F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches

We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in P^2, P^1x P^1 and the recently studied P^2(1,1,2), yield F-theory realizations of SUGRA theories with discrete gauge groups Z_3, Z_2 and Z_4. This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I_2-fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. We also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP_1 has a U(1)-gauge field induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q=3.


Introduction & Summary of Results
F-theory [1-3] is a non-perturbative formulation of Type IIB string theory with backreacted 7-branes, that is manifestly invariant under the SL(2, Z)-duality symmetry of the theory. String backgrounds constructed via F-theory are not only located in the heart of the web of string dualities, but also allow for the construction of phenomenologically appealing local GUT-models [4][5][6][7], which has recently rekindled a lot of interest into the subject. The basic idea of F-theory is to replace the axio-dilaton τ = C 0 +ie −φ , that is only defined up to SL(2, Z)-transformations, by a quantity, that only depends on the SL(2, Z)-equivalence class of τ . The canonical geometrical object with this property is a two-torus T 2 (τ ), whose complex structure is identified with τ . Thus, replacing τ by this auxiliary T 2 (τ ) provides an SL(2, Z)-invariant formulation of Type IIB. Non-trivial backgrounds of τ , which are sourced by 7-branes, on manifolds B are mapped under this replacement to torus-fibrations over B. In particular, for a supersymmetric and tadpole-canceling setup of 7-branes on a complex Kähler manifold B the total space of this T 2 (τ )-fibration is a Calabi-Yau manifold π : X → B.
Most of the torus-fibered Calabi-Yau manifolds X that have been studied are algebraic, that is they are realized as complete intersections in some ambient space. 1 In these constructions, the torus fiber over B is realized as an algebraic curve C of genus one. In addition, many examples considered in the literature are elliptically fibered, meaning that X has a section B → X, which was traditionally assumed to be holomorphic. These elliptically fibered Calabi-Yau manifolds have fruitful applications, e.g. for the construction of semi-realistic GUTs in global F-theory compactifications starting with [10,11] or the classification and study of 6D SCFTs [12][13][14].
Despite these successes, addressing open conceptual questions e.g. regarding the finiteness of the F-theory landscape 2 or which consistent 6D and 4D supergravity (SUGRA) theories can be realized in F-theory, 3 as well as the understanding of the geometric origins of discrete symmetries or analogous field theoretic mechanisms, crucial for the phenomenology of F-theory models, requires to broaden the class of Calabi-Yau manifolds X used for F-theory compactifications. In fact, using the well-developed map between the geometry of F-theory and SUGRA theories, see [20,21] for the complete map in 6D and [22] for results about certain topological terms in 4D 4 , one finds that the Calabi-Yau manifolds realizing many known consistent SUGRA theories, in particular those with U(1) symmetries [24] or discrete gauge groups, are still unknown [25]. 5 For the search of an F-theory realization of these theories it is crucial to construct new classes of Calabi-Yau manifolds X admitting new geometric features and to deduce the general SUGRA theories that arise in F-theory compactifications on these X. 6 There has been a lot of recent progress in systematically extending the set of Calabi-Yau 1 For recent advances on Calabi-Yau manifolds constructed as determinantal and Pfaffian varieties, see [8,9]. 2 See [15] and the recent [16] for a finiteness proof in related Type I compactifications. 3 F-theory compactifications to 8D are well-studied and classified, see e.g. the recent toric analysis of [17] and the classification of elliptic fibrations on K3-surfaces in [18]. Last subtleties in the understanding of the gauge group of a generic K3 have been understood in [19]. 4 SUGRA theories from string theory can also be constrained using tools from heterotic/F-theory duality [23]. 5 Of course it is a logical possibility that some of the SUGRA theories without a known F-theory realization are not consistent effective theories due to a violation of consistency constraints that are unknown at this point. 6 Compactifications of F-theory on Spin 7 manifolds, considered recently [26,27], have not yet produced SUGRA theories that cannot be obtained by a Calabi-Yau compactification. manifolds X that can be used for F-theory compactifications. The different approaches can be roughly sorted into two groups. The first group of approaches focuses on the classification and construction of all bases B that are admissible for F-theory [28][29][30]. The second group, to which this work belongs, focuses on generalizing the type of fiber C and the ways in which it can be fibered in a Calabi-Yau manifold X. There are three major extensions in this direction: • Elliptic fibers with an increasing number of rational points and their corresponding elliptically fibered Calabi-Yau manifolds X with a Mordell-Weil group (MW-group) of rational sections of increasing rank have been systematically constructed and studied [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. 7 The free part of the MW-group leads to U(1)-gauge fields in F-theory 8 [2] and the torsion part yields non-simply connected gauge groups [53].
• Elliptic fibrations X with a non-holomorphic zero section have been considered recently, starting with [37,39,43]. This permits the introduction of discrete degrees of freedom in the construction of the fibration of the elliptic curve over the base B yielding a finite number of strata in the moduli space of X.
• Algebraic curves C of one without any (rational) point have been used to construct genusone fibrations [54][55][56]. These are Calabi-Yau manifolds X which do not have a section, but only multi-sections. These models can be analyzed employing their associated Jacobian fibration J(X), which does exhibit a zero section, and its Weierstrass form.
The Calabi-Yau manifolds X we consider in this work invoke all three of these extensions. We study all F-theory compactifications on Calabi-Yau manifolds X F i with genus-one fiber C F i given as a hypersurface in the toric varieties associated to the 16 2D reflexive polyhedra, denoted by F i , i = 1, . . . , 16. 9 We refer to these Calabi-Yau manifolds as toric hypersurface fibrations. We determine the generic and intrinsic features of these X F i that are relevant to F-theory: the generic gauge group, the corresponding matter spectrum and the 4D Yukawa couplings corresponding to the codimension one, two and three singularities of X F i . These geometric results completely determine the 6D and non-chiral 4D SUGRA theories obtained by compactifying F-theory on Calabi-Yau threefolds and fourfolds without G 4 -flux. We prove completeness of our analysis of codimension one and two singularities by checking cancellation of all 6D anomalies. All these results are base-independent in the sense that they follow directly from the geometry of the fiber C F i . The only dependence on the base B enters through the choice of two divisors on B that label the possible Calabi-Yau fibrations of C F i [39].
We highlight the following interesting geometrical findings of our analysis of F-theory on the Calabi-Yau manifolds X F i : structure of X F i by means of Tate's algorithm [59][60][61] (see [62,63] for recent refinements) or upon addition of tops. The gauge groups G F i and G F * i associated to F i and its dual polyhedron F * i obey the rank relation rk(G F i ) + rk(G F * i ) = 6 . (1.1) • We consider three Calabi-Yau manifolds X F i , i = 1, 2, 4, without section. Their fibers are the general cubic in P 2 , the general biquadric in P 1 × P 1 and the general quartic in P 2 (1, 1, 2), respectively, where the latter is also studied in [54][55][56]. The fibrations X F i , i = 1, 2, 4, only have a genus-one fibration with a three-, a two-and a two-section, respectively. As a direct consequence of this absence of sections, F-theory has discrete gauge group factors given by Z 3 , Z 2 and Z 4 , respectively. We show that these Calabi-Yau manifolds, most notably the fibration of the cubic, X F 1 , have I 2 -fibers at codimension two that support singlet matter with charge 1 under the respective discrete gauge groups. We explain how the charge of all matter fields under these discrete groups are computed from the intersections of the multi-sections with the relevant codimension two fibers.
• We show that both X F 2 and X F 3 give rise to one U(1)-factor, namely G F 2 = U(1) × Z 2 and G F 3 = U(1). To this end, we show that unlike X F 2 , the Weierstrass form of the Jacobian J(X F 2 ) has one rational section, whereas already X F 3 has two sections: a toric and a non-toric one. In both cases, we determine the coordinates of all sections explicitly.
• For the first time, we find F-theory compactifications with matter of U(1)-charge three. This matter is supported at a codimension two locus of X F 3 with an I 2 -fiber where both the zero section and the non-toric rational section are ill-behaved and each "wrap" one irreducible fiber component.
We note that the 16 toric hypersurface fibrations X F i were considered in [41], were a thorough classification of their toric MW-groups was carried out. Further specializations of the X F i corresponding to toric tops [64,65] (see [66] for a systematic approach based on Tate's algorithm for elliptic fibers in Bl 1 P 2 (1, 1, 2)) permitted the engineering of toric F-theory models with certain gauge groups, in particular with an SU(5) GUT-group. Some 4D examples of chiral SU(5) GUTs were constructed in this manner [42,43]. Since we determine here the intrinsic gauge groups and the non-toric MW-groups, as well as the full matter spectrum and the Yukawa couplings of X F i , our approach is complementary to these previous works.
We have to remark that none of the fibrations X F i yield an SU(5) gauge factor in their lowenergy effective theories. Hence, strictly speaking, the intrinsic gauge symmetries associated to the toric hypersurface fibrations do not suffice to engineer SU(5) F-theory GUTs. There are, however, some arguments that challenge the simplest GUT picture in F-theory, and therefore, draw our attention towards alternative schemes which may be promising for particle physics models. In this spirit, we would like to briefly highlight some of the effective theories we obtain, which can potentially be used to construct promising particle physics models in F-theory. We find models with discrete symmetries and up to three U(1) factors. These additional symmetries can be used in order to forbid dangerous operators which would render the theory incompatible with observations, e.g. by mediating fast proton decay. In addition, we observe theories with interesting gauge groups and spectra. In fact, X F 11 precisely leads to an effective theory with the Standard Model gauge group and the usual representations 10 , and we further identify the trinification group for X F 16 as well as the Pati-Salam group for X F 13 . As we demonstrate explicitly, the matter spectra we obtain are very close to those one usually has in both of these theories.
In this paper we also work out the entire network of Higgsings relating the effective theories of F-theory on the toric hypersurface fibrations X F i . It is well-known that the toric varieties corresponding to the 16 2D reflexive polyhedra F i are related by blow-downs. Consequently, the Calabi-Yau manifolds X F i are related by the extremal transitions induced by these birational maps and a subsequent toric complex structure deformation. These transitions can be understood as Higgsings in the effective SUGRA theories arising from F-theory on the X F i : given two polyhedra F i and F i related by a blow-down as F i → F i , we explicitly determine the Higgsing that relates the effective theory of F-theory on X F i to that on X F i . The resulting diagram of all those Higgsings is given in Figure 1. Since this chain of Higgsings is only a sub-branch of the full Higgs branch of the effective SUGRA theories of F-theory on X F i , we refer to it as the toric Higgs branch. We check that both the charged and the neutral spectrum of the SUGRA theories in 6D match. This involves the computation of the number of neutral hyper multiplets, that in turn can be obtained from the Euler numbers of all X F i , which we also compute explicitly.
We point out some interesting observations about the Higgsing diagram in Figure 1: • All effective theories can be obtained by appropriately Higgsing the theories with maximal gauge groups and matter spectra obtained from F-theory on X F i , i = 13, 15, 16. We note that these are precisely the theories with non-trivial Mordell-Weil torsion.
• The network of Higgsings is symmetric around the horizontal line where the total rank of the gauge group is 3. Reflection along this line exchanges the polyhedron with its dual. This symmetry of Figure 1 reflects the rank condition (1.1). We emphasize that this symmetry maps theories with discrete gauge groups to theories with non-simply connected group, suggesting that multi-sections are somehow "dual" to MW-torsion.  Figure 1: The network of Higgsings between all F-theory compactifications on toric hypersurface fibrations X F i . The axes show the rank of the MW-group and the total rank of the gauge group of X F i . Each Calabi-Yau X F i is abbreviated by F i and its corresponding gauge group is shown. The arrows indicate the existence of a Higgsing between two Calabi-Yau manifolds.
two and three singularities are analyzed, and the number of their complex structure moduli is computed. The F-theory gauge group, matter spectrum and Yukawa couplings are extracted from these results. Section 4 is devoted to the study of the toric Higgs branch of F-theory compactified on the X F i . One particular Higgsing is discussed in detail in order to illustrate the relevant techniques. Here we also present the Higgsings leading to the effective theories with discrete gauge groups. We further elaborate on the details of the entire Higgsing chain in Appendices D and E. Our conclusions can be found in Section 5. This work contains additional Appendices on 6D anomalies (Appendix A), additional geometrical data of the X F i (Appendix B) and the explicit Euler numbers of all Calabi-Yau threefolds X F i (Appendix C).

Geometry & Physics of F-theory Backgrounds
In this section, we summarize the key geometrical properties of Calabi-Yau manifolds X that are genus-one fibrations over a base B which are relevant for the study of F-theory compactifications, see Sections 2.1 and 2.2. The structure of the 6D effective SUGRA theories obtained by compactifying F-theory on these manifolds is discussed in Section 2.3. Since we study in this work Calabi-Yau manifolds X with their genus-one fibers realized as toric hypersurfaces, we introduce the necessary elements of toric geometry in Section 2.4. Readers familiar with the tools and definitions presented here can safely skip this section and continue directly with Section 3.

Genus-one, Jacobian and elliptic fibrations with Mordell-Weil groups
We consider a smooth Calabi-Yau manifold π : X → B with general fiber given by an algebraic curve C of genus-one. C is a non-singular curve defined over a field K that is not necessarily algebraically closed. In particular, we can think of the fibration X as an algebraic curve C defined over the field K of meromorphic functions on the base B, which is clearly in general not algebraically closed. Thus, there are two qualitatively different situations to consider.

Curves with points
First, if the curve C has points with coordinates in K, then it is called an elliptic curve, which we denote by E, and X is called an elliptic fibration. The points on E form an Abelian group under addition: one point can be chosen as the zero point, denoted by P 0 , and the additional points P m , m = 1, . . . , r, (more precisely the differences P m − P 0 ) are the generators of the Mordell-Weil group of rational points of E. The Mordell-Weil theorem states that this group is finitely generated [71,72]. Thus, it splits into a free part isomorphic to Z r and a torsion subgroup, where the latter has been fully classified for K = Q by Mazur [73,74], see [71] for a review. 11 Every point on E gives rise to a section of the fibration X, i.e. rational maps from the base B into X. The section associated to P 0 is the zero section, denoted byŝ 0 : B → X, and the r rational points P m induce the rational sectionsŝ m : B → X. The set {ŝ m } can be seen to form a group, the MW-group of rational sections of X, by defining the addition of rational sections by addition of their corresponding points on E. The free part of the MW-group gives rise to Abelian gauge symmetry in F-theory [2] and its torsion part yields non-simply connected gauge groups [53], see also [46] for a recent discussion of MW-torsion. Every elliptic fibration X can be written in Weierstrass form (WSF) [75], i.e. as a hypersurface in the weighted projective bundle P (1,2,3) (O B ⊕ L 2 ⊕ L 3 ) over B of the form y 2 = x 3 + f xz 4 + gz 6 . (2.1) Here, O B is the trivial bundle on B and the line-bundle L is fixed by the Calabi-Yau condition of X as L = K −1 B , with K B denoting the canonical bundle of the base B. Then, the coefficients 11 For the field of meromorphic functions on B, there are more torsion subgroups possible than for K = Q [53].
f and g have to be sections of K −4 B and K −6 B , respectively. The map from the canonical presentation of X inherited from the canonical presentation of the elliptic curve E to the Weierstrass form (2.1) is birational. The zero sectionŝ 0 of X maps to the holomorphic zero section [z : x : y] = [0 : λ 2 : λ 3 ] in (2.1) and the rational sectionsŝ m map to rational sections in (2.1) with certain coordinates [z Pm : x Pm : y Pm ], that are rational expressions in K (we can clear denominators to obtain holomorphic coordinates).

Curves without points
Second, if the genus-one curve C does not have a point, the fibration π : X → B is without section. Such a fibration is referred to as a genus-one fibration [54]. Given a genus-one curve C, one can construct an associated elliptic curve E = J(C), that is the Jacobian of the curve C, i.e. the group of degree zero line bundles on C. The zero point on J(C) is the trivial line bundle. Thus, there exists an elliptic fibration π : J(X) → B associated to X with general fiber given by J(C). This implies that J(X) can be represented as a Weierstrass model (2.1). Furthermore, it is a key property for F-theory that the τ -function and the discriminant of X and J(X) are identical [54].
In this work, we consider concrete genus-one curves C with K-rational divisors of degree n > 1, respectively. 12 The corresponding fibration X does not have a section, but an n-section, that we denote byŝ (n) : B → X. Locally, at a point p on B, the function field K reduces to C and the n-sectionŝ (n) maps p to n points in the fiber C. Globally, however, upon moving around branch loci in B the individual points are exchanged by a monodromy action, so that only the collection of all n points together induce a well-defined divisor in X.
As we will see explicitly for concrete genus-one fibrations X, the map from X to the Weierstrass form (2.1) of J(X) can be obtained by an algebraic field extension L of K. This field extension is only necessary as an intermediate step, i.e. the final WSF (2.1) of J(X) is again defined over K. In Sections 3.2.1, 3.2.2 and 3.2.3 we explicitly work out the maps from X to the WSF of their Jacobian fibrations J(X), namely for the fibration of the cubic in P 2 , that has a three-section, and the two fibrations of the biquadric curve in P 1 × P 1 and the quartic curve in P 2 (1, 1, 2), that both have a two-section. 13 We note that genus-one fibrations X by the quartic in P 2 (1, 1, 2) have been considered recently in an F-theory context in [54][55][56].

Divisors on genus-one fibrations and their intersections
In F-theory we are particularly interested in Calabi-Yau manifolds X that arise as a crepant resolution of singular genus-one or elliptic fibrations. These resolved manifolds exhibit three different classes of divisors, that we discuss in the following.
The first set of divisors on X is formed by the vertical divisors, i.e. divisors that arise as pullbacks of divisors on B under the projection map π : X → B. Hence, there are h (1,1) (B) such divisors on X. We denote the preimage under π of a vertical divisor D on X by D b so that D = π * (D b ). Thus, D is a fibration of the curve C (or its degenerations) over the base D b .
The second class of divisors are the exceptional divisors of X. In more detail, if the discriminant ∆ = −16(4f 3 + 27g 2 ) of the WSF (2.1) of X or of its Jacobian J(X) vanishes to order higher than 1 at one of its irreducible components S b G I := {∆ I = 0} , I = 1, . . . , N , (2.2) in B, then the total space of the WSF is singular. These codimension one singularities are classified in [59,60]. In the resolution X the fiber over each S b G I splits into a number of rational curves whose intersection pattern often agrees with the Dynkin diagram of a Lie group G I . 14 The shrinkable irreducible components of the fiber at S b G I represent the simple roots of G I and are denoted by c G I −α i for i = 1, . . . , rk(G I ). Thus, X has a set of exceptional divisors D G I i given as the fibration of c G I −α i over S b G I for every I, to which we refer to as Cartan divisors of G I . The D G I i intersect the curves c G I −α i as where C G I ij denotes the Cartan matrix of G I . The F-theory gauge group is then given by the product of all G I , as discussed in Section 2.3.
Finally, the third class of divisors is induced by the independent sections and n-sections of the fibration of X. We denote the divisor classes of the zero sectionŝ 0 and the generators of the MW-group of rational sectionsŝ m by S 0 and S m , respectively. The class of a multi-section s (n) is denoted by S (n) . Then, the intersections of these divisors with the fiber f ∼ = C, E read The divisor classes that support Abelian gauge fields in F-theory [34,39,77], see also [22,31,78], are obtained from the Shioda map σ of the rational sectionsŝ m . To a given generator of the MW-groupŝ m the Shioda map assigns the divisor (2.5) Here π(·) denotes the projection of a codimension two variety in X to a divisor in the base B and [K B ] is the canonical bundle of B. The last term encodes contributions from non-Abelian gauge groups G I in F-theory with (C −1 G I ) i I j I denoting the inverse of the Cartan matrix C G I ij . The Shioda map (2.5) enables us to define the Néron-Tate height pairing of two rational sectionsŝ m ,ŝ n as ij is the coroot matrix of G I . We note that for evaluating this pairing in a concrete situation, the universal intersection relations prove useful (cf. [35,39,42,77] for details), whereas π(S m ·S n ) and π(S m ·S 0 ) are model-dependent. We note that in F-theory both the vertical divisor (2.2) and the matrix (2.6) of vertical divisors enter the coefficients of the Green-Schwarz terms [22,34,35] and are thus essential for anomaly cancellation, cf. Appendix A for details.

The spectrum of F-theory on genus-one fibrations
After the geometric preludes of sections 2.1 and 2.2, we are prepared to extract the spectrum of F-theory on a genus-one fibration X. The following discussion applies most directly to F-theory compactifications to 6D with effective theory given by an N = (1, 0) SUGRA theory. However certain statements directly generalize to 4D F-theory vacua without G 4 -flux.
For a more detailed derivation of some of the following results, that oftentimes require M-/F-theory duality, we refer [78][79][80] and references therein.

Codimension one singularities
All vector fields and certain hyper multiplets in F-theory arise from the singularities of the WSF of X that are induced by codimension one singularities of its fibration. Over a given irreducible discriminant component S b G I defined in (2.2), the fiber of X is reducible. We assume that there is a Lie group G I associated to this codimension one fiber of X. Then, the shrinkable holomorphic curves c G I −α in the fiber over S b G I represent all the positive roots of G I . By quantization of the moduli space of an M2-brane wrapping such a curve c G I −α one finds BPS-states transforming in one charged vector multiplet and 2g I charged half-hyper multiplets with charge-vector −α [81,82]. Another vector multiplet and 2g I half-hyper multiplets with charges +α are contributed by an M2-brane wrapping c G I −α with the opposite orientation. Here g I is the genus of the curve S b G I in B, that is computed as All these charged states become massless in the F-theory limit, when the volume of the class of the genus-one fiber of X is taken to zero. In this limit, these BPS-states fall into representations of the group G I as follows. First, we focus on the vector multiplets. All vector multiplets for every root α of G I are completed into one massless vector multiplet transforming in the adjoint representation adj(G I ) of G I . The additional vector multiplets are provided by the KK-reduction of the M-theory three-form C 3 along the harmonic (1, 1)-forms in X that are dual to the Cartan divisor D G I I of G I . Thus, every irreducible component (2.2) of the discriminant with respective codimension one fiber classified by a Lie group G I , I = 1, . . . , N , gives rise to a G I gauge symmetry in Ftheory [2,3,61]. Furthermore, if X has a MW-group of rank r, there are additional (1, 1)-forms on X, which are the duals of the divisors (2.5), that give rise to vector multiplets of Abelian gauge groups [3]. Thus, the total gauge group G X of F-theory on X is This discussion and the results of Section 2.2 imply further that the rank of G X can be directly computed in terms of the Hodge numbers h (1,1) (X) and h (1,1) (B) of X and B, respectively, as These results (2.9) and (2.10) hold in compactifications to eight 15 , six and four dimensions. Second, we turn to the massless half-hyper multiplets over S b G I . In fact, also these fields are completed into the adjoint representation adj(G I ) of G I . In order to see this, we first note that there are neutral hyper multiplets induced by the complex structure moduli of X. Their total number, denoted by H neut , is computed by the Hodge number h (2,1) (X) (or equivalently the Euler number χ(X)) of X as Then, for every group G I , the 2g I half-hyper multiplets with charges −α for all roots of G I combine with g I · rk(G I ) neutral hyper multiplets from (2.11) into g I hyper multiplets in the adjoint adj(G I ) of G I . Thus, the number of hyper multiplets transforming in adj(G I ) is given by (2.8) for every group G I . Let us emphasize that this discussion implies that h (2,1) (X) contains information about the gauge groups G I of X. Furthermore, also parts of the charged matter content from codimension two fibers are counted by h (2,1) (X), however of another theory related to the considered one by Higgsing. In Section 4, for F-theory compactified on all toric hypersurface fibrations X, we identify the part of h (2,1) (X) that comes from matter fields in all theories related by Higgsing.

Codimension two singularities
The rest of the charged spectrum of F-theory on X is encoded in the codimension two singularities of the WSF of X.
Non-Abelian charged matter is located at loci in S b G I , where the vanishing order of the discriminant ∆ of X enhances. These loci are typically complete intersections of S b G I with another divisor in B, that can be read off from ∆. The fiber of X at these codimension two loci contains additional shrinkable rational curves c that are not present in codimension one. These curves correspond to the weights of a representation R q , under the gauge group G X in (2.9), where q = (q 1 , . . . , q r ) denotes the vector of U(1)-charges. The Dynkin labels λ G I i of R are computed according to 12) and the m th U(1)-charge q m is computed using (2.5) as [34,77] We emphasize that these charges are automatically quantized, but not necessarily integers due to the usually fractional contribution form the last term in (2.13).
We note that in the presence of U(1)'s, we automatically have additional matter that does not originate from intersections of codimension one discriminant components. In fact, the WSF of X automatically has codimension two singularities for every rational sectionŝ m with coordinates [z Pm : x Pm : y Pm ] at the following locus in B: Pm + x 2 Pm = 0 , m = 1, . . . , r . (2.14) This can be seen by inserting [z Pm : x Pm : y Pm ] into (2.1), which implies a relation between f and g which allows for a factorization of (2.1) that reveals the presence of conifold singularities in the WSF of X precisely at (2.14), see e.g. [39,77] for details. In the crepant resolution X, there is a reducible I 2 -fiber with one isolated rational curve at the codimension two loci (2.14). The matter at the loci (2.14) are charged singlets 1 q with their U(1)-charges computed according to (2.13). This is clear as generically (2.14) does not intersect any discriminant component, which are the loci where the Cartan divisors D G I i are supported, so that (2.12) is trivial. In concrete applications, the complete intersection (2.14) describes a reducible variety in B supporting multiple singlets with different charges. Matter at a generic point of (2.14) has U(1)-charge one, whereas matter at non-generic points, i.e. points along which other, oftentimes simpler constraints vanish, too, has different U(1)-charges. From a technical point of view, we are interested in all prime ideals, denoted throughout the paper by I (k) , of the loci (2.14) for every m. These are obtained by a primary decomposition of the complete intersection (2.14), cf. [42] for details. The codimension two variety in B associated to I (k) is denoted by V (I (k) ), which is the standard notation in algebraic geometry for an algebraic set, i.e. the set of points in B so that all constraints in I (k) vanish. Then, we explicitly analyze the I 2 -fibers of X over all these irreducible varieties V (I (k) ) in order to compute the respective U(1)-charges via (2.13).
In general, the multiplicity of matter in the representation R q is given by the homology class of the corresponding codimension two locus in B. If the base B is two-dimensional, which is the case in compactifications to 6D, this is just a set of points and the multiplicity is the number of these points. In F-theory compactifications to 4D, the homology class of a codimension two locus is the class of the corresponding matter curve.
More specifically, the multiplicity of non-Abelian charged matter is computed easily as the homology class of the complete intersection with S b G I . However, the determination of the multiplicity of singlets 1 q is more involved since they are located on the varieties V (I (k) ) associated to the usually very complex prime ideals I (k) of the complete intersection (2.14). The respective matter multiplicities are then again given as the homology class of the variety V (I (k) ). It can be computed by the following procedure, see [39,42,44] for more details: we first compute the homology class of the reducible complete intersection (2.14). Given the list of its associated prime ideals I (k) , we then subtract the multiplicities (homology classes) of those matter loci V (I (k ) ), {k } ⊂ {k}, we already know. Here we have to take into account the order n k of the matter locus V (I (k ) ) inside the complete intersection (2.14). The order n k is computed using the resultant technique [39]. In all the cases considered below in Section 3, this strategy yields the homology classes of all singlets 1 q .
In summary, the 6D N = (1, 0) SUGRA theory obtained by compactifying F-theory on a Calabi-Yau threefold X has • a total number of vector multiplets V reading V = adj(G X ) = I dim(adj(G I )) + r , (2.15) where adj(G X ) and adj(G I ) denote the adjoint representations of G I and G X , respectively, and r denotes the MW-rank of X, • a total number of hyper multiplets H given by where we split into contributions H codim=2 , H codim=1 and H mod from codimension two fibers, from codimension one fibers over higher genus Riemann surfaces in B and from complex structure moduli of X (plus 1), respectively, • and a number of tensor multiplets T counted by For the second equality in (2.17) we have employed the identity Here we used in the last equality the Euler number χ(B) = (2+h (1,1) (B)) of a simply-connected base B with h (2,0) = 0 and the index formula for the arithmetic genus χ 0 (B) = 1, where c i (B), i = 1, 2, denote the Chern classes of B.

Codimension three singularities
For completeness we note that codimension three singularities of the WSF of a Calabi-Yau fourfold X support Yukawa points in F-theory compactifications to 4D. The codimension three singularities are at the points in the threefold base B of further enhancement of the vanishing order of the discriminant ∆. All such enhancement points are given as intersections of three matter curves in B, including self-intersections. Technically, given three matter curves V (I (1) ), V (I (2) ) and V (I (3) ) we have to check that the variety V (I (1) ) ∩ V I (2) ) ∩ V (I (3) contains a codimension three component in B. This is achieved by checking that the ideal I (1) ∪ I (2) ∪ I (3) is codimension three in the ring of appropriate polynomials on B, where we used the well-known equality k V (I (k) ) = V ( k I (k) ) for a family of algebraic sets V (I (k) ) [83].
As we see in Section 3, all gauge-invariant Yukawa couplings are realized for the case of toric hypersurface fibrations X F i . 13

Explicit examples: Calabi-Yau hypersurfaces in 2D toric varieties
All Calabi-Yau manifolds X considered in this work are constructed as fibrations of genus-one curves C that have a natural presentation as hypersurfaces in 2D toric varieties. These fibrations are automatically smooth, if the toric ambient spaces of the fiber C are fully resolved. In this section we present a very brief account on the construction of Calabi-Yau hypersurfaces in 2D toric varieties that are the basis for the rest of this work. For a more complete account, we refer to standard text books [84,85].
A toric almost Fano surface is associated to each of the 16 two-dimensional reflexive polyhedra F i , i = 1, . . . , 16, in a lattice N = Z 2 . 16 These 16 reflexive polyhedra are given in a convenient presentation in Figure 2 [17]. As indicated there, the polyhedra F i and F 17−i for i = 1, . . . , 6 are dual to each other, F * i = F 17−i , and the F i for i = 7, . . . , 10 are self-dual, 20) where ·, · is the pairing between N and M . For a given polyhedron F i , we denote the associated toric variety by P F i . Toric varieties are generalizations of weighted projective spaces [86]: to each integral point v k , k = 1, . . . , m + 2, except the origin of F i , we associate a coordinate x k in C. Next, we introduce the lattice of relations between the v k with generators (a) defined by Then, a smooth toric variety P F i is defined as the (C * ) m -quotient where the points x := (x 1 , . . . , x m+2 ) are not allowed to lie in the Stanley-Reisner ideal SR. The construction (2.22) provides a dictionary between the combinatorics of the polyhedron F i and the geometry of P F i . For example, the toric divisor group on P F i is generated by the divisors D k = {x k = 0} and the intersections of the D k are encoded in the SR ideal. A full basis of the divisor group on P F i can be obtained using the linear equivalences between the D k . Due to the relevance for the smoothness of a toric hypersurface fibration X, we stress here that points that are not vertices in F i correspond to exceptional divisors resolving orbifold singularities in P F i .
The polyhedra F 1 , F 3 , F 5 , F 7 describe the generic del Pezzo surfaces P 2 and dP i , i = 1, 2, 3, respectively, F 2 yields P 1 × P 1 , F 4 describes P 2 (1, 1, 2) and F 10 yields P 2 (1, 2, 3). In fact all other toric varieties P F i can be viewed as higher del Pezzo surfaces at a special point in their respective complex structure moduli spaces.  [17]. The polyhedron F i and F 17−i are dual for i = 1 . . . 6 and self-dual for i = 7 . . . 10.
Every toric variety P F i has an associated Calabi-Yau hypersurface, i.e. a genus-one curve C F i . It is defined as the generic section of its anti-canonical bundle K −1 P F i . The Calabi-Yau hypersurface in P F i is obtained by the Batyrev construction as the following polynomial [87] where q denotes all integral points in F * i and the a q are coefficients in the field K. We note that points v i interior to edges in F i are usually excluded in the product (2.23) because the corresponding divisors do not intersect the hypersurface C F i . However, when considering Calabi-Yau fibrations X F i of C F i , as in Section 3, these divisors intersect X F i and resolve singularities of X F i induced by singularities of its fibration, i.e. these divisors are related to Cartan divisors D G I i discussed above in Section 2.2.

Analysis of F-theory on Toric Hypersurface Fibrations
In this section we analyze the geometry of the Calabi-Yau manifolds X F i , that are constructed as fibration of the genus-one curves C F i over a generic base B. For each manifold we calculate the effective theory resulting from compactifying F-theory on it. We calculate the gauge group, the charged and neutral matter spectrum and the Yukawa couplings.
We start with a quick summary of some interesting results of this study. There are three polyhedra leading to manifolds X F i without a section: F 1 , F 2 and F 4 , see sections 3.2.1, 3.2.2 and 3.2.3, respectively. They yield the discrete gauge groups in F-theory. For three polyhedra we find associated gauge groups with Mordell-Weil torsion, giving rise to non-simply connected gauge groups: F 13 , F 15 and F 16 , see sections 3.6.1, 3.6.2 and 3.6.3, respectively. The analysis of the hypersurface X F 3 and the corresponding effective theory of F-theory, whose spectrum contains a charged singlet with U(1)-charge three, can be found in section 3.3.1.
We obtain the following list of gauge groups G F i of F-theory on the X F i : From this and as a simple consequence of (2.10), we see that there is the following rule of thumb for computing the rank of a gauge group G F i : given a polyhedron F i with 3 + n integral points without the origin, we have a gauge group with total rank n. Let us outline the structure of this section. In the first subsection 3.1 we briefly discuss the three different representations of genus-one curves C F i realized as toric hypersurfaces: the cubic, the biquadric and the quartic. There, we define the line bundles of the base B in which the coefficients in these constraints have to take values in order to obtain a genus-one fibered Calabi-Yau manifold. The functions f and g of the Weierstrass form (2.1) for the cubic, the biquadric and the quartic can be found in Appendix B. By appropriate specializations of the coefficients, one can obtain f , g and ∆ = 4f 3 + 27g 2 for all toric hypersurface fibration X F i .
In sections 3.2 to 3.6 we proceed to describe in detail each Calabi-Yau manifold X F i . In each case we first discuss the genus-one curve C F i realized as a toric hypersurface in P F i . We then construct the corresponding toric hypersurface fibration X F i and analyze its codimension one and two singularities from which we extract the non-Abelian gauge group and matter spectrum. If X F i has a non-trivial MW-group, we determine all its generators, their Shioda maps and the height pairing. For genus-one fibrations, we determine their discrete gauge groups. For completeness, we also determine the Yukawa couplings from codimension three singularities. In each case we show as a consistency check that the necessary 6D anomalies (pure Abelian, gravitational-Abelian, pure non-Abelian, non-Abelian gravitational, non-Abelian-Abelian and purely gravitational) are canceled implying consistency of the considered effective theories.
We organize the Calabi-Yau manifolds X F i into five categories: those with discrete gauge symmetries (Section 3.2), those with a gauge group of rank one and two but without discrete gauge groups (Section 3.3), those with a gauge group of rank three, whose fiber polyhedra happen to be also self-dual (Section 3.4), those with gauge groups of rank four and five without MW-torsion (Section 3.5) and those X F i with MW-torsion (Section 3.6). This arrangement is almost in perfect agreement with the labeling of the polyhedra F i in Figure 2 which facilitates the navigation through this section. We name the subsection containing the analysis of the specific manifold X F i by its corresponding fiber polyhedron F i .

Constructing toric hypersurface fibrations
In this section we explain the general construction of the Calabi-Yau manifolds X F i with toric hypersurface fiber C F i and base B. The following discussion applies to Calabi-Yau n-folds X F i with a general (n − 1)-dimensional base B. The cases of most relevance for F-theory and for this work are n = 3, 4. We refer to [39,42] for more details on the following discussion.
The starting point of the construction of the genus-one fibered Calabi-Yau manifold X F i is the hypersurface equation (2.23) of the curve C F i . In order to obtain the equation of X F i , the coefficients a q and the variables x i of (2.23) have to be promoted to sections of appropriate line bundles of the base B. We determine these line bundles, by first constructing a fibration of the 2D toric variety P F i , which is the ambient space of C F i , over the same base B, Here P B F i (D,D) denotes the total space of this fibration. The structure of its fibration is parametrized by two divisors in B, denoted by D andD. This can be seen by noting that all m + 2 coordinates x k on the fiber P F i are in general non-trivial sections of line bundles on B. Then, we can use the (C * ) m -action of the toric variety P F i to set m variables to transform in the trivial bundle of B. The divisors dual to the two remaining line bundles are precisely D,D.
Next we impose equation (2.23) in P F i (D,D). Consistency fixes the line bundles in which the coefficients a q have to take values in terms of the two divisors D andD. Then, we require (2.23) to be a section of the anti-canonical bundle K −1 P B F i , which is the Calabi-Yau condition. In addition, equation (2.23) imposed in P B F i (D,D) clearly describes a genus-one fibration over B, since for every generic point on B, the hypersurface (2.23) describes exactly the curve C F i in P F i . The total Calabi-Yau space resulting from the fibration of the toric hypersurface C F i is denoted by X F i in the following. It enjoys the fibration structure In principle, this procedure has to be carried out for all Calabi-Yau manifolds X F i associated to the 16 2D toric polyhedra F i . However, we observe that all the hypersurface constraints of the X F i , except for X F 2 and X F 4 , can be obtained from the hypersurface constraint for X F 1 , after setting appropriate coefficients to zero. This is possible because if F 1 is a sub-polyhedron of F i , then the corresponding toric variety P F i is the blow-up of P F 1 = P 2 at a given number of points, with the additional rays in F i corresponding to the blow-up divisors. However, adding rays to the polyhedron F 1 removes rays from its dual polyhedron F * 1 = F 16 . By means of (2.23), this removes coefficients from hypersurface equation for X F 1 , i.e. the hypersurface for X F i is a certain specialization of the hypersurface of X F 1 with some a q ≡ 0. We will be more explicit about this in the following subsection (Section 3.1.2).
Thus, we only have to explicitly carry out the construction of the toric hypersurfaces separately for the two Calabi-Yau manifolds X F 2 and X F 4 . The details of this are given in Sections 3.1.3 and 3.1.4.

Fibration by cubic curves: X F 1 and its specializations
We proceed to construct the Calabi-Yau manifold C F 1 → X F 1 → B with fiber given by the curve C F 1 in the toric variety P F 1 . In addition, we argue how the Calabi-Yau manifolds X F i , whose fiber polyhedron F i contains F 1 , can be obtained from X F 1 . The polyhedron F 1 and its dual are shown in Figure 3. The toric variety P F 1 , constructed using (2.22), is the well-known projective space P 2 . We introduce the projective coordinates [u : v : w] on P 2 . In terms of these coordinates, we can read off the SR-ideal from Figure 3 as The divisor group of P 2 is generated by the hyperplane class H. The Calabi-Yau onefold in P 2 is the degree three C F 1 in 3H. Its defining equation, constructed using (2.23) and Figure 3, is the most general cubic where the coefficients s i take values in the field K. Next, we follow the discussion of Section 3.1.1 to construct the toric hypersurface fibration X F 1 . We first construct the ambient space (3.1), which in the case at hand is a P 2 -fibration over the base B, (3.5) The two divisors parametrizing this fibration are S 7 and S 9 , cf. [39,42]. Upon imposing the constraint (3.4) and requiring the Calabi-Yau condition for X F 1 , we see that these two divisors are precisely the classes of the coefficients s 7 and s 9 , respectively. Indeed, as mentioned above, we can use the C * -action on P 2 to turn e.g. w into a section of the trivial line bundle of the base. Then, we choose the variables u and v as sections of the bundles This allows us to compute the anti-canonical bundle of the P 2 -fibration (3.5) using adjunction as Finally, we impose the Calabi-Yau condition on the constraint (3.4) for X F 1 which fixes the divisor classes of the coefficients s i . We summarize the divisor classes of the homogeneous coordinates [u : v : w] and the coefficients s i in the following tables: X F i as specialized cubics As mentioned in the previous subsection, the equations of the Calabi-Yau manifolds X F i with i = 2, 4 can be expressed as specialized versions of the cubic hypersurface equation (3.4) of X F 1 . In order to find the hypersurface equation for an X F i we begin by calculating the anticanonical class of the fibration P B F i (D,D) defined in (3.1). To this end, we first note that toric ambient spaces P F i are obtained from P F 1 by a certain number of blow-ups at points P j . Assuming the number of blow-ups is k, we have Each blow-up adds a P 1 with an associated new variable e j and divisor class E j . From the combinatorial point of view, this means that there is an additional C * -action on P F i . Next we note that the fibration P B F i (D,D) can be parametrized by the same base divisors S 7 and S 9 as the fibration (3.5), i.e. we identify D = S 7 andD = S 9 . Indeed, this is possible since we can use (C * )-actions, including the new (C * )-actions from the k blow-ups, to make the variables w and e j transform in the trivial bundle on B while maintaining the assignments 19 (3.6) for u and v. Employing these results we calculate the anti-canonical bundle of P B F i (S 7 , S 9 ), using the adjunction formula, yielding (3.10) Here,H denotes the pull-backH =π * (H) of the hyperplane class H on P 2 under the blow-down mapπ : P F i → P 2 . By abuse of notation, we will denote it throughout this work simply by H. It is to be observed that if the coefficient s i is present in the constraint of X F i , i.e. if it is not removed by the k blow-ups, its corresponding class [s i ] remains unaltered from the one given in Table ( 3.8). This relation of the hypersurface constraints of the X F i for i = 2, 4 and all the bundles entering it to the hypersurface equation (3.4) and bundles (3.8) of X F 1 will facilitate our following presentation. In particular, in the respective subsections on X F i for i = 2, 4 only the classes for the variables u, v, w and e j have to be given explicitly.

Fibration by the biquadric: X F 2
We construct the Calabi-Yau manifold C F 2 → X F 2 → B as the fibration of the curve C F 2 in the toric variety P F 2 over B. As mentioned before, its hypersurface equation cannot be described as a cubic. Thus, X F 2 has to be analyzed separately. The polyhedron F 2 and its dual are presented in Figure 4. The toric variety associated to it is P F 2 = P 1 × P 1 and we have introduced the projective coordinates [x : t] and [y : s] on the two P 1 's, respectively. The Stanley-Reisner ideal of P F 2 is given by There are two divisor classes on P F 2 , that we denote by H 1 and H 2 with respective representatives {x = 0} and {y = 0}. The Calabi-Yau onefold in P F 2 is the curve C F 2 in the class 2H 1 + 2H 2 . It is a biquadric of the form as can be shown using (2.23) and Figure 4. Here the b i denote coefficients in the field K.
In order to find X F 2 we proceed to construct P B F 2 (D,D), the fibration of P F 2 introduced in (3.1). It is possible to consistently parametrize this fibration in terms of the same divisor classes D = S 7 andD = S 9 as in (3.5). In the hypersurface constraint (3.12), they correspond to the classes of the coefficients b 7 and b 9 respectively. 17 This will facilitate the matching of the effective theories via Higgsings, as discussed in Section 4. Next, we use the (C * ) 2 -actions on P F 2 to achieve that the variables x and y transform in the trivial line bundle on B. The other two variables s and t take values in the following line bundles on B: With this assignment of line bundles to the coordinates on P F 2 , the anti-canonical class of P B F 2 (S 7 , S 9 ) is readily calculated as (3.14) Finally, we require that the hypersurface (3.12) is Calabi-Yau, which fixes the divisor classes of the coefficients b i in terms of S 7 , S 9 and [K −1 B ]. In summary, we obtain that the coordinates on P F 2 and the coefficients b i have the following divisor classes: section Divisor Class We emphasize that the classes of the coefficients b i , except for b 10 , agree with the classes of s i of the cubic X F 1 , c.f. (3.8), as expected.

Fibration by the quartic: X F 4
We proceed to construct the Calabi-Yau manifold C F 4 → X F 4 → B with general fiber given by the curve C F 4 in the toric variety P F 4 . As we have mentioned above, its hypersurface equation is not a special case of a cubic, which requires a separate analysis of X F 4 . The polyhedron F 4 and its dual polyhedron are shown in Figure 5. Its associated toric variety is P F 4 = P 2 (1, 1, 2) and we introduce the homogeneous coordinates [X : Y : Z : e 1 ]. The Stanley-Reisner ideal of this toric variety can be read off from Figure 5 as There are two divisor classes on P F 4 , that are denoted by H and E 1 with representatives {X = 0} and {e 1 = 0} respectively. We note that P F 4 automatically contains an exceptional divisor E 1 corresponding to the point interior to the edge of F 4 . The equation for the Calabi-Yau onefold C F 4 in P F 4 , which is a degree two curve, is the quartic equation in the class 4H − 2E 1 . It reads explicitly as we infer from (2.23) and Figure 5. The coefficients d i take values in the field K.
In order to find X F 4 we construct the fibration P B F 4 (S 7 , S 9 ), the fibration of P F 4 over B introduced in (3.1). Again, it is possible to parametrize the fibration by the identical divisors S 7 and S 9 as in the fibration (3.5) relevant for the X F 1 . 18 As before, we can use the two C *actions on P F 4 to make two of its homogeneous coordinates transform in the trivial bundles. However, it turns out that a convenient assignment of sections, avoiding fractions, is given by We use this to compute the anti-canonical bundle on P B F 4 (S 7 , S 9 ) as Imposing the Calabi-Yau condition on the constraint (3.17), we fix the classes of all coefficients d i . The assignments of divisor classes to the coordinates on P F 4 and the d i in summary read section Divisor Class Section Divisor Class We emphasize that the classes of the coefficients d i slightly differ from (3.8) in the cubic X F 1 due to the slightly different assignments (3.18) of classes to the coordinates.

Fibration with discrete gauge symmetry
In this section we analyze the toric hypersurface fibrations based on the fiber polyhedra F 1 , F 2 and F 4 . Since their fibrations do not have a section, but only multi-sections, they are genusone fibrations. We analyze the codimension one, two and three singularities of these models, employing also their respective associated Jacobian fibrations. We show that the effective theories of F-theory on these Calabi-Yau manifolds exhibit discrete gauge groups and include matter that is charged only under the respective discrete group.

Polyhedron
We consider the genus-one fibration X F 1 over an arbitrary base B with genus-one fiber C F 1 realized as the Calabi-Yau hypersurface in P F 1 = P 2 . The toric data of P F 1 = P 2 and the construction of the Calabi-Yau manifolds X F 1 have been discussed in Section 3.1.2. The hypersurface equation for X F 1 is given by (3.4) with the relevant divisor classes of the coordinates [u : v : w] and the coefficients s i summarized in (3.8).
The fibration π : X F 1 → B does not have a section, but only a three-section. Thus, X F 1 is only a genus-one fibration, cf. the general discussion in Section 2.1. In order to obtain the WSF of X F 1 , given the absence of sections of its fibration, we have to calculate the associated Jacobian fibration J(X F 1 ). The algorithm for computing J(X F 1 ) is well known in the mathematics literature, see for example [76], from where we calculate f and g, given explicitly in (B.1) and (B.2), and subsequently the discriminant ∆. The discriminant does not factorize, which shows the absence of codimension one singularities of X F 1 and therefore, the absence of non-Abelian gauge groups in the corresponding F-theory compactification.
The fibration X F 1 has a three-section that is given bŷ as follows from the Calabi-Yau constraint (3.4). We denote its divisor class, that agrees with H + S 9 + [K B ], by S (3) . Under the degree nine map from X 1 to its Jacobian this three-section is mapped to the canonical zero section z = 0 in the WSF of J(X F 1 ). However, in X F 1 , the three-sectionŝ (3) locally maps a point on the base B to three points on the fiber C F 1 . Globally, there exists a monodromy group that interchanges these three points, upon moving on the base B. This fact, together with the existence of I 2 -fibers in X F 1 at codimension two on which the mondromy group acts non-trivially, as we present next, and the results from Higgsing the U(1) gauge group in the effective theory associated to X F 3 , see Section 4.2, leads us to postulate the following discrete gauge group of X F 1 : In order to compute the charges of matter under this discrete group, we have to associate a divisor class to the three-section. As certain models with multi-section are related to models with multiple rational sections by conifold transitions, see [55,56], a natural proposal for such a divisor class is an expression similar to the Shioda map (2.5). We recall the three defining properties of a Shioda map summarized on page 21 in [34]. Imposing these conditions on the divisor class associated to (3.21), we obtain the following divisor class, We propose that matter charges under the discrete group Z 3 should be computed using this class. In fact, we demonstrate next, that the class (3.23) allows us to compute Z 3 -charges of matter-representations on X F 1 , that are consistent with 6D anomaly cancellation and the Higgsing from the model X F 3 , discussed in Section 4.2.
Charged and uncharged matter in X F 1 We proceed with determining the codimension two singularities of the WSF of J(X F 1 ). This analysis is most easily carried out directly in the smooth fibration X F 1 . The same techniques presented here will also be used in a slightly modified form for the analysis of the fibration X F 2 , X F 3 and X F 4 . We note that the same technique has been used recently in [54] and [56].
Finding the loci of I 2 -fibers using elimination ideals We are looking for loci of B that support I 2 -fibers in X F 1 . At these loci, the genus-one fiber C F 1 of X F 1 has to degenerate into two P 1 's, i.e. the hypersurface equation (3.4) has to factor into two smooth polynomials. For a smooth cubic the only factorization with this property is the one into a conic and a line, i.e. a factorization of (3.4) of the form where α j and β k are seven unknown polynomials on B. We note that we can assume s 1 = 0, because otherwise we would obtain a locus of codimension three or higher. Making a comparison of coefficients on both sides of (3.24), we obtain a set of constraints that defines an ideal in the ring K[s i , α j , β k ], where s i are the coefficients in (3.4). We denote this ideal by I (s i ,α,β) .

Multiplicity
Fiber Locus We emphasize that there are two more constraints, namely nine, in I (s i ,α,β,) than unknowns α j , β k , i.e. the system is over-determined. Thus, there only exists a solution for α j , β k satisfying (3.24), if two additional constraints on the s i are obeyed. This implies that the ideal I (s i ,α,β,) describes a codimension two locus of the s i . In order to obtain the constraints that the s i have to obey for the factorization (3.24) to exist, we compute the elimination ideal I (s i ) = I (s i ,α,β) ∩K[s i ], 19 where K[s i ] is the polynomial ring only in the variables s i . We compute I (s i ) , in the following abbreviated as I (1) ≡ I (s i ) , explicitly using Singular [88] and obtain an ideal with 50 generators. Furthermore, we calculate its codimension in the ring K[s i ] to be two. Thus, its vanishing locus V (I (1) ) describes a codimension two variety in B. In summary, we have shown that the factorization (3.24) corresponding to an I 2 -fiber in X F 1 happens at the codimension two locus V (I (1) ) in B.
We note that (3.24) is the only type of factorization that can occur. Thus, we do not expect any further codimension two fibers and corresponding matter representation in X F 1 . The spectrum of X F 1 is summarized in Table 1. Next we argue how to compute the charge of the matter located at V (I (1) ) under the discrete gauge group G F 1 = Z 3 . Due to the absence of a zero section on X F 1 , there is no preferred curve in the I 2 -fiber in Table 1. As can be observed from (3.24), the two P 1 's in this I 2 -fiber have intersection numbers one and two with the three-sectionŝ (3) 0 . By naively applying (2.13) using the divisor class (3.23), we compute the charges q = 1 and q = 2 for the two rational curves, respectively. Thus, it seems that there is no meaningful way to assign a discrete charge to the matter located at V (I (1) ). However, this seeming contradiction is resolved by noting that a 6D hyper multiplet of charge q = 1 is the same as one with charge q = −1. In addition, employing the discrete Z 3 symmetry, we have −1 = 2 mod 3, showing that a 6D hyper multiplet of charge q = 1 under a Z 3 symmetry is physically equivalent to one with charge q = 2. Thus, the matter at V (I (1) ) has charge q = 1 which is the same as q = 2 under the discrete gauge group G F 1 = Z 3 .
We proceed to calculate the multiplicity of V (I (1) ). Unfortunately given the size and number of polynomials in the ideal I (1) , we are unable to obtain its multiplicity geometrically with the available computing power. Instead, we invoke the results for its multiplicity that is obtained in Section 4.2 using the Higgs transition X F 3 → X F 1 . It is shown in Table 1 for completeness.
We complete the discussion of the matter spectrum of X F 1 by calculating the number of neutral hyper multiplets. We use (2.11) and the explicit expression for the Euler number χ(X F 1 ) of X F 1 in (C.1) to obtain Employing this together with the number of vector multiplets V = 0 and the charged spectrum in Table 1 we check cancellation of the 6D gravitational anomaly in (A.1).

Yukawa couplings in X F 1
We conclude this section by noting that there is only one gauge-invariant Yukawa coupling possible: Yukawa Locus Again, we cannot check for its presence explicitly due to the complexity of the ideal I (1) . 20

Polyhedron
Here, we analyze the genus-one fibration X F 2 constructed as a fibration of the Calabi-Yau onefold C F 2 in P F 2 = P 1 × P 1 . The toric data of P 1 × P 1 and the construction of the toric hypersurface fibration X F 2 have been presented in Section 3.1.3. The hypersurface constraint for X F 2 is given in (3.12) and the relevant divisor classes are summarized in (3.15). First, we note that the fibration π : X F 2 → B does not have a section, i.e. it is a genusone fibration. We obtain its WSF by computing its associated Jacobian fibration J(X F 2 ), employing again the straightforward algorithms from the mathematics literature [76]. The results for the functions f and g can be found in (B.3) and (B.4), from which the discriminant can be readily computed. The discriminant does not factorize, which again shows the absence of codimension one singularities. Thus, there is no non-Abelian gauge symmetry for this F-theory compactifcation.
The fibration of X F 2 has two independent two-sections, that are given bŷ where we used the hypersurface constraint (3.12) and the SR-ideal (3.11). We denote the two corresponding divisor classes, that agree with H 1 and H 2 , by S (2) 0 and S 1 , respectively. Analogous to the previous Section 3.2.1, we expect a discrete Z 2 gauge group associated to the two-section S (2) 0 , cf. the similar discussion in [55]. We will provide independent evidence for this by the analysis in Section 4.2 of the Higgsing of the effective theory of F-theory on X F 5 , that has a U(1) 2 gauge group, to the one arising from X F 2 . 20 The presence of this coupling can be deduced considering the Higgsing from X F3 to X F1 (see Section 4.2). Decomposing the states in X F3 in terms of states in X F1 , we observe that after Higgsing, the Yukawa coupling 1 1 · 1 1 1 2 in X F3 (see Table 7) gives rise to 1 1 · 1 1 · 1 1 in X F3 .

26
The role of the other two-sectionŝ (2) 1 , however, is less clear in the biquadric representation. Its meaning for F-theory is unraveled by transforming the biquadric (3.12) defining X F 2 into a cubic hypersurface and then by computing its Weierstrass form, which is precisely the WSF of the Jacobian fibration of X F 2 , as we show. This detour via the cubic yields a direct map to the Jacobian fibration J(X F 2 ), which allows us to follow the two-sectionŝ (2) 1 in (3.27).

Map to the cubic in
The curve C F 2 given as the biquadric (3.12) in P F 2 can be treated as the cubic in P F 5 after an appropriate change of variables. Indeed, by applying the transformation x → x + αt or y → y + βs, we can set the coefficient of the monomial s 2 t 2 (3.12) to zero for an appropriate α or β. We note that both α and β have to involve square roots of the coefficients b i in (3.12), i.e. the two variable transformations are only defined in a field extension. As we will see, this field extension will only be an intermediate step, since all square roots will drop out in the final result of our computation. After the change of variables, we obtain a polynomial of the following form: p = (s 1 y 2 +s 2 sy +s 3 s 2 )x 2 + (s 5 y 2 +s 6 sy +s 7 s 2 )xt + (s 8 y 2 +s 9 sy)t 2 , (3.28) where the redefined coefficientss i depend on the variables b i and are explicitly given in (B.5). We note thatp is precisely of the form of the cubic (3.73) in P F 5 after identifying Since the curvep = 0 is an elliptic curve, we can compute its WSF, in particular the functions f and g. Inserting the explicit expressions (B.5) for the sectionss i in (3.28) in terms of the b i in (3.12) into the expressions for f and g, we precisely recover (B.3) and (B.4) obtained from the WSF of the Jacobian fibration J(X F 2 ). Most notably, all square roots in the coefficientss i have dropped out, as claimed. Next, we note that the two-sectionŝ (2) 1 in (3.27) formally maps to the section in (3.28). Under the identification of coordinates (3.29), this is precisely the sectionŝ 1 of X F 5 given in (3.74). Inserting the explicit expressions (B.5) for thes i into the WS-coordinates (B.9) ofŝ 1 , we obtain We emphasize that all square roots in the coefficients b i in (B.5) have dropped out and we obtain completely rational WS-coordinates for the two-sectionŝ (2) In summary, we have shown for the first time that the associated Jacobian fibration J(X F 2 ) exhibits a rank one MW-group of rational sections generated by the section in (3.31), which is precisely the image of the two-sectionŝ (2) 1 in X F 2 under the map X F 2 → J(X F 2 ). This means that there is an associated Abelian gauge field in the F-theory compactified on X F 2 . We note that application of the same logic to the two-sectionŝ (2) 0 , which formally maps to the section s 2 defined in (3.74) in X F 5 , does not lead to a rational section of the Jacobian J(X F 2 ) since its WS-coordinates (B.10) after inserting (B.5) still contain square roots. Hence,ŝ (2) 0 does not yield an additional U(1)-factor, but corresponds to a discrete group Z 2 , as claimed.
Having proven the presence of a MW-group on J(X F 2 ), we compute the Shioda map of its generator. We note that the usual expression (2.5) has to be modified sinceŝ (2) 1 is a two-section. It can be shown that the following expression obeys all conditions listed in [34] that have to be obeyed by a Shioda map: Then we obtain the corresponding height pairing, using (2.6), as Here, we used the following intersections The first two equalities are just a translation of the SR-ideal (3.11) into intersection relations of divisor classes on X F 2 , employing (3.15) and (2.4) for n = 2. The third relation follows by noting that according to (3.27) the two two-sectionsŝ (2) 0 andŝ (2) 1 intersect precisely at b 10 = 0, whose class is ). We conclude by summarizing the full gauge group of the theory: We highlight again that the U(1) corresponds to a rational section in the Jacobian fibration J(X F 2 ), that is the image of the two-sectionŝ (2) 1 under the degree four map X F 2 → J(X F 2 ). As mentioned before, the discrete gauge Z 2 symmetry is induced by the two-sectionŝ (2) 0 . For the computation of charges of matter w.r.t. the Z 2 , we have to associate a divisor class to it. Imposing conditions on this divisor class similar to the one that have lead to the Shioda map (2.5) [34], we obtain We use this divisor class to successfully compute the Z 2 -charges of matter in the following.
Charged and uncharged matter in X F 2 Now that we know the gauge group of the theory, we proceed to derive first the matter representation and then the corresponding 6D matter multiplicities. As in Section 3.2.1, we use the elimination ideal technique to show directly the presence of three matter representations in X F 2 , namely 1 (1,+) , 1 (1,−) and 1 (0,−) , where ± denote the two possible Z 2 -eigenvalues. Then, we compute their multiplicities, where we also invoke the equivalent presentation of X F 2 as a quartic.
The first type of factorization of (3.12) corresponding to (2, 2) = (1, 1) + (1, 1) is given by Clearly, both factors are bilinear in [x : t] and [y : s], respectively, as required. As before, we can factor out b 1 because it must not vanish at a codimension two locus. We note that there are six unknown polynomials α j and β k and eight non-trivial constraints, as can be seen by a comparison of coefficients on both sides. Thus, the ideal of constraints is over-determined and imposes a codimension two condition on the coefficients b i for a solution to (3.37) to exist. The elimination ideal, that we call I (1) , obtained by eliminating the unknowns α j and β k from the ideal of constraints is generated by 50 polynomials. It is checked to be codimension two in the ring, as expected, proving the existence of the factorization (3.37) at codimension two. We denote the zero set of I (1) by V (I (1) ), which is the geometric codimension two locus in B.
Next, we note that each curve of the I 2 -fiber described by (3.37) has intersection one with both two-sectionsŝ (2) 0 andŝ (2) 1 . The U(1)-charge computed using (2.13) and (3.32) is zero. We also note that the representation has charge (−) under the discrete symmetry because the two intersection points ofŝ (2) 0 with the fiber are interchanged under a monodromy action. Formally, the charge under Z 2 is computed using (2.13) together with the divisor class (3.36), showing that both curves in the I 2 -fiber have Z 2 -charge (−). Thus, the representation at the locus V (I (1) ) is 1 (0,−) as shown in Table 2.
The second type of factorization of (3.12) into two polynomials of degrees (1, 0) and (1, 2), respectively, takes the following explicit form where α 1 and the β k are six unknown polynomials. We compute again the elimination ideal, denote I (2) , that is generated by eight polynomials in the b i and check that it is codimension two in the ring. The corresponding codimension two locus in B supporting this type of I 2 -fiber is denoted by V (I (2) ). The intersection pattern of the two-sections with the I 2 -fiber is shown in the second entry of Table 2. The U(1)-and Z 2 -charges readily follow as discussed before and we find the representation at this locus to be 1 (1,−) . Finally, the last type of factorization corresponds to a split of (3.12) into two polynomials of degrees (0, 1) and (2, 1). It can be written down explicitly and takes a similar form as (3.38). The codimension two elimination ideal corresponding to this factorization, denoted by I (3) , is generated by eight polynomials and its vanishing set is denoted by V (I (3) ). The intersection pattern of the two-sections with this type of I 2 -fiber is shown in the last entry of Table 2.

Multiplicity
Fiber Locus Using the charge formula (2.13) and the Shioda map (3.32), as well as (3.36) we show that the representation at V (I (3) ) is 1 (1,+) . As a confirmation of the completeness of our analysis of codimension two singularities of X F 2 supporting U(1)-charged matter, we recall that the codimension two locus supporting all I 2 -singularities associated to a U(1) is given by (2.14). In the case at hand we have to evaluate this constraint for the rational sections of J(X 2 ) with coordinates [x 1 : y 1 : z 1 ] given in (3.31). We calculate all associated prime ideals of the obtained complete intersection using Singular [88] and indeed find precisely the two prime ideals I (2) and I (3) corresponding to the two representations 1 (1,−) and 1 (1,+) found previously using the elimination ideal technique.
As a next step, we calculate the homology classes in B for the three codimension two loci supporting the I 2 -fibers, which determine, according to Section 2.3, the multiplicities of 6D charged hyper multiplets in the corresponding representations. We begin with the variety V (I (3) ), whose multiplicity we denote by x 1 (1,+) , supporting the representation 1 (1,+) . Its homology class is computed by taking two constraints of the ideal I (3) and computing the homology class of the complete intersection described by them. Then, we subtract (with their corresponding orders) those components that are inside this complete intersection but do not satisfy the other generators of the ideal I (3) . We obtain: (3.39) The multiplicity of V (I (2) ), denoted by x 1 (1,−) , can be calculated in a similar way. It is given in the third row of Table 2. As a consistency check, we calculate the sum of both multiplicities and it agrees with [y 1 ] · [f z 4 1 ] as it should, because the I (2) and I (3) are the two associated prime ideals of (2.14) for the section (3.31).
For the computation of the multiplicity of the variety V (I (1) ), denoted by x 1 (0,−) , we cannot carry out the previously mentioned algorithm, due to the size and complexity of the ideal I (1) . Instead, x 1 (0,−) is obtained by first calculating the multiplicity of all hyper multiplets charged under the discrete symmetry, namely the 1 (0,−) and 1 (1,−) , and then subtracting the number of hyper multiplets in the representation 1 (1,−) , that we already know.
We begin by noting that the total number of charge (−) hyper multiplets under the Z 2 symmetry was calculated geometrically in [55,56] for a genus-one fibration given by the quartic curve in P 2 (1, 1, 2). Indeed, we can directly use their results since we can transform the biquadric (3.12) to a quartic presentation. This quartic is to be obtained by taking the discriminant of the biquadric (3.12) with respect to y. To this end, we rewrite the biquadric in the suggestive form and then take the discriminant of this quadric in y (we also set s = 1). We construct a genusone curve as the double cover over this discriminant, which is then a quartic in [x : t] and a new variable w of weight two of the form Here we used the conventions of [55] in the last equality. The coefficients e i can be expressed in terms of the b i in (3.12) by a comparison of coefficients. In this form the reason for choosing the quadric w.r.t. to y in order to construct the quartic (3.41) is evident, because the two-sectionŝ (3.41). Using the results in [55,56], we calculate the multiplicities of all charge (−) hyper multiplets, both 1 (0,−) , 1 (1,−) , using the one-to-one correspondence between the loci of I 2 -fibers in (3.41) with the following complete intersection, c.f. equation (2.22) in [55], Its homology class is readily given as [4e 1 ] · [e 3 ], which has to agree as mentioned before with the sum x 1 (1,−) + x 1 (0,−) . Thus, the multiplicity x 1 (0,−) follows by subtracting the multiplicity The result is given in the second row of Table 2.
To complete the matter spectrum we calculate the number of neutral hyper multiplets. Using (2.11) and the explicit formula for the Euler number of X F 2 in (C.1) of Appendix C, we obtain Using this together with the charged matter spectrum in Table 2, the number of vector multiplets V = 1 and the height pairing (3.33) we confirm that all anomalies, including the purely gravitational one, are canceled.

Yukawa couplings in X F 2
We conclude this section by stating the geometrically realized Yukawa couplings. We find the single Yukawa coupling in Table 3, by checking explicitly that the corresponding varieties intersect at codimension three, i.e. that the ideal I (1) ∪ I (2) ∪ I (3) is codimension three in the ring generated by the coefficients b i . Table 3: Codimension three locus and corresponding Yukawa coupling for X F 2 .

Polyhedron
In this section we study the genus one-fibration X F 4 that is constructed as the toric hypersurface fibration of C F 4 in P F 4 = P 2 (1, 1, 2). The toric data of P 2 (1, 1, 2) as well as the construction of the toric hypersurface fibration X F 4 have been discussed in Section 3.1.4. The hypersurface constraint of X F 4 is shown in (3.17) and the relevant divisor classes can be found in (3.20). This model has recently received a lot of attention [54][55][56]. Here we provide additional insights in the nature of the Z 4 discrete gauge group of F-theory on X F 4 as well as in the computation of the charges of matter under this discrete group. We also check 6D anomaly cancellation, which requires knowledge of all multiplicities of 6D charged and uncharged hyper multiplets. We begin by noting that the fibration π : X F 4 → B does not have a section, but only twoand four-sections, i.e. X F 4 is a genus-one fibration, see Section 2.1. The three multi-sections induced by the ambient space P 2 (1, 1, 2) of the fiber arê where we used the SR-ideal (3.16). We denote the one independent divisor classes ofŝ (3.20), where E 1 is the class of the exceptional divisor on P 2 (1, 1, 2).
Since this fibration does not have a section one has to utilize its associated Jacobian fibration J(X F 4 ) in order to find its WSF. We readily compute the functions f and g in (2.1) using the algorithm in [76]. From the discriminant of this WSF, we find one I 2 -singularity over the divisors S b SU(2) = {d 9 = 0} ∩ B in B. Along this divisor, the constraint (3.17) factorizes as where q 3 is the polynomial that remains after factoring out e 1 . The corresponding I 2 -fiber is depicted in Figure 6. Due to the absence of a zero section, there is no preferred rational curve in this I 2 -fiber. Thus, a possible choice for the Cartan-divisor D 1 of the SU(2) is given by In summary, the gauge group of X F 4 is given by The discrete symmetry stems from the multi-sections. In order to calculate the charges under the discrete symmetry we have to orthogonalize the SU(2) such that it is not charged under the discrete symmetry. This is done in a similar way as for U(1)'s via the Shioda map (2.5). For the two two-sectionsŝ m with m = 1, 2, we propose as the appropriate divisor class to compute charges under the discrete gauge group Z 4 . Here we used that both two-sections intersect each node in Figure 6 precisely once. For the foursectionŝ 3 we note that the node corresponding to the simple root of SU(2) in Figure 6 is not intersected. Thus, the appropriate class for computing Z 4 charges based onŝ It is straightforward to check that the divisors (3.48) and (3.49) obey all properties of a Shioda map [34]. Next, we analyze the codimension two singularities of the WSF of J(X F 4 ) to determine the charged matter spectrum. We find two codimension two singularities leading to the matter representations and the corresponding codimension two fibers in X F 4 that are given in the first and second entry of Table 4, respectively. We have also added the adjoint matter at the divisor S b SU(2) = S 9 for completeness. We have checked the representation content at the two codimension two loci explicitly by computation of the Dynkin labels using (2.12) with D 1 given in (3.46) and using the charge formula (2.13) for (3.48) or (3.49). We note that the charges calculated from the two two-sections are half the integral charges computed from the foursection. However, both charges are physically equivalent since in the case of the two-section we have to calculate modulo two, whereas for the four-section, we calculate modulo four. In other words, we obtain charges in two different conventions. Here we choose the charge convention where all discrete charges are integer, which agrees with the charges computed using σ Z 4 (ŝ 3 ).

Multiplicity
Fiber Locus V (I (1) ), given by (3.50) V (I (2) ), given by (3.51) The codimension two locus supporting the representation 2 1 is given as the following complete intersection, that can be read off directly from the discriminant of J(X F 4 ): This ideal is easily checked to be prime. The discrete charge of the SU(2) doublet can be computed from the intersection of σ Z 4 (ŝ 3 ) with the irreducible fiber components, which are depicted in Table 4. There one observes that the four-section intersects one node three times and another one just once. Recall again that 6D hyper multiplets with discrete charges 1 and 3 are physically equivalent since a hyper multiplet contains two half-hypers with charges 1 and −1 and charge −1 is identified with charge 3 under the discrete Z 4 -symmetry.
The complete intersection supporting the representation 1 2 is obtained by directly searching for the loci of degeneration of the fiber of X F 4 to I 2 . One makes a general factorization ansatz of (3.17) and determines the ideal of constraints imposed by this factorization [55,56]. Then one eliminates the unknown variables introduced in this ansatz by computation of the elimination ideal, see Sections 3.2.1 and 3.2.2 for an explanation of this technique. The obtained elimination ideal, denoted by I (2) , is prime and codimension two in the ring of coefficients d i . It reads where we have indicated the quotient by the ideal The multiplicity of 6D hyper multiplets in the representation 2 1 is computed as the product of the classes of the two constraints in the complete intersection I (1) . The multiplicity of matter in the 1 2 representation is more involved since the locus described by I (2) , V (I (2) ), is one of three irreducible components of the complete intersection in the first ideal in (3.51). Using the resultant technique [39] we decompose this complete intersection into V (I (2) This allows us to obtain the multiplicity shown in Table 4 that we double check following the arguments in [55] explained around (3.42).
We complete the matter spectrum of X F 4 by the number of neutral hyper multiplets, which is computed from (2.11) using the Euler number of X F 4 in (C.1). It reads Finally, we use S b SU(2) = {d 9 = 0}, the charged spectrum in Table 4 and (3.52) together with the number of vector multiplets V = 1 to check that all 6D anomalies (A.1) are canceled.
We conclude this section with the computation of the Yukawa couplings. We find the single Yukawa coupling given in Table 5. In order to check that it is realized at codimension three in B we compute the associated prime ideals of the ideal I (1) ∪ I (2) . Indeed, we find that it is codimension three in the ring, as required for the existence of the Yukawa coupling.

Yukawa
Locus

Fibration with gauge groups of rank 1, 2 and no discrete gauge symmetry
In this section we analyze all toric hypersurface fibrations X F i with gauge groups of rank one and two, but without discrete gauge symmetries. They are constructed using the fiber polyhedra F 3 , F 5 and F 6 . Apart from X F 3 , that possesses a non-toric section, all other X F i considered here can be analyzed using techniques already developed e.g. in [39,42,44].
We construct a Calabi-Yau manifold, denoted X F 3 , as a fibration of the toric hypersurface in P F 3 = dP 1 over a base B. The polyhedron of F 3 along with a choice of projective coordinates as well as its dual polyhedron are depicted in Figure 7. The coordinate e 1 vanishes on the We have set e 1 = 1 for brevity of our notation. The zero section is indicated by the dot.
exceptional divisor E 1 of dP 1 and [u : v : w] are the pullback under the blow-down map dP 1 → P 2 of the P 2 -coordinates. The SR-ideal of dP 1 reads Using (2.23) we construct the Calabi-Yau manifold X F 3 as the hypersurface p F 3 = s 1 u 3 e 2 1 +s 2 u 2 ve 2 1 +s 3 uv 2 e 2 1 +s 4 v 3 e 2 1 +s 5 u 2 we 1 +s 6 uvwe 1 +s 7 v 2 we 1 +s 8 uw 2 +s 9 vw 2 , (3.54) in the ambient space (3.1), that in the case at hand is a dP 1 -fibration over B. The coordinates [u : v : w : e 1 ] and the coefficients s i take values in the line bundles in (3.8).
The Calabi-Yau manifold X F 3 is an elliptic fibration. This is clear because for a generic point on B there is one marked point P 0 on its fiber, which is the intersection of e 1 = 0 with (3.54). This point gives rise to a section of X F 3 , which we choose as the zero section. Its generic coordinates readŝ There always exists a second section of X F 3 , which generates a rank one MW-group. 21 We emphasize that this second section is not toric, i.e. not given as the intersection of a toric divisor in the fiber P F 3 with the hypersurface (3.54), in contrast to the zero section (3.55). This can be seen as follows. Without loss of generality, set e 1 = 1 in (3.54) and consider it as an elliptic curve E over a field K. Then construct the tangent t P to the point P 0 which now is at [u : v : w] = [0 : 0 : 1]. It is determined by requiring that along t P both p F 3 and its first derivative vanish at P 0 , i.e. that P 0 is a point of intersection two of E and t P . It is described by (3.56) Since (3.54) is a curve of degree three, every line has to intersect it at three points. Thus, t P = 0 intersects E at a third point, denoted by P 1 , which is automatically rational. It gives rise to a rational section of X F 3 , with generic coordinateŝ Thus, the elliptic fibration X F 3 indeed has a rank one MW-group with a non-toric generator, as claimed. The Shioda map (2.5) of the sectionŝ 1 reads where S 1 , S 0 are the divisor classes of the rational sectionsŝ 1 andŝ 0 . This result allows us to compute the height pairing of the sectionŝ 1 . We obtain where we employed (2.6) along with the self-intersection (2.7) for the sectionŝ 1 as well as This follows by noting that π(S 0 · S 1 ) is the locus in B where the coordinates (3.55) and (3.57) of the two sections agree, which happens at z 1 := s 7 s 2 8 − s 6 s 8 s 9 + s 5 s 2 9 = 0, that is precisely the z-coordinate ofŝ 1 in the WSF, cf. (3.61). The divisor class of z 1 is read off from (3.8).

Weierstrass form and gauge group
We can apply Nagell's algorithm to the cubic (3.54) with respect to the point P 0 to obtain a birational map to its WSF. We plug the coordinates of the rational section (3.57) into this map to obtain its coordinates in WSF, Here p 8 (s) and p 12 (s) are two homogeneous polynomials in the coefficients s i of degree eight and twelve, respectively. We have written out only one monomial in x Q 1 and y Q 1 , respectively, in order to be able to determine their divisor classes. We refer the reader to (B.8) in Appendix B for the explicit and lengthy expressions for p 8 (s) and p 12 (s). Furthermore, we determine the functions f , g and the discriminant ∆ of the WSF for X F 3 . They are given by specializing (B.1) as s 10 = 0. We observe that there is no factorization of ∆ indicating the absence of codimension one singularities and a non-Abelian gauge group. Thus, the full gauge group on X F 3 is given by the single U(1) associated to its rank one MW-group, We emphasize again that the generator (3.57) of the MW-group of X F 3 is not toric.

Charged and uncharged matter
Since the Calabi-Yau manifold X F 3 has a non-trivial MW-group, it automatically has I 2 -fibers at codimension two in B, that support U(1)-charged matter. We first summarize the charged matter spectrum of X F 3 before we discuss its derivation in detail. The full charged matter spectrum is shown in Table 6, which includes the U(1)-charges and the multiplicities of 6D charged hyper multiplets, as well as a schematic presentation of the corresponding reducible fibers and the full expressions for the codimension two loci.

Representation
Multiplicity Fiber Locus Table 6: Charged matter representation under U(1) and codimension two fibers of X F 3 .
The starting point for the derivation of the matter spectrum of X F 3 is, as discussed in Section 2.3, the complete intersection (2.14) in B with the WS-coordinates (3.57) of the section s 1 inserted: We show that (3.63) is a reducible variety with three irreducible components supporting matter with charges one, two and three. The corresponding codimension two loci are denoted V (I (1) ), V (I (2) ) and V (I (3) ), respectively, with I (1) , I (2) and I (3) denoting the corresponding prime ideals, cf. Table 6. In order to strictly prove that these three varieties are all irreducible components of the complete intersection (3.63), we have to compute all its associated prime ideals. Unfortunately, this is unfeasible with the available computing power and computer algebra programs, due to the high degree of the two polynomials in (3.63). However, we explain that X F 3 has three possible types of I 2 -fibers corresponding to the three possible factorizations of (3.54) and that these factorization happen precisely at the codimension two loci V (I (1) ), V (I (2) ) and V (I (3) ).
Thus, we claim that the corresponding ideals I (1) , I (2) and I (3) are the only associated prime ideals of (3.63). We will further substantiate this claim by checking 6D anomaly cancellation at the end of this section as well as by reproducing the spectrum of X F 3 by Higgsing the effective theories of X F 5 and X F 6 , see Section 4.
We begin by analyzing the fiber at the first two codimension two loci in Table 6. These are precisely the loci where the coordinates (3.57) of the sectionŝ 1 are ill-defined, since they are forbidden by the SR-ideal (3.53). This indicates, that the sectionŝ 1 does not mark a point on the elliptic fiber of X F 3 , but does wrap an entire P 1 . Since the rational section is non-toric, determining the wrapped P 1 is slightly more involved than usual, as we demonstrate next.
First, we consider the locus V (I (3) ) = {s 8 = s 9 = 0}, which we readily check to obey (3.63). At this locus the constraint (3.54) factorizes as (3.64) Clearly, V (I (3) ) is the only codimension two locus where this factorization can occur. We immediately observe that the zero sectionŝ 0 defined by (3.55) has wrapped the entire rational curve e 1 = 0 in (3.64). The rational sectionŝ 1 can be identified at this locus by recalling the definition of the point (3.57) as the second intersection point of the tangent to P 0 with E. However, at s 8 = s 9 = 0 the curve (3.64) is singular (after setting e 1 = 1) precisely at P 0 . Thus, every line through P 0 is automatically tangential at P 0 . This simply means that P 1 has become the entire singular fiber at s 8 = s 9 = 0, since given any point on (3.64) (for e 1 = 1) we can construct a tangent at P 0 that passes through that point. Thus, at s 8 = s 9 = 0 the sectionŝ 1 wraps the rational curve described by the parenthesis in (3.64). The resulting fiber at V (I (3) ) is shown in the second column of Table 6. We readily compute using the charge formula (2.13) that the U(1) charge of the matter is indeed q = 3 and its multiplicity is given by [s 8 ] · [s 9 ], which after using (3.8), yields the result shown in Table 6. We emphasize that this is the first occurrence of matter with charge q > 2 in models with Abelian gauge symmetry in F-theory. Second, we consider the locus V (I (2) ). The complete intersection in V (I (2) ) shown in Table 6 has two irreducible components, one of which given by V (I (3) ), that we forbid by requiring (s 8 , s 9 ) = (0, 0), and a second one described by a prime ideal I (2) with ten generators. 22 The variety V (I (2) ) supports matter of charge two. We can check this locally by solving the complete intersection inside V (I (2) ) e.g. for s 3 and s 6 and by plugging this solution into (3.54). Indeed, the fiber splits into a line and a non-singular quadric q 2 (e 1 u, e 1 v, w), Furthermore, we prove that V (I (2) ) is the only locus that can yield an I 2 -fiber of this type by computing the elimination ideal of the ideal of constraints necessary for the factorization (3.65). We see that the zero section (3.55) is well-defined at V (I (2) ) and passes through the line. However, the rational section (3.57) is ill-defined. This is clear because the line in (3.65) is precisely the tangent t P at P 0 defined in (3.56) and since the sectionŝ 1 is defined as the intersection of t P with E. Thus, the sectionŝ 1 at V (I (2) ) wraps the entire rational curve given by the line in (3.65). Again we use (2.13) to show that the U(1)-charge is q = 2, as claimed in Table 6. The multiplicity of a 6D hyper multiplet in the representation 1 2 is given by the homology class of V (I (2) ). It is computed by first computing the homology class of the complete intersection in V (I (2) ) in Table 6 using (3.8) and by subtracting the class of the unwanted component V (I (3) ) with the appropriate order. We determine it to be six using the resultant technique of [39], which precisely yields the multiplicity in the third row of Table 6. 22 As all prime ideals in this work, it is computed by the primary decomposition function in Singular [88].
Finally, we turn to the codimension two locus V (I (1) ) supporting matter of charge one. In order for the charge formula (2.13) to produce charge one for an I 2 -fiber, bothŝ 0 andŝ 1 have to be regular and pass through different rational curves in the I 2 -fiber. This can only happen for a factorization of (3.54) of the form (we can set e 1 = 1) with q 2 (u, v, w) denoting a quadric without the monomial w 2 . We note that all coefficients d i , i = 1, 2, 3, have to be non-vanishing since d 1 = 0, d 2 = 0 or d 3 = 0 lead to a factorization in (3.66) that cannot happen at codimension two. We see thatŝ 0 intersects the quadric q 2 = 0 andŝ 1 intersects the line, as required for matter with charge one. Furthermore, we compute the elimination ideal, denoted by I (1) , of the ideal of constraints necessary for the factorization (3.66). It is prime and of codimension two in the ring, that means that the factorization (3.66), indeed, occurs in codimension two in B. In addition, we check that the complete intersection (2.14) is inside the ideal I (1) and that I (1) is in turn not contained in I (3) or I (2) , as required. Thus, we identify I (1) as the third and last associated prime ideal of (3.63).
Under the well-motivated assumption that I (3) , I (2) and I (1) are the only associated prime ideals of (3.63), we determine the multiplicity of the 1 1 -matter as follows. First, we determine the orders of the loci V (I (3) ) and V (I (2) ) in the complete intersection (2.13). Using the resultant technique of [39] and random integers for some of the s i we find the orders 81 and 16 for these loci, respectively. Then, we subtract their multiplicities with these orders from the class of the complete intersection (3.63) and obtain, using (3.8), the multiplicity in the last row of Table 6.
The matter spectrum of X F 3 is completed by the number of neutral hyper multiplets H neut . Employing (2.11) and the Euler number χ(X F 3 ) of X F 3 given in (C.1), we obtain Finally, we check anomaly-freedom of the full 6D SUGRA theory. To this end we use (3.59), the charged spectrum in Table 6 and (3.67), together with V = 1, to show that all relevant anomalies of the 6D SUGRA theory in (A.1) are canceled. There is another quantum consistency condition the spectrum in Table 6 has to pass. In order to have an effective theory that makes sense also in a quantum gravity model, it has been argued in [89] that all charges allowed by Dirac quantization have to be present in the spectrum. Indeed, it is clear from the multiplicity formulas in Table 6 (e.g. by evaluation for a concrete base B, see Section 4.1.3) that if matter with a maximal charge q is present in the spectrum, also matter with all lower charges q < q is automatically there, as required.
For completeness, we include a discussion of the Yukawa couplings. Forming the union of the ideals and computing their codimension to be three in the polynomial ring K[s i ], we find the two Yukawa couplings given in Table 7.

Yukawa
Locus An alternative perspective: X F 3 from X F 5 by an extremal transition There is a second perspective on X F 3 that provides an alternative explanation for the presence of the rational point (3.57) and that will be useful for the understanding of the Higgs transition in Section 4. The following can be skipped on a first reading, as it is not important for the main thread of this work. We begin by noting that (3.54) becomes singular if we tune the complex structure so that s 4 ≡ 0. The induced I 2 -singularities occur at codimension two and can be resolved by the blow-up in the fiber at u = w = 0. The Calabi-Yau manifold after this extremal transition is precisely X F 5 , that we discuss below in Section 3.3.2. It has been shown that X F 5 has a rank two Mordell-Weil group [38,39].
In the singular fibration with all exceptional divisors blown down, the three rational points on the fiber C F 5 are the three intersection points with the line u = 0. One point agrees with the origin (3.55) of X F 3 . We denote the other two points by Q 1 , Q 2 . This implies that the point Q 1 + Q 2 is precisely given by (3.57), in the limit s 4 ≡ 0. Indeed, the group law on a cubic curve is defined so that the point Q 1 + Q 2 is found by first constructing the third intersection point of the line through Q 1 and Q 2 and then by forming the line through that point and the origin P 0 . This line again has a third intersection point with the curve, which is defined to be Q 1 + Q 2 In our situation, the line through Q 1 and Q 2 is u = 0. Thus, the third intersection point of u = 0 with E is the origin P 0 . Consequently, the point Q 1 + Q 2 is the second intersection point of the tangent through P 0 with the elliptic curve. In fact, it can be checked by performing this addition on the fiber of X F 5 explicitly that the coordinates of the point Q 1 + Q 2 on the fiber of X F 3 agree with the coordinates (3.57) after setting s 4 ≡ 0. Furthermore, we compute the Weierstrass coordinates of Q 1 + Q 2 that also agree with (3.61) after setting s 4 ≡ 0. 23 This is not surprising since we recall that the P 1 in X F 3 has been constructed as the second intersection of the tangent (3.56) to P 0 . Thus, we see that the sectionŝ 1 can be understood as the sum of the sectionsŝ 1 +ŝ 2 on X F 5 , which survives the extremal transition X F 5 ↔ X F 3 , i.e. the complex structure deformation associated to switching on s 4 . In contrast, the individual sectionsŝ 1 andŝ 2 on X F 5 do not map to rational sections on X F 3 . As consequence, cf. Section 4, the U(1)-charges of matter in X F 3 are given by the sum of the U(1)-charges q 1 + q 2 on X F 5 .
We can make these statements even more explicit by mapping X F 3 to X F 5 . The shift precisely cancels the monomial proportional to s 4 in (3.54). Clearly, this requires an extension of the field of meromorphic functions on B by the square root s 2 7 − 4s 4 s 9 . Thus, this map is certainly not birational. After this shift, we precisely obtain the hypersurface of X F 5 , cf. (3.73) 23 The coordinates of Q 1 + Q 2 in WSF are obtained by inserting its coordinates into the birational map from X F5 to its WSF. We note that the result agrees with the WS-coordinates of Q 1 + (−Q 2 ), not Q 1 + Q 2 , where '+' denotes here the addition in the WSF of X F5 . with s 1 , s 5 , s 8 and s 9 unchanged. If we insert this variable transformation into the expressions forŝ 1 orŝ 2 in (3.74), we introduce square roots, i.e. these sections do not map to rational sections on X F 3 . However, if we insert (3.69) into the coordinates forŝ 1 +ŝ 2 on X F 5 , we precisely reproduce (3.57), i.e. all square roots cancel. Furthermore, we can re-derive the Weierstrass coordinates (3.61) ofŝ 1 on X F 3 by first computing the Weierstrass coordinates of Q 1 + Q 2 and then inserting (3.69). In addition, f and g of the WSF of X F 3 can be obtained from the WSF for X F 5 by insertion of (3.69).

Polyhedron
The toric hypersurface fibration X F 5 is constructed as the fibration of the elliptic curve in P F 5 = dP 2 . As it is completely analyzed in [39,43], we only state the results here for completeness.
The toric diagram of F 5 along with a choice of homogeneous coordinates as well as its dual polyhedron are depicted in Figure 8. In the monomials corresponding to the integral points of F 12 by (2.23) we have set e i = 1, ∀i. The toric variety P F 5 is the blow-up of P 2 , cf. Section 3.1.2, The Stanley-Reisner ideal of P F 5 is given by SR = {we 2 , wu, ve 1 , e 2 e 1 , vu} . (3.72) By use of (2.23) the hypersurface equation for X F 5 in the dP 2 -fibration (3.1) is given by p F 5 = s 1 e 2 2 e 2 1 u 3 + s 2 e 2 2 e 1 u 2 v + s 3 e 2 2 uv 2 + s 5 e 2 e 2 1 u 2 w + s 6 e 2 e 1 uvw + s 7 e 2 v 2 w + s 8 e 2 1 uw 2 + s 9 e 1 vw 2 , (3.73) where the sections s i take values in the line bundles shown in (3.8). We see that (3.73) can also be obtained from (3.4) by the specialization s 4 = s 10 = 0 and the map (3.70).
There are three rational sections of the fibration of X  1) and (B.2), respectively, after setting s 4 = s 10 = 0. After using this to calculate the discriminant we do not find any codimension one singularities. Then the total gauge group of X F 5 is Thus, the corresponding Shioda maps (2.5) forŝ 1 andŝ 2 read which allows us to compute the corresponding height pairing (2.6) as To determine the 6D spectrum of charged hyper multiplets we analyze the codimension two singularities of the WSF of X F 5 . There are six singularities leading to the matter representations and the corresponding codimension two fibers in X F 5 given in the first and third column of Table 8. The detailed derivation of these results can be found in [38,39,43,44].

Yukawa
Locus 1 (−1,−2) · 1 (0,2) · 1 (1,0) s 7 = s 8 = s 9 = 0 1 (0,2) · 1 (1,−1) · 1 (−1,−1) s 3 = s 7 = s 9 = 0 We consider an elliptically fibered Calabi-Yau manifold X F 6 with an arbitrary base B and general elliptic fiber given by the elliptic curve E in P F 6 . The toric data of P F 6 is summarized in Figure 9, where the polyhedron F 6 along with a choice of homogeneous coordinates as well as its dual polyhedron F 11 are shown. For brevity, we have set e i = 1, ∀i, in the monomials that are associated to the integral points of F 11 by (2.23). Figure 9: Polyhedron F 6 and its dual F 11 . The zero section is indicated by the dot.
We note that P F 6 is the blow-up of P 2 , cf. Section 3.1.2, at two points, defined by The homogeneous coordinates on the fiber after this blow-up are [u : v : w : e 1 : e 2 ] and take values in the line bundles associated to the following divisor classes: Here H denotes the pullback of the hyperplane class on P 2 and the E i are the exceptional divisors of the blow-up (3.79). The Stanley-Reisner ideal of P F 6 then reads SR = {uv, ue 1 , we 1 , we 2 , ve 2 } . where we chooseŝ 0 as the zero section. The corresponding points on E are denoted P 0 and P 1 , respectively. We compute the Weierstrass form (2.1) of (3.82) using Nagell's algorithm. The WScoordinates of the sectionŝ 1 are given by (B.8) after setting s 9 = 0. Furthermore, the functions f and g take the form of (B.1) and (B.2), respectively, after setting s 9 = s 10 = 0. From this the discriminant ∆ is readily computed. This allows us to find all codimension one singularities of the WSF of X F 6 . We find one I 2 -singularity over the divisor S b SU(2) = {s 8 = 0} ∩ B in B. Along this divisor the constraint (3.82) factorizes as where q 3 is the polynomial that remains after factoring out e 1 . This is clearly an I 2 -fiber, cf. Figure 10, giving rise to an SU(2) gauge group. In summary, the gauge group of X F 6 is The rational curve of the I 2 -fiber in Figure 10 that is intersected by the zero sectionŝ 0 is the affine node and the other rational curve, c −α 1 , corresponds to the simple root of SU(2).
Thus, the class of the SU(2) Cartan divisor in X F 6 , which is the fibration of c −α 1 over S b SU(2) , reads (3. 86) This can be seen by noting that [s 8 ] is the class of the complete I 2 -fiber fibered over the base divisor S b SU(2) , whereas [e 1 ] is the class of the affine node fibered over S b SU (2) . With these results, we compute the Shioda map (2.5) of the sectionŝ 1 as Here S 0 , S 1 denote the divisor classes of the sectionsŝ 0 ,ŝ 1 , respectively, and we use which can be deduced from Figure 10. Using (3.87), we compute the height pairing (2.6), where we use (2.7) as well as that follows since the coordinates (3.83) of the two sections agree at s 7 = 0. Next, we analyze the codimension two singularities of the WSF of X F 6 to determine the charged matter spectrum. Here, the corresponding representations under the gauge group are determined following the general procedure outlined in Section 2.1 for the computation of Dynkin labels and U(1)-charges. We find five codimension two singularities. Four of these lead to the matter representations and the corresponding codimension two fibers in X F 6 given in the first and third column of Table 10, respectively. At the remaining locus s 8 = s 2 6 − 4s 5 s 7 = 0, the fiber is of Type III , i.e. it is a degenerate version of the I 2 -fiber in Figure 10 with the two P 1 's intersecting in one point. Thus it does not support any additional matter.
The spectrum of charged singlets is determined starting from the complete intersection (2.14) for the sectionŝ 1 , see (3.91) for its explicit expression. By computing its primary decomposition, we find two associated prime ideals, denoted I (3) and I (4) , corresponding to two different matter representations 1 2 and 1 1 . We observe that the ideal I (3) describes precisely the locus, where the section 3.83 is ill-defined and has to acquire a fiber component.
There is one subtlety since the constraint (3.82) does not factorize further at the locus V (I (2) ). In order to see the I 3 -fiber of the representation 2 1/2 we have to compute the associated prime ideals of (3.82) at the locus V (I (2) ). Indeed, we find three prime ideals with the right intersections and, they thus, correspond to the three irreducible components of an I 3 -fiber. Two fiber components are described by prime ideals generated by more than three constraints.
The multiplicities of the charged hyper multiplets are presented in Table 10. These have been computed following Section 2.3. For the case of the representations 2 −3/2 , 2 1/2 and 1 2 , the multiplicities can be directly computed from (3.8) and Table 10, as the corresponding varieties V (I (1) ), V (I (2) ) and V (I (3) ) are irreducible complete intersections. In contrast, V (I (4) ) Representation

Multiplicity
Fiber Locus with x Q , y Q , f given in (3.91)   (2)×U(1) and corresponding codimension two fibers of X F 6 . The adjoint matter is included for completeness.
supporting the 1 1 -matter is not a complete intersection. However, note that the varieties V (I (4) ), V (I (1) ) and V (I (3) ) are the three irreducible components of the complete intersection (2.14) for the sectionŝ 1 . Using its WS-coordinates, given by (B.8) for s 9 = 0, it reads Thus, the homology class of this complete intersection minus the classes of V (I (1) ) and V (I (3) ) (with their respective orders inside (3.91)) yields the multiplicity of the 1 1 -matter. The spectrum of charged matter is completed by the matter in the adjoint representation 3 0 given in the last row of Table 10. We recall that it does not originate from codimension two fibers of X F 6 , but is present if the divisor S b SU(2) is a higher genus curve in B, cf. Section 2.3. The multiplicity of charged hyper multiplets in the adjoint is calculated using (2.8).
We complete the matter spectrum of X F 6 by the number of neutral hyper multiplets, which is computed from (2.11) using the Euler number (C.1) of X F 6 . It reads The number T of tensor multiplets is given by (2.17) and we have V = 4. Finally, we use S b SU(2) = {s 8 = 0}, (3.89), the charged spectrum in Table 10 and (3.92) to check, following Appendix A, that all 6D anomalies are canceled.
We conclude this section by analyzing codimension three singularities of the WSF of X F 6 . This determines the Yukawa points in a compactification to 4D. All geometrically allowed Yukawa couplings of the charged matter spectrum of X F 6 are given in Table 11. In order to check the last Yukawa coupling in Table 11, we compute the minimal associated primes of I (2) ∪ I (4) . Indeed, it has a codimension three associated prime, which confirms the presence of the Yukawa coupling. We emphasize that all Yukawa couplings allowed by gauge symmetry are indeed realized.

Fibrations with gauge groups of rank 3: selfdual polyhedra
In the following section we analyze all toric hypersurface fibrations constructed from the four self-dual polyhedra F 7 , F 8 , F 9 and F 10 . The rank of the gauge group of all these models is three and the rank of the MW-group assumes all values from zero to three. We encounter one novelty in the analysis of codimension two fibers in X F 8 and X F 10 . There we find matter representations from non-split fibers at codimension two. The Calabi-Yau manifold X F 10 is also a generalization of the Tate form, allowing for non-trivial coefficients of the monomials x 3 and y 2 . The vanishing loci of these coefficients support SU(2) and SU(3) gauge groups, respectively.

Polyhedron
We consider the elliptically fibered Calabi-Yau manifold X F 7 with base B and general elliptic fiber given by the elliptic curve E in P F 7 . The toric diagram of P F 7 = dP 3 is depicted in Figure 11, where the polyhedron F 7 along with a choice of homogeneous coordinates as well as its dual polyhedron F 7 are shown. For brevity, we have set e i = 1, ∀i, in the monomials associated to the integral points in the dual polyhedron by (2.23). Figure 11: The toric diagram of polyhedron F 7 and its dual. The zero section is indicated by the dot.
The toric variety P F 7 is the del Pezzo surface dP 3 , that is the blow-up of P 2 , cf. Section 3.1.2, at three points with blow-down map defined by u → e 1 e 3 u , w → e 1 e 2 w , v → e 2 e 3 v . (3.93) The homogeneous coordinates on the fiber after this blow-up are [u : v : w : e 1 : e 2 : e 3 ] and take values in the line bundles associated to the following divisor classes: The Stanley-Reisner ideal of P F 7 is given by SR = {uw, ue 2 , uv, e 1 e 2 , e 1 v, e 1 e 3 , wv, we 3 , e 2 e 3 } . In total there are six rational sections of the elliptic fibration of X F 7 with four of them being linearly independent [41]. Their coordinates arê Thus we compute the Shioda map (2.5) of all rational sectionsŝ m , m = 1, 2, 3, as where we use the following intersection relations: π(S 1 · S 0 ) = S 7 , π(S 2 · S 0 ) = 0 , π(S 3 · S 0 ) = S 9 , π(S 1 · S 2 ) = 0 , π(S 1 · S 3 ) = 0 , π(S 2 · S 3 ) = 0 .
Next, we analyze the codimension two singularities of the WSF of X F 7 to determine the charged matter spectrum. We find ten codimension two singularities, which lead to the matter representations and the corresponding codimension two fibers in X F 7 given in the first and third column of Table 12, respectively. Given all these codimension two fibers and the positions of the rational sections, we readily compute the U(1)-charges using (2.13). Here, the starting point of the analysis, that has led to the complete charged matter spectrum in Table 12, are the three complete intersections (2.14) evaluated for the three rational sectionsŝ 1 ,ŝ 2 andŝ 3 . Then, we determine all their minimal associated prime ideals using Singular [88], which precisely produces all the ten ideals I (k) , shown in Table 12.
The multiplicities of the charged hyper multiplets are presented in Table 12. These have been computed following Section 2.3 as the homology classes of the respective codimension two varieties V (I (k) ), k = 1, . . . , 10. In the cases where V (I (k) ) is not a complete intersection, the multiplicities are calculated using the resultant technique similarly as described in Section 3.3.3.
To complete the matter spectrum of X F 7 we compute the number of neutral hyper multiplets from (2.11) using the Euler number χ(X F 6 ) given in (C.1). It reads We note that there are T tensor multiplets and V = 3 vector multiplets. These results together with the charged spectrum in Table 12 and (3.102) as well as the height pairing (3.101) allows us, following Appendix A, to check cancelation of all 6D anomalies.
To conclude this section we analyze the codimension three singularities of the WSF of X F 7 . This determines the Yukawa points in a compactification to 4D. The geometrically allowed Yukawa couplings of X F 7 are given in Table 13. Here we confirm the presence of each Yukawa coupling by checking that the intersection of the relevant varieties is codimension three in B.

Polyhedron
In this section, we consider the elliptically fibered Calabi-Yau manifold X F 8 over an arbitrary base B and with general elliptic fiber given by the elliptic curve E in P F 8 . In Figure 12 the toric diagram of F 8 and of its dual polyhedron are depicted. For brevity, we have set e i = 1, ∀i, in the monomials that are associated by (2.23) to the integral points of the dual polyhedron. Figure 12: The toric diagram of polyhedron F 8 and its dual. The zero section is indicated by the dot.
The toric variety P F 8 is P 2 , cf. Section 3.1.2, blown-up at three non-generic points. The blow-down map reads u → e 1 e 2 2 e 3 u , v → e 1 v , w → e 3 w . (3.103) The homogeneous coordinates on the fiber after this blow-up are [u : v : w : e 1 : e 2 : e 3 ] and take values in the line bundles associated to the divisors given by: The Stanley-Reisner ideal of P F 8 reads SR = {uw, uv, ue 1 , e 3 v, e 3 e 1 , e 3 e 2 , we 1 , we 2 , ve 2 } . where the classes of the sections s i are given in (3.8). We note that restriction of (3.4) as s 4 = s 9 = s 10 = 0 and application of the map (3.103) also leads to (3.106).
There are two rational sections of the fibration of X  We compute the Weierstrass form (2.1) of (3.106) using Nagell's algorithm. The WScoordinates of the sectionŝ 1 are given by (B.8) after setting s 4 = s 9 = 0. Additionally, the functions f and g take the form of (B.1) and (B.2), respectively, after setting s 4 = s 9 = s 10 = 0. This allows us to find all codimension one singularities of the WSF of X F 8 . We find two I 2singularities over the divisors S b SU(2) 1 = {s 7 = 0} ∩ B and S b SU(2) 2 = {s 8 = 0} ∩ B in B. Along these divisors the constraint (3.106) factorizes as SU (2) (3.108) Here q 2 , q 3 are the polynomials of degree n in [u : v : w] that remain after factoring out u and e 1 , respectively. These are clearly I 2 -fibers, cf. Figure 13, giving rise to two SU(2) gauge groups. In summary, the gauge group of X F 8 is This allows us to compute the Shioda map (2.5) of the sectionŝ 1 as where we use that the two sections in (3.107) do not intersect, implying π(S 1 · S 0 ) = 0, and which follows from Figure 13. We compute the height pairing (2.6) ofŝ 1 using (2.7) as In order to determine the charged matter spectrum we analyze the codimension two singularities of the WSF of X F 8 . For the singlets we compute the associated prime ideals of the complete intersection (2.14) associated toŝ 1 . We find seven codimension two singularities.

55
Five of these lead to the matter representations and the corresponding codimension two fibers in X F 8 given in the first and third column of Table 14, respectively. The representations under G F 8 have been determined following the general procedure outlined in Section 2.3. The remaining two loci, s 7 = s 2 6 − 4s 3 s 8 = 0 and s 8 = s 2 6 − 4s 5 s 7 = 0, support Type III singularities, which do not lead to additional matter.

Representation
Multiplicity Fiber Locus (1, 1) 1    We note the following subtlety. The fiber supporting the matter in the representation (2, 1) 0 is non-split in the sense of [61]. 24 This means that the constraint (3.106) of the elliptic fiber at the codimension two locus V (I (4) ) does not fully factorize, as one expects, over the field K of meromorphic functions on B. It only factorizes in a field extension where certain square roots of the coefficients s i are allowed. In fact, the fiber we obtain at the locus V (I (4) ) allowing only for factorizations in K is a line and a singular conic. However, a singular conic describes two lines, i.e. the conic has to be factorized into two linear constraints describing two lines. This factorization requires introducing square roots of some combinations of the s i . Geometrically, this means that these lines are interchanged by a codimension three monodromy (that occur only on threefold bases B). 25 The two lines of the non-split fiber that are interchanged by this monodromy are the dashed P 1 's in the fourth row of Table 14.
The multiplicities of the charged hyper multiplets are presented in Table 14 in the second row. These have been computed following Section 2.3. Since the locus of the representation (1, 1) 1 is not a complete intersection we compute its multiplicity as described in Section 3.3.3 by subtraction of the locus V (I (3) ) with its appropriate order.
The number of neutral hyper multiplets completes the matter spectrum of X F 8 . It is computed from (2.11) using the Euler number (C.1) of X F 8 and reads The base-dependent number T of tensor multiplets is given by (2.17) and we have V = 7. Finally, we use this together with S b , the charged spectrum in Table 14 and (3.114) to check cancelation of all 6D anomalies in (A.1).
To obtain the Yukawa points in a compactification to 4D we analyze codimension three singularities of the WSF of X F 8 . All Yukawa couplings of the charged matter spectrum of X F 8 are given in Table 15. Clearly, all relevant loci here are codimension three.

Polyhedron
Here, we consider the elliptically fibered Calabi-Yau manifold X F 9 over a base B and with general elliptic fiber given by the toric hypersurface C F 9 in P F 9 . The toric data of P F 9 is depicted in Figure 14. Here the polyhedron F 9 along with a choice of homogeneous coordinates as well as its dual polyhedron is shown. For brevity, we have set e i = 1, ∀i, in the monomials associated to the integral points of the dual by (2.23).

Figure 14:
The toric diagram of polyhedron F 9 and its dual. The zero section is indicated by the dot.
The toric variety P F 9 is obtained from P 2 , cf. Section 3.1.2, by the three non-generic blow-ups u → e 1 e 2 e 3 u , w → e 1 w , v → e 2 e 2 3 v .
There are four rational sections on X  We chooseŝ 0 as the zero section andŝ m , m = 1, 2, as the generators of the MW-group of X F 9 .
Employing Nagell's algorithm we compute the Weierstrass form (2.1) of (3.118). The WScoordinates of the sectionsŝ 1 andŝ 2 are given by (B.9) and (B.10), respectively, after setting the appropriate sections s i to zero. To get the functions f and g we specialize (B.1) and (B.2) as s 4 = s 8 = s 10 = 0. Using f and g we can compute the discriminant ∆ to find all codimension one singularities of the WSF of X F 9 . We find one I 2 -singularity over the divisor S b SU(2) = {s 9 = 0} ∩ B in B. Along this divisor the constraint (3.118) factorizes as SU (2) : p F 9 | s 9 =0 = e 2 · q 3 , (3.120) where q 3 is the polynomial that remains after factoring out e 2 . This is an I 2 -fiber, cf. Figure 15, giving rise to an SU(2) gauge group. In summary, the gauge group of X F 9 is Here S 0 , S m denote the divisor classes of the sectionsŝ 0 ,ŝ m , m = 1, 2, respectively, and we used the relation (3.125) Turning to the charged matter spectrum we analyze the codimension two singularities of the WSF of X F 9 . The matter in non-trivial representations of the non-Abelian part of G F 9 follows directly from the discriminant with s 9 = 0, whereas the charged singlets are most easily seen Representation Multiplicity Fiber Locus V (I (2) ) := {s 5 = s 1 = 0} V (I (3) ) := {s 2 s 2 7 + s 2 3 s 9 − s 3 s 6 s 7 = 0 s 5 s 3 − s 7 s 1 = 0}\V (I (1) )  from the primary decomposition of the complete intersections (2.14) corresponding to the two sectionsŝ 1 ,ŝ 2 , respectively. We find eight codimension two singularities. Seven of these lead to the matter representations and the corresponding codimension two fibers in X F 9 given in the first and third column of Table 16, respectively. Here, the corresponding representation under the gauge group are determined following the general procedure explained in Section 2.1. At the remaining locus s 9 = s 2 6 − 4s 5 s 7 = 0, the fiber is of Type III, cf. the discussion in Section 3.3.3. Thus it does not support any additional matter.
We note that at the locus V (I (2) ) the linearly dependent section in (3.119) is singular. At the locus V (I (7) ) corresponding to matter in the representation 2 (0,−1/2) the elliptic curve (3.118) does not naively factor into three rational curves. To correctly derive its splitting one needs to compute the associated prime ideals of the elliptic fiber at this locus. We find three ideals corresponding to three rational curves, which indeed intersect as an I 3 -fiber.
The multiplicities of the charged hyper multiplets are presented in Table 16. These have been computed following Section 2.3. The multiplicities of V (I (3) ) and V (I (4) ) must be calculated by appropriately subtracting the multiplicities of the loci V (I (1) ), V (I (5) ) and V (I (6) ), respectively, as described in Section 3.3.3 and indicated in Table 16.
We complete the matter spectrum of X F 9 by the number of neutral hyper multiplets, which is computed from (2.11) using the Euler number χ(X F 9 ) given in(C.1). It reads The number of tensor multiplets T can be obtained by (2.17) and we have V = 5 vector multiplets. To check that the anomalies are canceled we use S b SU(2) = {s 9 = 0}, (3.125), the charged spectrum in Table 16 and (3.126), following the discussion in Appendix A.
We conclude this section by analyzing codimension three singularities of the WSF of X F 9 determining the Yukawa points in a compactification to 4D. In Table 17 all geometrically allowed Yukawa couplings of the charged matter spectrum of X F 9 are given.

Polyhedron
u → e 1 e 2 2 e 3 3 u , v → e 1 e 2 e 3 v .  (3.130) Here the sections s i take values in the line bundles associated to the divisor classes shown in (3.8). The hypersurface equation (3.130) can also be obtained by the specialization s 7 = s 9 = s 10 = 0 and the map (3.127) applied to (3.4).There is one rational section of the fibration of X F 10 . Its coordinate isŝ

Comparison with the Tate form
Before proceeding with the analysis of X F 10 , let us pause to compare with the standard elliptic fibration with fiber in P 2 (1, 2, 3), that is the Tate form. We emphasize that (3.130) describing X F 10 can be viewed as a two-fold generalization of the standard Tate form of an elliptic fibration studied e.g. in [2,3,91] , which is produced in the special case s 8 = 1 and s 4 = 1.
First, we identify the usual projective coordinates [z : x : y] on P 2 (1, 2, 3) and the Tate coefficients a i . They read Using this, we see that (3.130) is indeed in Tate form. However, we note that there are two additional coefficients, namely s 8 and s 4 , that do not have an analog in the standard Tate form, because they correspond to the coefficients of y 2 and x 3 , that are typically set to one. As we see below, at the vanishing loci of these sections we find a SU(3)-and a SU(2)-singularity, respectively. Thus, allowing for non-trivial s 4 , s 8 , is the first of the two aforementioned generalizations of X F 10 , compared to the standard Tate form.
In addition, consistently imposing s 4 = s 8 = 1 fixes the degrees of freedom in constructing the fibration of the elliptic curve C F 10 over the base B. Indeed, setting s 4 = s 8 = 1 requires their divisor classes to be trivial, [s 8 ] = 0, [s 4 ] = 0. This fixes S 7 and S 9 according to (3.8) as Thus, the fibration X F 10 is completely fixed in terms of the canonical bundle K B of the base B.
As we see from (3.128), the coordinates u and v transform as a section of the line bundles K 3

B
and K B , respectively. Using the C * -action on P F 10 , this is equivalent, employing (3.132), to Thus, we see that by relaxing s 4 = s 8 = 1, we also get more freedom, parametrized in the divisors S 7 and S 9 , in constructing the fibration of C F 10 over a given base B. This is the second generalization of X F 10 in contrast to the standard Tate model.

Higher codimension singularities & the spectrum of F-theory on X F 10
We consider in the following the most general elliptic fibration X F 10 with general, non-trivial coefficients s 4 and s 8 . In order to compute the Weierstrass form (2.1) of the general hypersurface equation where q 2 and q 3 are the remaining polynomials after factoring out u and e 1 e 2 , respectively. The reducible fibers at these loci are depicted in Figure 17. Thus, the total gauge group of X F 10 is Next, we turn to the charged matter spectrum, which is obtained by analyzing the codimension two singularities of the WSF of X F 10 . All loci of codimension two singularities directly follow from the behavior of the discriminant and the representation content under the gauge group G F 10 is determined following the general procedure outlined in Section 2.1. We find five codimension two singularities. Three of these lead to the matter representations and the corresponding codimension two fibers in X F 10 in the first and third column of Table 18, respectively. We note that the fiber corresponding to matter in the representation (2, 1) is non-split, cf. the discussion in 3.4.2. The two nodes that are identified by codimension three monodromies are drawn with dashed lines in Table 18. At the locus s 4 = s 2 6 − 4s 3 s 8 = 0 the fiber is of Type III and at the locus s 6 = s 8 = 0 it is of Type IV , i.e. the fiber is a degeneration of the I 3 -fiber at the locus s 8 = 0, where the three P 1 's intersect in one point. Thus both loci do not support any additional matter. The matter in the adjoint representations has been added to Table 18 for completeness.
In the second column of Table 18 the multiplicities of the charged hyper multiplets are presented. They have been computed following Section 2.3, directly from the classes of all varieties V (I k ), k = 1, 2, 3. This is straightforward, employing (3.8), as these varieties are irreducible complete intersections.
Finally, the number of neutral hyper multiplets is computed from (2.11) using the Euler number (C.1) of X F 10 . It reads The number T of tensor multiplets is given by (2.17) and we have V = 11. We check that all 6D anomalies, cf. Appendix A, are canceled using S b SU(2) = {s 4 = 0} ∩ B, S b SU(3) = {s 8 = 0} ∩ B, the charged spectrum in Table 18 and (3.138).

Multiplicity
Fiber Locus V (I (1) ) :=   (2) and corresponding codimension two fibers of X F 10 . The adjoint matter is included for completeness.
We conclude this section by the analysis of codimension three singularities of the WSF of X F 10 and the corresponding Yukawa points in compactifications to 4D. We find one possible Yukawa coupling of the charged matter spectrum of X F 10 , which is given in Table 19.

Fibrations with gauge groups of rank 4, 5 and no MW-torsion
In this section we analyze toric hypersurface fibrations based on the fiber polyhedra F 11 , F 12 and F 14 . These are the fibrations that give rise to F-theory models with simply-connected gauge groups of maximal rank among all toric hypersurface fibrations, that is four and five. Most outstanding here is X F 11 that exhibits the gauge group and the matter representations that coincide precisely with that of the Standard Model.

Polyhedron
We construct an elliptically fibered Calabi-Yau manifold X F 11 with an arbitrary base B and general elliptic fiber given by the elliptic curve E in P F 11 . The toric data of P F 11 is encoded in Figure 18, where the corresponding polyhedron F 11 , a choice of homogeneous coordinates as well as its dual polyhedron F 6 are shown. In the monomials that are associated to the integral points of F 6 according to (2.23), we have set e i = 1, ∀i, for brevity of our notation.
The Stanley-Reisner ideal of P F 11 can be read off from Figure 18 where the sections s i take values in the line bundles associated to the divisor classes in (3.8).
The elliptic fibration X F 11 has three rational sections. Two of these are linear independent, that means the MW-group of X F 11 has rank one. The coordinates of the sections read where q 2 , q 3 are the homogeneous polynomials in [u : v : w] of degree two and three that remain after factoring out e 1 and e 2 u. The corresponding I 2 -and I 3 -fibers are depicted in Figure 19.
In summary, the total gauge group of X F 11 is This enables the computation of the Shioda map (2.5) of the sectionŝ 1 . It reads which follows since the sectionŝ 1 does not intersect the zero section, see (3.143), implying π(S 1 · S 0 ) = 0 , (3.148) and from the intersections ofŝ 0 andŝ 1 with the codimension one fibers in Figure 19, yielding The data of the MW-group is completed by the height pairing (2.6) ofŝ 1 . It is computed as where we use the universal intersection relation (2.7) as well as (3.148). Next, we turn to the codimension two singularities of the WSF of X F 11 to calculate its charged matter spectrum. Here, all representations under the gauge group are determined using the methods outlined in Section 2.1. The non-Abelian representations readily follow from the discriminant, whereas the charged singlets are determined from the primary decomposition of the complete intersection (2.14) for the sectionŝ 1 . Using this, we find seven singularities in codimension two. Five of these lead to the matter representations and the corresponding codimension two fibers in X F 11 given in the first and third column of Table 20, respectively. The remaining singularities at s 3 = s 2 6 − 4s 2 s 9 = 0 and s 6 = s 9 = 0 are of Type III and IV , respectively. Since they are just degenerations of the codimension one fibers in Figure 19 without additional P 1 's, they do not yield further matter representations. The adjoint representations in the last two rows of Table 20 are shown for completeness.
The multiplicities of the charged hyper multiplets that are presented in Table 20 are straightforwardly computed from the homology class of all complete intersections V (I (k) ), k = 1, . . . , 5.
We complete the matter spectrum of X F 11 by the number of neutral hyper multiplets, which is computed from (2.11) using the Euler number χ(X F 11 ) given in (C.1). It reads  Table 20 and (3.151) we check cancelation of all 6D anomalies in (A.1), following the discussion of Appendix A.
We conclude our analysis with the Yukawa couplings of the charged matter spectrum of X F 11 , corresponding to the codimension three singularities of its WSF. All Yukawa points of X F 11 are presented in Table 21.

Multiplicity
Splitting Locus    (2) × U(1) and corresponding codimension two fibers of X F 11 . The adjoint matter is included for completeness.
We compute the Weierstrass form (2.1) of (3.155) using Nagell's algorithm. The WScoordinates of the sectionsŝ 1 andŝ 2 are given by (B.9) and (B.10), respectively, in the limit s 3 = s 4 = s 8 = s 10 = 0. Similarly, we obtain the functions f and g from (B.1) and (B.2) using this specialization. From this the discriminant ∆ is readily computed. The factorization of ∆ shows the presence of two At these loci, the constraint (3.155) factorizes as where q 3 , q 3 are the remaining polynomials after factoring out e 1 and e 2 . The corresponding I 2 -fibers are depicted in Figure 21. In summary, the total gauge group of X F 12 is Analogous to Section 3.3.3, we obtain the divisor classes of the Cartan divisors of X F 12 as (1, 2) (−1,−1/2) V (I (6) ) := {s 9 = 0 s 5 s 2 2 − s 6 s 2 s 1 + s 7 s 2 1 = 0} V (I (7) ) := {s 7 = 0 V (I (8) ) := {s 9 s 2 − s 7 s 5 = 0 Here S 0 , S m denote the divisor classes of the sectionsŝ 0 ,ŝ m , respectively, and we used π(S 1 · S 2 ) = π(S 1 · S 0 ) = π(S 2 · S 0 ) = 0 , (3.161) which follows directly from (3.156) as well as which can be deduced from Figure 21. Employing these intersection relations along with (2.7), we obtain the height pairing (2.6) as To obtain the charged matter spectrum we analyze the codimension two singularities of the WSF of X F 12 . The corresponding representation under the gauge group are determined following the general procedure outlined in Section 2.1. As before all non-trivial representations of the non-Abelian part of G F 12 are easily read off from the discriminant. The charged singlets are obtained by the primary decompositions of the two complete intersections (2.14) associated to the sectionsŝ 1 andŝ 2 . We find ten codimension two singularities, eight of which lead to the matter representations and the corresponding codimension two fibers in X F 12 given in the first and third column of Table 22, respectively. At the remaining loci, namely s 7 = s 2 6 − 4s 2 s 9 = 0 and s 9 = s 2 6 − 4s 5 s 7 , we find Type III singularities and thus no additional matter, cf. Section 3.3.3 for more details. We note that the matter locus V (I (3) ) agrees with the singular locus of the dependent rational section in (3.156). For completeness, matter in the adjoint representation of G F 12 is also given in the last two rows of Table 22.
The number of neutral hyper multiplets completes the matter spectrum of X F 12 . It is computed from (2.11) using the Euler number (C.1) of X F 12 . It reads There are a base-dependent number T of tensor and V = 7 vector multiplets. Finally, we use S b SU(2) 1 = {s 7 = 0}, S b SU(2) 2 = {s 9 = 0}, (3.163), the charged spectrum in Table 22 and (3.164) to confirm that all 6D anomalies in (A.1) are canceled.
We conclude with the list of all codimension three singularities of the WSF of X F 12 and the corresponding Yukawa points in Table 23.

Polyhedron
Consider the elliptically fibered Calabi-Yau manifold X F 14 with base B and general elliptic fiber given by the elliptic curve E in P F 14 . In Figure 22 the toric data of P F 14 is summarized in terms of its polyhedron F 14 , a choice of homogeneous coordinates as well as its dual polyhedron F 3 with all monomials (shown in the patch e i = 1, ∀i) corresponding to its integral points.
As a prerequisite for the analysis of the singularities of X F 14 , we compute its Weierstrass form (2.1). This is obtained by applying Nagell's algorithm to (3. where q 2 , q 3 and q 3 are the polynomials that remain after factoring out w, e 1 e 4 and e 2 . The fibers at these three codimension one loci are depicted in Figure 23. In summary, the gauge group of X F 14 is given by The divisor classes of the Cartan divisors are calculated in a similar way as in Section 3.3.3. Using these results, we compute the Shioda map (2.5) of the sectionŝ 1 as Here S 0 , S 1 denote the divisor classes of the sectionsŝ 0 ,ŝ 1 , respectively, and we have employed which follows from (3.169) as well as which can be deduced from Figure 23. Employing (3.173), we compute the height pairing (2.6), using these results and the intersection (2.7), as Next, we determine the spectrum of charged matter by investigating the codimension two singularities of the WSF of X F 14 . As before, all matter representations under the gauge group G F 14 are obtained by application of the techniques discussed in Section 2.1. Again, the nontrivial representations under the non-Abelian part of G F 14 easily follow from the discriminant ∆, while the charged singlets require the primary decomposition of the locus (2.14) for the sectionŝ 1 . We find nine codimension two singularities in X F 14 . Six of these lead to the matter representations and the corresponding codimension two fibers in X F 14 given in the first and third column of Table 24, respectively. The remaining loci, namely s 1 = s 2 6 − 4s 5 s 7 = 0, s 9 = s 2 6 − 4s 5 s 7 = 0 and s 6 = s 7 = 0, support two type III and one type IV fiber, respectively, and thus do not support further representations. The adjoint representations of G F 14 are shown in the last three rows of Table 24 for completeness.
We note that the fiber corresponding to the representation (2, 1, 2) 0 is non-split, cf. Section 3.4.2 for a more detailed discussion. We have indicated the fibers that are exchanged by codimension three monodromies by dashed lines in Table 24.
The multiplicities of the charged hyper multiplets are presented in the second column of Table 24. They are computed directly from all complete intersections V (I (k) ), k = 1, . . . , 5.
The matter spectrum of X F 14 is completed by the number of neutral hyper multiplets, which can be computed from (2.11) using the Euler number (C.1) of X F 14 . It is given by (1, 3, 2) 1/6 S 7 S 9 V (I (6) ) :=   We note that there are a base-dependent number T of tensor and V = 15 vector multiplets. As a consistency check we confirm cancelation of all 6D anomalies following Appendix A, employing the divisors S b SU(2) 1 , S b SU(2) 2 , S b SU(3) , (3.176), the spectrum in Table 24 and (3.177). Finally, we list all codimension three singularities of the WSF of X F 14 and the corresponding Yukawa points of an F-theory compactification to 4D in Table 25.

Fibrations with gauge groups of rank 5 and 6 and MW-torsion
In this section we study the toric hypersurface fibrations constructed from the fiber polyhedra F 13 , F 15 and F 16 . These are the three toric hypersurface fibrations that have non-trivial Mordell-Weil torsion and give rise to non-simply connected gauge groups in F-theory. The Calabi-Yau manifold X F 13 has Mordell-Weil group Z 2 , X F 15 has Mordell-Weil group Z ⊕ Z 2 and the fibration X F 16 has Mordell-Weil group Z 3 [41,46]. We confirm these findings by explicitly working out the WSF of these toric hypersurface fibrations, which are shown to precisely take the standard form of WSF's with these MW-torsion groups, cf. [53].
The influence of the MW-torsion on the spectrum of F-theory was discussed recently in [46]. There, the models considered in this section were also studied, but under the assumption of a holomorphic zero section. Here, we relax this condition which results in additional gauge groups and matter representations. This has interesting consequences for the phenomenology of these models, because we find that the gauge groups and matter representations of X F 13 and X F 15 are completed precisely into the ones of the Pati-Salam and trinification model, respectively.

Polyhedron
Consider the elliptically fibered Calabi-Yau manifold X F 13 with base B and general fiber given by the elliptic curve E in P F 13 . The toric diagram of the fiber polyhedron F 13 as well as a choice of homogeneous coordinates and its dual polyhedron are depicted in Figure 24, where we have set e i = 1, ∀i, in the monomials that are associated to the integral points of F 4 by (2.23).
We note that P F 13 is the blow-up of P 2 at five non-generic points, that is defined by u → e 1 e 2 e 3 e 4 e 2 5 u , w → e 1 e 5 w , v → e 2 e 2 3 e 3 4 v . and take values in the line bundles associated to the divisor classes given by: where the divisor classes of the s i are given in (3.8).
There are two seemingly rational sections of the fibration of X F 13 . However, there is one torsional relation between them, which reveals that the MW-group is pure torsion, namely Z 2 [41]. The coordinates of our choice for the zero sectionŝ 0 and the section of order two arê The presence of MW-torsion restricts the matter spectrum realized in F-theory [53]. Indeed, the torsion acts on the gauge group, turning it into a non-simply connected group, which reduces its weight lattice, i.e. the realized representations. In [46] this has recently been understood in terms of a geometric k-fractional refinement of the coweight lattice. In particular, it has been argued that the MW-torsion Z 2 of X F 13 forbids the presence of fundamental matter in this model. We will confirm these findings in the following explicit analysis.
We begin by computing the Weierstrass form (2.1) of (3.181). As an intermediate step we use the birational map of X F 5 to the Tate form given in [39] in the limit s 5 = s 7 = s 8 = 0. We obtain the local Tate coefficients (B.14) from which we readily compute the functions f and g, that are given in (B.16). We note that the same WSF arises from the global Tate model given (B.15), which precisely agrees with the Tate form of a model with Z 2 MW-torsion as argued in [53], confirming the presence of Z 2 MW-torsion in X F 13 .
Using these results, we readily compute the discriminant ∆, which allows us to find all codimension one singularities of the WSF of X F 13 . We find two I 2 -singularities over the divisors S b SU(2) 1 = {s 1 = 0} ∩ B and S b SU(2) 2 = {s 3 = 0} ∩ B in B as well as an I 4 -singularity over the divisor S b SU(4) = {s 9 = 0} ∩ B in B. Along these divisors the constraint (3.181) factorizes as where q 2 , q 3 and q 2 are the polynomials that remain after factoring out v, e 1 and ue 2 e 3 . The corresponding fibers are depicted in Figure 25 and give rise to two SU(2) and one SU(4) gauge groups. There is another potential codimension one singularity of the WSF of X F 13 , where the fiber of X F 13 splits into two P 1 . However, as it is shown in [46], the torsional MW-group identifies these two P 1 's, so that the fiber in the quotient space is a single singular P 1 . Thus, Figure 25: Codimension one fibers of X F 13 . The crosses denote the intersections with the zero section.
there is no additional gauge symmetry and the gauge group of X F 13 is   (2) 2 )/Z 2 and corresponding codimension two fibers of X F 13 . The adjoint matter is included for completeness.
As before, cf. Section 3.3.3, we calculate the classes of the Cartan divisors of X F 13 as  Next, we calculate the charged matter spectrum of X F 13 , which requires the analysis of all its codimension two singularities. We directly read off from the discriminant of X F 13 the loci of six codimension two singularities. Four of these lead to the matter representations in the first column of Table 26 that are determined, using the techniques discussed in Section 2.1, from the corresponding codimension two fibers in X F 13 given in the third column of the same table. The remaining loci s 1 = s 2 6 − 4s 2 s 9 = 0 and s 3 = s 2 6 − 4s 2 s 9 = 0 are both of type III , that we first encountered in Section 3.3.3, and, thus, do not support additional matter representations. The three adjoint representations in the last three rows of Table 26 are shown for completeness.
We find three singularities which support the bi-fundamental representations and one singularity leading to an anti-symmetric representation of SU (4), but no fundamental representation. This has been anticipated before, due to the action of MW-torsion on the gauge group G F 13 given in (3.184). In addition, we find that the fibers at the loci V (I (1) ) and V (I (4) ), that correspond to the (2, 2, 1) and the (1, 1, 6) representation, are non-split, cf. Section 3.4.2 for more details. The P 1 's drawn with a dashed line are interchanged by codimension three monodromies.
The spectrum of matter of X F 13 is completed by the number of neutral hyper multiplets, which is computed from (2.11) using the Euler number given in (C.1) of X F 13 . It reads The number T of tensor multiplets is given by the base-dependent expression (2.17) and we have V = 27 vector multiplets. In order to check that all 6D anomalies are canceled we use the divisors S b SU(2) 1 , S b SU(2) 2 and S b SU(4) as well as the charged spectrum in Table 26 and (3.186). Indeed, we find that all 6D anomalies in (A.1) are canceled.
We conclude this section with the list of all geometrically allowed Yukawa couplings of the charged matter spectrum of X F 13 , that is given in Table 27.

Polyhedron
We construct an elliptically fibered Calabi-Yau manifold X F 15 over a base B and with general fiber given by the elliptic curve E in P F 15 . The toric data of P F 15 is encoded in Figure 26, that shows the polyhedron F 15 , our convention for projective coordinates as well as the dual polyhedron F 2 , with the monomials (in the patch e i = 1, ∀i) associated to its integral points. We note that P F 15 is obtained from P 2 by the five non-generic blow-ups defined by u → e 1 e 2 e 3 e 4 u , w → e 1 e 2 4 e 5 w , v → e 2 e 2 3 e 5 v . Their divisor classes are given by: The fibration X F 15 has four seemingly rational points with one linear and one torsional relation between, showing that the MW-group is Z ⊕ Z 2 [41]. Their coordinates arê where we chooseŝ 0 as the zero section andŝ 1 as the generator of the free part of the MW-group.
Next, we compute the Weierstrass form (2.1) of (3.190). Again, we use the birational map from X F 5 in [39] to first obtain the local Tate coefficients (B.21), which determine the WSF (B.23). The equivalent global Tate model in (B.22) is precisely of the form of an elliptic fibration with Z ⊕ Z 2 MW-group, that has been studied in [53]. The WS-coordinates of the sectionŝ 1 are given by (B.8) after setting s 1 = s 3 = s 4 = s 8 = 0 and the torsion point is given in (B.24).
These results allow us to compute the discriminant ∆ of X F 15 . We find four I 2 -singularities over the divisors S b Along these divisors the constraint (3.190) factorizes as where q 2 , q 2 , q 3 and q 3 are the polynomials that remain after factoring out w, v, e 1 and e 2 , respectively. The corresponding codimension one fibers in X F 15 are shown in Figure 27. In summary, the gauge group of X F 15 is  In order to calculate the Cartan divisors of X F 15 we use a similar logic as in Section 3.3.3. We obtain the following divisor classes: ( 3.194) With these results at hand, we compute the Shioda map (2.5) of the sectionŝ 1 as Here S 0 , S 1 denote the divisor classes of the sectionsŝ 0 ,ŝ 1 , respectively, and we used π(S 1 · S 0 ) = 0 , (3.196) which directly follows from (3.191) as well as     where we used (2.7) and (3.196). Next, we turn to the analysis of the codimension two singularities of the WSF of X F 15 and the determination of the charged matter spectrum. As before, all representations are determined from the codimension two fibers in X F 15 following the procedure presented in Section 2.1. All codimension two singularities are easily seen from the discriminant ∆. We find nine codimension two singularities. Seven of these lead to the matter representations and the corresponding codimension two fibers in X F 15 given in the first and third column of Table 28, respectively. At the remaining two loci s 2 = s 2 6 − 4s 5 s 7 = 0 and s 9 = s 2 6 − 4s 5 s 7 = 0, the fiber is of Type III , cf. Section 3.3.3, which means that there are no additional matter representations. Again, we observe the absence of fundamental matter which is consistent with the action of the MWtorsion on the gauge group in (3.193). The spectrum of charged matter is completed by the matter in the adjoint representations (3, 1, 1, 1) 0 , (1, 3, 1, 1) 0 , (1, 1, 3, 1) 0 and (1, 1, 1, 3) 0 given in the last four rows of Table 28.
The total matter spectrum of X F 15 is completed by the number of neutral hyper multiplets, which is computed from (2.11) using the Euler number χ(X F 15 ) given in (C.1). It reads The number T of tensor multiplets is base-dependent, cf. (2.17), and we have V = 24 vector multiplets. Finally, we use this together with the divisors S b SU(2) I , I = 1, . . . , 4, (3.198), the charged spectrum in Table 28 and (3.199) to check that all 6D anomalies in (A.1) are canceled.
Finally, we present our analysis of codimension three singularities of the WSF of X F 15 and the corresponding Yukawa points in a compactification to 4D in Table 29.
There are three rational sections of the fibration of X F 16 with two torsional relations between them which shows that the MW-group is Z 3 [41]. The coordinates of the sections arê where we chooseŝ 0 as the zero section.
In order to compute the WSF (2.1) of (3.203), we first compute the Tate form using the birational map from X F 5 [39] in the limit s 2 = s 3 = s 5 = s 8 = 0. The global Tate coefficients are given in (B.28), which is precisely of the form of an elliptic fibration with MW-group Z 3 [53]. The WSF is given in (B.29) and the WS-coordinates of the torsional section are given in (B.30).
We readily compute the discriminant ∆, which allows us to find all codimension one singularities of the WSF of X F 16 . We find three I 3 -singularities over the divisors The hypersurface constraint (3.203) factorizes along these divisors as where q 1 , q 3 and q 3 are homogeneous polynomials in [u : v : w] that remain after factoring out vw, e 1 e 4 and e 2 e 3 . The corresponding fibers are depicted in Figure 29. In summary, the gauge group of X F 16 is   The determination of the charged matter spectrum requires the knowledge of the codimension two singularities of the WSF of X F 16 . Again, we then extract the corresponding representation data by application of the general recipe outlined in Section 2.1. By investigation of the discriminant of X F 16 we readily find three codimension two singularities that lead to bi-fundamental representations. These and their corresponding codimension two fibers in X F 16 are listed in the first and third column of Table 30   codimension two singularities at s 6 = s 9 = 0, s 6 = s 7 = 0 and s 1 = s 6 = 0, that, however, yield fibers of Type IV in X F 16 , that do not support any additional matter. Again, we do not find fundamental matter, confirming the restrictions imposed on the spectrum of X F 16 by the MW-torsion. The spectrum of charged matter is completed by the matter in the adjoint representations (8, 1, 1), (1, 8, 1) and (1, 1, 8) given in the last three rows of Table 30. We recall that they do not originate from codimension two fibers of X F 16 , but are present if the divisors S b G are higher genus curves in B. The multiplicity of charged hyper multiplets in the adjoint is given by (2.8).
We complete the matter spectrum of X F 16 by the number of neutral hyper multiplets, which is computed employing the Euler number (C.1) of X F 16 from (2.11). It reads (3.208) The base-dependent number T of tensor multiplets is given by (2.17) and we have V = 24 vector multiplets. Finally, we use this together with the divisors S b SU(3) I for I = 1, 2, 3, the charged spectrum in Table 30 and (3.208) to confirm cancelation of all 6D anomalies (A.1).
By analyzing the codimension three singularities of the WSF of X F 16 , we finally calculate all Yukawa couplings of the charged matter spectrum of X F 16 , that are given in Table 31.

The Toric Higgs Branch of F-Theory
In Section 3 we discussed in great detail the geometric derivations of the gauge groups and matter spectra of all genus-one fibrations X F i based on the 16 polyhedra. Here we show that the effective SUGRA theories obtained from these fibrations are not isolated from each other but connected by means of the Higgs mechanism. This section is devoted to the study of those transitions and the resulting network of theories summarized in Figure 1. As we have noted earlier, this network is nothing but the field theoretic realization of the network of extremal transitions relating the X F i that are induced by blowing up/down in the toric ambient varieties P F i of their genus-one fibers. This network, to which we refer to as the toric Higgsing diagram, is a powerful consistency check for the results of Section 3 and exhibits some remarkable features: • The Higgsing diagram is symmetric with respect to the horizontal axis corresponding to the self dual polyhedra, all of which have a gauge group rank equal to three.
• The rank of the gauge groups of a polyhedron and its dual sum up to six.
• Every toric hypersurface fibration can be reached upon a chain of Higgsings starting from one of the three manifolds X F 13 , X F 15 and X F 16 which exhibit non trivial MW-torsion.
• Both the analysis on the geometrical side (see Sections 3.2.1, 3.2.2 and 3.2.3) and the Higgsings (see Section 4.2) lead to the conclusion that the MW-torsion in the X F i with fibers in F 13 , F 15 and F 16 , manifests itself as discrete gauge symmetries in the X F i with fibers in their respective dual polyhedra F 1 , F 2 and F 4 .
In the following we discuss the above features and the Higgsing diagram in more detail. In order to illustrate the relevant features of the Higgsing we focus on a particular sub-branch of the Higgsing diagram which we depict in Figure 30. This includes the transition of the effective theory derived from F-theory on X F 9 to that on X F 5 . This transition is convenient in order to discuss certain (unphysical) redefinitions of the divisor classes that are sometimes needed in order to match the field theoretic results with the geometrical computation. In this example we also describe the matching between the gauge group generators before and after the Higgsing,  Figure 30: A subbranch of the Higgsing chain which we use to illustrate certain features relevant for all Higgsings. We use the Higgsing from X F 9 → X F 5 to demonstrate the type of bundle and charge redefinitions which are needed in order to match a Higgsed model with the geometrical computations. The Higgsings to X F 1 , X F 2 and X F 4 are used to confirm the presence of discrete gauge symmetries.
and how this information can be inferred from the toric diagram (see Section 4.1). After these redefinitions we obtain a perfect match of the massless spectrum of the effective theory after Higgsing with that obtained from the geometrical computation. While our results apply for any generic two-dimensional base B, we also discuss the Higgsing for the specific case of a P 2 base.
Here we comment on specific boundary strata of the moduli space where a specific Higgsing might not be possible but different equivalent Higgsings are. In section 4.2 we focus on the theories corresponding to the genus-one fibrations X F 1 , X F 2 and X F 4 . The field theoretical Higgsings imply the presence of discrete gauge symmetries Z 3 , Z 2 and Z 4 which confirms the results of the geometrical computations. In Section 4.3 we discuss the full chain of Higgs transitions. There we summarize the relevant redefinitions of the gauge group generators and divisor classes needed to match the spectrum obtained in the geometrical computations of Section 3.

Toric Higgsing: an example
We are interested in Higgs transitions relating two supersymmetric vacua in a 6D N = 1 SUGRA theory. This requires that the vacuum expectation value (VEV) of the Higgs field triggering this transition must preserve supersymmetry. This is guaranteed by imposing flatness of the D-term potential along the direction of the VEV. As it has been observed in [92], for the case of a Higgs field that is not in the adjoint representation, at least two hyper multiplets with identical quantum numbers have to acquire a VEV in order to achieve a D-flat direction. Furthermore, the Higgs mechanism of interest here has to relate two toric hypersurface fibrations. Transitions of this type have a clear geometric interpretation in terms of the toric diagrams. This can be seen as follows. First recall that the coefficients s i which appear in the hypersurface constraint of X F k correspond to integral points in the dual polyhedron F * k , see (2.23). As we have seen in Section 3, many X F k exhibit hyper multiplets at codimension two loci of the form {s i = s j = 0}, where s i and s j are neighboring vertices in the dual polyhedron F * k connected by an edge. If the Higgs fields are of this type, they lead to a toric Higgsing, i.e. the resulting theory is associated to a new toric hypersurface fibration X F k , k = k. 26 Here the polyhedron F k is obtained from F k by blowing up the dual polyhedron F * k precisely at the edge connecting the vertices corresponding to s i , s j , respectively, and taking its dual. In F k , this corresponds to the blow-down associated to removing the corner that is dual to the aforementioned edge in F * k . Note that, since the Higgs fields in the toric breaking are never in the adjoint representation, the toric Higgsing is not rank preserving.
To exemplify this, consider the Calabi-Yau manifold X Considering the group theoretical breaking that could be induced by these fields, we see that taking the Higgs fields in any of the singlet representations in (4.1), for instance 1 (1,2) leads to SU(2) × U(1), which coincides with the gauge group of the fibration X F 6 , while a VEV in the 2 (−1,−1/2) or 2 (1,3/2) (see (4.2)) leads to U(1) 2 , i.e. the gauge group expected for X F 5 .
In Figure 31 (a) we depict the polyhedron which is dual to F 9 on the right and highlight in different colors the edges corresponding to the fields in (4.1) and (4.2). They have to be blown up, i.e. subdivided by a new ray, for each of the possible toric Higgsings. In the actual polyhedron of F 9 , we indicate the vertices that are dual to these edges and get cut off in the toric Higgsings. In the following we consider the Higgsing from X F 9 to X F 5 in more detail.
In the case of Higgs fields in the representation 2 (−1,−1/2) , the Higgsing corresponds to a removal of the lower left corner in the polyhedron F 9 and to a blow-up at its dual edge, which is the edge between s 5 and s 9 , in its dual polyhedron (see Figure 31 (b)). After that, we obtain the toric diagram of F 5 as given in Section 3.3.2. Similarly, if we pick VEVs in the 2 (1,3/2) representation, we observe that after the Higgsing, the resulting polyhedron and its dual both have to be reflected along the horizontal axis (see Figure 31 (c)) in order to recover Figure 31 (b). Thus, the obtained effective theories after these two Higgsings are physically equivalent.
However, we note that the geometrical computations leading to the spectrum of X F 5 have been made for the polyhedron given in Figure 31 (b). Thus, whenever we perform a Higgsing with the Higgs in the representation 2 (1,3/2) (leading to Figure 31 (c)) we have to transform certain divisor classes in order to match the multiplicities of hyper multiplets resulting from the Higgsing with those found geometrically. More general, the "Higgsed" polyhedron of F 9 can be brought to the canonical form (i.e. the one used for the computation of the matter spectrum and multiplicities) by means of an SL(2, Z) transformation, which acts simultaneously on the polyhedron and its dual. This transformation determines how to transform the divisor classes in order to recover precisely the effective theory obtained by the geometric computation on X F 5 , as we demonstrate next.

Matching the charged spectrum
In order to match the charged spectrum in the Higgsed theory arising from F-theory on X F 9 with that on X F 5 , we first have to relate the generators of the gauge groups before and after the Higgsing. In X F 9 , the U(1) generators are (see (3.119)) with D 1 being the class of the SU(2) Cartan divisor given in (3.122) as D 1 = S 9 − [e 2 ]. The Shioda maps in X F 5 are given by (see (3.76)) Let us consider first the canonical Higgsing induced by VEVs in the 2 (−1,−1/2) . As shown in Figure 31 this corresponds to blowing down the divisor e 3 = 0. We see that after setting its divisor class [e 3 ] = 0, the following relations hold Since the vertical divisors [K −1 B ], S 7 and S 9 do not contribute to the U(1) charges, these equations allow us to make contact with the charges in X F 5 (which we denote by Q 1 and Q 2 ). Indeed, recalling that Dynkin labels and U(1)-charges are computed according to (2.12) and (2.13), respectively, we translate (4.5) into the charge relation 27 with Q 1 , Q 2 being the U(1)-charges and T 3 = 1 2 D 1 the Cartan generator of the SU(2) in X F 9 . For the Higgsing with fields in the 2 (1,3/2) representation we can proceed in a similar manner. In this case, according to Figure 31, one has to set the divisor class [u] = 0. In addition, one must take into account that the toric diagram is reflected with respect to the canonical one (compare Figure 31 (c) with Figure 8, including the location of the zero section). This implies that the classes of [e 3 ], [v] and [e 1 ] in X F 9 get mapped to [u], [e 1 ] and [v] in X F 5 , respectively. Hence, the Shioda maps (4.3) for X F 9 , written in terms of divisor classes on X F 5 , read Writing both (4.4) and (4.7), in terms of the exceptional divisors E 1 , E 2 as well as the hyperplane class H using (3.8), we find the following relations among them from which it follows that the U(1)-charge redefinition in this case is given by The charge formulas in both cases agree with the field theory expectations i.e. there is a complete gauge singlet in the decomposition of the Higgs field into representations of the residual gauge symmetry. The decomposition of the states for both cases is given in Table 32. We observe that, indeed, all charged states in X F 5 have been reproduced.
In order to match their multiplicities as well, we first recall the basic fact that a hyper multiplet consists of two half-hypers transforming in representations conjugate to each other. Thus, in order to compute the resulting multiplicities after Higgsing, one has to take into account all states transforming under the representation of interest together with their complex conjugates. For example, note that in the branching induced by 2 (−1,−1/2) , the states 2 (0,−1/2) and 1 (1,1) decompose as thus, the multiplicity of hyper multiplets in the representation 1 (0,1) after the Higgsing must be computed as the sum of the multiplicities of 2 (0,−1/2) and 1 (1,1) in X F 9 . Similarly, note that every Higgs doublet decomposes into a neutral and a charged singlet. In computing the multiplicity of such charged singlets after Higgsing, one has to take into account that two of these are absorbed as longitudinal components of the massive W bosons from the broken SU(2).
In the case of the Higgsing induced by a VEV in the 2 (−1,−1/2) representation of X F 9 , we can directly compare the resulting multiplicities with the geometric result of Section 3.3.2.
VEV: 2 (−1,−1/2) VEV: 2 (1,3/2) However, in the case where we turn on VEVs for the fields in the representation 2 (1,3/2) , the multiplicities only match after performing a redefinition of divisor classes. Indeed, we note that on the dual polyhedron, the reflection 28 relating Figure 31 (c) with Figure 8 enforces an exchange of the sections s 5 and s 1 in X F 9 with s 7 and s 3 in X F 5 , respectively. From (3.8) we see that this effectively amounts to a shift in the bundles S 7 and S 9 from X F 9 to X F 5 , which is given by (4.11) Using the shift (4.11) for the second Higgsing, we find that in both cases (either VEVs in 2 (−1,−1/2) or 2 (1,3/2) ), there is a perfect agreement with the spectrum in Table (8) of the toric hypersurface fibration X F 5 .

Matching of the neutral spectrum: Higgsing & Euler numbers
So far we have matched only the charged spectrum of the Higgsed theory of X F 9 with that computed geometrically on X F 5 . In this section we work out the counting of complex structure moduli, that determine the number of neutral hyper multiplets by (2.11), before and after Higgsing. We show that the mismatch of the Hodge numbers h (2,1) (X F 5 ) − h (2,1) (X F 9 ) ≥ 0 precisely agrees with the amount of massless neutral singlets contributed from the Higgs multiplets.
First, let us discuss the geometric side of the matching. For a given Calabi-Yau manifold X, the amount of complex structure moduli can be inferred from its Euler number χ(X) as with h (1,1) (X) given by where rk(G X ) is the rank of the total gauge group G X of X. Thus, in an extremal transition from a toric hypersurface fibration X F i to X F j , with the same base B, the change in h (2,1) reads (4.14) For the specific Higgsing X F 9 → X F 5 , their Euler numbers are given according to (C.1) as so that the difference in their Hodge numbers h (2,1) is given by Here the −1 is the contribution from the change in the rank of the gauge group. Next, in the corresponding field theories we consider the canonical Higgsing, that is induced by a Higgs in the representation 2 (−1,−1/2) . The multiplicity of hyper multiplets in the repre- Table 16. This is also the number of new uncharged singlets produced in the Higgsing X F 9 → X F 5 . However, out of those neutral singlets, exactly one gets massive, as can be seen from the D-term potential. 29 This is also closely related to Goldstone's theorem: As three vectors have been lifted (an entire SU(2) is broken), three hyper multiplets must be removed from the massless spectrum. Two of these hyper multiplets are charged, as we discussed in the previous section, while the third one must be neutral, since it provides the longitudinal component of a massive U(1). In fact, only the simultaneous removal of three vectors and three hyper multiplets makes it possible for the purely gravitational anomaly in (A.1) to cancel after the Higgs mechanism.
Thus, there are precisely S 9 (2[K −1 B ] − S 7 ) − 1 massless singlets after the Higgsing, which precisely agrees with (4.16). In other words, we observe that the neutral massless hyper multiplets resulting from the Higgs mechanism become the complex structure moduli that were gained in the transition X F 9 → X F 5 . In this work, we explicitly confirm the matching of the complex structure moduli for all toric Higgs transitions between two toric hypersurface fibrations .

Allowed regions for base P 2
While we considered in the previous sections the Higgsings for fibrations over an arbitrary two dimensional base, we regard it appropriate to devote some time on a concrete example with base P 2 in order to discuss some subtleties that might arise.
Choosing the base fixes the range of allowed divisor classes for S 7 and S 9 [39]. Expanding these divisors and the canonical class K P 2 in terms of the hyperplane class H B on P 2 , we have S 7 = n 7 H B , S 9 = n 9 H B , K P 2 = −3H B , (4.17) 29 In the case of an adjoint Higgs, all neutral hyper multiplets remain massless as the D-term is exactly zero.

96
with n 7 and n 9 being positive integral coefficients. The effectiveness condition on all divisor classes [s i ] in (3.8) that occur for a given manifold X F i imposes constraints on the allowed values for n 7 and n 9 . These allowed values depend on the choice of the fiber. For the case of X F 9 and X F 5 the allowed regions are depicted in Figure 32. A choice of a point in this diagram constitutes a consistent fibration. Note that the allowed region for X F 5 is fully contained inside that of X F 9 . Thus, there are compactifications X F 9 inside the blue region in Figure 32, for which the transition to X F 5 is not possible because the effectiveness condition for some coefficient s i in X F 5 would be violated. Toric Higgsing X F9 → X F5 allowed Allowed strata for X F9 where the toric Higgsing with 2 (−1,−1/2) is not possible X F5 ≡ X F7 , adjoint Higgsing allowed n 9 Figure 32: Allowed regions of S 7 and S 9 for X F 9 (blue and purple regions) and X F 5 (purple region) with base B = P 2 .
Indeed, this can also be seen from the field theory perspective, as for the points outside the allowed region for X F 5 , the multiplicity of Higgses in the representation 2 (−1,− 1 2 ) , that reads is smaller than two. Recall that in order to have a D-flat potential we need at least two Higgs fields in the same representation to acquire a VEV. 30 We also observe that the Higgs mechanism is possible for all points (n 7 , n 9 ) in the interior of the allowed region of X F 5 . However, for certain points on the boundary of the allowed region for X F 5 we see that the amount of doublets does not suffice for a supersymmetry preserving Higgsing. These points are • 0 ≤ n 7 ≤ 3, n 9 = 0: Here we see that s 9 belongs to the trivial bundle. Since in X F 9 the locus of the SU(2) singularity is precisely {s 9 = 0}, cf. (3.120), it is removed and the gauge group of X F 9 at these points equals that of X F 5 , namely U(1) × U(1). In addition, we see that the spectra of X F 5 and X F 9 in Tables 8 and 16, respectively, match perfectly. 30 Note that in this region, a different Higgsing with 2 (1, 3 2 ) is possible.
Hence we are at points where the strata of the moduli spaces of the two theories overlap and a transition among them is trivial.
• n 7 = 6, n 9 = 3: At this point there are no states in X F 9 which transform in the 2 (−1,− 1 2 ) , so that the toric Higgsing is again not possible. Note also that since s 9 does not belong to the trivial bundle, the SU(2) factor is part of the gauge symmetry of the effective theory. However, in the hypersurface constraint for X F 5 , the sections s 1 , s 5 and s 8 transform in the trivial bundle. Hence, at this particular point one can shift the toric coordinates u, v and w in order to globally set the section s 1 = 0 [42], resulting in a non-toric U(1). This shows that the effective theory of X F 5 coincides with that of X F 7 precisely at the point n 7 = 6, n 9 = 3. Since the rank of the gauge groups of X F 9 and X F 7 coincide, no toric Higgsing is possible. However, on X F 9 , one sees that there is one hyper multiplet in the adjoint of SU (2). In fact one can use this field to induce an adjoint (non-toric) Higgsing which leads precisely to the effective theory of X F 5 at n 7 = 6, n 9 = 3. Indeed, we have explicitly computed that at this particular stratum in moduli space the numbers of complex structure moduli in X F 7 and X F 9 coincide. Similarly, we have confirmed that the entire matter spectrum in X F 7 is reproduced (with the correct multiplicities) after the adjoint breaking from X F 9 . We omit all the details and just state the corresponding U(1) redefinitions, in terms of the generators in X F 9 : Looking at the Calabi-Yau constraint for X F 9 we also observe that, by a shift in the toric coordinates, we can set s 1 = 0. Hence, the hypersurface constraint for X F 7 only contains the additional monomial s 8 w 2 u which is absent in the one of X F 9 . At n 7 = 6, n 9 = 3 the coefficient s 8 is just a constant, cf (3.8), i.e. one degree of freedom. On the field theory side, this degree of freedom corresponds precisely to the single adjoint Higgs on X F 9 .

Higgsings to theories with discrete gauge symmetries
From the analysis carried out in Section 3, we observe that the presence of discrete gauge symmetries is exclusive to the polyhedra F 1 , F 2 and F 4 . On the field theory side, we can use the Higgs mechanism to track the discrete symmetries as well, since these correspond to surviving remnants of broken U(1) symmetries. In fact, one can use the Higgsing diagram to show that the only Higgs mechanism for which the U(1)s are broken to a discrete subgroup, are those leading precisely to F 1 , F 2 and F 4 , as expected geometrically. In this section we want to discuss in some detail those Higgsings leading to the toric hypersurface fibrations with discrete symmetries. To this end, we focus on the possible Higgs branches of X F 6 , X F 5 and X F 3 . In many of the transitions considered here, the charge of the Higgs fields does not allow us to directly infer whether or not there is a non-trivial discrete symmetry. Consider for example the Higgsings from X F 9 → X F 6 , with the toric Higgses given in (4.1). One possibility is to have VEV fields in the representation 1 (1,2) . In principle one might think that, given the charge of the Higgses, there is a discrete remnant of the second U(1). However, this leftover symmetry is trivial, since there is an SL(2, Z) transformation which maps the charge of the Higgses from (1, 2) to (1, 0). As discussed in Section 4.1, the field 1 (1,0) allows for a geometrically equivalent breaking, where it becomes clear that no discrete symmetries are present in X F 6 as there the U(1) charge is minimal.
More general, if we break a U(1) n gauge symmetry by the VEV of a field φ, there is no discrete symmetry left provided the existence of an SL(n, Z) which makes its charge minimal. In other words, after the SL(n, Z) transformation of the VEV field, the charge (q 1 , q 2 , . . . , q n ) takes the form (1, 0, . . . , 0). In this new basis it is obvious that no discrete gauge group is left after the breaking. In other cases a seeming discrete group can be embedded into an unbroken U(1) or the center of a non-Abelian gauge group, showing again the absence of a discrete group.

X F 6 Higgs branches
From the spectrum in X F 6 given in Table 10 we see that there are two possible toric Higgsings, depending on whether the Higgs fields are taken in the representation 2 −3/2 or 1 2 . In the first case the SU(2) × U(1) symmetry in X F 6 is broken to a single U(1), so that this Higgs branch leads to X F 3 . In the second case the SU(2) symmetry remains unbroken, as expected from a Higgsing to X F 4 . The splitting of the states in either cases proceeds according to Table 33. Table 33: Branching of representations under the possible Higgsings of X F 6 to X F 3 and X F 4 .
Using these branchings of the representations on X F 6 into representations of G F 3 , we can compute the multiplicities of the multiplets after Higgsing using Table 10. They read (4.20) The above multiplicities agree with our geometrical result for the spectrum of X F 3 , see Table 6. Note also that naively, due to the non-primitive U(1)-charge of the Higgs field 2 − 3 2 , we expect a surviving discrete Z 3 symmetry. However, this symmetry is contained in the surviving U(1) symmetry in X F 3 , i.e. there is no discrete gauge group on X F 3 as expected geometrically.
In contrast, we see that the U(1)-charge of the VEV 1 2 triggering the transition X F 6 → X F 4 is non-minimal. Thus, we expect a discrete gauge symmetry to be left unbroken, in addition to the SU(2) gauge factor. The decomposition of representations for this Higgsing is given in Table 33. There we see that the doublet after the Higgsing carries a half integral charge with respect to the discrete gauge factor. Hence, one has to rescale all charges by a factor of two, so that all charges become integral. Thus, we see that the resulting discrete symmetry is in fact Z 4 . Note that in this case, the discrete factor is of physical relevance, since we can not embed it into the local gauge group of X F 4 . This implies that in a four dimensional theory, there will be gauge invariant couplings which are absent due to selection rules imposed by the Z 4 -symmetry.
The multiplicities of charged states following from Tables 33 and 10 are given by (4.21) This precisely agrees with the geometrically obtained spectrum of X F 4 in Table 4. We emphasize that charges of the matter states w.r.t to the Z 4 , that we have obtained by Higgsing X F 6 , precisely coincide with those computed by intersections with the four-section in (3.49).

X F 5 Higgs branches
In X F 5 there are two possible toric Higgsing to X F 3 . The Higgs fields in that case are in the representations 1 (−1,−2) or 1 (1,−1) , cf. Table 8. The branching of the representations of X F 5 into representations of G F 3 are shown in Table 34. The resulting spectrum matches that in (4.20), cf. Table 6, up to redefinitions of S 7 and S 9 , that correspond to the transformations needed in order to bring the resulting polyhedron to its canonical form in Figure 7.
VEV: 1 (−1,−2) VEV: 1 (−1,1) VEV: 1 (0,2) Similarly, the polyhedron allows for an additional toric Higgsing from X F 5 to X F 2 which is triggered by VEVs in the representation 1 (0,−2) . This leaves the first U(1) unbroken, together with a remnant discrete Z 2 symmetry from the second U(1). The multiplicities of the charged matter in X F 2 that are deduced by Higgsing from X F 5 are obtained from the group-theoretical 100 branchings shown in Table 34 and the spectrum of X F 5 in Table 8. They read which precisely matches the geometrical result in Table 2.

X F 3 Higgs branches
In the spectrum of X F 3 , the singlet 1 3 allows for a toric Higgsing. In this transition, the U(1) symmetry gets broken to a Z 3 subgroup. The decomposition of representations of X F 3 for this Higgsing is shown in Table 35. As mentioned before, 6D hyper multiplets in the representation 1 2 under the discrete group Z 3 are equivalent to hyper multiplets in the representation 1 1 . Hence, there is only one type of charged hyper multiplet in X F 1 . This is in agreement with the geometrical computation (see Section 3.2.1), where one sees a single codimension two locus supporting an I 2 -fiber. From the Higgsing we can read of the multiplicity of this charged state as

The complete Higgsing chain
Having summarized the relevant features of the toric Higgsing procedure, we devote this section to a complete account on all possible toric Higgsings, that are summarized in Appendix D, Tables 36-38. In these tables we indicate in the first column, which toric hypersurface fibrations X F i → X F j , i = j, are to be related. Then, in the second column, we state the possible toric Higgsings and which fields are to be identified as the Higgs fields that acquire a VEV in the transition. We note that all toric Higgsings between the same two toric hypersurface fibrations are physically equivalent. In the third column the U(1)-generators on X F j are expressed in terms of the U(1)-generators and Cartan generators on X F i . We have checked explicitly in all Higgsings that all matter charges of the fibration X F j determined in Section 3 are obtained. In addition, in some Higgsings, the unbroken non-Abelian gauge group factors are interchanged in the Higgsing process. In these cases, the change of the order of non-Abelian factors is indicated in the third column of the tables in Appendix D. In addition in most of the cases, an SL(2, Z)-transformation on the "Higgsed" polyhedron is necessary in order to bring it into the canonical form used for the geometric computations in Section 3. These transformations determine a unique redefinition of the divisor classes, similar as in (4.11), that is necessary in order to compare the matter multiplicities of the representations obtained after the Higgsing with the ones obtained by inspecting the geometry of X F j . The relevant redefinitions are shown in the last column of the tables in Appendix D.
The decompositions of the representations on X F i under the group G F i into representations of the unbroken gauge group G F j after Higgsing can be found in Appendix E for all canonical toric Higgsings. We have checked in all cases that the matter spectra obtained by Higgsing the theory on X F i to the one corresponding to X F j agree with those of Section 3, which provides another non-trivial check of the geometric analysis presented there.
We conclude this section with one final observation. As highlighted before, we observe that for every transition between two toric hypersurface fibrations X F j → X F k there exists a dual transition between X F * k → X F * j . This symmetry of the Higgs diagram in Figure 1 can be directly understood from the interpretation of the toric Higgsing on the level of the fiber polyhedron and its dual as we will explain in the following.
As pointed out in Section 4.1, a toric Higgsing acts exactly as a blow-down in the original polyhedron F j and a blow-up F * j in its dual polyhedron: with j > k. Next we consider the inverse of the above process: We take F * k as the starting polyhedron with F k as its dual. From the diagram (4.24), we know that there exists a blow-up map from F * j to F * k . However, now we take its inverse map as the blow-down from F * k to F * j . The same can analogously be done for the dual polyhedron F k whose blow-up map is obtained from the inverse of the blow-down map in (4.24). Consequently, we arrive at the following map for the dual Higgs transition (4.25) The above relation holds for every toric Higgsing and hence shows, that every Higgs transition has indeed a dual counterpart. However in general we observe more equivalent transitions between higher polyhedra than in their dual counterparts. In the geometry this reflects the fact, that in polyhedra with a larger area, there are more ways to embed subpolyhedra. On the field theory side this corresponds to less representations, that can be used for the Higgsing. An example is the transition X F 15 → X F 12 which can be equally realized by cutting any of the four vertices in the square of F 15 . However the dual transition F 5 → F 2 can only be achieved by deleting the unique node and hence there is only one Higgsing possible.

Conclusions
In this work we have analyzed F-theory compactifications on all toric hypersurface fibrations X F i . In these manifolds the genus-one fibers are given as a hypersurface in any of the 2D toric varieties associated to the reflexive polyhedra F i depicted in Figure 2. For each of these 16 Calabi-Yau manifolds we have computed the full MW-group (cf. with the results of [41] for the toric MW-group) and determined all codimension one, two and three singularities and the corresponding reducible fibers in the crepant resolutions X F i . Our work presents the first complete analysis of all these aspects. In the course of our study, we have encountered some codimension two fibers which are non-split and others whose complete splitting is visible only by computing their associated prime ideals. In addition, we have computed the Hodge numbers of the X F i in the case of Calabi-Yau threefolds. All these geometric results determine the gauge groups, matter representations and Yukawa couplings of the effective SUGRA theories of Ftheory on these manifolds. We have shown that these effective theories are anomaly-free in 6D, which proves in turn the completeness of our analysis of codimension one and two singularities of all these models [93].
The gauge groups we have found range from rank zero to six with up to three U(1)-factors corresponding to a rank three MW-group of rational sections. The Calabi-Yau manifolds X F 1 , X F 2 and X F 4 , that are constructed as fibrations of the cubic in P 2 , the biquadric in P 1 × P 1 and the quartic 31 in P 2 (1, 1, 2), respectively, do not have a section and lead to F-theory models with discrete gauge groups Z 3 , Z 2 and Z 4 . We have established a relationship between the order of the multi-section in these Calabi-Yau manifolds and the order of the discrete gauge group. Furthermore, we have shown the existence (and computed the multiplicity) of I 2 -singularities that support matter charged only under these discrete groups. We have also a proposal for a "Shioda-map" of multi-sections, that allowed us to consistently compute the charges of all matter fields under these discrete gauge groups. In addition, by an explicit computation of the respective generators of their rank one MW-groups, we have demonstrated that X F 2 and X F 3 (which is constructed as a fibration of the elliptic curve in dP 1 ) both yield effective theories with one U(1)-gauge field. Most notably, we have found the first F-theory realization of charged singlets with U(1)-charge q = 3 in X F 3 . Furthermore, the non-trivial consistency conditions imposed on the U(1)-charge lattice of a model in quantum gravity [89] have been checked.
We emphasize that the non-toric nature of the generator of the MW-group of the fibration X F 3 was key to obtaining this U(1)-charge. We expect that the presence of non-toric sections can lead to more exotic U(1)-charge assignments of matter than those that occur in toric cases. Such a situation can be desirable as these exotic charge assignments could serve to control the phenomenology of particle physics models constructed in F-theory.
Besides these geometrical advances, we have shown that those extremal transitions between all toric hypersurface fibrations X F i , which are induced by toric blow-downs in the toric varieties P F i , can be described by a corresponding Higgs mechanism in the effective theories of F-theory on the X F i . Although this correspondence between geometry and physics is expected to hold in general, also in chiral F-theory compactifications to 4D 32 , we considered here the 6D case. 31 As remarked already, this case has been subject of recent attention in [54][55][56]. 32 As 4D chirality is induced by G 4 -flux in F-theory, this requires the understanding of the behavior of G 4 -flux We have explicitly worked out the full network of all toric Higgs transitions, shown in Figure  1, in the 6D SUGRA theories of these F-theory models. Cutting off a vertex in a polyhedron F i corresponds to a blow down in P F i , which implies the removal of a corresponding divisor in X F i . After this blow-down, a new monomial can be added to the hypersurface constraint of X F i , resulting in a new Calabi-Yau manifold X F i . In the effective theory of F-theory on X F i , this transition corresponds to giving a VEV to a particular matter field along a D-flat direction and a consequent breakdown of the gauge symmetry of the theory. For all extremal transitions, we have identified the relevant matter field that has to acquire a VEV and matched the effective theory after Higgsing with the one obtained geometrically from F-theory on X F i . As an explicit example, for a specific choice of the base B = P 2 , we have described the transition X F 9 → X F 5 for all strata in moduli space, that are labeled by (n 7 , n 9 ), cf. Figure 32. There we have found that a D-flat Higgsing in the effective theory is only possible for those points (n 7 , n 9 ), that are allowed for both X F 9 and X F 5 . In this context, we have also commented on the different gauge groups that appear on the boundary of the allowed region for (n 7 , n 9 ) in X F 5 .
We have found that the full toric Higgs network in Figure 1 is beautifully mirror symmetric under the exchange of a polyhedron F i and its dual F * i . 33 This gave rise to a number of interesting observations: The toric hypersurface fibrations X F i and X F * i always have the same amount of U(1)-symmetries and the ranks of their gauge groups, G F i and G F * i , always sum up to six. Indeed, this sum rule of the rank of the gauge group is a direct consequence of the sum rule for the volumes of F i and its dual F * i . 34 The duality between X F i and X F * i is realized also on the level of the Higgs transitions, i.e. for every toric Higgs transition X F i → X F i , there is a dual Higgs transition X F * i → X F * i . In addition, we have observed that this duality maps discrete symmetries in F-theory to Mordell-Weil torsion.
Finally, let us highlight some features of toric hypersurface fibrations, which make them attractive for particle physics applications. First, recall that the presence of discrete symmetries is a desirable feature in field theoretic constructions beyond the standard model. 35 So far, all efforts towards embedding the standard model in F-theory have been based on compactifications with a zero section, which are typically free of discrete symmetries. 36 Since discrete symmetries arise naturally in genus one-fibrations with multi-sections, this type of compactifications constitutes a promising new arena for engineering semi-realistic particle physics models.
In addition, we have found concrete toric hypersurface fibrations, that directly realize the gauge group and representations of the Standard Model (X F 11 ), the Pati-Salam model (X F 13 ) and the trinification model (X F 16 ). Even more interestingly, we have found that the Standard Model can be obtained via toric Higgsings from the Pati-Salam or the trinification model, both of these models being at the same time the two theories with the maximal non-Abelian gauge during extremal transitions in Calabi-Yau fourfolds, which is discussed in [94]. The results of [95] will be crucial for carrying out any quantitative analysis in this context. 33 We note that a similar observation has been made for elliptically fibered toric K3 surfaces in [96]. 34 We thank Albrecht Klemm for explaining this fact to us. 35 The reader is referred to [97][98][99][100][101] for a selection of discrete symmetries which have been invoked in order to forbid certain unappealing operators in supersymmetric models. 36 Instead of discrete symmetries, the phenomenology of these models is kept under control by virtue of additional U(1) symmetries. Discrete symmetries can arise by a manual breakdown of these additional U(1)s. This possibility has been studied in e.g [102,103]. groups among all toric hypersurface fibrations.

Outlook
For future directions it would be interesting to use the effective theories we have obtained for particle physics applications. Since the Pati-Salam and trinification model are the two maximal toric enhancements of the Standard Model, as we have seen, they are natural candidates for toric unified model building in F-theory. It would be fascinating to work out the phenomenological implications of this observation.
Clearly, the results of this paper most directly apply to six dimensional or to non-chiral four dimensional compactifications of F-theory. Thus, a natural and most straightforward extension of this work, that would also be crucial for phenomenological applications, is the inclusion and construction of G 4 -flux as well as the computation of the chiral indices of all the matter representations found in all toric hypersurface fibrations X F i , following the recipe and techniques described in [42].
The beautiful realization of mirror symmetry in the Higgs network and related observations like the sum rule for the gauge groups and the connection between Mordell-Weil torsion and discrete symmetries are topics of further investigation. It would be exciting to understand all these observations by unveiling a common structure underlying all toric hypersurface fibrations, for example a master gauge group into which all gauge groups G F i could be embedded. This might require a departure from the toric framework.
There has been a lot of recent progress in obtaining matter representations in F-theory using deformations or, in physical terms, the Higgs mechanism [104,105]. It would be very interesting to see how the results about the Higgs network of the toric hypersurface fibrations obtained here can be worked out using the deformation techniques employed there.
Finally, it would be interesting to use the toric hypersurface fibrations studied here also for compactifications of M-theory to engineer 3D N = 2 gauge theories and to study their Coulomb-branches and phase structures, see [106][107][108][109] (and also the seminal works [110,111]) for recent detailed analyses of the phase structure of 3D SU(N )-gauge theories for all N ≤ 5.

A Anomaly Cancellation Conditions in 6D
In this appendix we summarize the consistency relations that have to be obeyed by an anomalyfree 6D SUGRA theory. We follow the conventions and notations of [24,39,80], to which we also refer for further details.
There are three qualitatively different types of anomalies, the pure gravitational anomalies, the mixed gauge-gravitational anomalies and the pure gauge anomalies. Depending on the number of gauge group factors, mixed anomalies between different gauge group factors are present. A theory is referred to as anomaly-free if all one-loop anomalies are canceled by the contributions from the anomalous variations of Green-Schwarz (GS) counter-terms.
For an effective SUGRA theory in 6D, the anomaly cancellation conditions read: x qm,qn,q k ,q l q m q n q k q l = b (mn · b kl) (Pure Abelian) F m F n trF 2 κ : R,qm,qn x R,qm,qn q m q n A R = b κ λ κ · b mn (Non-Abelian-Abelian) Here, we have given the terms in the 6D anomaly polynomial, whose coefficients are the respective anomalies. The Ricci tensor is denoted by R and the field strengths of the non-Abelian and Abelian gauge field for the gauge group factor G κ and the m th U(1) are denoted by F κ and F m , respectively. The overall number of hyper, vector and tensor multiplets is denoted by H, V and T , respectively and the variables 'x · ' denote the multiplicities of certain charged hyper multiplets: x R , x R,qm and x R,qm,qn are the number of hyper multiplets in the representation R, in the representation R with charge q m under U(1) m and in the representation R with charges {q m , q n } under U(1) m ×U(1) n , respectively; x qm,qn and x qm,qn,q k ,q l denote the number of matter hyper multiplets with charges (q m , q n ) and (q m , q n , q k , q l ) under U(1) m × U(1) n and U(1) m × U(1) n × U(1) k × U(1) l , respectively. In the contributions from the GS counter-terms, a, b κ and b mn are the anomaly coefficients. These transform as vectors of SO(1, T ), and are determined by the underlying microscopic theory. In our F-theory compactification these coefficients can be readily interpreted in terms of geometrical objects. We have a = [K B ] , b κ = S b Gκ , b mn = −π(σ(ŝ n ) · σ(ŝ m )) , where K B is the canonical divisor of B, S b Gκ is the divisor on B defined in (2.2) supporting the non-Abelian group G κ and π(σ(ŝ n ) · σ(ŝ m )) is the Néron-Tate height pairing defined in (2.6). Under these identifications, the inner product in (A.1) is replaced by the intersection pairing on the base B.
In addition, in the anomalies (A.1), we have made use of the following group theoretical relations between traces in different representations R: Here 'tr' denotes the trace with respect to the fundamental representation, while tr R is the trace for a given representation R. For κ = SU(N ) with N > 3, the group theory factors in (A.3) assume the following values: 37

Representation Dimension
For the specific case of SU(2) and SU(3), the coefficients A R coincide with those given in the table. In contrast to that, the coefficients B R and E R are equal to zero in both cases. The actual coefficient C R can be computed using the values for B R and C R in the above table, as C R + 1 2 B R , for N = 2, 3. Finally, the coefficient λ κ in (A.1) corresponds to the group normalization constant defined by λ κ = 2c κ /E adj κ , where c κ is the dual Coxeter number for the group G κ and E adj κ is E R for the adjoint representation. For G κ = SU(N ), we have λ κ = 1.

B Additional Data on Toric Hypersurface Fibrations
In this appendix we provide the explicit expressions for f and g of the WSF of the Jacobian fibrations of X F 1 , X F 2 and X F 4 . Additionally, we present the explicit WS-coordinates of the rational sections of X F 3 , X F 5 and X F 7 . The functions f , g as well as the WS-coordinates of the rational sections of all other toric hypersurface fibrations X F i can be obtained by specializing the ones presented here. Finally, we derive the Tate form, the WSF and the WS-coordinates of the generators of the MW-torsion of the toric hypersurface fibrations X F 13 , X F 15 and X F 16 .

WSF of J(X F 1 )
Here we explicitly write out the polynomials f and g of the WSF of the Jacobian fibration J(X F 1 ). The discriminant ∆ is calculated straightforwardly from these quantities but is omitted here due to the length of its explicit form. The functions f , g in the WSF of (3.4) read

C Euler Numbers of the Calabi-Yau Threefolds X F i
In this section we present the explicit expressions for the Euler numbers of all Calabi-Yau threefolds X F i that are constructed as toric hypersurface fibrations over an arbitrary two-fold base B with their fibrations parametrized by two divisors S 7 and S 9 , see Section 3.1.1. The Euler numbers are computed using the presentation of the vertical cohomology ring of X F i as a quotient ring in its divisors, see [48,112], and the adjunction formula. For a detailed explanation in an F-theory context and many explicit examples, we refer the reader to [42].
The following table contains our results for the Euler numbers of X F i , where we denote the first Chern class of the base B by c 1 , implicitly invoke Poincaré duality between divisors and forms and suppress the integral over B: Table 38: Data of toric Higgs transitions for F-theory compactified on X F 6 − X F 3 .

E Group Theoretical Decomposition of Representations
In Tables 39, 40