Chiral spectrum of the universal tuned $(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1))/\mathbb{Z}_{6}$ 4D F-theory model

We use the recently developed methods of 2108.07810 to analyze vertical flux backgrounds and associated chiral matter spectra in the 4D universal $(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1))/\mathbb{Z}_{6}$ model introduced in 1912.10991, which is believed to describe the most general generic family of F-theory vacua with tuned $(\text{SU}(3) \times \text{SU}(2) \times \text{U(}1))/\mathbb{Z}_{6}$ gauge symmetry. Our analysis focuses on a resolution of a particular presentation of the $(\text{SU}(3) \times \text{SU}(2) \times \text{U}(1))/\mathbb{Z}_{6}$ model in which the elliptic fiber is realized as a cubic in $\mathbb{P}^2$ fibered over an arbitrary smooth threefold base. We show that vertical fluxes can produce nonzero multiplicities for all chiral matter families that satisfy 4D anomaly cancellation, which include as a special case the chiral matter families of the Minimal Supersymmetric Standard Model.


Introduction
One of the hallmarks of the Standard Model is the presence of chiral fermions transforming in representations of the gauge symmetry group (SU(3) × SU(2) × U(1))/Γ, where Γ ∈ {1, Z 2 , Z 3 , Z 6 } is a possibly trivial discrete subgroup of the center of SU(3) × SU(2) × U(1). Chiral fermions charged under the Standard Model gauge group have been realized in numerous string theory constructions-see [1] for a review of recent developments. The purpose of this paper is to analyze the chiral matter spectrum and aspects of the Yukawa interactions of the universal tuned F-theory model with gauge group (SU(3) × SU(2) × U(1))/Z 6 introduced in [2].
There are obvious motivations for finding specific, dynamically stable solutions of string theory that reproduce all observed aspects of the Standard Model. However, rather than aiming for a complete string-theoretic construction of the Standard Model, we instead take a top-down approach and focus on sets of solutions in the space of string vacua that exhibit the gross features of the Standard Model, specifically the gauge group and allowed representations in the chiral matter spectrum. One reason for this approach is that taking a bird's eye view can give us a comprehensive picture of the set of candidate string vacua, which may help to address broad questions about how the fundamental interactions of the Standard Model can be reproduced by a self-consistent theory of quantum gravity (many examples of which we expect to come from string theory). Some obvious questions of this sort are how common or uncommon various realizations of the Standard Model are in the space of string theory solutions with given supersymmetry, and what kinds of physics beyond the Standard Model are associated with these different realizations. Another, related, question is what, if any, constraints are imposed on the kinematics of Standard Model-like theories, and whether or not these constraints can teach us anything about more general constraints imposed by consistency with quantum gravity that we might expect to be manifest at low energies. The set of string theory vacua with prescribed low-energy data is sometimes referred to as part of the string landscape, while low-energy effective theories with similar kinematic structure that do not admit a UV completion in string theory (and therefore, perhaps not in quantum gravity more generally) are said to belong to the swampland [3,4]. Adopting this terminology, the questions described above can be rephrased as questions about, respectively, the prevalence of Standard Model-like vacua within the string landscape, and identification of the "boundary" between the landscape and the swampland as viewed from the perspective of Standard Model-like constructions. Taking the broad approach of constructing large classes of Standard Model-like theories also provides the more pragmatic opportunity of comparing how natural or phenomenologically-relevant different classes of such constructions may be, and for identifying more specific features of observed physics that may arise in subsets of the classes of constructed models.
The questions described above could in principle be explored in the context of various different types of constructions belonging to different branches of string theory. In this paper, however, we focus exclusively on Standard Model-like constructions in F-theory [5][6][7]. Ftheory provides a remarkably effective set of tools for exploring the landscape of 4D N = 1 string vacua. The reason for this is that F-theory relates nonperturbative type IIB flux compactifications on compact Kähler threefolds with 7-branes to the geometry of singular elliptically-fibered Calabi-Yau (CY) fourfolds, and this relationship has in turn led to the development of a systematic procedure for constructing 4D N = 1 string vacua with desired kinematics simply by tuning the mathematical properties of the elliptic CY singularities using celebrated results in algebraic geometry. An enormous number of string vacua have been constructed following this procedure, and although F-theory is dual in certain regimes to other constructions such as heterotic string compactifications, F-theory is believed to give the broadest global picture currently available of the supersymmetric string landscape in terms of a unified space of elliptic CY fourfolds that are connected through various topologychanging transitions (see, e.g., [8][9][10][11]).
An extensive literature on 4D F-theory flux compactifications has been produced over the past fifteen years, with most papers on standard model-like F-theory vacua focusing on GUT models whose gauge group is broken, giving the Standard Model gauge group at low energies, see, e.g., [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], and [29] for a review of the extensive literature on SU(5) F-theory GUT constructions. The topic of this paper is a somewhat less extensively explored type of construction, namely F-theory vacua with exact (SU(3) × SU(2) × U(1))/Γ gauge symmetry and no larger geometric GUT symmetry. Specifically, we study the family of F-theory vacua engineered by the universal tuned (SU(3) × SU(2) × U(1))/Z 6 model described in [2]. The (SU(3) × SU(2) × U(1))/Z 6 model describes a universal class of F-theory Weierstrass models with tuned 1 (SU(3) × SU(2) × U(1))/Z 6 gauge symmetry that has geometrically generic matter for this gauge group, 2 which includes the matter representations of the Minimal Supersymmetric Standard Model (MSSM) and three additional exotic matter representations; these representations can combine into three independent anomaly-free families (one of which is that of the MSSM). Note that a subclass of these models, which arise naturally through a toric fiber ("F 11 ") construction [32], contains the exact matter spectrum of the MSSM with no exotics; relatedly, chiral matter in F 11 models was investigated in [33,34], and vector-like matter in these models (which we do not address here) has been explored in [35][36][37][38].
One of the primary questions we address in this paper is whether the tuned (SU(3) × SU(2) × U(1))/Z 6 model over an arbitrary base that allows this tuning naturally contains chiral matter in the other two allowed families, or whether the other two families are for some reason forbidden by F-theory geometry and hence belong to the swampland. To this end we use the recently developed approach of [39] to determine the lattice of vertical fluxes of the (SU(3) × SU(2) × U(1))/Z 6 model defined over an arbitrary smooth base and in the presence of a non-trivial flux background. By computing the full set of linear dependences of the vertical fluxes, we show that all three independent families of anomaly-free chiral matter are realized, and hence that the linear constraints imposed by F-theory geometry on chiral multiplicities precisely match those imposed by 4D anomaly cancellation. Furthermore, we examine aspects of the quantization of the fluxes (and hence, of the chiral indices) both in general and for specific choices of F-theory base, and comment on the implications for the numbers of allowed chiral families.
More precisely, we compute an explicit resolution of a presentation of the (SU(3) × SU(2) × U(1))/Z 6 model in which the elliptic fiber is realized as a general cubic in P 2 . We use this resolution to confirm that the gauge symmetries, matter representations, and Yukawa interactions anticipated in [2] are realized geometrically by the singular elliptic CY fourfold associated to the (SU(3) × SU(2) × U(1))/Z 6 model. We then use the methods of [39] to show that the lattice of vertical fluxes preserving the 4D local Lorentz and gauge symmetry, which gives at least a subset of the full set of chiral multiplicities realized by the F-theory model, spans the full linear space of anomaly-free chiral spectra transforming under (SU(3)×SU(2)× U(1))/Z 6 . The analysis carried out here is performed in a (mostly) base-independent fashion so that the results can be applied to a large number of F-theory vacua with a wide range of pactification, corresponding to fixing some specific complex structure moduli, as opposed to one that arises on a rigid divisor as a generic feature of the F-theory base geometry, or which arises from breaking a larger such rigid group. In this sense, the standard SU(5) F-theory GUT constructions reviewed in [29] also rely on tuning of the SU(5) structure, while the more recent flux-breaking GUT constructions of [27,28] and the direct SM construction of [30] use rigid gauge factors, which may be more prevalent in the landscape. 2 The notion of genericity for matter representations in 6D and 4D F-theory was defined and explored in [31].
Note that while universal tuned Weierstrass models will exist for other choices of global gauge group structure associated with other quotient factors Γ, the Standard Model chiral matter is only generic for Γ = Z6. Note also that generic matter types are associated with the geometry of general Weierstrass tuning of any given gauge group; while in 6D this definition can be naturally understood as a feature of the low-energy theory (where generic matter is associated with the largest-dimensional branch of the moduli space for fixed gauge group), this distinction is less transparent in 4D, and other constructions such as that of [28] do not seem to share the same natural structure of generic matter in 4D.
bases. Note, however, that not all base geometries admit tuning of this gauge structure. In particular, at large Hodge numbers, there are many rigid gauge factors and it can become difficult to find room in the geometry for tuning additional features. The remainder of this paper is structured as follows: In Section 2, we give a brief summary of the main results of the paper. In Section 3, we outline the basic setup and describe explicitly the resolution of the universal tuned (SU(3) × SU(2) × U(1))/Z 6 model. In Sections 4 to 6, we study the explicit resolutions of singularities at codimension one, two, three, corresponding to the gauge factors, matter representations, and Yukawa interactions. In Section 7, we analyze the chiral matter multiplicities in the presence of (vertical) fluxes using the formalism of [39], and in Section 8 we consider explicit examples. In Section 9 we discuss some more detailed aspects of the quantization of chiral multiplicities. We conclude in Section 10 with a discussion of the results and further directions for future work.

Summary of results
Our analysis of the chiral spectrum of the tuned G SM = (SU(3) × SU(2) × U(1))/Z 6 model is divided into two parts. First, we resolve the singular F-theory background X 0 in order to study the kinematics of the corresponding low-energy effective 4D theory for an arbitrary choice of base and characteristic data where the tuned G SM structure is possible. Second, we switch on a generic vertical flux background and use the geometry of the resolved model to compute the corresponding chiral excesses induced in the 4D spectrum. We summarize the results of these two parts of the analysis in the subsections below.

Resolution of singularities and 4D kinematics
Let X 0 → B be the singular elliptic CY variety defining the (SU(3) × SU(2) × U(1))/Z 6 model, where B is the base of the elliptic fibration. In Section 3, we use well-established methods 3 to study the geometry of a particular resolution X 5 → X 0 in order to explore the kinematic structure of the low-energy effective 4D theory. Although for the purposes of computing the chiral spectrum following the approach of [39] it is sufficient to simply resolve the singularities of X 0 and use the structure of the codimension-one singular fibers of X 5 to write the intersection numbers of X 5 in terms of a basis of Cartan divisors, we nevertheless provide a comprehensive description of the singular fibers over the codimensionone, codimension-two, and codimension-three components of the discriminant locus in B. Our analysis of the singular fibers of X 5 shows that the 4D effective theory exhibits the g locus fiber Table 1. Singular elliptic fibers of the resolution X 5 → X 0 (see Eq. (3.10)) of the (SU(3) × SU(2) × U(1))/Z 6 model over codimension-one and codimension-two components of the discriminant locus ∆ = 0; these singular fibers correspond to (resp.) gauge symmetries and charged matter representations. The nodes of the graphs in the right-most columns represent irreducible components of the singular elliptic fibers, while the edges of the graphs correspond to points of intersection between pairs of irreducible components. Each irreducible component is birational to P 1 . A blue (red) '×' in the center of a node indicates that the curve represented by that node intersects the zero section (generating section) at a point away from the points of intersection with other P 1 components. A blue node is an exceptional P 1 wrapped by the zero section; such curves correspond in the M-theory frame to primitive BPS particles whose KK central charges are approximately the same scale as their Coulomb branch central charges. (Note that we express the loci in terms of the parameters of the resolution X 5 , so that d 0 e 1 = 0, b 1 e 2 e 3 = 0 are, respectively, the SU(2), SU(3) gauge divisors Σ 2 , Σ 3 ⊂ B.) gauge symmetries and charged matter spectrum listed in the left-hand columns of Table 1.
The gauge symmetries and charged matter spectrum we find agree with the analysis of the singular limit X 0 in [2], and hence our analysis provides a check of the predictions made in [2] regarding the gauge symmetries and matter spectrum of the (SU(3)×SU(2)×U(1))/Z 6 model. Our analysis also includes detailed geometric information relevant for further investigation of structures such as the Yukawa interactions of the theory-see Table 2.

Vertical flux backgrounds and chiral matter spectrum
We next use the resolution X 5 → X 0 to compute the multiplicities of 4D N = 1 chiral multiplets in the (SU(3) × SU(2) × U(1))/Z 6 model induced by a non-trivial vertical flux background. The multiplicities of 4D N = 1 chiral multiplets can be expressed in terms of the chiral indices χ r = n r − n r * , where n r , n r * are the numbers of chiral and anti-chiral multiplets transforming in the complex representations r, r * . We direct the reader to Section 7.2 for a detailed review of the terminology and concepts discussed below. Computing the chiral multiplicities for a 4D F-theory compactification in terms of the geometry of a resolved elliptic CY is a well-studied problem in F-theory, reviewed in [42]. Base-independent analyses of this problem for specific gauge groups have also been worked out in, e.g., [43][44][45][46]. In [39], we developed a base-independent analysis, in which the intersection numbers of a resolved elliptic CY X are used to compute the integral pairing on the vertical homology subgroup, After homological equivalences on vertical cycles are modded out, this nondegenerate intersection pairing M red was found to be independent of the choice of resolution in many cases, and conjectured to be resolution-independent in general. If true, the resolution-independence of the matrix M red suggests that the integral pairing on vertical cohomology is intrinsic to the singular F-theory limit that directly encodes information about the low-energy 4D theory; this is physically natural since all physical features such as the chiral spectrum should be resolution-independent, but this kind of cohomology of singular spaces is not fully understood mathematically. Other recent work aimed at a direct characterization of the physics of F-theory models in terms of the singular geometry described by type IIB string theory includes [47][48][49]. The chiral multiplicities of any theory of interest can be computed from simple linear algebra using M red , where schematically Θ IJ = M (IJ)(KL) φ KL in terms of a (homologically redundant) basis of surfaces S IJ =D I ∩D J , φ IJ parameterizes the allowed (vertical) fluxes, and the chiral multiplicities can be related to certain Θ IJ , while other Θ IJ are constrained to vanish by various symmetry principles. For example, for an F-theory Tate model with a nonabelian gauge group (defined over an arbitrary base, and with generic complex parameters, with no additional Kodaira singularities at higher codimension), the reduced matrix M red takes the schematic form in a reduced basis S 0α , S αβ , S iα , S ij where 0 represents the zero section, indices α correspond to base divisors, and i correspond to Cartan elements of the gauge group; the matrix elements denoted by " * " are undetermined and the non-zero matrix elements are given by intersection products of divisors D α in the base B, where K is the canonical class of the base. A rational change of basis removes all the * elements except the bottom right block, which becomes a matrix M phys relating chiral multiplicities to the subset of flux parameters φ IJ that correspond to symmetry-preserving F-theory fluxes (note that all other flux parameters are forced to vanish in order to satisfy the constraints Θ Iα = 0, which are necessary to preserve 4D Poincaré and gauge symmetry). This simple schematic structure becomes more complicated for models with additional abelian U(1) factors, as described in [39]. In particular, the presence of additional sections complicates the form of the reduced matrix (2.3), and makes it harder to describe the general solution of the constraints as easily. This is described in some detail for a simpler model with gauge group (SU(2) × U(1))/Z 2 in [39]; working through the details of this for the more complicated universal (SU(3) × SU(2) × U(1))/Z 6 model is a central part of the work in this paper. The matrix M red representing the integral pairing for the resolved (SU(3) × SU(2) × U(1))/Z 6 model is displayed in Table 5 and Table 6 in two particular homologically independent bases for S IJ , which are useful in complementary aspects of the analysis. This reduced matrix is used in Section 7 to describe the chiral matter content of the model with fluxes.
An interesting question one can ask related to the landscape of F-theory compactifications is whether or not all linearly independent anomaly-free combinations of chiral matter can be realized in F-theory, i.e., whether F-theory imposes further linear constraints on the available chiral spectrum beyond those associated with anomalies. (Note that we expect that all F-theory models will satisfy anomaly constraints; the geometric manifestation of these constraints has been studied in, e.g., [46,[50][51][52]). We address this question by geometrically computing the possible combinations of chiral matter multiplicities that can be realized in the (SU(3) × SU(2) × U(1))/Z 6 model and comparing them to the full list of possible chiral matter combinations compatible with 4D anomaly cancellation, namely the set of all integer values for the chiral indices χ r (where r is one of the eight complex representations appearing in the left-most column of the right hand table in Table 1) subject to the constraints Since there are eight possible chiral matter representations and five constraint equations (above), this leaves behind a three-dimensional linear space of anomaly-free chiral matter multiplicities. One basis for the three linearly-independent families of anomaly-free matter is displayed in equal to zero reduces this three-dimensional family of anomaly-free chiral multiplicities to a one-dimensional family corresponding to a single generation of the MSSM chiral matter spectrum. 5 We compare the anomaly-free chiral matter spectra satisfying Eq. (2.4) to the chiral indices of the resolved (SU(3) × SU(2) × U(1))/Z 6 model induced by a non-trivial vertical flux background; see Eq. (7.47). Consistent with the findings of [39], namely that all anomaly-free linear combinations of matter can be realized in F-theory (without considering multiplicities) for the models considered there, we indeed find that the full set of three independent linear families of possible chiral matter multiplicities are realized in the (SU(3) × SU(2) × U(1))/Z 6 model in the presence of a non-trivial vertical flux background for a generic choice of base. We comment on the precise quantization of the chiral indices imposed by the geometry of specific F-theory vacua (i.e., a specific choice of base and associated topological data) in Section 9. In addition to the quantization constraints, tadpole constraints, which we do not consider here, will also limit the range of chiral matter multiplicities available in any given model.

Basic setup and resolution
The universal tuned (SU(3) × SU(2) × U(1))/Z 6 model was identified in [2] by un-Higgsing a U(1) model with charge q = 4 matter introduced in [53]. The charge q = 4 U(1) model of [53] was constructed as a singular hypersurface of an ambient projective bundle Y 0 whose fibers are isomorphic to P 2 with homogeneous coordinates [u : v : w]. The hypersurface equation is 4 We refer to charged matter transforming in representations that do not belong to the MSSM as "exotic" matter, although these are still generic features of the class of universal tuned (SU(3) × SU(2) × U(1))/Z6 F-theory models. 5 Another specific linear combination of the anomaly-free families (with multiplicities of each family multiplied by 2, 0, 1 respectively), has no matter charged under the SU(3) factor and indeed the resulting multiplicities correspond precisely to those of the (SU(2) × U(1))/Z2 model [32] whose chiral spectrum is explored in [39]. Table 3. Multiplicities of the three anomaly-free families of generic chiral matter in 4D models with (SU(3) × SU(2) × U(1))/Z 6 gauge group as follows: In terms of the above construction, the un-Higgsing U(1) → (SU(3) × SU(2) × U(1))/Z 6 is achieved by restricting to a special locus in complex structure moduli space on which the sections a 1 = s 3 vanish: so that We now discuss this construction in more detail. First, observe that the above elliptic fibration is equipped with two rational sections: zero section: The coefficients of the cubic (in u, v, w) monomials appearing in the homogeneous polynomial p 0 in Eq. (3.3) are (the pullbacks to Y 0 of) sections of various line bundles over B. We may equivalently express these line bundles in terms of their first Chern classes, each of which can in turn be expressed as a linear combination of the following divisor classes: We refer to the above four divisor classes as the characteristic data of the elliptic fibration Y 0 → B. Let us describe these divisor classes in further detail: • K is the canonical class of B; • Σ 2 , Σ 3 are the su(2), su(3) gauge divisors wrapped by 7-branes; • and Y is the divisor class of the locus in B over which the zero section and the generating section associated with the u(1) factor intersect.
The classes of the zero loci of the other sections appearing in p 0 besides d 0 , b 1 , s 2 are (3.6) The ambient space of the elliptic fibration, Y 0 , can be viewed as the projectivization of a rank-two vector bundle: To ensure that the homogeneous polynomial p 0 is a well-defined section of the line bundle we assign the following choice (unique up to an overall additive constant) of divisor classes 6 L i = c 1 (L i ) to the hyperplanes u, v, w = 0 in the Chow ring of Y 0 (3.9) Although the hypersurface equation Eq. (3.3) differs from the standard Weierstrass form presented in [2] (with the two equations being related by Nagell's algorithm 7 ), the singular locus of X 0 is nevertheless still given by the zero locus of the discriminant ∆ = 4f 3 + 27g 2 .
In this sequence of blowups, the variable t appearing in the defining equation for X 4 is defined as follows: t := s 2 v + e 1 e 2 e 3 e 4 s 1 u . (3.11) The first three blowups appearing in Eq. (3.10) (counting from the right) are conceptually straightforward, as they are restricted to codimension-one components of the discriminant locus in B. However, the fourth and fifth blowups appearing in Eq. (3.10), whose corresponding blowdown maps are (respectively) X 4 → X 3 , X 5 → X 4 , are somewhat unintuitive in that they appear to blow up the entire base B. However, it turns out that these choices of blowups produce new divisors with precisely the desired geometry: the proper transform of each blowup is topologically a small blowup of the base B at the points over which the elliptic fibers become singular, and hence is a smooth birational representative of a rational section (see Eq. (3.4)). 9 The proper transform of X 0 under the above composition of blowups, X 5 , is the complete intersection  The above homogeneous coordinates are identified under the rescalings , e 4 → e 4 λ 4 , e 5 → e 5 λ 5 , t → λ 0 λ 5 t (3.14) 9 To see why the fourth and fifth blowups in Eq. for λ i ∈ C * . One can verify via explicit computation in the affine open sets of X 5 that the rank of the Jacobian [[∂ i p 5,j ]] does not (in general) reduce over any codimension-two loci in B, indicating that X 5 is smooth through codimension-two in B without additional tunings of the parameters defining the (SU(3) × SU(2) × U(1))/Z 6 Weierstrass model. 10 We abuse terminology and refer to X 5 as "smooth", a "resolution", etc., keeping in mind that X 5 may contain singular fibers over certain codimension-three loci; these codimension-three singularities do not affect the main results of this paper. We next analyze the geometry of X 5 to identify the kinematics of the low-energy effective 4D N = 1 supergravity theory describing the (SU(3) × SU(2) × U(1))/Z 6 model at long distances. Unless otherwise stated, throughout our analysis we work over an arbitrary smooth base B.

Codimension one: gauge symmetry
In this section, we analyze the geometry of the singular elliptic fibers of X 5 over codimension one loci of the discriminant locus in B, in order to verify that our resolution correctly exhibits the expected gauge algebra characterizing the 4D N = 1 theory describing F-theory compactified on the singular fourfold X 0 .

u(1)
In F-theory compactifications pure u(1) gauge symmetries are identified with the Mordell-Weil (MW) group of rational sections of the elliptic fibration [7,[56][57][58], where the zero section (i.e., a choice of zero element of the MW group, extended fiber-wise over B) corresponds to the Kaluza-Klein (KK) u(1) and all other sections generating the free part of the MW group correspond to additional u(1) factors. 11 Therefore, pure abelian gauge symmetries are not determined by codimension one singularities in B, although they are nonetheless associated to smooth divisorsDā of X 5 ; see Eq. (7.2) for a description of the proper transforms of the zero and generating sectionsD 0 ,D 1 under the composition of blowups leading to X 5 . Note that sinceD 0 is rational, rather than holomorphic, the resolution X 5 describes a phase of M-theory in which some of the primitive BPS particles in the 3D spectrum carry nontrivial u(1) KK charge; this point is explored further in Section 5.1. 12 Since 3D BPS particles 10 Note that certain choices of base B can impose additional specializations on the Weierstrass model parameters that can in principle lead to additional singularities appearing over higher codimension components of the discriminant locus. 11 More precisely, in the dual M-theory picture, in the vicinity of a divisor of a smooth elliptic CY, it is possible to locally expand the M-theory 3-form in a basis of harmonic forms ωi as C3 = A I ∧ ωI + · · · , where A i are abelian gauge fields and the harmonic forms ωi are Poincaré dual to the divisorsDi. See Section 4.3 of [42] and references therein for further discussion. 12 See, e.g., [40] for a discussion of the interplay between rational sections and KK charges in the context of 6D F-theory compactifications.
arise from M2-branes wrapping holomorphic curves C, these quantized u(1) KK charges can be computed in terms of the intersection pairingD0 · C. Similarly, the addtional u(1) charges of BPS particles can be determined by computing the intersection pairingsD1 · C.

su(3) ⊕ su(2)
In contrast to u(1) gauge symmetries, simple nonabelian gauge symmetries g s are associated to singularities of the elliptic fibers over codimension one components of the discriminant locus ∆ = 0 in B. Since the discriminant ∆ of the singular Weierstrass model Eq. (3.3) exhibits the following behavior, we can see there is an I 2 singularity overe d 0 = 0 and an I 3 singularity over b 1 = 0 in X 0 (see, e.g., [59]). We use our explicit description of the resolution X 5 → X 0 in Eqs. (3.12) and (3.13) to verify that both singular fibers are split in the sense of arithmetic geometry and indicate the presence of (resp.) su(2) and su(3) gauge symmetries. We describe the singular fibers in more detail below. In the following discussion, for convenience, we study the singular fibers and their intersections in the affine open set e 4 e 5 = 0, away from possible points of intersection with the rational sectionsD 0 ,D 1 . The singular elliptic fibers F over the locus d 0 e 1 = 0 split into the following irreducible rational curves: Setting d 0 = 0, we learn that the irreducible curve F 0 is the normalization of the singular cubic curve in P 2 with coordinates [u : v : w] (after appropriate coordinate redefinitions), where the singularity u = w = 0 has been replaced with the exceptional curve F 1 . Setting e 1 = 0, one can see that F 1 is a smooth conic in P 2 with coordinates [u : w : d 0 ]. The intersection F 0 ∩ F 1 consists of two distinct points, hence the generic fibers are type I 2 and we associate F 0 , F 1 to, respectively, the affine and non-affine nodes of the affine su(2) Dynkin diagram. In codimension one,D 0 intersects F 0 in a point, whereasD 1 intersects F 1 in a point. Next we turn our attention to the fibers over the locus b 1 e 2 e 3 = 0: Analogous to F 0 , we associate F 0 to the affine node and F 2 , F 3 to the other two nodes of the affine su(3) Dynkin diagram. Making the redefinitions w → uw/(e 3 e 5 ), e 2 → e 2 /e 3 , v → e 1 e 4 uv, we learn that F 0 can be described locally as a smooth conic in a subspace birational to P 2 with coordinates [e 2 : v : w] with the points [0 : 1 : 0] and [s 2 v : s 1 e 2 : 0] removed. Next, we consider the curve F 2 . Making coordinate redefinitions u → u/(e 1 e 4 ), e 3 → e 3 /(e 1 e 4 e 5 w), we find that F 2 is the sum of a base and fiber curve in F 1 with homogeneous coordinates [v : Finally, in the case of the curve F 3 , after making the coordinate redefinitions w → vd 0 w/(e 4 e 5 ), u → vd 0 u/e 4 we learn that the class Figure 1. Schematic diagram of the singular elliptic fibers over the codimension-one su(2) (left) and su(3) (right) loci of the discriminant locus in B. Over the su(2) locus d 0 e 1 = 0, the singular elliptic fiber F is Kodaira type I 2 , and admits the decomposition F = F 0 +F 1 ; over the su(3) locus b 1 e 2 e 3 = 0, the singular elliptic fiber is Kodaira type I split 3 , and admits the decomposition F = F 0 + F 2 + F 3 . Note that all irreducible components F a , F b are rational curves. The blue × indicates that F 0 , F 0 intersect the zero sectionD 0 in a point, whereas the red × symbol indicates that F 1 , F 3 intersect the generating sectionD 1 in a point.
of F 3 is a sum of the base and fiber curve classes in F −1 with homogeneous coordinates It is straightforward to verify that the three pairwise intersections F 0 ∩ F 2 , F 0 ∩ F 3 , F 2 ∩ F 3 are all points, and hence the fibers over b 1 e 2 e 3 = 0 are of Kodaira type I split 3 . In codimension oneD 0 intersects F 0 in a point, whereaŝ See Figure 1 for a schematic depiction of the codimension-one singular fibers.

Codimension two: local matter
In this section we study the geometry of the codimension-two singular fibers (i.e., collisions of codimension-one singular fibers including those described in Section 4) associated to local matter representations is an irreducible quaternionic representation of the nonabelian gauge algebra g s and w1 = q is the u(1) charge. We show, in particular, that the codimension-two singularities are consistent with the matter spectrum discussed in [2].

Kaluza-Klein charges
As was mentioned briefly in Section 4.1, a special feature of the resolution X 5 is that the zero sectionD 0 is rational, rather than holomorphic, which is closely related to our realization of the F-theory vacuum X 0 as a hypersurface of a general P 2 fibration as opposed to a Weierstrass model y 2 z = x 3 + f xz 2 + gz 3 (the Weierstrass model, by contrast, is naturally equipped with a holomorphic zero section x = z = 0). It turns out that the zero sectionD 0 is isomorphic to a small blowup of the base B at various points and is hence only birational to B. This can be seen by noting that the locus s 2 = 0, over which the zero and generating sections intersect (and hence over which our choice of zero section fails to be holomorphic), has non-trivial intersection with the discriminant locus. Thus, the singular elliptic fibers of X 0 over the codimension-two locus ∆ = s 2 = 0 have singular points that are blown up as part of the resolution X 5 → X 0 ; the proper transforms of these points are irreducible P 1 components of I 2 singular fibers over ∆ = s 2 = 0. Since these exceptional P 1 's are by definition extremal generators of the Mori cone of X 5 , they necessarily have non-trivial intersection withD 0 ∼ = Bl {∆=s 2 =0} B, and thus it follows that the central charges of the primitive 3D BPS particles corresponding to M2 branes wrapping these exceptional P 1 's carry non-zero u(1) KK charge. These 3D BPS particles' KK central charges are in fact small in comparison to their Coulomb branch central charges (see Eq. (7.43)), so there is no mass hierarchy between BPS particles with zero KK charge and BPS particles with non-zero KK charge. While the presence of light particles with non-trivial KK charge clearly does not alter the 4D spectrum, this does imply that we cannot simply read off the 4D kinematics from the naive 3D BPS spectrum in the usual manner. 13 Thus, we need to be able to use the geometry of X 5 to precisely specify the contribution of the KK charges to the 3D Coulomb branch dynamics, so that we can disentangle the KK spectrum from our analysis. We discuss how to do this below. We first focus on the zero section. Rescaling the homogeneous coordinates appropriately and defining w → e 2 uw, e 4 → e 4 /u, we can describe the restriction of the zero section to the locus s 2 = 0 by the complete intersection We thus expect the intersection ofD 0 ∩Ŷ with each of these codimension one loci to describe the following collection of exceptional P 1 s: Setting e 1 = 0 and working in the affine open set s 1 e 2 e 3 = 0, after appropriately rescaling and defining new variables in a subspace birational to P 1 × P 1 with homogeneous coordinates [e 4 : in a subspace birational to a Hirzebruch surface 14 In all three cases, we see that the zero section wraps a full rational curve, i.e., the elliptic fiber splits as F → F + F , with F , F ∼ = P 1 . As discussed, since these exceptional curves F lie inD 0 , the primitive BPS particles corresponding to M2 branes wrapping F carry non-trivial KK charge given byD0 · F = 0.
Having specified the geometric origin of the light BPS particles in the 3D spectrum with non-trivial u(1) KK charges, we next turn our attention to local matter charged under su(3) ⊕ su(2) ⊕ u(1).

(1, 1) w1
The pure u(1) charged matter loci were found in [2] to corresponds to singular fibers of X 0 over the following components of the discriminant locus ∆ = 0: In the above equation To see that the fibers in the resolution X 5 degenerate as expected over the total transforms of both loci described above, we work in the open set d 0 e 1 b 1 e 2 e 3 = 0. For the charge q = 1 locus, we further restrict to the open set s 2 = 0 and solve for d 2 , s 8 in the equation V q=1 = 0. Doing so, we find that We learn also that the zero section intersects F + q=1 in a point, while the generating section intersects F − q=1 in a point. Moreover, it is straightforward to verify that the intersection F + q=1 ∩ F − q=1 consists of two distinct points, hence we see the fibers over V q=1 = 0 are type 14 Fn denotes the Hirzebruch surface P(O ⊕ O(n)) → c, with divisor classes f, c satisfying f 2 = 0, f · c = 1, c 2 = −n.
I 2 , as anticipated, where F + q=1 as the affine component of the affine su(2) Dynkin diagram describing the intersection structure of the enhancement. The enhancement of the elliptic fiber to an I 2 Kodaira fiber indicates the presence of charged matter and since this enhancement occurs away from the components of the discriminant locus associated to nonabelian gauge symmetry, we conclude this matter can only be charged under the u(1) of the 4D gauge algebra. The u(1) charges can be checked by computing intersection productsD1 · F ± q=1 . We treat the charge q = 2 locus in a similar manner. Setting s 1 = s 2 = 0, Eq. (3.12) splits into two components, p 5,2 (s 1 = s 2 = 0) = e 5 t = 0. It follows that the fibers over this locus degenerate: Notice that the above definitions imply that the zero section wraps the exceptional curve F − q=2 . To study the geometry of the degenerated fibers more explicitly, we use local descriptions of the fibers in affine open sets, as we now describe.
For F + q=2 , Making the redefinitions u → u/(e 1 e 2 e 4 ), w → w/(e 1 e 3 e 4 ), we can describe F + q=2 locally as a smooth conic in P 2 with homogeneous coordinates [u : v : we 5 ] with the point [0 : 1 : 0] removed. For F − q=2 , we make the redefinitions w → e 2 e 4 uw/e 3 , e 4 → e 4 /(e 1 e 2 u) so that F − q=2 can be described locally as a sum of base and fiber curves in The generating sectionD 1 intersects F − q=2 in a point, but does not intersect F + q=2 . Moreover, the intersection F + q=2 ∩ F − q=2 consists of two distinct points, hence the fibers over the charge q = 2 locus are type I 2 . Once again, the charges can be determined explicitly by computing the intersection productsD1 · F ± q=2 .

(1, 2) w1
The residual discriminant locus intersecting the codimension-one I 2 locus d 0 = 0 is thus according to Tate's algorithm [59] we can anticipate the presence of localized matter charged under su(2) for all codimension-two loci except for d 0 = 4b 1 d 1 s 2 − s 2 6 = 0. According to the analysis of [2], we anticipate the following representations In this subsection we focus on the local matter away from the collision of I 2 and I split 3 fibers; see Figure 2. Bifundamental matter, which is localized at the collision of the I 2 and I split 3 fibers, is discussed in Section 5.5.
Judging from Eq. (5.11), we expect the I 2 fiber to enhance to I split 3 over ∆ (a) = 0. Unfortunately, an explicit algebraic description of the degeneration of the elliptic fibers over ∆ (a) = 0 is difficult to obtain so we simply verify that the discriminant ∆ F 0 of the cubic polynomial defining F 0 in Eq. (4.3) vanishes on this locus. In particular, we find (5.14) In the above equation, (· · · ) denotes factors of b 1 , e 2 , e 3 responsible for the degeneration leading to bifundamental matter; we analyze this degeneration in detail in Section 5.5. Schematically, we write Unfortunately, without an explicit algebraic description, it is not precisely clear howD 0 ,D 1 intersect the irreducible components F ± 0 of the elliptic fiber over this locus. However, given that the BPS particles corresponding to M2 branes wrapping F ± 0 have much larger KK central charges than Coulomb branch central charges, we can further anticipate thatD 0 only intersects either F + 0 or F − 0 in a point, rather wrapping the entire curve. Turning next to s 2 = 0 and keeping in mind p 5,2 (e 1 = s 2 = 0) = te 5 = 0, we find The above equation describes a pair of curves intersecting transversally. A local algebraic description of F t 1 can be obtained by restricting the description of F 1 (see Eq. (4.3)) to the locus s 2 = 0. Making appropriate coordinate redefinitions, F e 5 1 can be described locally as a sum of a base and fiber curve in subspace birational to F 1 with coordinates [u : are distinct points and hence the degeneration over s 2 = 0 enhances the I 2 fiber to an I split 3 fiber, indicating matter transforming in the 2 of su(2) Note thatD 0 wraps the entire curve F e 5 1 , and bothD 0 ,D 1 intersect F t 1 in distinct points.

(3, 1) w1
The residual discriminant intersecting the I split Figure 2. Schematic depiction of the degeneration of the I 2 Kodaira fibers over the codimension-two loci corresponding to local matter transforming in the representations (1, 2) 1/2 (over d 0 e 1 = ∆ (a) = 0) and (1, 2) 3/2 (over d 0 e 1 = s 2 = 0). Note that a blue (respectively, red) × in the center of a node corresponding to an irreducible P 1 component of the singular elliptic fiber indicates thatD 0 (respectively,D 1 ) intersects the said P 1 in a point. The blue node is wrapped entirely byD 0 .
Following Tate's algorithm, we expect to see localized matter charged under su(3) at all codimension-two components of the discriminant locus except for b 1 = s 6 = 0. 15 The analysis of [2] anticipates the following su(3) charged matter representations (see Figure 3): Over the locus s 8 s 2 2 − s 5 s 2 s 6 + s 1 s 2 6 = 0, we find that the fibers in the resolution X 5 behave as follows: Substituting w → w/(e 4 e 5 ), e 2 → e 2 /(e 4 u) we find in the open set s 2 = 0 that F ± 0 can be given a local algebraic description as a pair of lines in P 2 with coordinates [e 1 e 3 e 2 : v : e 1 e 3 w] and the point [1 : 0 : 0] removed. Given this description, it is straightforward to verify that the pairwise intersections 15 The locus b1 = s6 = 0 is an IV locus, and does not support any charged matter [2].
indicating matter in the 3. Note thatD 0 intersects F 0 − in a point along Similarly, over the locus d 0 e 1 s 2 8 + d 2 s 2 6 − d 1 s 8 s 6 = 0 the curve F 2 degenerates as follows: Making the coordinate substitutions e 3 → e 3 /(e 4 e 5 w), u → u/e 4 we find that in the open set s 6 s 8 = 0 that F ± 2 can be described locally as as a pair of lines in F 1 (the homogeneous coordinates for this F 1 are described in Section 4.2). The pairwise intersections Finally we come to the locus s 2 = 0. The component F 3 degenerates as follows: Making coordinate redefinitions t → vt, u → d 0 vu, w → d 0 vw, we find that the irreducible components of the above degeneration can be described by . Note thatD 0 wraps F e 5 3 andD 1 intersects F e 5 3 in a pointD 0 ∩D 1 away from the intersection F t 3 ∩ F e 5 3 .

(3, 2) w1
In this section we describe the collision of I 2 , I split 3 fibers over the codimension-two locus b 1 = d 0 = 0: where F ab denotes the curve obtained by setting e a = e b = 0 for a, b = 0, or d 0 , b 1 = 0 for (resp.) a, b = 0. The analysis of [2] indicates that su(3) ⊕ su(2) charged matter is localized at the collision of the I 2 , I split 3 singularities (see Figure 4):  Kodaira fibers over the codimensiontwo loci corresponding to local matter transforming in the representations ( Note that a blue (respectively, red) × in the center of a node corresponding to an irreducible P 1 component of the singular elliptic fiber indicates thatD 0 (respectively,D 1 ) intersects the said P 1 in a point. The blue node in the bottom-most graph is wrapped entirely byD 0 . Figure 4. Schematic depiction of the collision of I 2 and I split 3 Kodaira fibers over the codimension-two locus corresponding to local matter transforming in the representation (3, 2) −1/6 (over d 0 e 1 = b 1 e 2 e 3 = 0). Note that a blue (respectively, red) × in the center of a node corresponding to an irreducible P 1 component of the singular elliptic fiber indicates thatD 0 (respectively,D 1 ) intersects the said P 1 in a point. a description as curve of bi-degree (1,1) , e 2 → ve 2 /e 4 u allows us to describe F 10 locally as a line in P 2 with homogeneous coordinates [e 3 e 2 : e 3 w : d 0 ]. For F 12 , making the redefinitions u → vu/e 4 , e 3 → ve 3 /(e 4 e 5 w), we find that F 12 can described as a curve of bi-degree (1, 1) in Finally, for F 13 , making the redefinitions e 4 → d 0 ve 4 /u, w → uw/e 5 and rescaling the coordinates appropriately, we find that F 13 can be described by the equation where the F 1 in the above equation has homogeneous coordinates [ It is straightforward to verify that the only intersections F 00 ∩ F 02 , F 00 ∩ F 10 , F 02 ∩ F 12 , F 10 ∩ F 13 , F 12 ∩ F 13 are distinct points, hence we find that the elliptic fiber is type I split 5 . Note thatD 0 intersects F 00 in a point, whileD 1 intersects F 13 in a point.

Codimension three: Yukawa interactions
In this section, we study the degenerations of the elliptic fibers over codimension-three loci in B that are characterized by triple intersections of matter curves, Figure 5. Cartoon depiction of the various junctions of triples of matter curves C R , C R , C R ∈ B associated to candidate Yukawa interactions appearing in the low-energy 4D N = 1 action. Each codimension-three point in B in the above figure (represented by a black dot) is labeled by the triple of u(1) charges w1, w 1 , w 1 of the three representations r, r , r (see Eq. (6.4)) contracted into a gauge singlet term in the low-energy Lagrangian; note that each triple labels one of a pair of possible complex conjugate interaction terms. The red, blue, and purple curves correspond to matter curves associated to non-trivial su (3), su (2), and su(3) ⊕ su(2) representations, respectively. By contrast, the dashed lines denote matter curves corresponding to pure u(1) charges. Note also that the (1, 1) −1 matter curve intersects the (1, 2) 1/2 in a tangent point.
Since the orders of vanishing of the sections f, g, ∆ can further increase at these codimensionthree loci, the singularity type of the elliptic fibers can enhance; we refer to this behavior as a degeneration of the elliptic fibers because analogous to the degenerations of the elliptic fibers over codimension-one or codimension-two components of the discriminant locus, these singularity enhancements also manifest themselves in the resolved geometry as the splitting of one or more of the components of the elliptic fibers into multiple irreducible P 1 components. However, there is an important distinction to be made here, in that the elliptic fiber degenerations over loci of the form Eq. (6.1) do not produce additional generators of the cone of effective curves. This is because the degenerations are of the schematic form at a given point in the locus Eq. (6.1). In the above equation, F R i is an irreducible component of the elliptic fibers (and hence a generator of the cone of effective curves) over a generic point the codimension-two locus C R i , and thus we see that the classes of any apparently "new" curves that appear as a result of these degenerations turn out to be linear combinations of existing curve classes. Eq. (6.2) implies that at each degeneration point in the resolved geometry, there resides a localized interaction in the M-theory effective action that mediates the splitting of a BPS state corresponding to an M2-brane wrapping F R 1 into a pair of BPS states corresponding to M2-branes wrapping F R 2 and F R 3 [43]. The representation-theoretic version of this statement that characterizes the F-theory limit is that degenerations of the form Eq. (6.2) indicate the existence of maps which produce gauge singlets from the weights of the matter representations in the low-energy effective action. In other words, these codimension-three singularities correspond to Yukawa interactions appearing in the low-energy effective 4D action [12][13][14] (see also Section 6 of [42] and references therein). An important check that such interactions indeed can appear in the low-energy effective action is to identify an explicit elliptic fiber degeneration in the resolution of the singular F-theory background. 16 In the case of the (SU (3)×SU (2)×U (1))/Z 6 model, we anticipate the existence of Yukawa interactions corresponding to the following gauge singlets: To ensure that the couplings are gauge singlets separately for each gauge factor su(3), su(2), u(1) we require that the nonabelian irreps are either singlets or pairwise complex conjugates and that the u(1) charges sum to zero. 17 In more detail, we compute the full list of possible Yukawa interactions by identifying all solutions to the equation w1 + w 1 + w 1 = 0, where w1, w 1 , w 1 are a subset of u(1) charges in the low-energy (anti-)chiral spectrum, and then excluding all candidate solutions for which the nonabelian factors cannot be combined into a gauge singlet. With the exception of the possible gauge singlet described in Footnote 17, it turns out that all possible Yukawa interactions consistent with the matter representations of the (SU(3) × SU(2) × U(1))/Z 6 model arise as elliptic fiber singularity enhancements over codimension-three loci in B, assuming generic characteristic data. In the following, we discuss one of a pair of complex conjugate Yukawa interactions associated to each codimension-three singularity. 16 Note that while there are also Yukawa type couplings that include (non-chiral) adjoint representations of the nonabelian factors, associated with non-localized matter fields in the bulk of the 7-branes, we focus here on localized Yukawa couplings associated with nonabelian fundamental fields and the charge combinations of local matter described in the previous section. locus b1e2e3 = s8s 2 2 − s5s2s6 + s1s 2 6 = 0. However, the only possible double intersection would occur at tangent point s 2 6 = 0, which lies along the IV locus b1e2e3 = s6 = 0, which does not support any charged matter, hence there is no additional Yukawa interaction.

(3
First, consider the case (w1, w 1 , w 1 ) = ( 4 3 , 2 3 , −2). We use the description of the (3, 1) − 4 3 degeneration given in Eq. (5.21), in which we already see the degeneration F 3 → F t 3 + F e 5 3 . Over the locus s 1 = 0 an additional exceptional curve F e 5 0 appears due to the intersection of the zero sectionD 0 with the affine component F 0 of the singular elliptic fiber, leading to the degeneration (6.10) The new intersection points are F e 5 0 ∩ F t 0 , F e 5 0 ∩ F e 5 3 , F t 0 ∩ F 2 , hence we find an enhancement I split 4 → I split 5 . Notice thatD 0 now wraps both F e 5 3 , F e 5 0 , whileD 1 still intersects F e 5 3 in a point.
Next, consider the case (w1, w 1 , w 1 ) = (− 2 3 , − 1 3 , 1). We work with the description of the Here, consistent with the local matter representation (3, 1) − 1 3 , for which we see the degeneration F 2 → F + 2 + F − 2 , we find that the elliptic fiber decomposes as  The new intersections points are Using the projective scaling symmetry of the ambient space and making the coordinate redefinitions u → u/(e 1 e 4 ), e 3 → e 3 /(e 1 e 4 e 5 w), d 0 → d 0 /e 1 , the new irreducible curves can be expressed as (6.14) The new points of intersection are F + 2 ∩ F − 2 , F + 2 ∩ F 0 , F − 2 ∩ F t 3 and therefore we see an enhancement of the form I split 4 → I split 5 . See Figure 7 for a schematic depiction of the singular fibers over these special points in B. Figure 7. Singular fibers associated to Yukawa interactions of the form (3 * , 1) w1 ×(3, 1) w 1 ×(1, 1) w 1 .
The first case is (w1, w 1 , w 1 ) = ( 1 2 , − 3 2 , 1). In this case, we borrow the description of the fiber degeneration F 1 → F t 1 + F e 5 1 that occurs over the codimension-two locus associated to local matter transforming in the representation (1, 2) 3 2 . In codimension three, we see the degeneration where now, in contrast to the discussion in Section 5.3, the polynomial ∆ (a) factors over the locus s 2 = 0:   The new points of intersection are The next case is (w1, w 1 , w 1 ) = ( 1 2 , 3 2 , −2). This Yukawa interaction can be analyzed in a manner analogous to that of the previous case, with the key distinction being that we now solve ∆ (a) | s 2 =0 = 0 by setting s 1 = 0. Here, it is illuminating to denote both degenerations by F 0 → F t 0 + F e 5 0 , F 1 → F t 1 + F e 5 1 : where the points of intersection are We thus see an enhancement of the form I split We solve the above equations by eliminating d 2 , s 8 . Over this codimension-three locus, using the degeneration F 0 → F + 0 + F − 0 inherited from the codimension-two enhancement associated -30 -with local matter in the representation (1, 2) 1 2 described in Eq. (5.15), we see the further . See Figure 8 for a schematic depiction of the singular fibers over these special points in B.

Intersection theory, 3D Chern-Simons couplings, and chiral matter multiplicities
In the preceding sections, we have studied the geometry of the resolution X 5 . This analysis confirms that the singularities of X 0 are consistent with the expected kinematics of the lowenergy 4D theory [2], and provides a geometric grounding for further investigation of the structure of the matter spectrum and Yukawa interactions in the theory. In this section, we shift our attention to computing the chiral excesses introduced to the 4D matter spectrum by switching on a non-trivial flux background. Since we closely follow the approach described in [39], we omit here many technical details of the methodology involved and mostly focus on describing the results of using these methods to analyze the (SU(3)×SU(2)×U(1))/Z 6 model; the interested reader should refer to [39] for details of the approach that are not spelled out here explicitly, and to [42] and further references in these two papers for further background and the earlier literature on chiral matter analyses in F-theory.

Quadruple intersection numbers and the reduced intersection matrix
The analysis of [39] provides a framework for understanding chiral multiplicities from fluxes in terms of a reduced intersection pairing matrix on the vertical part of the middle cohomology of the resolution X 5 → X 0 . This reduced intersection matrix can be computed from the quadruple intersection numbers of divisors in X 5 . One of the central results of [39] was the observation that the intersection pairing matrix appears to be independent of resolution, despite the fact that the quadruple intersection numbers are resolution-dependent. If this result indeed always holds, it suggests that the reduced intersection pairing matrix is the underlying resolution-independent mathematical structure that encodes information about the chiral matter spectrum in 4D F-theory compactifications. In the following discussion, we adopt this viewpoint and turn our attention towards computing the reduced intersection matrix associated to X 5 as a means to recover the chiral spectrum of the (SU(3) × SU(2) × U(1))/Z 6 model. The resolution X 5 has a basis of divisorŝ D I=0,1,α,i 2 ,i 3 , (7.1) Figure 8. Singular fibers associated to Yukawa interactions of the form (1, 2) w1 × (1, 2) w 1 × (1, 1) w 1 .
which includes a rational zero sectionD 0 and rational generating sectionD 1 associated to the u(1) gauge factor,D as well as an su(2) Cartan divisorD i 2 =D 3 , and su(3) Cartan divisorsD i 3 =D 4 ,D 5 appearing as irreducible components in the pullback of the gauge divisors Σ s from B to X 5 : 3) The u(1) KK and u(1) divisors (i.e., the images ofD 0 ,D 1 under the Shioda map [61]) are given byD0 Note that our convention for ordering the indices i s of the Cartan divisorsD is matches the following presentation of the su(2) ⊕ su(3) Cartan matrix: Now that we have identified a suitable basis of divisors, our next task is to compute their quadruple intersection numbers. We use the adaptation of the pushfoward technology of [62] to compute the pushforwards W IJKL of the quadruple intersection numberŝ D I ·D J ·D K ·D L (7.7) (here, · denotes the intersection product in the Chow ring of X 5 ) to the Chow ring of B. In more detail, we first represent the pullbacks of the divisor classesD I as linear combinations of the divisors D α , H, E i = [e i ] restricted to the class X 5 of the hypersurface X 5 ∈ Y 5 : In the above equation, the class X 5 is given by so that in terms of the projection π : Y 5 → B , (7.11) the pushforwards of the quadruple intersection numbers can be expressed as The above formulas allow us to represent all intersection products as products of divisor classes in the Chow ring of Y 5 . We then use the fact that the projection map π can be represented as a composition of maps, where f i are the blowups appearing in Eq. (3.10) and the is the canonical projection Y 0 → B in Eq. (3.7), to represent the pushforward map π * acting on elements of the Chow ring of Y 5 as the composition π * = * • f 1 * • · · · • f 5 * . (7.14) Since the action of * , f i * on any formal power series in the appropriate Chow ring is known explicitly, this decomposition enables us to compute the pushforward of any quadruple intersection number-see Appendix E of [39] for more details.
For convenience, we follow the approach of [63] and arrange the intersection numbers into the generating function eĴ = exp(ϕ ID I ) · X 5 . The intersection numbers can be extracted from Z ϕ by computing derivatives: (7.18) The pushforward of the generating function e J under the map π * encodes all of the intersection numbers of X 5 : where the functions Z 1 , . . . , Z 7 are given by (7.20) Following the analysis of [39], we organize the pushforwards of the quadruple intersection numbers into a matrix of intersection pairings: where we view the matrix M (IJ)(KL) as acting on an integer lattice Λ S spanned by the elements S IJ . The components of M (IJ)(KL) are displayed in Table 4. Although the matrix M (IJ)(KL) may appear at face value to simply be a trivial repackaging of the intersection numbers, as discussed in [39] we expect the nondegenerate part of this matrix to be independent of resolution (up to a choice of basis), despite the fact that complete set of intersection numbers does not share this feature. The matrix M (IJ)(KL) has a non-trivial null space which is spanned by (at least) the following vectors: The quotient of the lattice Λ S by the complete set of null vectors is isomorphic to the nondegenerate lattice H vert 2,2 (X 5 , Z) defined by the pairing M red . While in simple cases (such as the base P 3 ), the set Eq. (7.23) is a complete basis of the nullspace of M , there may be further null vectors, associated, for example, with fluxes φ α3 D α = D, with D · Σ 2 = 0. We write a general abstract form of M red in Table 5 and Table 6, in two different choices of bases, with the understanding that some further null vectors may need to be removed to define the reduced basis for some bases, as discussed above. While we have computed this from a particular resolution, as in the cases studied in [39] we expect this form of M red to be resolution-independent. We discuss the relationship between M-theory flux backgrounds and the middle cohomology subgroup H vert 2,2 (X 5 , Z) in more detail in the following subsection. In the above equation, G 4 = dC 3 is the field strength of the M-theory 3-form C 3 and is expected to satisfy the shifted quantization condition G 4 − c 2 /2 ∈ H 4 (X 5 , Z) in consistent M-theory vacua [64], and S IJ :=D I ·D J (see Eq. (7.21)) are so-called "vertical" 4-cycles whose homology classes generate the subgroup H vert 2,2 (X 5 ). While G 4 could in principle be any element of H 4 (X 5 , Z), because the middle cohomology of smooth elliptic CY fourfolds X admits an orthogonal decomposition [65,66] H 4 (X) = H 2,2 vert (X) ⊕ H 4 hor (X) ⊕ H 4 rem (X) , (7.25) for the purpose of computing Eq. (7.24) we can restrict our attention to vertical flux backgrounds, i.e., G 4 ∈ H 2,2 vert (X 5 , Z/2), possibly at the expense of being able to determine the precise quantization of the chiral indices (further discussion of the quantization issue appears in Section 9). We follow this approach, and for convenience we parametrize the vertical part of G 4 by its Poincaré dual, where we view φ KL ∈ Λ S as an integer 18 vector living in the integral lattice Λ S of vertical flux backgrounds spanned by the 4-cycles S IJ . This convenient choice of parametrization leads to the following simple linear algebraic expression for the space of vertical fluxes:

Symmetry constraints
We are specifically interested in M-theory fluxes that lift to 4D F-theory fluxes preserving local Lorentz and (SU(3) × SU(2) × U(1))/Z 6 gauge symmetry. In the above equation (7.27), these symmetry constraints can be imposed by restricting φ KL to live in the sublattice Λ C ⊂ Λ S of vertical flux backgrounds that preserve 4D Poincaré and (SU(3)×SU(2)×U(1))/Z 6 symmetry, i.e., flux backgrounds whose corresponding fluxes satisfy [67,12] Θ Iα = 0 . (7.28) We defer explicitly solving the constraints Θ Iα = 0 to Sections 7.5 and 7.6, where we give two complementary approaches to analyzing the lattice of constrained fluxes. For now, we focus on computing the constraints imposed on the vertical fluxes Θ IJ by the geometry of the (SU(3) × SU(2) × U(1))/Z 6 model defined over an arbitrary base, when these symmetry conditions are imposed. These constraints on the vertical fluxes will reveal whether or not the full set of chiral matter combinations compatible with 4D anomaly cancellation can be realized in a generic F-theory compactification. In [39], it was observed that the homology relations associated with the nullspace of the matrix M (IJ)(KL) correspond to linear constraints on the vertical fluxes Θ IJ , due to the symmetry of M . Thus, an arbitrary null vector ν = ν IJ S IJ in the span of the vectors described in Eq. (7.23) implies the existence of a null relation among fluxes,  In Section 7.4, after equating the F-theory fluxes Θ IJ to 3D Chern-Simons couplings, we translate the above linear relations into relations among chiral indices and compare the resulting relations to the 4D anomaly cancellation constraints to see whether or not F-theory geometry is more restrictive then the physical anomaly cancellation conditions.

Integrality of the second Chern class
The shifted quantization condition G 4 − c 2 /2 ∈ H 4 (X 5 , Z) implies that when c 2 /2 is not an integer class, G 4 must also be a half-integer class. This in turn implies that φ = PD(G 4 ) cannot take integer values. Note, however, that φ can be fractionally quantized in some cases even when c 2 /2 is an integer class, as discussed further in Section 9, although in such situations integer values of φ are still possible.
To clarify some of the issues involved here, let us define 19 two sublattices of H 4 (X 5 , Z): The quotient in the first line of the above equation is by the nullspace of M , i.e., by the equivalence relation In general,Λ vert is an overlattice of Λ vert . While Λ vert is easily computed from the set of integral divisors, as far as we know there is not yet any systematic way of determiningΛ vert for an arbitrary CY fourfold. When c 2 is even, but c 2 /2 is not contained in Λ vert , then integer values of φ are still allowed. When c 2 lies in Λ vert but is not even in H 4 (X 5 , Z), however, then φ must have some half-integer components. Half-integer elements of φ can modify certain important aspects of the theory, such as the precise quantization of the chiral indices or even the enforcement of the symmetry constraints (7.28), although as discussed in more detail below we do not encounter the latter issue for the constructions here. For this reason, it is important to determine for what choices of characteristic data c 2 /2 fails to be an integer class.
One way to check if c 2 /2 is not an integer class is to compute χ/24 where χ is the Euler characteristic of the smooth CY fourfold X 5 . If χ/24 fails to be an integer, this indicates that c 2 /2 is not an integer class. In fact, from a physical standpoint, in such situations it is important that c 2 /2 is not an integer class in order to maintain the integrality of the M2-brane tadpole, When χ/24 fails to be an integer, the fractional part of χ/24 is precisely canceled by contributions to the integral X 5 G 4 ∧ G 4 coming from the fractional part of G 4 (i.e., c 2 /2), see, e.g., [68].
We can use the pushforward technology described in the previous subsection to evaluate 19 The expression H vert 2,2 (X5, Z) is defined in [39] as the first of these objects, but may in other places in the literature be defined as the latter; we define these distinct symbols to avoid confusion and ambiguity. the pushforward of the Euler characteristic to B: 20 This expression for the Euler characteristic may be useful in various aspects of further analysis of these constructions, for instance aspects involving the tadpole constraint (Eq. (7.34)). It is also useful to have an explicit expression for c 2 . For example, using the fact that the hyperplane class is we can represent the image of c 2 of in the lattice H vert 2,2 (X 5 , Z) as Notice that the above expression for c 2 is expanded in the reduced basis S Iα , S 03 , S 05 , S 34 of H vert 2,2 (X 5 , Z). From this expression we can immediately see that for all our constructions, c 2 /2 cannot be an element of Λ vert , since it contains the components (3S 03 + S 05 − S 34 )/2. As discussed above, however, this does not definitively indicate that c 2 /2 cannot be an integral element of H 4 (X 5 , Z), as it may live inΛ vert .
Another way to check if c 2 /2 is not an integer element of cohomology is to compute c 2 2 /4. If this is not integer-valued, then c 2 /2 cannot be an integer class, since the intersection product on integer cohomology takes integer values. The matrix M red , of intersection pairings of any two elements belonging to H vert 2,2 (X 5 , Z) is presented in Table 6 using the basis relevant for Eq. (7.37). Using the explicit expression given there for M red , we find that one-fourth of the square of the second Chern class is given by (7.38) 20 Note that the total Chern class c(X5) can be expressed as a formal power series in Dα, H, Ei-see Appendix E of [39]).
Whenever the characteristic data K, Σ 2 , Σ 3 , Y are such that the above expression is noninteger valued, c 2 cannot be an even class. In Section 8.1 we show explicitly that there are many cases where this occurs, so that integer flux parameters φ IJ are not possible in those cases.

(7.41)
In order to match the above expressions with our M-theory vacuum, we must supply as input the signs of the central charges of 3D BPS particles in the specific phase of the vector multiplet 21 The "gauge" basis indicesĪ take the valuesĪ =0,1, α, 3, 4, 5, whereD0,D1 are defined in Eq. (7.4). See also Appendix A. 22 It is not entirely clear that every crepant resolution corresponds to a 3D phase in which the system Eq. (7.40) is invertible-see Appendix G of [39]. 23 Our conventions are as follows. Let g be a semisimple Lie algebra. We may view a collection of real Coulomb branch moduli ϕ i as the components of a real 3D vector multiplet scalar expanded in a basis of simple coroots, ϕ = ϕ i α ∨ i where i = 1, . . . , rank(g). Similarly, a weight w transforming in a representation r of g may be expanded in a basis of fundamental weights, w = wiω i . The fundamental weights are canonically dual to the simple coroots and we denote their inner product by α ∨ i · ω j = δ j i , hence ϕ · w = ϕ i wi. We define ϕ0 = 1/rKK. ������   00  01  11  0α  1α  αβ  03  13  α3  33  04 14  α4  34  44  05  15  α5  35  45  55 00 Table 4. Matrix of pushforwards of quadruple intersection numbers M (IJ)(KL) = S IJ · S KL (recall that S IJ :=D I ·D J ) associated to the resolution X 5 of the (SU(3) × SU(2) × U(1))/Z 6 model. The entries of the uppermost (leftmost) row (column) are the indices IJ of the corresponding classes S IJ . We work in the basisD I=0,1,α,is whereD 0 ,D 1 are the rational sections,D α is the pullback of a divisor in B, andD i2=3 ,D i3=4,5 are (resp.) the su(2), su(3) Cartan divisors. We omit the · notation indicating the intersection product for brevity and use the shorthand Table 5. Matrix elements M red(IJ)(KL) for the reduced intersection pairing M red of the lattice H vert 2,2 (X 5 , Z). We compute the matrix elements M red(IJ)(KL) by quotienting by the equivalence relation ∼, where φ ∼ φ iff M (IJ)(KL) (φ − φ ) KL = 0, assuming additional null vectors have been removed to get reduced bases for S αβ , S Iα , etc.. Note that the indices used in this reduced basis are IJ = Iα, 33, 34, 55. We use the same notation as in Table 4, namely we omit the · notation indicating the intersection product for brevity and use the shorthand Table 6. Matrix elements M red(IJ)(KL) in alternative basis. Compared to the basis used in Table 5, we find this basis to be more convenient in some approaches for computing the chiral indices-see Section 7.5. moduli space described by the resolution X 5 . These signs can in principle be computed geometrically by using the Kähler classĴ = ϕĪDĪ to compute the volumes vol(C w ) =Ĵ ·C w = ϕĪ wĪ of all primitive holomorphic and anti-holomorphic curves C w in the Mori cone of X 5 (and its negative), as the 3D BPS particle spectrum descends from M2 branes wrapping these curves. 24 Since the volumes vol(C w ) are equal to the BPS central charges, partitioning the Mori cone generators {C w } into holomorphic and anti-holomorphic (based on whether vol(C w ) is positive or negative) determines the full set of signs associated to X 5 . Rather than following the above approach, we use a shortcut. Namely, we identify the signs appearing in Eq. (7.41) by formally matching intersection numbers of the form WĪJK with 5D CS couplings kĪJK in which the coefficients have been appropriately "promoted" to the classes of matter curves C R ∈ B; see Section 5 of [39] for a more detailed discussion of this procedure.
To simplify the task of computing the 5D CS couplings, we first focus on the KK charges. To this end, we use the fact that Y is the pushforward of the intersection of the zero section with the generating section to conclude that the only elementary BPS particles with nontrivial KK charge are those transforming in the representations (1, 1) , as these are the only local matter representations whose associated matter curves in B have the schematic form (7.42) We then proceed to compute 5D Chern-Simons couplings according to the standard matching procedure detailed in Section 5.2 of [39]. Following this approach, we find that the resolution Eq. (3.10) that leads to X 5 corresponds to the signs in Table 7 and KK masses in the parametric regime with all other primitive BPS particles having vanishing KK charge. This data can in turn be used to evaluate the one-loop exact expressions for the 3D Chern-Simons couplings presented in Eq. (7.41).
In the following subsection, we describe the results of inverting the linear system Eq. (7.40).

Chiral indices
Our goal in this section is to match the geometric fluxes with the 3D Chern-Simons (CS) terms in order to determine the chiral indices of the (SU(3) × SU(2) × U(1))/Z 6 model. For ease of comparison, we first convert the fluxes to the gauge basisĪ =0,1, α, i s using Eq. (A.5) (recall that the indices0,1 correspond, respectively, to the abelian gauge factors U(1) KK , U(1)). In the gauge basis, we find that the linear relations Eq. (7.30) can be expressed as:  We adopt the convention that the Coulomb branch modulus ϕ1 dual to the u(1) factor is nonpositive in the Coulomb branch, in contrast to the Coulomb branch moduli ϕ i2=3 , ϕ i3=4,5 associated to nonabelian Cartan u(1)s, which are non-negative.
By identifying the 3D CS terms with the geometric fluxes in Section 7.4 (see Eq. (7.24)), we learn that the following constraints are satisfied by the chiral multiplicities in the (SU(3) × SU(2) × U(1))/Z 6 model: After rearrangement, these constraints exactly match those in Eq. (2.4). The constraints can also be used to write the chiral indices in terms of a minimal subset of the fluxes: In terms of the parameterization in Table 3, we see that the multiplicities of the first and second families are given by −Θ 55 and Θ 33 respectively, and the number of Standard Model generations is given by Θ 34 + 2(Θ 33 + Θ 55 ). When there is no exotic matter, the number of generations of MSSM matter is then simply Θ 34 . Note that in the chiral multiplicity formulae of Eq. (7.47), we can freely replace the fluxes Θ 33 , Θ 55 with −Θ 03 , −Θ 05 , using Eq. (7.44) and the fact that Θ0 i = Θ 0i in the symmetryconstrained space where Θ αI = 0. This allows us to easily move back and forth between the bases of Table 5 and Table 6.

Chiral spectrum
We now use the formulation of M red in Table 6 to solve the constraint equations Θ αI = 0 and compute formulae for the chiral multiplicities, where Θ ∼ M red φ is expressed in terms of flux parameters φ IJ . This can be done in a systematic fashion even though, as mentioned above, in some bases null vectors may remain for Table 6. However, unlike F-theory models without U(1) factors, it is not straightforward to obtain a completely general solution by solving the constraint equations for all flux parameters φ αI . As discussed in [39], the main obstruction to obtaining completely general solutions in models with U(1) factors is imposing the constraints Θ 1α = 0 . (7.48) Despite these apparent obstructions, in the case of the (SU(3) × SU(2) × U(1))/Z 6 model we nevertheless find that we are able to circumvent the difficulties associated with solving the above constraints in full generality. The approach described in this subsection thus leads to a general solution to the constraints Θ Iα = 0, along with general expressions for the associated fluxes and chiral multiplicities, but at the cost of leaving the solution for the variable φ 1α implicit. In Section 7.6, we specialize to specific bases and follow an approach that more closely parallels the approach used in [39] for purely nonabelian theories.
To illustrate our systematic solution, we spell out the first constraints explicitly. We begin by solving the constraint Θ αβ = 0. Note that the space of homologically nontrivial S αβ has the same dimension h 1,1 (B) as the space of, e.g., S 0α by Poincaré duality. From Table 6, we see that this set of constraints simply imply that We next solve the equations Θ 0α = 0 for the flux parameters φ αβ , giving Next, consider the equation for Θ α4 : Since φ α5 D α is always combined with a factor of Σ 3 , we can use this equation to replace φ α5 everywhere it appears, and although not all the parameters φ α5 are fixed through this equation, those that are not are null vectors. Note that the constraint equations (7.49) to (7.51) are all consistent even when c 2 is not an even class, so that the associated symmetries can be preserved even in such cases. For example, from Eq. (7.37) we see that φ 1α and φ 0α each are half-integer flux parameters precisely when (Σ 3 + Y ) α is odd and c 2 is not even, so Eq. (7.49) is consistent even in these cases. Similar arguments hold for the other symmetry constraints, using in particular the fact that c 2 (B) + K 2 is always even [68]. Continuing in this fashion, we can solve the Θ α3 = Θ α5 = 0 constraints, giving Note that this implies some extra nontrivial integral constraints on some of the flux parameters.
Putting this all together, the remaining constraint Θ 1α = 0 states that is the (scaled) height pairing divisor. Before imposing this set of constraints, the remaining fluxes are given by ) .

(7.56)
Note that this matches with the expectation that the exotic matter multiplicities associated with Θ 03 , Θ 05 explicitly can only be nonzero when Σ 2 · Y, Σ 3 · Y are non-vanishing as expected from Table 1, while the possibility of matter charged under both the SU(3) and SU(2) factors controlled by Θ 34 can only be non-vanishing if Σ 2 · Σ 3 is a nontrivial curve in the base. The constraints imposed by Eq. (7.53) reduce the number of independent fluxes φ 1α , φ 03 , φ 05 , φ 34 to three independent parameters that give the fluxes Eq. (7.56) and some null vectors associated with parameters in A. One way to think about these constraints is as a geometric condition on curves. The curves Σ 2 · Σ 3 , Σ 2 · Y , Σ 3 · Y span a space of dimension at most three in the linear space of curves on the base. We can think ofH as a linear map from the space of divisors on B to the dual space of curves on B. The set of possible flux configurations can be determined from the intersection of the image of this mapH with the space of curves spanned by the primary pair intersections listed above. These constraints can be thought of as constraints on the parameters in A or on the φ 03 , φ 05 , φ 34 . For example, if all three of the primary pair intersections are independent and the image ofH contains this full space (in which case, the curves are all contained in the divisorH), then we can take as the basic set of three parameters any set of all or some of the parameters φ 03 , φ 05 , φ 34 , and a complementary set of parameters in A.
In some sense, Eq. (7.56) along with the constraint Eq. (7.53) give the simplest general formulation of the available fluxes for an arbitrary base geometry. We can proceed somewhat more explicitly, however, if we assume that the primary pair intersection products appearing in the fluxes are non-vanishing. In particular, if Σ 2 · Σ 3 is a non-vanishing curve in the base, then we can simply solve Eq. (7.53) for φ 34 Σ 2 · Σ 3 and plug into Eq. (7.56). This gives a general formula for the chiral multiplicities (7.57) Note that there appear to be more than three independent flux parameters here since there are multiple parameters in A; these linearly independent extra parameters, however, correspond to a combination of null vectors of the full matrix M red and directions that are constrained by Eq. (7.53) as discussed above. For generic characteristic data, there are no further linear dependencies on these multiplicities, and we can realize all three linearly independent sets of chiral matter for many bases with arbitrary characteristic data. Note, however, that since the flux parameters on the LHS of Eq. (7.52) must be (half-)integer, and there must be an integer (or half-integer) parameter φ 05 satisfying Eq. (7.53), there are nontrivial constraints on the choices of flux parameters appearing in Eq. (7.57). We illustrate these features explicitly with some examples in the following section. The general formula for the multiplicities Eq. (7.57) holds whenever Σ 3 · Σ 2 is a nontrivial curve, even if one of the other primary pair intersection products vanishes such as Σ 2 · Y = 0, in which case Θ 03 = 0. In the cases where Σ 3 · Σ 2 = 0, we cannot have jointly charged SU(3) × SU(2) matter, and the theory simplifies significantly, essentially to the combinations of anomaly-free matter expected for gauge groups (SU(2) × U(1))/Z 2 (as described in more detail in [39]) and (SU(3) × U(1))/Z 3 . The multiplicities in such a case can be similarly computed assuming, e.g., that Σ 2 · Y is non-vanishing, in which case we can solve Eq. (7.53) for φ 03 Σ 2 ·Y and compute the resulting multiplicities in a similar fashion to the above analysis.
divisor W1q with the vertical curves of B. Without a more general approach to the analysis, M + Cna can only be computed explicitly for a specific choice of base, and thus has to be worked out on a case-by-case basis. We can plug in the values of W IJKL presented in Table 4 into the above expressions in order to explicitly evaluate the fluxes ΘÎĴ = M C(ÎĴ)(KL) φKL. We perform this analysis explicitly for the base P 3 in the next section.

Examples
We next turn our attention to some examples with constrained choices of characteristic data and/or a specific choice of base B.

B = P 3
We describe the specific case B = P 3 , K = −4H, Σ 2 = n 2 H, Σ 3 = n 3 H, Y = yH to illustrate features of the general theory, which we analyze in several complementary ways.

(8.3)
In order for c 2 /2 to be an integer class (and note that this is a necessary, but not sufficient condition), we at least require that the term in square brackets in the second line of the above expression is a multiple of 4. Even for the F 11 model, there are many examples where this condition cannot be satisfied and hence the flux must be half-integer quantized. In the case of the F 11 model over B = P 3 , we must set y = 0, so that the term in brackets reduces to n 3 (n 2 + n 3 )(3n 2 + 2n 3 ) . . While these expressions all have a factor of h in the denominator, the fact that all flux backgrounds φÎĴ must be (half-)integral guarantees that this factor will be cancelled properly for any set of allowed integer fluxes, such that corresponding fluxes ΘÎĴ are then integral.

Polynomial expressions and chiral multiplicities
In the base P 3 , the triple intersection products in the base simply become products of numerical integer factors. Thus, from the general form of M red , in a basis of fluxes (we use the index 2 here for the divisor H) the reduced matrix of Table 6 is simply −n 3 y n 2 n 3 y 1 n 2 0 n 3 −4 n 2 y n 3 y 0 −n 2 y 0 −n 2 y 0 0 n 2 y (4 − y − n 2 )n 2 y −n 2 n 3 y 0 −n 3 y 0 0 0 −n 3 y n 3 y −n 2 n 3 y (4 − y − n 3 )n 3 y 0 0 0 −n 2 n 3 −n 2 n 3 n 2 n 3 0 0 0 Note that with this ordering we see the characteristic structure of the upper left 5 × 5 block, which is resolution-independent and encodes the basic structure of the base geometry and the nonabelian gauge factors. In this case the analysis of Section 7.5 applies, and the space of curves on B is onedimensional (spanned by H 2 ), so all curves in Eq. (7.53) lie in the same one-dimensional space and we can solve the constraint for any of the parameters involved. For variety, we solve here explicitly for the φ 2I variables except φ 12 and use φ 05 instead to solve Eq. (7.53), giving Θ 03 = n 2 8 (n 2 + 32 − 4y)yφ 03 − (48 − 3n 2 − 4n 3 )φ 12 + n 3 (4y − n 2 )φ 34 , The requirement that all φ IJ take integer (or, appropriately, half-integer) values provides additional constraints that guarantee that these expressions related to chiral multiplicities are all integers. These constraints all follow from the condition 8n 3 yφ 05 = −9n 2 yφ 03 + hφ 12 + n 2 n 3 φ 34 , (8.14) which is Eq. (7.53). Generally, the resulting chiral multiplicities are all nonzero, so that all three independent families of chiral matter are allowable, though the multiplicities only take certain combinations of integer values when the flux parameters φ IJ are (half-)integral.
In order to make a direct comparison with the rational solutions described in Section 8.1.2, we need to use the constraint Eq. (8.14) to eliminate the parameter φ 12 in the expressions for Θ 03 , Θ 34 , Θ 05 given in Eq. (8.13). For generic choices of characteristic data, the scaled height pairing divisor h = 0, so in any such case we impose the constraint We also make use of the relations implied by the restriction of the null vectors Eq. (7.23) to the subspace Λ C defined by the condition Θ 1α = 0. These relations enable us to account for the redundancy in the set of parameters φ 03 , φ 33 , φ 34 , φ 05 , φ 45 , φ 55 used to parametrize the solutions of Section 8.1.2. Specifically, we find that before eliminating the null directions, we may make the identification The constrained null vectors also imply relations among the fluxes, in particular the last 3 relations of Eq. (7.30). This in turn implies that when comparing the solutions in Section 8.1.2 to Eq. (8.13), we should identify, e.g., the coefficient of (φ 03 −φ 33 ) in Θ 33 in Section 8. which is precisely (minus) the expression for Θ 33 whose coefficients are presented in Eq. (8.8).
As another class of examples, we can look at the chiral multiplicities that are possible without exotic matter representations. Setting Θ 03 = Θ 05 = 0 in Eq. (8.12), the expression for Θ 34 simplifies to For example with n 2 = n 3 = y = 1, the number of generations of Standard Model matter is given by −27φ 12 /2. The condition from Eq. (8.18) that Θ 05 = 0 implies that φ 12 is a multiple of 2 (whether φ 03 is an integer or of the form (2k + 1)/2), so the number of generations is a multiple of 27. Imposing also the condition Θ 03 = 0 gives the general solution  21) and the constraint Eq. (7.53) reduces to Imposing the constraint we can alternatively write where the parameters in A are constrained such that Eq. (8.22) can be satisfied for (half-)integer φ 34 ; in particular, parameters in A with a nonzero image under the map induced byH from divisors to curves in any direction other than Σ 2 · Σ 3 are fixed to vanish, while parameters in A associated with divisors such that D ·H = 0, D · Σ 2 · Σ 3 = 0 can contribute.

Specialized "SU(5)" F 11 models
Specializing further to the case Σ 2 = Σ 3 = −K, as studied in [34], which are associated with a hidden broken SU(5) unification group [2] as all the gauge branes lie on the same divisor, we haveH = −5K, Σ 2 · Σ 3 = K 2 , and with the constraint 5AK = −φ 34 K 2 , in agreement with the results of [34]. For example, if K = kK with k ∈ Z andK,K 2 a primitive divisor and primitive curve respectively, then we can parameterize A = aÃ whereÃ is the Poincaré dual divisor toK 2 , and we have Θ 34 = −ak 2 , with the quantization condition 5a = −φ 34 kK 3 . Taking the special case of the base P 3 (i.e., n 2 = n 3 = k = 4,K 3 = 1), we have, as in [34]: In this case, all parts of c 2 are manifestly even except for a shift by S 03 + S 05 + S 34 , so if c 2 is an even class the flux parameter φ 34 is integral, while if c 2 is an odd class, φ 34 is of the form (2k + 1)/2. The quantization constraint shows that 4|a, and 5|φ 34 when φ 34 is an integer, and 2|a, 5|2φ 34 when φ 34 is a half-integer. We discuss the question of whether c 2 is even in the next section. As another example, for the base B = P 2 × P 1 , we have K =K, K 3 = 18, and Θ 34 = −a = 18φ 34 /5, with similar quantization conditions.

Quantization of multiplicities
The question of precisely which chiral multiplicities are allowed for a given model is somewhat subtle. We give a few brief comments here that address some further aspects of which fluxes and corresponding chiral multiplicities may be possible in a broad class of flux compactifications. For convenience here, in parts of the discussion we take G Z = G 4 + c 2 (X 5 )/2 ∈ H 4 (X 5 , Z) to be an integer class, even when c 2 is not even; when c 2 is not even, fluxes must be suitably shifted by a half-integer class to G 4 . The analysis here expands in particular on related discussions in [34,39]. Note that aside from the specific examples at the end of this section, most of the discussion of quantization here is relevant for general classes of F-theory models, independent of the specific gauge group and matter structure.
Recall from Eq. (7.31) and Eq. (7.32) the definitions of the lattices Λ vert ,Λ vert . The homology lattice H 4 (X 5 , Z) has a unimodular intersection pairing. Thus, for any surface S that is a primitive 27 element of this lattice, in particular for any primitive surface inΛ vert , there exists some integer flux G Z ∈ H 4 (X 5 , Z) such that S G Z = 1. This would suggest that chiral multiplicities can always be quantized in arbitrary integer units, and that essentially any model would allow three generations of anomaly-free chiral matter (leaving aside the issue, for the moment, of non-even c 2 ). There are several reasons, however, why such a flux may not be possible or may not correspond to a physical vacuum of interest. First, it may be the case that a vertical surface S IJ = D I ∩ D J ∈ Λ vert is primitive in Λ vert but is an integer multiple of an element ofΛ vert and H 4 (X 5 , Z). In particular, a (vertical) matter surface S r can be primitive in Λ vert but not primitive inΛ vert ; in such a situation, since S r = mS , with S primitive inΛ vert , the associated chiral multiplicities χ r = Sr G Z will always be multiples of m, and unit multiplicity, for example, (or multiplicity 3 if m = 3) is not possible. Second, it is possible that the specific G Z that is dual to a surface S that is primitive inΛ vert and H 4 (X 5 , Z) breaks Poincaré or gauge invariance by giving a nonzero S Iα G Z . If neither of the conditions just summarized are the case, then, recalling the decomposition Eq. (7.25), the flux G Z that is dual to a primitive vertical matter surface S may have a fractional component in the vertical cohomology and a fractional part in the horizontal or remainder cohomology, and satisfy the desired properties. Completely determining which of these scenarios is realized depends on a full understanding of H 4 (X 5 , Z) and its intersection form. Note also that while in many models the matter surfaces lie in Λ vert , this need not be the case in general; in any specific model, a clear analysis of matter surfaces and/or the relationship between chiral multiplicities and integrated fluxes Θ IJ is needed for precise statements regarding allowed multiplicities. Finally, we must incorporate the possibility of non-even values of c 2 into the preceding discussion. Note that when c 2 is not even, as in many examples discussed in the preceding section, the chiral multiplicities associated with integrating over matter curves are still integral, S c 2 ∈ Z, in which case the allowed multiplicities are odd multiples of some basic factor.
Because of the orthogonal decomposition (7.25), and the fact that the intersection product on H 4 (X 5 , Z) is unimodular, the set of possible fluxes S G Z through vertical surfaces, and hence the set of chiral multiplicities, can be characterized by considering φ in the projection of H 4 (X 5 , Z) toΛ vert , which is the dual lattice dual(Λ vert ) (i.e., the set of points in H vert 2,2 (X 5 , C) with integer inner product under M red with all points inΛ vert ). WhenΛ vert = Λ vert , then this dual space is characterized by the set of necessary conditions stated in [34], which are that S (G 4 + c 2 (X 5 )/2) must be integer-valued for any allowed flux G 4 and vertical surface S ∈ Λ vert . 28 However,Λ vert could contain extra integral points not realized in Λ vert , in which case these necessary conditions would not be sufficient as there would be points in dual(Λ vert ) that are not in dual(Λ vert ) and hence do not correspond to the set of physically-allowed fluxes. Furthermore, fluxes associated with some points in dual(Λ vert ) may break gauge or Poincaré invariance. The information of Λ vert and M red is not generally sufficient to uniquely determinē Λ vert . For example, the lattice defined by the inner product diag (4,4) can alternatively be embedded in a self-dual Euclidean 2D lattice by extending to an overlattice that includes one half of each generator, or can represent the complete integral 2D subspace of the unimodular lattice E 8 along a pair of orthogonal primitive vectors of length 4.
The upshot of this discussion is that to determine the possible chiral multiplicities for a given model, one must consider the possible integer overlatticesΛ vert of Λ vert ; for each such overlattice the quantization condition is then determined by identifying all elements of the dual space dual(Λ vert ) that leave Poincaré and gauge invariance unbroken. Determining which is the proper overlattice requires a complete determination of H 4 (X 5 , Z), which is beyond the scope of this paper but will be addressed in future work. Note also that consistent chiral multiplicities associated with points φ v in dual(Λ vert ) that are not inΛ vert can be allowed only if non-vertical components of flux are turned on. In such cases, the integrality of the D3-brane tadpole from φ v can only be violated when the vertical component of flux is considered; thus, that condition (which was used in [34]) is not actually necessary for an allowed multiplicity, but rather amounts to the condition that the flux φ v lies in an allowed overlatticeΛ vert . Moreover, when flux is quantized properly with G Z ∈ H 4 (X 5 , Z), the D3-brane quantization condition is automatically satisfied. The uncertainty of which overlattice is realized byΛ vert also means that unless there is a clear signal such as non-integrality of Eq. (7.38) indicating that c 2 /2 is not even, there may be some overlatticesΛ vert that contain c 2 /2, in which case c 2 is even, and other overlattices that do not, in which case c 2 is not even-this further complicates the issue of precisely determining the flux quantization and associated chiral multiplicities.
As a specific example of the kinds of overlattice conditions that may arise, let us consider the quantization of chiral multiplicities in the simplest model considered above in Section 8.2.2, with base P 3 , and Σ 2 = Σ 3 = −K = 4H, Y = 0. In this case, M red from Eq. (8.12) simplifies to (dropping the null directions φ 03 , φ 05 , and moving φ 12 to the second slot) When such a flux is allowed, it gives a chiral multiplicity Θ 34 = p. The flux in Eq. (9.2) is in the dual lattice dual(Λ vert ) whenever p is integral. We now consider the issues of even/non-even c 2 and possible overlattices forΛ vert in this specific case. First, note that c 2 /2 = c + (0, 0, 0, 0, 0, 0, 1/2) , where c ∈ Λ vert . Thus, whether c 2 is even depends upon whether S 34 /2 ∈Λ vert . Note that Eq. (7.38) is integral in this case, so we cannot rule out the possibility that c 2 is even. Furthermore, there are several overlattices of Λ vert that contain c 2 /2. There is a maximal overlattice Λ max over that includes the generators (S 01 − S 12 )/2, S 23 /2, S 24 /2, S 25 /2, S 34 /8, and which is self-dual. IfΛ vert = Λ max over then c 2 is even, the minimum allowed p in Eq. (9.2) is p = 16, the quantization is controlled by integer values of a in Eq. (8.25) and the chiral multiplicity must be a multiple of 16. There is another overlattice of Λ vert that contains only the additional vector c 2 /2. The dual of this overlattice consists of all vectors of the form Eq. (9.2) with even p. If this overlattice isΛ vert , then the chiral multiplicity must be a multiple of 2, but non-vertical flux must be included for any multiplicity that is not a multiple of 32. Finally, ifΛ vert = Λ vert then c 2 is odd, since c 2 /2 is not inΛ vert . Because c 2 ∈ dual(Λ vert ) in this case, the flux associated with chiral multiplicity is quantized in units of 1, but non-vertical flux must be included for any chiral multiplicity that is not of the form χ = 32(2k + 1), and, e.g., a model with χ = 3 will appear to violate the D3-brane tadpole integrality condition unless the non-vertical flux is properly included). There are a variety of other possible overlattices, such as other sublattices of Λ max over that contain Λ vert , some of which contain c 2 /2 and others of which do not; we do not go through all the possibilities here. Thus, even in this simple case there are many possibilities forΛ vert , each of which gives distinct quantization conditions when restricted to vertical or more general integral fluxes. It would be interesting to gain further detailed information about H 4 (X 5 , Z) to determine the precise form ofΛ vert and definitively determine the precise quantization condition on chiral multiplicities for these models.

Conclusions
In this paper we have performed a detailed analysis of the geometry and flux-induced chiral matter of the universal tuned G SM = (SU(3) × SU(2) × U(1))/Z 6 Weierstrass model identified in [2]. This model admits three linearly independent families of chiral matter transforming under the Standard Model gauge group G SM . All three families can be produced by F-theory fluxes, so the observed Standard Model chiral matter structure is realized in a subset of models where one or two discrete parameters are tuned to vanish. Tuning the parameter Y = 0 gives the class of "F 11 " models identified and studied in [32][33][34] and related works.
The analysis of this paper provides a framework for more detailed study of further phenomenological features of these tuned Standard Model-like models, including aspects of Yukawa interactions, chiral and non-chiral spectra, and moduli stabilization. This class of models complements other types of F-theory constructions such as the tuned SU(5) GUT models [12][13][14][15] that have been studied for several decades, or the more recently constructed models that arise from flux breaking of a rigid E 6 or E 7 symmetry [27,28]. Comparing how these different classes of F-theory Standard Model constructions differ in detailed phenomenology and in frequency in the landscape may provide insights into what features of beyond the Standard Model physics are correlated with the most typical or natural realizations of the Standard Model in F-theory.
While in this paper we have focused on the specific class of models with tuned gauge group G SM , we have also developed here further the theoretical toolkit for analyzing the geometric structure and chiral matter content of broad classes of F-theory compactifications, extending the framework developed in [39]. In particular, we have extended the analysis of the middle intersection form on vertical cohomology to include theories with a U(1) factor, which complicates the structure of the matrix M red substantially, and we have given a more thorough analysis of quantization questions related to the multiplicity of chiral matter and allowed fluxes for large classes of F-theory models. One interesting aspect of this is that in the particular class of models studied here we were able to completely solve the flux constrained equations associated with Poincaré and gauge symmetries, although part of the solution was implicit. It would be interesting to investigate whether this kind of solution is possible for more general models with U(1) gauge factors. Regarding the quantization of chiral multiplicities, the analysis here highlights the importance of attaining a full understanding of the structure of H 4 (X 5 , Z) for F-theory models of interest. facility and the MIT International Science and Technology Initiatives (MISTI) MIT-Africa-Imperial College seed fund program for hospitality and support during some stages of this project.