ã 67 , which is a coefficient of a term higher than o(z 6 ) in g, contains terms written by only the polynomials which are needed to express f res and g res . This means that, to achieve the singularities, it is not sufficient to constrain the terms of o(z 4 ) or lower in f and of o(z 6 ) or lower in g, but we also need to impose conditions on the higher order terms, which are supposed to be independently tuned to describe the instanton bundle of the "other" E 8 gauge group of the dual heterotic theory. Thus, in the same way as in the G = SU (8) case (4.23) discussed in section 4.2, the map from CY (1) 3 to a RES-fibered geometry changes the singularity. The other solution can be found as follows. From the form of (7.2), we can see that it is solved for A 21 by requiring that all the terms have a common factor of p 2+n q 2 6 . This is possible if 17 p 2+n = r n−4 q 6 , h n = r n−4 s 4 .
The solution is given by This CY 3 , which we call CY 3 , can be mapped to a RES-fibered geometry, where the singularity is unchanged. Mapping f and g to f and g and setting the polynomials (5.19) contained in f and g , one obtains the RES-fibered geometry with G = SU (4) × SU (2) × SU (2). The explicit form of f res , g res and ∆ res are given in Appendix C.6. In this way, we obtain the geometry for the non-Cartan Mordell-Weil lattice E(K) = A * 1 ⊕ A * 1 ⊕ 1/4 in Table 1.
Among them, we find that setting s 4 = 0 (7.9) yields a CY 3 with G = SU (6) × SU (2). We then map f and g to f and g , and set the polynomials (6.13) contained in f and g to zero. It gives a RES-fibered geometry with G = SU (6) × SU (2). The explicit forms of f res , g res and ∆ res are given in Appendix C.7. In this way, we obtain the geometry for the non-Cartan Mordell-Weil lattice E(K) = A * 1 ⊕ 1/6 in Table 1.

Other cases
We expect that the other tunings give the other models in the SU (8) series, but we have not obtained corresponding RES-fibered geometries yet. For example, setting h n = r n−4 s 4 (7.12) is expected to give a CY 3 with G = SU (4)×SU (4), but it leads to unresolvable singularities. For the enhancement SU (3) × SU (2) × SU (2) → SU (3) × SU (4) discussed in section 6.3, we were able to deform such singularities to resolvable ones. In the present case, however, we could not find how to deform them suitably. Since we have not obtained a CY 3 yet, we do not have a RES-fibered geometry, either. There are two possibilities. First, there are such geometries but we just haven't found them yet. Second, one can prove that there are no such geometries in these cases. 21 It would be interesting to find which is correct, but we leave it for future work. Also, we could not find the way of realizing the enhancement . We also leave this issue for future investigations.

Conclusions
In this paper, we studied how the non-Cartan Mordell-Weil lattices are realized in the sixdimensional heterotic/F-theory duality. We focused on the SU (7) series and SU (8) series given in Table 1.
On the heterotic string side, we give a procedure to derive the massless matter spectrum by using index theorem. Except No.45, the gauge bundle H is a product of a semi-simple group and one or more U (1) group(s). The total 12 + n instantons are distributed into each group factor. We found that the U (1) instanton numbers are a multiple of some particular non-one integer r and the resulting number of each matter is integral. We showed that this integer r is determined by the lattice spacing of the U (1) direction in the E 8 lattice, which is orthogonal to the lattice of the semi-simple part of H and the singularity lattice T . For rank(E(K)) > 2 cases, more than one choices of the number of U (1) direction(s) in H are possible for the given MW lattice. In these cases, more than one matter spectra are obtained for a gauge symmetry G. We found that if the number of the U (1) factor(s) is larger, the spectrum is finer.
On the dual F-theory side, we examined how to construct a geometry for a given non-Cartan MW lattice E(K). We first construct CY 3 with singularity G of the lattice T for the given E(K). It starts from G = SU (2) × SU (2) × SU (2) case, and successive unHiggsing processes are carefully performed. Then we construct the map from these CY 3 to RESfibered threefolds. In order that the singularity G is unchanged by the map, we generically have to do slight modifications on the geometry. As a result, we obtained the explicit forms of RES-fibered geometries for all the SU (7) series and for some cases in the SU (8) series (G = SU (4) × SU (2) × SU (2) (No.22) and G = SU (6) × SU (2) (No.29)). Note that the geometries for the SU (8) series we have constructed cannot be obtained from the (Higgsing of the) known CY 3 with G = SU (8) of [66], because mapping that CY 3 to a RES-fibered geometry inevitably changes the singularity. We found that the F-theoretic spectrum derived from each of these geometries is completely identical to the heterotic spectrum with a specific distribution of instantons.
We have not succeeded to construct CY 3 's (and hence RES-fibered threefolds) for SU (8) series other than the above two cases yet. It would be an interesting problem to understand whether there are no such geometries in principle, or we merely have not found them yet for some technical reason; we leave it for future investigations. Also, each RES-fibered geometry for SU (7) and SU (8) series we have constructed in this paper corresponds only to a particular distribution of instantons. It is expected that the deformation to the generic distributions can be done similarly to the case of G = SU (2) × SU (2) × SU (2), though it is technically much more elaborated. We also leave this issue for future work.

A.1.2 The case when
(A.11)  2n The spectrum in this case is summarized in Table 14, where This coincides with the result shown in Table.12 if (4, 3) and (4, 3) are identified and the replacement is made. Again, the spectrum for SU (2) × U (1) × U (1) is more general than that for SU (2) × SU (2) × U (1) as k is restricted to even in the latter. The orthogonal decomposition of the junction lattice in this case is 14) this agrees with the fact that the U (1) instanton number is a multiple of 6.
Assuming the instantons in 12 + n − r and r in SU (2) and U (1) 1/6 , we obtain the spectrum of No.29 as shown in Table 18.
Charged matter spectrum can be read from these expansions. The result is