High U(1) charges in type IIB models and their F-theory lift

We construct models with U(1) gauge group and matter with charges up to 6, in the context of type IIB compactifications. We show explicitly that models with charges up to 4 can be derived from corresponding models in F-theory by applying the Sen weak coupling limit. We derive which type IIB models should be the limit of charge 5 and 6 F-theory models. Explicit six dimensional type IIB models with maximal charge 5 and 6 are constructed on an algebraic K3 surface that is the double cover of $\mathbb{CP}^2$. By using type IIB results we are also able to rediscover the F-theory charge 4 model in a straightforward way.

1 Introduction F-theory [1][2][3] constitutes a convenient framework to oversee the String Theory Landscape in various dimensions. In fact, provided the known dualities relating F-theory to type IIB string theory, heterotic string theory as well as M-theory, F-theory compactifications are representative of string theory vacua. In particular, F-theory is expected to be a good arena to study how effective field theories that look consistent at low energy may get obstructions when completed with quantum gravity.
There has been more than a decade long search for consistent F-theory models that could be of relevance for particle physics, starting by the pioneering work of [4,5]. This has led to significant progress towards understanding formal aspects of the theory related to the possible gauge symmetries (Abelian and non-Abelian, continuous and discrete), the type of matter that is allowed as well as the interaction terms. Regarding the moduli sector and the cosmological applications the theory still faces critical challenges. For this reason, the Landscape analysis in F-theory has concentrated essentially on issues related to particle physics.
A challenge for F-theory is to construct explicit models that are beyond the known landscape and, in doing so, identifying possible obstructions to realize lower dimensional effective theories. Regarding the non-Abelian symmetries, F-theory has an advantage over type IIB string theory in that it allows to obtain exceptional groups. Many of the constructed models contain subgroups of E 8 , but systematic surveys show that gauge symmetries including as many as O(100) E 8 factors are possible in four dimensions [6,7]. Concerning the Abelian gauge symmetry sector in F-theory, significant progress in understanding the corresponding proper global setup has led to fully fledged models with up to three U(1) factors [8][9][10][11][12][13]. Similarly, it has been possible to obtain the Weierstrass form for globally consistent models with Z 2 , Z 3 and Z 4 discrete gauge symmetries [11,[14][15][16][17][18].
A pressing question deals with what types of massless multiplets are possible in the effective theories resulting from F-theory. In dimesions higher than six, supersymmetry is constraining enough to ensure that the matter multiplet representations must not be bigger than the adjoint. In 6D and 4D one is only at the mercy of the anomaly cancellation conditions and these do not exclude the possibility for light exotic matter beyond the adjoint representation [19][20][21][22][23][24][25][26]. Common representations that arise in F-theory models are the fundamental, the two index antisymmetric and the adjoint. However, in recent constructions it has been possible to obtain a three index symmetric and antisymmetric representations of various SU(N) gauge groups [27][28][29][30]. The problem becomes more severe when considering Abelian gauge symmetries: is there an upper bound on the maximum U(1) charge of a massless state? This question was raised in [26,31,32]. So far, in F-theory it has been possible to construct globally consistent models with fields with U(1) charges only up to Q = 3 and Q = 4 [15,33].
The above considerations are essentially an invitation to the explicit construction of F-theory models that lie at the frontier between the Landscape and the Swampland. In this paper we are going to approach the problem of constructing models with U(1) symmetries exhibiting massless states with large U(1) charges. Since U(1) symmetries are due to the presence of extra sections of the elliptic fibration, one expects the Weierstrass form describing these models to be highly specialized. U(1) models with charges |Q| > 2 are described in terms of so called non unique factorization domains (non-UFD) [29]. The non-UFD description has been used to extend the charge Q = 3 model found in [15], and to obtain a Weierstrass model with Q = 4 [33]. We will show that these models are relatively easy to construct in the dual type IIB theory, in the perturbative limit. Thanks to this correspondence we will be able to straightforwardly extend the construction to charge Q = 5 and Q = 6 models in type IIB. Their F-theory lift is non-trivial, but in principle doable and we will leave this for a future project.
To understand the type IIB duals of F-theory models with one massless U(1) and states with high charge, we apply the Sen weak coupling limit [34]. This approach was intiated in [35], where we obtained the weak coupling limit for a variety of globally consistent F-theory models, including one with gauge group U(1) and charge up to Q = 3 and the Z 3 models of [15]. In this work we proceed more systematically towards the analysis of the features of these models, including the U(1) construction with charge Q = 4. Part of the systematics has to do with a way to construct divisors in the base B of the elliptic fibration that are guaranteed to split into two divisors in the type IIB double cover Calabi-Yau (CY) X. Thanks to these techniques, we are able to obtain a family of type IIB models with a single massless U(1) symmetry and maximum charges that can reach up to Q = 6. Thanks to type IIB we are also able to rediscover the F-theory charge Q = 4 model of [33] in a straightforward way: in type IIB it is easy to understand which deformation of the D7-brane loci leads from a model with Z 3 symmetry to the charge Q = 4 model; applying the same deformation to the Z 3 F-theory model of [15], one immediately obtains the charge four model of [33]. This paper is organized as follows, in Section 2 we describe the generalities of the double cover Calabi-Yau manifolds and the techniques necessary to construct D7-brane configurations with one massless U(1) and high charge spectrum. In Section 3 we discuss the features of type IIB models with higher U(1) charges, and present the configurations of branes and orientifolds leading to a single massless U(1) symmetry. We focus on the case where we have two U(1) D7-branes and one orientifold odd axion available to make one U(1) massive, leaving the other U(1) massless. We show that, under certain assumptions, in this setup the homology relations among the various 7-brane cycles allow models with maximum charges Q = 3, 4, 5, 6. In section 4 we describe the models with maximal charges Q = 3 and Q = 4 from the perspective of type IIB and F-theory. In Section 5 we present the type IIB versions for the charge Q = 5 and Q = 6. For the case of charge 6 model we present an explicit K3 compactification. We devote Section 6 to present our conclusions and prospects. We present some complementary material in the appendices.

Type IIB manifold vs F-theory base
F-theory is defined on a manifold Y that is an elliptic fibration over a base manifold B. The perturbative type IIB limit is defined on a double cover X of the base manifold B. In this section, we will describe tools that allow to directly connect the two descriptions.

Double covers
Our starting point will be a Kähler manifold B and a line bundle L ⊗2 on B. Take a section b 2 of L ⊗2 . The double cover X of B branched over the locus b 2 = 0 is given by the following hypersurface in the total space of the line bundle L [36,37]: where ξ is a coordinate along the fiber. If the line bundle L is the anticanonical bundle of B, i.e. L =K B , then X is a Calabi-Yau space.
By construction, X is symmetric under the involution The map (2.4) induces a map between the Poincaré dual cohomologies: f * : H 2 (X) → H 2 (B). If b 2 in (2.1) is a sufficiently generic polynomial, then this map is one-to-one, i.e. b 2 (X) = b 2 (B). If this is the case, then any pair of divisors D and I(D) are in the same homology class. On the other hand, if b 2 presents a specific factorization, there can be more divisors in DivX than in DivB. 2 In the last case, there will be a divisor D B of B such that f * (D B ) splits into two divisors D X and I(D X ) that are in different homology classes in X. Correspondingly, b 2 becomes a square when restricted to the locus D B . If this happens globally, one can write b 2 as where s 6 is a section of the line bundle L, s L is a section of another line bundle L L on B and s R is consequently a section of L ⊗2 ⊗L −1 L . This factorization produces a singularity on X at ξ = s 6 = s L = s R = 0. This is a conifold singularity when X is a three-fold. However, in this paper we will consider type IIB compactifications to 6 dimensions; hence X is a (complex) two-dimensional space in a three-dimensional ambient space and then there are generically no solution to the four equations. For this reason, in this paper we will assume X to be smooth. Most of our results will be valid also for 4D compactifications, if one considers base manifolds B such that the locus {s 6 = s L = s R = 0} is empty [38].
Let us consider the divisor D B = {s L = 0} in B. f * (D B ) splits on X into the loci {ξ−s 6 /2 = s L = 0} and {ξ+s 6 /2 = s L = 0}, that are in different homology classes [38][39][40]. Let us call the corresponding divisors D L,− and D L,+ . They are image to each other under the involution ξ → −ξ and by construction their homology classes satisfy the following relation in where [s L ] is the homology class of the locus {s L = 0} in X.
In the following, we will need information about the intersection numbers. As before, we take X to be a (complex) two-dimensional hypersurface in the ambient space A that is the total space of the line bundle L on B. If P is a polynomial in the coordinates of B, then we have where the double intersections are always meant in X if not specified. We keep this convention also in the following. Moreover, we have and, using (2.6),

Splitting divisors from matrices
The existence of a new divisor class can be detected also from the fact that if b 2 takes the form (2.5), then X has a 2 × 2 matrix factorization (MF)(see [41] for more details on an analogous model), 3 i.e. there exist matrices (M,M) such that One can then define with L an arbitrary line bundle). All the involved line bundles are naturally defined on the base manifold B and can be easily lifted to X by the map f * .
If X is smooth, L M is a line bundle on X: in fact detM = ξ 2 − b 2 and hence at ξ 2 = b 2 the matrix rank goes down to one, generating a one-dimensional cokernel. 4 The first Chern class of a line bundle L is the Poincaré dual of the divisor where a generic section of L vanishes. The line bundle L M is given as the cokernel of the map M. For a 2×2 MF (M,M), one has an isomorphism cokerM ∼ =imM given by the mapM restricted to cokerM. 5 Hence the locus where a section of L M vanishes is the same as the locus where a section of imM vanishes, i.e.
where p = (p 1 , p 2 ) is a section of the vector bundle W 2 in (2.12). The homology class of D p can be computed in the following way: one can deform the generic divisor D p by setting p 2 = 0. Then the idealM p = 0 splits into the union of p 1 = 0 in X and the divisor D L,− , i.e. at the level of homology classes: (2.14) The equation ξ 2 − b 2 = 0 has an inequivalent MF (M ,M ) that is obtained from (2.11) by taking ξ → −ξ. Following the same steps as before, we can then construct a line bundle L M , whose first Chern class is 3 For a nice review on MF in physics see [42]. For application of MF in similar contexts see also [43,44]. 4 If the locus {ξ = s 6 = s L = s R = 0} is non-empty, then on this points the matrix rank goes down by two units and at that point the cokernel dimension jumps from one to two. Such a L M is called a non-trivial irreducible Maximal Cohen Macaulay (MCM) module. 5 The space cokerM is given by W 2 /imM. But over ξ 2 = b 2 the exactness of the sequence implies that imM =kerM. Hence, by definition, kerM = 0 when the mapM is restricted to cokerM and hence it is an isomorphism between cokerM and imM. whose homology class is (2. 16) In particular, when (p 1 , p 2 ) = (1, 0) we have D p = D L,− and D p = D L,+ .
Notice that M is written also in terms of A 1 and B 0 but with ξ → −ξ. It is then easy to see that the divisors D p and D p intersect each other over two loci: The first one is on top of the fixed point locus of the involution I, while the other may intersect the fixed point locus but its points are generically not fixed.
The two divisors D p and D p are mapped to each other by the involution I. Hence they will be projected down by f to the same divisor D B,p of B: We now prove that D B,p is described by the equation 6 • We first prove that all points of D B,p satisfy (2.19): such points are pulled-back by f * to points either of D p or of D p ; over these loci A 1 p = ∓ξB 0 p. Hence p · A 1 p = ∓ξ p · B 0 p = 0 as B 0 is antisymmetric.
• We then prove that any points satisfying (2.19) belong to D B,p : over these points, A 1 p is orthogonal to p, i.e. it is proportional to B 0 p. This means that B 0 p is an eigenvector of A 1 B 0 (B 2 0 = −1). We also have (A 1 B 0 ) 2 = −det(A 1 )1. Hence the eigenvalues of A 1 B 0 are ± √ b 2 . If we pull-back these points, they will belong either to D p or to D p (since on X we have b 2 = ξ 2 ).
The formula (2.19) gives then an algebraic expression for a divisor D B,p of B that, once lifted to the double cover X, splits into two components, one the image of the other under the inolution I. 7 6 Notice that the submanifold (2.19) in B is singular at p 1 = p 2 = 0. This is not surprising. The locus {p 1 = p 2 = 0} is the intersection locus of the divisor D p and D p away from the fixed point locus. On the other hand, the two divisors of X join each other in B, forming a connected divisor D B,p . The two branches of D B,p still intersect transversally at p 1 = p 2 = 0, hence generating a singularity. This also happens when D and D are in the same homology class. 7 It is easy to see why the equation (2.19) splits when intersected with ξ 2 = b 2 . In fact, b 2 =-detA. Hence, on X the determinant of A is a square and then the quadratic form in (2.19) factorizes into two factors (that are exchanged by taking ξ → −ξ).

Splitting divisors of higher degree
The procedure outlined above can be used to construct other pairs of algebraic cycles mapped to each other by the involution I: Take the line bundle L ⊗2 M ; thanks to the isomorphism given byM restricted on cokerM, this line bundle is isomorphic to im(M⊗M). The vanishing locus of a generic section is then given bỹ that is equivalent to the block diagonal form up to operation of summing or subtracting lines or columns and changing the order of lines and columns. Notice that the element 2(ad − bc) is proportional to the determinant of M. Let us apply this to M =M. Notice that in the block-diagonal form the relevant part ofM ⊗M is the 3 × 3 block (since detM vanishes on X). We can moreover separate the parts linear in ξ as The subscript "2" signals the fact that A 2 is homogeneous of degree 2 in s L , s 6 , s R . On the other hand B 1 is of degree 1.
The two divisors mapped to each other by I are then Their divisor classes can be derived as above and are The two divisors D (2) q and D (2) q intersect each other over two loci: The points of the second locus are generically away from the fixed point locus. To write the second equation we have used the fact that One can then find the divisor of B that splits into D Following similar considerations as for the previous case, we found that it is given by (2.27) By analogous considerations, one can construct divisors D The two divisors are then Once again there is a relation between A 3 and B 2 : and their homology classes are is given by the equation (2.32) One can in principle continue with this procedure to obtain divisors in homology classes n[D L,∓ ] + [P ]. We give the result for n = 4 in Appendix A.

Odd divisor classes
The involution I in (2.2) splits the second cohomology of X into even and odd elements: The even elements, as we said, are in one-to-one correspondence with the divisors of B (b 2 (B) = b 2 + (X)). The odd elements can be written as differences between divisors (or Poincaré dual two-forms) that are mapped to each other by I. Of course, to obtain non-trivial elements, one needs that such divisors are in different classes. The divisors constructed in Section 2.2 are of this type. We can then associate an odd class in H 2 (X) with the matrix factorization (2.11): (2.33) Changing p does not affect the class of D − in H 2 (X), as can be seen from (2.14) and (2.16): If we take differences of connected algebraic divisors D If the space X admits further MF's, we can associate an independent odd class to each of them.
3 Type IIB limit of F-theory 3.1 F-theory models with U(1) gauge group F-theory is defined on a CY manifold Y that is an elliptic fibration over a Kähler manifold B. If the fibration has a section (called the "zero section") the space Y can be described by a Weierstrass model, i.e. by the equation where f and g are sections ofK ⊗4 B andK ⊗6 B respectively (K B is the canonical line bundle of the base manifold B), and x, y and z sections of (K B ⊗ H) ⊗2 , (K B ⊗ H) ⊗3 and H (H is the line bundle which z belongs to). The elliptic curve degenerates over the zero locus of the discriminant ∆ = 4f 3 + 27g 2 : this gives the location of the 7-branes.
If the Weierstrass model is smooth, the effective lower dimensional theory has no gauge group nor matter. In this paper, we are interested on the simplest gauge group, that is U(1). This is realized when the elliptic fibration Y has one extra section. When this happens, Y develops singularities along codimension-2 loci in the base B, where states charged under the U(1) gauge group live. The charge of states localized at different loci are typically different.
So far, models with charges up to 4 have been constructed as global F-theory compactifications [8,15,33,45]. The Weierstrass model descriptions of such configurations are birationally equivalent to smooth manifoldsỸ that are hypersurfaces in an ambient space P B F i that is the fibration of a toric two-dimensional variety P F i over the base manifold B. They are described by the equation [15,46], The toric variety P F i is defined over the polytope F i , with homogeneous coordinates z 1 , ..., z k , one per each non-zero lattice vector v ∈ F i , and w being lattice vectors in the dual polytope F * i (including the origin). The polynomial in (3.2), called Batyrev polynomial, defines a hypersurface X F i ∈ P B F i elliptically fibered over the base B. The accompanying coefficients s w are taken as sections of line bundles over the base manifold B and and can be seen (locally) as polynomials on the base manifold's coordinates. The birational map allows to write f and g in (3.1) in terms of the sections s w : the particular expression of f and g brings all the information about which configuration one has, i.e. one can deform the s w , by choosing a different generic section in the same line bundle, but the gauge group and which charged spectrum one has, does not change. Instead, if some of s w 's are identically zero or have very specific factorized forms, then the gauge group or the charged spectrum can change. This is what happens for example for the charge 4 models [33]: As we will see in Section 4.4, one can start from a model with a Z 3 discrete symmetry in the form (3.2) and deform some of the s w 's to specific sections of the corresponding line bundle; these s w 's will be written in terms of sections a 1 , b 1 , d i of new line bundles on the base manifold B generating a model with gauge group U(1) and charge 4 matter. Now choosing different generic sections s w , a 1 , b 1 , d i in the new line bundles does not chage the gauge group and matter spectrum. In the following the sections defining the gauge group and the matter sector will be called s κ (so, for example, in [33] s κ = s w , a 1 , b 1 , d i ).

Sen limit
In this paper we are interested in the weak coupling limit of F-theory comapactifications with U(1) gauge group. This limit, first studied by Sen [34], is a limit in the complex structure moduli space. For this reason, it is a delicate limit: complex structure deformations can change the gauge group and the matter spectrum of the F-theory model; on the other hand, for a weak coupling limit one means studying the F-theory 7-brane configuration under consideration but in the perturbative type IIB language. Hence, the weak coupling limit should not change the gauge group and the matter spectrum. As we have said above, the information about the 7-brane configuration is encoded in a choice of line bundles over B and corresponding generic sections s κ , in terms of which f and g are expressed. So, the Sen limit should not deform the polynomials s κ .
In order to see how the Sen limit works, one can first reparameterize f and g in (3.1) as In F-theory, the type IIB axio-dialaton τ (and thus the string coupling) varies over the base manifold B. The SL(2, Z) invariant function j(τ ) is in fact given by f 3 /∆. Sen found a limit that sets the string coupling small almost everywhere in B: if one scales and then the string coupling becomes small: In fact j(τ ) As it can be seen from (3.6), in this limit the codimension-1 loci of the base where the 7-branes lie are described by the two zeroes of the discriminant: From the monodromies of τ around these loci, one can find that b 2 = 0 and ∆ E = 0 describe respectively an O7-plane and a D7-brane, i.e. one has only perturbative objects.
The type IIB compactification manifold must be a double cover of the base B branched over the O7-plane locus, i.e. it is given by the equation ξ 2 = b 2 in (2.1). The involution I that sends ξ → −ξ is the orientifold involution. When the Weierstrass model is smooth, the locus ∆ E = 0 describes one brane that is invariant with respect to the involution ξ → −ξ; if the b i have a proper special form, then ∆ E can factorize so that there is more than one stack of D7-branes. As we will see, one can also have pairs of branes and their orientifold images.
Let us start with a model of the form (3.2) (or specializations thereof). There is a birational map to the Weierstrass model that gives f and g as functions of s κ . After a choice of b 2 , that will also be a function of s κ (it is not a coincidence the name we gave to the polynomials in (2.5)), one can derive the expressions for b 4 and b 6 in terms of s κ : using These will be functions of the s κ 's as well. As we said above, in the weak coupling limit we should not deform the polynomial s κ as they bring the information about the 7-brane configuration (gauge group and matter spectrum). Hence the Sen limit should be implemented by scaling (some of) the s κ , i.e. 9 such that we realize the scaling (3.5) for the b i 's. 10 Of course, the s κ that are in b 2 should not scale. If the F-theory model we started with has several gauge groups, at weak coupling There is a general observation regarding the weak coupling limits of toric hypersurface fibers. The consistent scalings leading to a perturbative type IIB model can be obatined with scalings of the form (3.8) with n κ = 1 for all the points κ lying along a facet in the dual polytope, while leaving all other sections invariant under the scaling. This occurs for all of the 16 2D hypersurfaces considered in [15]. Take for example the polytope F 3 = dP 1 . One has then four weak coupling limits as indicated in Figure 1. In reality, from the type IIB perspective such limits lead to only two inequivalent brane setups.

Massive and massless U(1)'s
In type IIB compactification, the U(1) symmetries live on the worldvolume of single D7branes. If the D7-brane locus is invariant under the orientifold involution, the U(1) gauge boson is projected out; hence a U(1) symmetry is present if there is a pair of a D7-brane and its orientifold image. If the loci of these two branes are in the same homology class, then the U(1) gauge boson is massless. If the loci are in different homology classes, the gauge boson gets a mass through the "geometric" Stückelberg mechanism [48][49][50], by eating an axion. This axion comes from the reduction of the RR 2-form C 2 along an odd two-form of the double cover CY. If there are several massive U(1)'s in the compactification, some combinations of them may be massless. For example, if we have two massive U(1) gauge bosons and h 1,1 − (X) = 1, then one combination of them will eat the only one axion, while the orthogonal combination will stay massless.
The charge corresponding to the massless gauge boson will be a linear combination of the charges of all the U(1)'s living on the D7-branes.
Let us see how one can find the massless U(1) generator in 6D compactifications (for the 4D case see [51][52][53]). On the D7-brane worldvolume, the coupling that gives mass to the gauge bosons is given by: where F is the six-dimensional gauge boson field strength and C 6 is the dual of the RR twoform C 2 . The C 6 potential can be expanded as ∈ H 1,1 − (X) and c α 4 six dimensional four-forms (dual in 6D to axionic scalar fields). Plugging this expansion in (3.9) we obtain D7 is the divisor wrapped by the D7-brane in the compact space and D (+) ) is the even (odd) component under the orientifold involution. The imagebrane gives the same coupling term (it has opposite odd components and its field strength is −F). An invariant brane does not have such a coupling, as its odd component is zero.
If we have N massive U(1) branes, we will have the following term in the six dimensional effective action −,eff , all the U(1) gauge bosons get a mass by Stückelberg mechanism [48][49][50].

Minimal models with high charges
We want to construct a model in type IIB with one massless U(1) and with states that have high charge under this U(1).
The easiest way to realize a massless U(1) is to take a pair of a D7-brane and its image in the same homology class (see [8] for an F-theory realization and [43,54] for the weak coupling limit). If there are no other branes, there will be only a state at the intersection D7 ∩ D7 that will have unit charge. On the other hand, if there is another (invariant) 12 brane, there will be a state with charge 1 at D7 ∩ D7 inv and a state with charge 2 at the intersection of the D7-brane with its image, away from the orientifold locus, i.e. at (D7 ∩ D7 ) \ (D7 ∩ O7) (see [45] for an F-theory realization and [35] for its weak coupling limit).
To obtain U(1) models with charges higher than 2, one needs to introduce massive U(1) D7-branes in a CY double cover with h 1,1 = 0. Since we want to end up with only one massless gauge boson, we need that the number of massive U(1) D7-branes is one unit bigger than the number of axions to be eaten, that is equal to h 1,1 −,eff . The minimal choice is h 1,1 −,eff = 1. We will see that under this assumption we will construct the weak coupling limit of all the high charge F-theory models known so far. We will then consider 11 The divisors D (−) D7i are not independent of each other in general. 12 If there is another pair, it must be massive, otherwise we would have two massless U(1)'s. the case when the double cover CY is given by the equation As explained in Section 2, for generic sections s 6 , s L , s R of the corresponding line bundles on B, this space has an odd 2-form dual to the divisor D − , whose class can be represented by the difference D p − D p , where D p , D p are given in (2.13). In this situation, we obtain one massless U(1) from two would-be geometrically massive U(1)'s: one combination of them will eat the axion associated to D − whereas the orthogonal combination remains a massless U(1). In type IIB we then have two pairs of massive D7 brane/image-brane, plus a possible invariant D7-brane (see Figure 2). Let's assume that the odd brane divisor classes satisfy then the six-dimensional coupling will be (with the normalization X ω (−) ∧ D − = 1) Hence the combination b F 1 − a F 2 is massive and the orthogonal combination a F 1 + b F 2 is massless. In particular, the charge associated with the massless U(1) will be a combination of the two massive U(1) charges: where a, b are integers and κ is the greatest common divisor of the U(1) charges aQ 1 +bQ 2 of the states in the configuration. In Table 1 we report the massless U(1) charge for each state in the configuration we have chosen. Table 1: U(1) charges for generic a, b (see (3.14)).
We will consider only configurations in which the different intersections provide matter with charges taking all the integral values between 1 and Q max , where Q max is the maximal charge realized in the model. In this way we will automatically satisfy the completeness conjecture. It in fact states that an effective field theory with a U(1) gauge symmetry is only consistent with quantum gravity provided that all charges Q ∈ Z appear at some level in the mass spectrum [55,56]. 13 Notice that in Table 1 there are six different states. Hence if we want that all the values of Q are filled up to Q max , then we can only construct models with the highest charge up to 6. Hence with the chosen configuration we have models with Q max = 3, 4, 5, 6: Charge 6: a = 3, b = 2 and non-empty In principle, in the cases with a = 2 and b = 1 the invariant brane is not necessary to realize all charges up to Q max , as the charges of D7 i ∩ D7 i are already realized in other intersections. We will see in the following that the invariant brane is necessary when a + b is odd. We will also see at the end of Section 3.5 why we have secretly considered the bound a + b ≤ 8. All the values of a, b satisfying this bound and that we did not consider correspond to models where some of the charge values is not populated by actual massless states. We do not consider models with a = b, that always give states with zero charge.

6D compactification and D7-brane setup
We consider 6D type IIB compactifications on a K3 surface X with an orientifold projection. 14 Holomorphic involutions on K3 were classified by Nikulin [58] in terms of three integer parameters (r, a, δ). In particular r is the number of K3 two-cycles that are even under the involution, i.e. b + 2 (X) = r and b − 2 (X) = 22 − r (remember that b 2 (K3) = 22). The fixed point locus is always given (except for two special cases) by the disjoint union of a genus g curve and k spheres, that have the following expressions in terms of r and a: In our case, the fixed point locus ξ = 0 that lives in the homology class c 1 (L) =K B (whereK B is the anticanonical class of the quotient B, pulled back to X). We consider cases where we have only one connected O7-plane. This means that k = 0, i.e. r = a. 15 The genus of the O7-locus is The 6D effective theory has N = 1 supersymmetry. The low energy spectrum will be made up by the gravity multiplet, V vector multiplets, T tensor multiplets and H hypermultiplets. The number of vectors is given by the number of non-invariant D7-branes (if the D7-brane worldvolume is invariant, the corresponding gauge field is projected out by the orientifold). The number of tensors is given by b + 2 (X); considering that one tensor sits in the gravity multiplet, we have 14 For some aspects of type IIB 6D models related to F-theory constructions see [57]. 15 When k = 0, the O7-planes wrap rigid two-sphere in K3 with gauge group SO (8). The corresponding F-theory lift will have Non-Higgsable Clusters with non-abelian D 4 singularities. Since we are interested in abelian gauge groups, we consider only involutions with k = 0. tensor multiplets, that includes also b + 2 (X) − 1 Kähler moduli. There are then 2b − 2 (X) further complex scalars that organize in H bulk = b − 2 (X) = 12 + 1 2K 2 B hypermultiplets: • 1 volume modulus complexified by C 4 along the volume form of X; In all the models we consider in this paper, we will consider configurations with two pairs of (massive) brane/image-brane, say D7 1 /D7 1 and D7 2 /D7 2 , and (in most cases) an invariant brane D7 inv .
Let us begin considering the pairs of brane/image-brane. Given the consideration in Section 3.4, the divisors wrapped by the branes are in the classes where the corresponding loci are given by (see Section 2) with x = (x 1 , ..., x a+1 ) and y = (y 1 , ..., y b+1 ), and M ⊗k red the (k + 1) × (k + 1) non trivial block of M ⊗k .
The number of open string moduli for each pair is given by the genus of the surface D D7 i , i.e.
Plugging (3.19) and (3.20) into (3.23) and using the relations (2.7), (2.8) and (2.9) one obtains The number of charged hypermultiplets at the inersection of branes coming from different pairs is given by the intersection numbers The number of states at the intersection of a brane with its image is instead given The invariant brane can be obtained by recombining a pair of brane/image-brane. The recombination can be described by a Higgs mechanism: a field living at the intersection of the brane with its image gets a non-zero vev, the vector multiplet living on the D7-brane gets a non-zero mass by eating one of the charged hypermultiplets. The number of open string moduli hypermultiplets of the invariant brane is then given by In what follows we will always consider invariant branes obtained by recombining a pair of brane/image-brane wrapping a divisors in the classes [D L,± ] + [w 2 ], i.e. it will wrap the locus We then have The invariant brane intersects the branes D7 1 and D7 2 , giving respectively [D D7 inv ]·[D D7 1 ] and [D D7 inv ] · [D D7 2 ] hypermultiplets: i.e. this configuration satisfies the gravitational anomaly cancellation condition, as it should be for a D-brane configuration with all D-brane charges canceled. The same occurs for the mixed U(1)-gravitational as well as the pure U(1) anomaly. The details of the anomaly computation can be found in Appendix B.
We finish this section with the D7-brane tadpole cancellation condition. In the case when D7 is present, we have

Type IIB models with maximal charge 3 or 4
In this section we will consider models with charge 3 (I) and (II) and model with charge 4 (I). As we will see the first two come from the Sen limit of the same F-theory model.

Charge 3 (I) and charge 4 (I) models
We start with a = 2 and b = 1 and an invariant brane, that realizes the charge 3 (I) and charge 4 (I) models. The values of a and b imply that where D D7 is the divisor class wrapped by the D7 brane. The D7 1 locus in the quotient is P D7 1 = 0 where P D7 1 is given by (2.19), i.e.
The only difference between Q max = 3 and Q max = 4 models is that in the first case the intersection D7 1 ∩ D7 1 is empty outside the O7-plane locus. This means that, while for the Q max = 4 model the vector (p 1 , p 2 ) is generic, for the Q max = 3 model it is given by either (1, 0) or (0, 1).
The invariant brane must satisfy the constraint that it has double intersection with the orientifold plane ξ = 0, i.e. on top of the O7-plane it must split as a brane/image-brane pair [59,60]. In particular, we require that the polynomial ∆ E (where ∆ E = 0 is the full D7-brane locus in (2.27)) reduces to a square on top of the O7-plane b 2 = 0 [59,60]. Since s L , s R , s 6 appear linearly in P D7 1 (for generic p 1 , p 2 ) and quadratically in P D7 2 (for generic q 1 , q 2 , q 3 ), then they should appear to an odd power in P D7 inv , up to a polynomial that vanishes if b 2 = 0. If we make the minimal choice (in which s 6 , s L and s R appear linearly), we have an invariant brane wrapping the locus (3.31). 16 The full D7-brane locus is then ∆ E = 0, where The charges of the states and their locations are reported in Table 2. We notice that in type IIB there are two loci corresponding to charge 2 and two corresponding to charge 1. As we shortly see, loci with the same charge recombine away from weak coupling, giving a unique locus with charge 1 and a unique locus with charge 2 in F-theory.

Charge 3 (II) model
We now consider the charge 3 (II) model, i.e. a = 3, b = 1. In this case there is no invariant brane. We then have The D7 1 locus in the quotient B is P D7 1 = 0 where P D7 1 is again given by (2.19), i.e.
The D7 2 locus in the quotient is P D7 2 = 0 where now P D7 2 is given by (2.32) , i.e. P D7 2 = r 2 1 s 3 L − r 1 s 6 (r 4 s 2 6 + s L (−r 3 s 6 + r 2 s L )) + r 1 s L (3r 4 s 6 − 2r 3 s L )s R + s R (r 2 2 s 2 L + s R (−r 3 r 4 s 6 + r 2 3 s L + r 2 4 s R ) + r 2 (r 4 s 2 6 − r 3 s 6 s L − 2r 4 s L s R )) . (4.7) 16 With a different choice we would have added more coefficients r i , with a higher restriction on their degrees and the degrees of the coefficients p j , q k . This choice will give the most generic such situation; specializing these coefficients, one can realize the configuration with higher powers of s L,6,R .

The full D7-brane locus is then
(4.8) The charges of the states and their locations are reported in Table 3.We notice that in type IIB there are two loci corresponding to charge 1. These will recombine away from weak coupling.

Charge 3 type IIB model from F-theory
In F-theory the charge three model was described in [15]. The weak coupling limit was performed in [35]. We summarize the result here.
The F-theory fourfold can be described as a toric hypersurface fibration based on the toric ambient space P F 3 = dP 1 as shown in Fig. 3.

Section
Line Bundle O(E 1 ) Figure 3: The polytope F 3 and its dual. The table on the right provides the line bundle classes for the coordinates in P F 3 .
The hypersurface equation is p F 3 = 0 with p F 3 = s 1 u 3 e 2 1 + s 2 u 2 ve 2 1 + s 3 uv 2 e 2 1 + s 4 v 3 e 2 1 + s 5 u 2 we 1 + s 6 uvwe 1 + s 7 v 2 we 1 + s 8 uw 2 + s 9 vw 2 . (4.9) The line bundles of the s i are fixed by choosing two arbitrary classes, in this case S 7 = [s 7 ] and S 9 = [s 9 ], and by requiring that all the monomials of (4.9) are sections of the same line bundle O(3H − E 1 + 2S 9 − S 7 ). After mapping p F 3 to the Weierstrass form we obtain f , g and ∆, which can be taken from [15] and are also reported in Appendix B of [35].
The fourfold described by (4.9) has two sections of the elliptic fibration: the (birationally equivalent) Weierstrass model zero section S 0 , and an extra section S 1 . This gives a massless U(1) gauge symmetry in the low dimensional effective theory.
There are no codimension-one singularities. At codimension-two one finds three I 2 fibers corresponding to states charged under the U(1) symmetry. The loci for the corresponding states are given in Table 4.  Let us now discuss the type IIB limit of this model. As pointed out already in Section 3.2, from the facets of the dual polytope we can deduce four types of -scalings leading to consistent weak coupling limits. Out of these four, only two are inequivalent from the point of view of the brane setups one realizes in type IIB, both of these lead to the same gauge gauge group and matter spectrum of the parent F-theory model.
Applying the limit → 0 to the matter loci in Table 4, one sees that the charge 3 locus become the corresponding locus in type IIB, while the charge 1 (charge 2) locus splits into the two charge 1 (charge 2) loci of the type IIB model [35]. We will see how this mechanism works explictly in the next example. For the charge 3 (I) model the computations are reported in [35] Sen limit to the charge 3 (II) model One can take a different weak coupling limit, by choosing the following scaling 1, 2, 3, 4) . We then find the type IIB charge 3 (II) model, with the identifications (s L , s 6 , s R ) = (s 5 , s 6 , s 7 ), (p 1 , p 2 ) = (s 9 , s 8 ), (r 1 , r 2 , r 3 , r 4 ) = (s 4 , s 3 , s 2 , s 1 ). The compete match of codimension two loci between the F-theoretic and the type IIB model is summarized in Appendix C. In this case the intersections D7 1 ∩ D7 2 and D7 2 ∩ D7 2 recombine away from weak coupling.

Charge 4 type IIB model from F-theory
The first U(1) F-theory model with massless matter charged up to Q max = 4 was derived in [33]. It can be viewed as a specialized version of the torus hypersurface equation (p F 1 = 0) in the toric ambient space P F 1 [15]. The polynomial equation is given as follows The divisor classes of the coordinates in P F 1 as well as those for the sections s i , a i and d i are given in Table 5. The model is therefore described in terms of three base divisors: is the coordinate of the extra section S 1 of the elliptic fibration, responsible for the massless U(1).
The matter spectrum is given in Table 6, we stick to the notation of [33] and refer to it for further details. For our purposes it suffices to specify the expression (4.17) Representation Locus Let us now consider the following weak coupling limit We then have with the irreducible components ∆ i given by The first locus corresponds to a pair of massive U(1) brane/image-brane of the form (4.2), the second is an invariant brane of the type (3.31) and the third is a pair of brane/imagebrane of the type (4.3). Hence we find the configuration predicted in Section 2.4, with the identifications (s L , s 6 , s R ) = (s 3 , s 6 , s 8 ), (p 1 , p 2 ) = (b 1 , a 1 ), (w 1 , w 2 , w 3 ) = (s 2 , s 5 , s 1 ) and The match of the charged loci with the type IIB ones is provided in Appendix D.

Charge 4 in F-theory from a Z 3 model through type IIB
In [33], the author found the charge 4 model by Higgsing a non-generic U(1) × U(1) model [61]. In this section we show that this model can be obtained also from the model in [15], i.e. a fourfold with Z 3 discrete symmetry. We will use the weak coupling limit to achieve this result.
At weak coupling, one may wonder which restriction on the sections r i makes the U(1) brane locus (2.32) factorize into two massive branes, one with equation (4.2) ((p 1 , p 2 ) = (b 1 , a 1 )) and one with equation (4.3) ((q 1 , q 2 , q 3 ) = (d 2 , d 1 , d 0 )), i.e. one needs to find the expressions of r 1,...,4 in terms of p 1,2 and q 1,2,3 that solve the equation This problem has a definite answer, that in the F-theory model notation is The resulting D7-brane configuration is the one realizing the charge 4 (I) type IIB model. This result is just obtained in the perturbative limit of the Z 3 model. If one applies the same restriction (4.24) to the equation (4.21), one gets the equation (4.25) that is exactly the F-theory charge 4 model found in [33].
This computation shows how the weak coupling limit can help to construct explicit complicated models in a simple way. We leave for the future the application of this method to higher charge models.

Charge and 6 models in type IIB
As we have seen in Section 3.4, there are not many possibilities to construct a model with maximal charge 5 or 6 when h 1,1 −,eff = 1. There is only one charge 6 model and in principle two charge 5 ones. Actually only the charge 5 (II) model can be constructed in full generality. The charge 5 (I) model requires a brane/image-brane pair wrapping the locus (2.32) with zero intersection away from the orientifold locus; this is not as easy to realize as when they wrap a locus like (2.32); only for specific base manifolds this may be realized. For this reason we will explicitly describe only the charge 5 (II) model and the charge 6 one.

Charge 5 (II) model in type IIB
In this case we have the following homology relation among the odd cycles: which is consitent with the choice a = 1, b = 4 for the generator of the massless U(1). Note that the intersections among D D7 1 and D D7 1 are required to vanish away from the orientifold locus. This is possible whenever we specialize the vector p in (2.19) to be of the form (p 1 , p 2 ) = (1, 0) such that the brane locus in the weak coupling discriminant reads 17 The locus for the brane D D7 2 and its image must be of order four in s L , s 6 and s R , and it can be written as, The matrix A 4 can be obtained as the non-trivial 5 × 5 block insideM ⊗4 and it is given in (A.1). Equation (5.3) finally reads with the D7 2 locus given by P D7 2 = 0. The invariant brane locus is given by P D7 inv = 0, with the polynomial P D7 inv given in (2.32). The D7-brane configuration is therefore described by ∆ E = P D7 1 P D7 2 P D7 inv . Note that from the weak coupling perspective we require 11 sections of line bundles on B in order to describe the charge five model: s 6,L,R , w 1,2,3 , t 1,...,5 . These must become the sections s κ defining the F-theory model.
The charges of the fields for the type IIB perspective are given in Table 7. Table 7: U(1) charges for the charge 5 (II) model. The state at D7 1 ∩ D7 1 is absent.
As a final remark, notice that in this model we have the peculiarity that all the matter loci exhibit different charges under the massless U(1) therefore in the F-theory uplift we expect no recombination phenomena occurring for the codimension two loci.

Charge 6 model in type IIB
The charge six model at weak coupling is obtained with a = 3, b = 2 and κ = 1. We will have the following relation among the odd divisors: The D7-brane configuration is given by ∆ E = P D7 1 P D7 2 P D7 inv with P D7 1 given by (2.27), i.e.

(5.6)
The invariant brane is at P D7 inv = 0 with P D7 inv given in terms of the base sections (w 1 , w 2 , w 3 ) according to (3.31). For completeness we summarize the charges for the fields in Table 8. Note that all different intersections have different charges under the massless U(1) symmetry and therefore we expect no recombination for these loci in the F-theory lift. In this case we need 13 base sections in order to fully describe a charge 6 model at weak coupling: s 6,L,R , q 1,2,3 , r 1,...,4 and w 1,2,3 . We claim that these will be the s κ sections describing the charge 6 F-theory model.

Explicit examples with charge 5 and 6
We now proceed to construct explicit 6 dimensional charge 5 and charge 6 models in perturbative type IIB. We choose the two-dimensional base manifold to be B = P 2 , with homogeneous coordinates [x 1 , x 2 , x 3 ]. The double cover CY two-fold is described by (3)). Hence the O7-plane is in the class [O7] = 3H. We choose also the line bundle L L =K P 2 , so that we have L R =K P 2 as well. This means that the classes of s 6 , s L , s R are i.e. they are homogeneous polynomials of degree 3 in the homogeneous coordinates x i . 18 These polynomials can be taken independent of each other. 19 The CY X is a K3 surface and the orientifold involution is one of the Nikulin involution with k = 0. In the present example, the fixed point locus ξ = 0 is connected and has genus The only involution compatible with g = 10 and k = 0, according to Nikulin, is (r, a, δ) = (1, 1, 1). Accordingly we have only one even two-cycle (as we expect from b 2 (P 2 ) = 1).
Since b 2 (K3) = 22, we have several odd divisor classes. Choosing b 2 = s 2 6 4 − s L s R , we have restricted the complex structure in a way to make algebraic two image two-cycles in different homology classes (without generating singularities in K3).

Charge 5 model
Let us consider first the model with charge 5. In order to work out the homology classes of the brane stacks D7 1 /D7 1 , D7 2 /D7 2 and D7 inv we notice that the choice of L L implies that the divisor class for the sections t i in ( Effectiveness of these divisor classes implies that ν is an integer greater or equal than two. The classes of the weak coupling discriminant loci are therefore given by: D7-tadpole cancellation implies that and therefore the only two possibilities for (β, ν) are (1, 2) and (0, 3). The matter multiplicities for these states are given in Table 9.
The type IIB analogous of the Nèron-Tate height pairing that enters the anomaly cancellation in 6D (see Appendix B) is given by (5.14)

Models with incomplete spectra beyond charge 6
In the models we constructed so far, we have demanded that all charges between 1 and a given maximum charge Q max appear in the spectrum. As we said, since one can have at most six intersections with the D7-brane configuration studied so far, the maximum charge is six. However the bound a + b ≤ 8, derived by the effectiveness of the divisor classes, allows in principle higher charges.
Take for example the case of two U(1) branes without an invariant brane. As discussed around equation (3.38) this system is bounded by the constraint a + b ≤ 8 and hence we will have additional options beyond the ones indicated in Section 3.4: these are going to have only four types of massless charged matter. The possibilities are shown in Table 11 where we can see that a model with charge 7 is possible.  A similar analysis can be done for the cases in which the invariant brane is present. In that case one would be able to get maximum charge 10.  We find no consistency conditions that would prevent these models in type IIB, even though the absence of some charge in the spectrum sounds odd in light of F-theory constructions. To realize generic models with Q max > 6 and with all charges up to Q max , one needs to increase the number of (massive U(1)) D7-brane stacks.

Conclusions and future directions
In this paper we have faced the problem of constructing 6D F-theory models with U(1) gauge group and matter fields with high charge. These models typically have massless states with integral charges from 1 to a maximal value Q max . Our approach was to construct these models in the type IIB perturbative limit of F-theory. If a model with high charge exists in type IIB, it must have a consistent F-theory lift.
We have worked out a method to easily construct type IIB models with high charge. We verified that for Q max = 3, 4 these are exactly what one obtains by taking the Sen weak coupling limit of the existing F-theory models with Q max = 3, 4. We then described type IIB models with Q max = 5, 6. We have built explicit examples, where the base manifold B is P 2 . These are consistent string theory models exhibiting a massless U(1) symmetry with massless hypermultiplets with charges up to five and six. This proves therefore that massless states with U(1) charges as high as six are part of the 6D string theory landscape. Unfortunately, while it is relatively easy to go from an F-theory model to its weak coupling limit, it is not straightforward to lift a type IIB model to F-theory. Nevertheless, the knowledge of what should be its Sen limit, can help towards the realization of the corresponding model in F-theory. We plan to approach this issue in the future.
So far we have explored only models with Q max ≤ 6. This has been realized by two pairs of U(1) D7 brane/image-brane: one combination of the two U(1) gets a mass by eating a C 2 axion, while the orthogonal combination remains massless. To obtain higher charges one needs to generalize the construction presented in this paper, adding more D7branes with massive U(1)'s and allowing more C 2 axions to be eaten by the massive gauge bosons, i.e. h 1,1 −,eff > 1 in the notation of Section 3.3. Algebraically one can realize it by a more specific form of b 2 that admits more than two inequivalent matrix factorizations. This will lead, without any obstruction, to models with Q max > 6. It would be nice to see if there is an upper bound for Q max in type IIB. In [32], it was shown that models with U(1) symmetries with higher charges can in principle be obtained from models with exotic non-Abelian matter by means of Higgsing. Along the same line, it has been shown that SU(N) models with exotic matter could lead, upon Higgsing to U(1) models with charges Q ≤ 21 in six dimensions. It would be nice to find a similar bound in type IIB U(1) models (even though with different techniques, as for example three-index antysymmetric states are not realized in perturbative type IIB).
Another way to obtain models with high U(1) charge is to consider models with gauge symmetry U(1) n . As the number of U(1)'s increases, the number of charged massless fields increases as well. Higgsing models with multiple U(1)'s could lead to single U(1) models with higher charges. For example, the U(1) × U(1) model of [61] has multiplets with charges (−1, 1) and (−2, −2) among others. A vev in (−1, 1) makes the field (−2, −2) to pick a charge 4 along the diagonal massless U(1) [33]. This Higgsing can be done in type IIB as well as in F-theory. Understanding which deformations realize the Higgsing may be easier in type IIB in some cases. Applying then the same deformation to F-theory models may lead to the desired high charge realizations.
One may apply the same reasoning to models with discrete symmetry. F-theory models exhibiting only discrete symmetries are described by genus one fibrations. In this paper we have considered the Z 3 model of [15] and applied the weak coupling limit, obtaining a Z 3 model in type IIB, realized by a pair of brane/image-brane with massive U(1) and an invariant brane. It was easy to see under which deformation the brane/image-brane system splits into the two brane/image-brane system realizing the Q max = 4 model. Applying the same deformation to the corresponding Z 3 F-theory model, we were able to straightforwardly obtain the Q max = 4 F-theory model of [33]. This method can in principle be applied to obtain a charge Q max > 4 model in F-theory. Models with discrete symmetries Z 4 [11] and Z n with n ≤ 5 [62] are already present in literature.
A Divisors of order 4 in s 6 , s L , s R Here we provide the explicit matrix form of A 4 and B 4 :

B Anomaly cancelation in 6D models
In Section 3.5 we computed the number of charged and neutral hypermultiplets. We distinguish the two cases with and without invariant brane: • If there is the invariant brane, then The tadpole cancellation condition • When there is no invariant brane we have The D7-tadpole cancellation condition gives and we obtain H = 14 + 29 2K 2 .
In both cases this matches with the anomaly cancellation condition H = V + 273 − 29T . Remember that in our setup V = 2 where one vector multiplet is massless, while the other gets a mass by eating one (axionic) hypermultiplet. The total number H that we computed includes such eaten hypermultiplet. If we count only massless hypermultiplets we should substract 1 from both sides of the anomaly cancellation condition, and the match will be still valid.
Next we consider the mixed U(1) gravitational anomaly as well as the pure U(1) anomaly. The corresponding anomaly cancellation conditions read where b for example can be derived from the CS couplings D7 i C 4 ∧ F i ∧ F i . These conditions are analogous to the ones we typically have in F-theory setups with a slight difference in a factor 1/2 on the right hand side of Eqs. (B.6) and (B.7) due to the fact that we are computing intersections on the double cover manifold X instead of the F-theory base B (while on the left hand side we are summing only on the projected spectrum: for example we are not counting both states from D7 1 ∩ D7 2 and states from D7 1 ∩ D7 2 ). The divisor b is the analogous of the Nèron-Tate height pairing −π(σ(S 1 ) · σ(S 1 )) [63], with σ(S 1 ) being the Shioda map of the section S 1 associated to the massless U(1) symmetry, i.e. the extra section in addition to zero section S 0 .
Again we consider two different cases depending on whether the invariant brane is present or not.
Therefore, at weak coupling the locus {y 1 = f z 4 1 + 3x 2 1 = 0} becomes {Az 2 1 = Bz 3 1 = 0}. Since we have to subtract V (I (2) ) ∪ V (I (3) ), then z 1 = 0 and we obtain that the charge one locus is fully captured by {A = B = 0}. Considering the primary decomposition of this locus in the double cover, one obtains three codimension two irreducible components: the first one is We then found that the charge one locus of the F-theory threefold, splits into the two charge one loci that are expected in the corresponding type IIB model.

D Matching Charged Loci in Charge 4 Model
In this appendix we illustrate the matching of the codimension two loci for the charged matter in F-theory and type IIB. Recalling the identifications (s L , s 6 , s R ) = (s 3 , s 6 , s 8 ), (p 1 , p 2 ) = (b 1 , a 1 ), (w 1 , w 2 , w 3 ) = (s 2 , s 5 , s 1 ) and (q 1 , q 2 , q 3 ) = (d 2 , d 1 , d 0 ), one can see that for the discriminant locus ∆ 1 (see (4.20)) the splitting into brane/image brane is governed by (4.2). For the discriminant locus ∆ 2 is an invariant brane (see (3.31)) and and similarly for ∆ 3 that splits in the Calabi-Yau X accoding to (4.3). Note that in this case D7 1 is described in terms of (p 1 , p 2 ) = (a 1 , b 1 ). According to (2.18), the brane and its image intersect away from the orientifold over the locus {a 1 = b 1 = 0}, so that the charge 4 locus in type IIB corresponds to D7 1 ∩ D7 1 , in accordance with table 2.
• The charge three locus in F-theory is described by the variety with α and β given in (4.16). We notice that after taking the scaling a 1 → a 1 and b 1 → b 1 , the polynomials α and β are homogeneus of degree two in and hence the locus does not suffer modifications at weak coupling.One can show that in the Calabi-Yau X the ideal I (3) decomposes into three prime ideals: The first one is given by (D.1) and it should be removed. The remaining two are image to each other under ξ → −ξ. One of them is equal to the intersection of D7 1 and D7 2 up to codimension three loci (the other is its orientifold image): therefore at weak coupling, the charge 2 ideal becomes {βA, βB}. This ideal decomposes into three codimension 2 ideals when intersected with the type IIB CY X.
The first one corresponds to the intersection of D7 2 and its image away from the orientifold locus: Hence, the splitting of the F-theory matter locus coincides with the expectations from the type IIB side (see Table 2).
The expressions for these, in terms of the base sections a 1 , b 1 s i , d i , are very long and can be found in a Mathematica notebook (Charge4Model.nb) as part of the ancillary files of the arXiv post 21 of ref. [33]. Following an analogous procedure as the one outlined in this and in the previous appendix one can apply the weak coupling limit to this locus and find that it splits in type IIB to the intersections D7 2 ∩ D7 2 and D7 2 ∩ D7 inv .