The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications

The four-form field strength in F-theory compactifications on Calabi-Yau fourfolds takes its value in the middle cohomology group $H^4$. The middle cohomology is decomposed into a vertical, a horizontal and a remaining component, all three of which are present in general. We argue that a flux along the remaining or vertical component may break some symmetry, while a purely horizontal flux does not influence the unbroken part of the gauge group or the net chirality of charged matter fields. This makes the decomposition crucial to the counting of flux vacua in the context of F-theory GUTs. We use mirror symmetry to derive a combinatorial formula for the dimensions of these components applicable to any toric Calabi--Yau hypersurface, and also make a partial attempt at providing a geometric characterization of the four-cycles Poincar\'e dual to the remaining component of $H^4$. It is also found in general elliptic Calabi-Yau fourfolds supporting SU(5) gauge symmetry that a remaining component can be present, for example, in a form crucial to the symmetry breaking ${\rm SU}(5) \longrightarrow {\rm SU}(3)_C \times {\rm SU}(2)_L \times {\rm U}(1)_Y$. The dimension of the horizontal component is used to derive an estimate of the statistical distribution of the number of generations and the rank of 7-brane gauge groups in the landscape of F-theory flux vacua.


Introduction
Work in string theory has traditionally focussed on the study of Calabi-Yau threefolds, as they are relevant to compactification of strings theories to four dimensions. From a mathematical point of view, it is very natural to ask about the properties of Calabi-Yau manifolds in (complex) dimensions other than three. Besides the omnipresent torus, two-and four-dimensional Calabi-Yau manifolds have subsequently acquired a central position within string theory 1 .
Two-dimensional Calabi-Yau manifolds, more commonly called K3 surfaces, form a single connected family and have a long history in mathematics (see e.g. the classic [2]). As the simplest non-trivial Calabi-Yau manifolds, they also have long been used to compaction string and supergravity theories. Their full relevance to string theory has only been appreciated with the advent of string dualities [3]. Besides appearing in relation to mirror symmetry [4], it was also in the context of string dualities, and in particular compactifications of F-and M-theory, that Calabi-Yau fourfolds became an intense object of interest, [5,6,7,8,9,10,11,12] is a partial list of early papers on the subject.
What is common to all Calabi-Yau n-folds is that their Kähler and complex structures are measured by integrating two special harmonic differential forms: the Kähler form J ∈ H 1,1 and the holomorphic top-form Ω ∈ H n,0 over appropriate cycles. However, we may already point out a crucial difference between Calabi-Yau threefolds on one side and K3 surfaces and Calabi-Yau fourfolds on the other side: whereas (powers of) J and Ω 3,0 live in different cohomology groups in the case of Calabi-Yau manifolds of odd complex dimensions, Ω n,0 and J n/2 share the middle cohomology for Calabi-Yau manifolds of even complex dimensions. For K3 surfaces this observation is tightly connected with the concept of polarization, where both J and Ω 2,0 are confined to lie in mutually orthogonal subspaces of H 2 . The Torelli theorem for lattice-polarized K3 surface states that the complex structure of K3 surface with a lattice W ⊂ H 2 being algebraic is parametrized by the period domain M * = P Ω ∈ [W ⊥ ⊂ H 2 ⊗ C] | Ω ∧ Ω = 0, Ω ∧ Ω > 0 .
For Calabi-Yau fourfolds, however, there is no such convenient Torelli theorem. Consider a family of Calabi-Yau fourfolds π : Z −→ M * (i.e.,Ẑ p := π −1 (p) for p ∈ M * is a Calabi-Yau fourfold); Pic(Ẑ p ) for a generic choice of p ∈ M * plays a role similar to the lattice Wpolarization in the case of K3 surfaces. Now, for any two classes in Pic(Ẑ p ) for a generic p ∈ M * with representatives η 1 and η 2 , we may form a vertical cycle η 1 ∧ η 2 , which defines a class in H 2,2 (Ẑ p ; R) ∩ H 4 (Ẑ; Z). This subspace (for p not in any one of the Noether-Lefschetz loci within M * ) is called the primary vertical component of H 2,2 , and is denoted by H 2,2 V (Ẑ p ). It is identified within H 4 (Ẑ; Z) defined in topology, independently of the choice of the "generic" p ∈ M * . The period integral ΩẐ p takes its value in H 4 (Ẑ; C) and satisfies the obvious constraints Ω ∧ Ω = 0 and Ω ∧ Ω > 0-(*), but the image of the period map does NOT occupy all the complement of the primary vertical component. That is, the period domain satisfying (*) may be contained in a space much smaller than References [13,4] introduced another subspace H 4 H (Ẑ; C) ⊂ H 4 (Ẑ; C) called the primary horizontal component (see section 2 for a brief review). The period map is locally injective for Calabi-Yau fourfolds and maps M * into an m := h 3,1 (Ẑ p ) = dim C M * subvariety of P[H 4 (Ẑ; C)]; as we will elaborate on in the next section, M * is in fact mapped into the projectivization of the horizontal component, P H 4 H (Ẑ; C) . The middle cohomology H 4 (Ẑ; C) of a Calabi-Yau fourfold is decomposed into where the decomposition is orthogonal under the intersection pairing. Unless the H 4 RM (Ẑ; C) component vanishes, the primary horizontal subspace H 4 H (Ẑ; C) is smaller in dimension than the non-vertical subspace (1).
There is another context-flux compactification of F-theory-where one is interested in the decomposition (2) of the middle cohomology above. An ensemble of flux vacua is specified by specifying a subspace of when this subspace is affine, the vacuum index distribution over the moduli space M * is given by a concise analytic formula [14,15]. As discussed already in [16], and refined further in section 2 in this article, we can see that any pair of topological four-form fluxes whose difference belongs to the real part of the primary horizontal subspace H 4 H (Ẑ; R) share the same symmetry group from 7-branes in their effective theories below the Kaluza-Klein scale. This motivates us to choose the affine subspace as some form of shift of H 4 H (Ẑ; R). Because the analytic formula of vacuum index density involves the dimension of the affine subspace, it is of interest in physics application of F-theory to know the dimension of H 4 H (Ẑ; R). It is also a question of interest what kind of space H 2,2 RM (Ẑ; R) is, and the role it plays in phenomenology applications. Mirror symmetry indicates that [13,4] H 4 H (Ẑ; C) ∩ H 2,2 (Ẑ p ; C) =: H 2,2 H (Ẑ p ; C) and H 2,2 V (Ẑ m ; C) have the same dimensions for a mirror pair (Ẑ,Ẑ m ). This article shares the idea of using the dimension h 2,2 V of the mirrorẐ m to find the dimension h 2,2 H of the original geometryẐ with [17,18]. References [17,18] used the intersection ring of the vertical subspace of the mirrorẐ m not just to determine the dimension h 2,2 H (Ẑ) = h 2,2 V (Ẑ m ), but also to compute period integrals ofẐ.
We combine this intersection ring in the mirrorẐ m with a stratification ofẐ m and a long exact sequence of morphisms of mixed Hodge structure, a combination of techniques that Ref. [19] used in order to derive the formula for h 1,1 and h k−1,1 of a toric-hypersurface Calabi-Yau kfold. With this approach, we are not only able to determine the dimensions h 2,2 V (Ẑ m ) = h 2,2 H (Ẑ), h 2,2 V (Ẑ) = h 2,2 H (Ẑ m ) and that of the remaining component, but also to construct the cycles representing the remaining component, study their geometry, and discuss their roles in physics applications.
We start with an introductory discussion in section 2, which motivates the study of the space of non-vertical four-cycles (primary horizontal cycles in particular) in the context of the landscape of flux vacua in F-theory compactifications. Using mirror symmetry, we derive a combinatoric formula for h 2,2 V , h 2,2 H , and h 2,2 RM , the dimensions of the space of vertical, horizontal and remaining (i.e. non-vertical and non-horizontal) cycles for Calabi-Yau fourfolds obtained as hypersurfaces of toric varieties in section 3. Already in this simple class of Calabi-Yau fourfolds, the remaining component is found to be non-zero in general. We provide several examples in section 4.
A non-zero H 2,2 RM (Ẑ; C) already occurs for a family of Calabi-Yau fourfoldsẐ = K3 × K3 = S 1 × S 2 with a lattice polarization W 1 ⊂ H 2 (S 1 ; Z) and W 2 ⊂ H 2 (S 2 ; Z). Here, (5) where ρ i = rank(W i ) [16]. In this example, the Poincaré duals of H 2,2 RM (Ẑ) are not represented by algebraic four-cycles. Section 5 shows another example of a family π : Z −→ M * where h 2,2 RM = 0 and provides some more intuition for the geometry relevant to cycles in h 2,2 RM . This family is a simple example within the class of Calabi-Yau fourfolds motivated by [20,21] for F-theory compactification where SU(5) unification symmetry is broken down to that of the Standard Model SU(3) c × SU(2) L × U(1) Y without the hypercharge U(1) Y vector field becoming massive. The remaining component H 2,2 RM (Ẑ) in this family contains forms that are Poincaré dual to four-cycles that are non-vertical, but still algebraic over generic points in moduli space. This observation plays an important role in the discussion of section 2, which discusses the relevance of the real primary horizontal subspace H 4 R (Ẑ; R) ⊂ H 4 (Ẑ; R) in the context of landscape of flux vacua in F-theory compactification.
In section 6, we discuss how the abundance of flux vacua depends on the unification group, and the number of generations. Under rather general assumptions, the dependence on the number of generations is found to factor from the distribution and to be given by a Gaussian quite generically for any choice of base manifold. We also develop an estimate for the dependence of the number of flux vacua on the gauge (unification) group, generalizing earlier attempts in [16]. We estimate that the abundance of flux vacua with e.g. gauge group SU (5) is suppressed by a factor of roughly e O(1000) when compared to models with no non-abelian gauge group. One can think of this surpression as a fine-tuning wildly surpassing the fine-tuning of 10 −120 needed to explain the smallness of the cosmological constant. Appendix B contains a further elaboration of how such estimates may be obtained. Finally, we discuss several open problems reserved for future investigation in section 7.
The appendices A and C mostly contain supplementary material and applications. Appendix A reviews details of the geometry of F-theory GUTs with gauge group SU(5) relevant for section 5. Appendix A.3 explains how to compute the Hodge diamond of exceptional divisors in such a geometry by using stratification. Appendix C reviews the construction of chirality-inducing fourform flux, and computation of the D3-tadpole from the geometry and this flux. The numerical results in the appendices B and C are used as input in the discussion in section 6.
A letter [22] by the same authors is focused on a part of subjects discussed in this article, and is addressed to broader spectrum of readers. It covers the subject in sections 2 and 6 and uses the results in section 4.4.1 and the appendices B and C, while the material in sections 3 and 5 are not discussed in the letter [22].

Ensembles of F-Theory flux vacua and the primary horizontal subspace
Supersymmetric compactification of F-theory to 3+1-dimensions is specified by a set of data (X, B 3 , J, G (4) ), where π X : X −→ B 3 is an elliptic fibration with a section, J ∈ H 1,1 (B 3 ; R) a Kähler form on B 3 and a G 4 a four-form flux in [H 4 (X; Z)] shift . Once G (4) is given topologically, the superpotential W ∝ X Ω X ∧ G (4) determines the vacuum expectation value of the complex structure of X, B 3 , π X etc. For an ensemble of fluxes in [H 4 (X; Z)] shift , therefore, we obtain an ensemble of low-energy effective theories in 3+1-dimensions, called a landscape.
Flux compactification not only determines the values of low-energy coupling constants, but also the gauge group. Once the complex structure of the elliptic fibration π X : X −→ B 3 is determined by the mechanism above, we know the configuration of 7-branes (i.e. the discriminant locus of π X ). Now, remember that we usually classify low-energy effective theories in their algebraic information such as gauge group, matter representations and unbroken symmetry first, in their topological information such as the number of generations next, and then finally in their values of the coupling constants. It is thus desirable to be able to classify flux vacua in a landscape also in the same way, sorting out first in the algebraic information, secondly in the topological data and finally in the moduli data [16]. This requires to work out what kind of topological flux G (4) results in an effective theory with a given algebraic and topological information.
In order to address this problem, it is useful to consider a family of elliptic fibred Calabi-Yau fourfolds characterized as follows [16]. First, let us choose (B 3 , [S]) and an algebra R. We specify only the topology of an algebraic three-fold B 3 , and let [S] is a divisor class in Pic(B 3 ). R is one in the A-D-E series, to be used for a unification group of one's interest. The family π : X −→ M R * is that of smooth Calabi-Yau fourfolds X with an elliptic fibration π X : X −→ B 3 with a section, so that there is a locus of singular fibres of type 2 R along a divisor S of B 3 that belongs to the class [S]. The restricted moduli space M R * parametrizes the complex structure of such fourfolds X. For a generic point p ∈ M R * , the corresponding fourfold X p := π −1 (p) has the property that H 1,1 (X p ; Q) ∩ H 2 (X; Q) is generated by divisors in the base, the zero-section σ, and {Ĉ i } i=1,··· ,rank(R) called Cartan divisors. Such a family over the restricted moduli space M R * is a useful notion to express the flux vacua distribution, as we see in (13,14).
There are a couple of physical conditions to be imposed on the fluxes. First of all, the flux preserves the SO (3,1) symmetry in the effective theory on 3+1-dimensions if and only if the flux G (4) does not have a component that has either "two legs in the T 2 fibre" or "no legs in the T 2 fibre" [23]. This condition is best paraphrased as σ ∧ η ∧ G (4) = 0, where σ is (the differential form Poincaré dual to) the zero section, and η, η 1,2 are divisors on B 3 , see [24,25,26,27,28]. The D-term condition (equivalently N = 1 supersymmetry condition, primitiveness condition) is that G (4) → J ∧ G (4) = 0 ∈ H 6 (X; R).
shift . The subspace of H 4 (X; Z) without the shift by c 2 (T X)/2 satisfying the condition (6,7) is denoted by [H 4 (X; Z)] Lor.prim. . shift to preserve the same symmetry group within the 7-brane gauge group R, their difference needs to satisfy where iĈ i :Ĉ i → X is the embedding of the Cartan divisors (generators) {Ĉ i } i=1,··· ,rank(R) . Thus, we formulate an ensemble of fluxes leading to effective theories with a common unbroken symmetry group as G (4) tot = G (4) scan + G (4) fix | G (4) scan ∈ H scan ; (9) we choose G (4) fix and H scan such that G (4) scan = 0 is contained in H scan . G (4) fix must be in [H 4 (X; Z)] Lor.prim.
shift , while H scan needs to be a sub-set of the cohomology group H 4 (X; Z) satisfying all the conditions (6,7,8). Because all of these conditions 3 are linear in G (4) , we always take H scan to be a sub-group of H 4 (X; Z).
We now argue that the real primary horizontal subspace H 4 H (X; R) ⊂ H 4 (X; R) satisfies all of the conditions (6,7,8) modulo ⊗R, so that hence we can take H scan to contain all of H 4 H (X; R) ∩ H 4 (X; Z). To see this, we take a little moment to provide a brief review on the definition of H 4 (X; C), and to make clear what we mean by the real primary horizontal subspace. For any point p ∈ M R * , we can define a subspace = H 4,0 (X p , C) ⊕ H 3,1 (X p ; C) ⊕ H 2,2 H (X p ; C) ⊕ H 1,3 (X p ; C) ⊕ H 0,4 (X p ; C) ⊂ H 4 (X; C).
The cohomology group H 4 (X; C) = C ⊗ H 4 (X; Z) is topological; there is a canonical identification between H 4 (X p ; C) and H 4 (X p ; C) for p, p in a small neighbourhood in M * -called flat structure-and the reference to p ∈ M * is suppressed in the expression above. The (p, q)-Hodge components with the canonical identification relatively to H 4 (X; C), however, vary over p ∈ M R * .
The fact that the Picard-Fuchs equations on the period integral are closed in the subspace specified above, however, implies that the space (10) remains the same in H 4 (X; C) for p ∈ M * (at least locally) under the canonical identification (flat structure). This invariant subspace is called the primary horizontal subspace H 4 F (X; C). Reference to p ∈ M R * is suppressed because of this invariance. The remaining component in the decomposition (2), H 2,2 (X p ; C), should therefore be independent of p ∈ M * under the canonical identification. This is why reference to p ∈ M * is completely dropped in (2); the decomposition (2) is topological.
The four-form flux in M-theory/F-theory compactification is real-valued, while the primary horizontal subspace H 4 H (X; C) is complex-valued. Noting, however, that the complex conjugation operation is compatible with the canonical identification (topological tracking) of H 4 (X p ; C) for p ∈ M * , we also have a decomposition of the real part H 4 (X; R): just like in (2). The "primary horizontal subspace" component of H 4 (X; R) is what we call the real primary horizontal subspace H 4 H (X; R). Now, let us see that the four-forms in the real primary horizontal subspace H 4 R (X; R) satisfy all the conditions (6,7,8) modulo ⊗R. First, noting that σ ·η and η 1 ·η 2 in (6) form only a subset of generators of the vertical four-cycles, and that all the elements of the horizontal component are orthogonal to those in the vertical component in (2), it is straightforward to see that the four-forms in H 4 H (X; R) satisfy (6). Secondly, in order to verify the primitiveness condition, consider the (h 3,1 + 1) = (m + 1)dimensional variety occupied by the complex line C[Ω Xp ] (this is a C × -cone over the period domain). Any four-form G in this variety satisfies the condition (7), because G is a (4, 0)-form for some choice p ∈ M * of complex structure on X, and J ∧ G would have become a (5, 1)-form under that complex structure, if it were non-zero. The absence of such a Hodge component in a Calabi-Yau fourfold X implies that J ∧ G = 0. Since all kinds of four-forms in H 4 H (X; C) are obtained by taking derivatives of such G with respect to the local coordinates in M * , we find that all the four-forms in the (real) horizontal primary subspace satisfy the primitiveness condition (7).
Finally, we can make a similar argument to show that the four-forms in H 4 H (X; R) satisfy the condition (8). Consider again the set of four-forms realized in the form of Ω Xp for some p ∈ M * . Its pull-back to any one of Cartan divisorsĈ i must be a (4, 0)-form on a complex three-foldĈ i under the complex structure of p ∈ M * induced onĈ i ⊂ X p . Thus, the pull-back of such a four-form vanishes in H 4 (Ĉ i ; C). Taking derivatives with respect to the local coordinates of M * , we find that the four-forms in H 4 H (X; R) satisfy the condition (8). References [14,15,29] studied how the flux vacua are distributed in the complex structure moduli space for ensembles of fluxes of the form (9). One of the two key ideas is to replace the vacuum distribution by the vacuum index distribution so that the problem becomes easier. (z,z) are local coordinates of M * , W ∝ X Ω Xp ∧ G (4) and L * is the upper bound of the D3-brane charge allowed for the fluxes G (4) in H scan . The other idea is to make a continuous approximation to H scan , which is to treat the subgroup H scan ⊂ H 4 (X; Z) as a vector space H scan ⊗ R, and to replace the sum by an integral over the vector space H scan ⊗ R. Under the continuous approximation, the vacuum index density dµ I is of the form [14,15] where ρ I is an (m, m) form on M * . It is given as the Euler class of some real rank R = 2m vector bundle on M * [29], and can be put in the explicit form where ω is the Kähler form on M * derived from K = − ln[ Ω∧Ω], and L a line bundle satisfying c 1 (L −1 ) = ω/(2π), whenever the vector space H scan ⊗ R contains the real primary horizontal subspace H 4 H (X; R) [14,15,29,16]. Having reminded ourselves of how H scan is used, let us return to the question of how we should take G (4) fix and H scan . The problem we set in this article (and also in [16,22]) is to classify the fluxes in [H 4 (X; Z)] Lor.prim.
shift into sub-ensembles of the form (9) so that each subensemble corresponds to the ensemble of low-energy effective theories with a given set of algebraic (or algebraic and topological) information. G (4) fix is used as a tag of the subensemble. Apart from [16], the H 2,2 RM (X; R) component has not been carefully discussed (at least from the perspective of the geometry of Calabi-Yau fourfolds) in the context of F-theory phenomenology to out knowledge. In order to address this problem, therefore, it is necessary to reopen the case and understand carefully how the low-energy effective theories are controlled by the fluxes in each one of the components of the decomposition (11).
First of all, it is already known that h 2,2 RM can be non-zero in a family of X = K3 × K3 [16], as we have already reviewed in the introduction. We will prove in section 5 that the class of families of elliptic fibred Calabi-Yau fourfolds for F-theory compactification with SU(5) unification results in h 2,2 RM = 0 precisely in the cases that are well-motivated for a mechanism of SU(5) symmetry breaking in [20,21]. Despite the presence of the H 2,2 RM (X; R) component in the middle cohomology, however, we have confirmed (see also the detailed discussion in the appendix A.4 of this article) that the matter surfaces for 10 + 10 representations and alsō 5 + 5 representations of SU(5) belong to topological classes of H 2,2 V (X; Q), confirming [25]. This means that the net chirality of these representations is controlled by the flux in the H 2,2 V (X; R) component.
It has also been indicated that fluxes in H 2,2 RM (X; R) may break the symmetry of the 7brane gauge group R based on the family X = K3 × K3 [16]. Certainly this family is not suitable for realistic compactifications of F-theory in that all the matter fields in the relatively light spectrum are in the adjoint representation of R; this family may also be somehow special in that h 2,0 (X) = 0. However, we have confirmed that the flux in the H 2,2 RM (X; R) component may indeed break the symmetry R; the hypercharge flux of [20,21] turns out to be precisely in this category.
Given all the observations above we conclude that H scan ⊗ R should be chosen such that it contains all of the horizontal component H 4 H (X; R) and possibly a part of H 2,2 RM (X; R). Clearly, H scan ⊗ R should not contain the entire H 2,2 RM (X; R) for sub-ensemble of fluxes (9) to correspond to an ensemble of effective theories with a given set of algebraic and topological information. An example of the family of X = K3 × K3 suggests strongly that we should choose (H scan ⊗ R) to be precisely the horizontal component, because any flux in H 2,2 RM (X; R) breaks some of the 7-brane gauge group and Higgses away vector bosons from the low-energy spectrum in this example. We remain inconclusive about the case of families for realistic F-theory compactification with (B 3 , [S], R) (with R = A 4 , D 5 etc.), however, because the discussion in sections 3 and 5 provides only partial understanding of the geometry of the H 2,2 RM (X; R) component. This material is enough, however, to conclude that the formula (13,14) of [14,15]  We have so far assumed that smooth Calabi-Yau fourfolds X with flat elliptic fibration are used for F-theory compactifications with fluxes; it should be reminded, however, that this is a belief rather than a fact that such smooth models X may be used instead of Weierstrass models X s , which are singular in for compactifications leading to effective theories with unbroken non-Abelian gauge groups. There may be alternative and/or equivalent formulations of fluxes that lead to the same physics end results. Even when we pursue the direction of using smooth models X, there can be more than one choices of such a smooth model X. All of those different resolutions, however, should describe low-energy physics of the same vacuum with an unbroken unified symmetry. Thus, no matter how fluxes in F-theory are formulated, all the observable physics consequences should not depend on the choice of resolutions. The dimension of the primary horizontal component ((H scan ⊗ R) to be more precise) should be regarded as one of such physical consequences of F-theory, and hence K = dim R [H 4 H (X; R)] must not depend on the choice of the smooth model X for a singular X s .
There is a general argument that is good enough to make us consider that the dimensions of h 2,2 V (X; R), h 2,2 H (X; R) and h 2,2 RM (X; R) are resolution independent. It seems reasonable to assume that, for a singular Calabi-Yau fourfold (X s ), any crepant resolution (X) gives rise to the same complex structure moduli space. If this holds, the dimension of the space of primary horizontal (2, 2) cycles, h 2,2 H (X; R), as well as h 3,1 (X) cannot depend on the resolution. By mirror symmetry, the same statement must be true also for the space of primary vertical (2,2) cycles where we assume that all the different resolutions X for X s share some mirror geometries X m with a common complex structure moduli space. If furthermore the Euler characteristic is invariant under which resolution is used, it follows that also h 2,2 cannot be resolution dependent. This is because h 2,2 is not an independent Hodge number but related to all others by (76). From the independence of h 2,2 H (X; R) and h 2,2 V (X; R) on which resolution is used, it then follows that also h 2,2 RM (X; R) is independent of the resolution. The two assumptions we have made are obvious for a Calabi-Yau manifold X given as a hypersurface (or, more generally, complete intersection) in a toric ambient space: for a fixed singular Calabi-Yau manifold, different crepant resolutions correspond to different triangulations of the N-lattice polytope whereas all Hodge numbers, and, in particular the complex structure moduli space depend on the combinatorics of the N -and M -lattice polytopes, but not on triangulations. We will give a more specific proof of this for the expressions we derive in the hypersurface case in section 3.6.3.
More generally, our assumptions follow if different resolutions correspond to different cones in the extended Kähler moduli space, so that they are connected by flop transitions, which are known to leave both the complex structure moduli space and the Euler characteristic invariant. This state of affairs is realized in the geometries which are the main motivation for the present work: F-theory on Calabi-Yau fourfolds supporting non-abelian gauge groups [30,31,32,33,34,35] Let us leave a few remarks at the end of this section, in order to clarify what the vacuum index density distribution dµ I is for, as well as what it is not (yet) for. These are more or less known things, and we include this discussion in this article only as reminder.
The first remark is that a family π : X −→ M R * for some choice of symmetry R and topology of (B 3 , [S]) is used to describe the distribution of a subensemble of flux vacua that is inclusive in nature. Even though the scanning component of the flux, G scan , is chosen from the real primary horizontal component H 4 H (Ẑ; R) ⊂ H 4 (Ẑ; R), it may eventually end up with the Poincaré dual of a algebraic cycle as a result of dynamical relaxation of the complex structure moduli due to the superpotential W ∝ Ω ∧ G (4) . The delta function δ 2m (DW, DW ) for a given G (4) scan eventually has a support on a point p in a Noether-Lefschetz locus of M R * , see [24,16,18] and references therein. For some choice of G (4) scan (and its corresponding p ∈ M R * , the rank of H 1,1 (X p ; R) ∩ H 2 (X p ; Z) may even be enhanced, 4 not only the rank of H 2,2 (X p ; R) ∩ H 4 (X p ; Z) is. There may be an enhanced gauge symmetry forming a hidden sector in addition to the gauge group R, or the symmetry R is enhanced to another R containing R, for the effective theory corresponding to such a choice of G (4) scan . Such a subensemble of fluxes with (H scan ⊗ R) = H 4 H (X; R) for (B 3 , [S], R) therefore contains not just effective theories with the 7-brane gauge group being precisely R, but also those with an extra factor of gauge group, or those with the 7-brane gauge group larger than R. In this sense, the subensemble (and the distribution dµ I ) captured on the moduli space M R * is an inclusive ensemble. With the expectation that only very a small fraction of such an inclusive ensemble has a hidden sector or enhanced symmetry, however, we will refer to vacua in such an ensemble as those with symmetry R; the fraction of such vacua can be estimated by using the discussion in section 6 and the appendix B, and we see the fraction is exponentially small indeed.
Secondly, let us record our understanding of the issue of potential instabilities. Given a topological flux G fix , we may not be able to find a solution to DW = 0 within the restricted moduli space M R * . Such a flux, if there is any, has been removed automatically from the ensemble by the time the continuous approximation H scan −→ H scan ⊗ R is introduced and the distribution dµ I is cast into the form of (13). When the flux space integral is carried out, the delta-function picks up contributions only from fluxes that satisfy DW = 0 somewhere in M R * . This argument may be, however, only about the instability issue associated with DW = 0 for moduli along M R * , i.e., R-singlet moduli. Instability associated with physical "moduli" transverse to M R * may be captured by the effective superpotential W = S tr[ϕ ∧ F ] of [36,37]. Thirdly and finally, obviously the Kähler moduli J has been (and will be) treated in this work as if they were given by hand in H 1,1 (B 3 ; R), but their stabilization also has to be studied separately. The distribution (13,14) on the moduli space of complex structure M * needs to be convoluted with other data, including stabilization of Kähler moduli and dynamical evolution of cosmology in the early universe. There is nothing to add in this article to this well-known zoomout picture, aside from reminding ourselves that it may make sense to study the distribution over the complex structure moduli space separately from Kähler moduli stabilization when there is a separation of scales between the stabilization of two distinct sets of moduli. In this case the two problems can be treated separately and then be combined, rather than facing the mixed problem.
3 H 2,2 of a Calabi-Yau fourfold hypersurface of a toric variety Suppose thatẐ is a smooth variety and {Ŷ i } a set of its divisors. We can introduce the following stratification toẐ: The first stratum, Z :=Ẑ \ ∪ iŶi , is the only one that has the same dimension asẐ; we call it the primary stratum. For the pairẐ and Y :=Ẑ \ Z, there is a long exact sequence (25). It was with this exact sequence in combination with the mixed Hodge structure on these cohomology groups that Ref. [19] derived the formula of h 1,1 (Ẑ) and h n−2,1 (Ẑ) of a Calabi-Yau (n − 1)-fold hypersurface of a toric n-fold, and showed the beautiful mirror correspondence.
We find that the same combination -stratification (15) and the mixed Hodge structureis also very useful in studying various properties of algebraic cycles inẐ even whenẐ is not necessarily realized as a hypersurface of a toric variety. Section 5 of this article uses this combination to study the space of vertical cycles of general fourfoldsẐ, whereas we can derive stronger results in the caseẐ is a Calabi-Yau (n − 1)-fold hypersurface of a toric n-fold. In the latter case, which is discussed in the present section, the Hodge-Deligne numbers of the cohomology of the primary stratum can be easily computed by following [38].
By carefully examining the contributions to H 2,2 (Ẑ) and using mirror symmetry, we are able to isolate the pieces h 2,2 The results are given in section 3.6.

General setup and known results
Let us start by fixing our notation and reviewing some helpful facts; subtleties associated with singularity resolution are summarized in section 3.1.1. Reviews on toric geometry may be found in [39,40].
A toric variety P n Σ can be constructed from a fan Σ, which is composed of strictly convex polyhedral cones in a lattice N := Z ⊕n . The dual lattice is denoted by M = Z ⊕n and the pairing between them for ν ∈ N andν ∈ M is denoted by ν,ν ∈ Z. We also use the notation N R := N ⊗ R and M R := M ⊗ R. For a given fan Σ, Σ(k) stands for the collection of all the k-dimensional cones. Σ(0) = { 0 ∈ N }. We denote the k-skeleton ∪ k i=0 Σ(i) by Σ[k]. By definition, an n-dimensional toric variety contains an open algebraic torus (C * ) n = T n . From this perspective, a fan Σ gives information on how T n is compactified. In particular, the fan determines a stratification into algebraic tori of lower dimension: simplicity cones of Σ are in oneto-one correspondence with strata of P n Σ such that a k-dimensional cone σ ∈ Σ(k) corresponds to a stratum of P n Σ isomorphic to (T) n−k . We denote such strata by T σ . A lattice polytope is the convex hull in N R (or M R ) of a number of lattice points of the lattice N (or M ). For a lattice polytope ∆ in M the polar (or dual) polytope ∆ (in N ) is defined as If ∆ is a lattice polytope as well, it follows that the ∆ is also the polar of ∆ and the two are called a reflexive pair. An elementary property which follows from reflexivity is that the origin is the only integral point which is internal to the polytope. Faces of an n-dimensional polytope ∆ are denoted by their dimensions 5  ) to indicate such a pair of dual faces. Note that an n-dimensional polytope can be considered as its own n-dimensional face. We indicate that a face Θ a is on a face Θ b by writing Θ a ≤ Θ b . We let (Θ) stand for the number of integral points on a face Θ and * (Θ) for the number of integral points interior to Θ. The p-skeleton of ∆, i.e. the union of all faces of ∆ which have dimension p or less is denoted by ∆ ≤p .
Given a reflexive polytope ∆ we can easily construct a fan Σ by forming the cones over all faces of ∆. The one-to-one correspondence between (k − 1)-dimensional faces { Θ [k−1] } and k-dimensional cones Σ(k) in N R (resp. between {Θ [k−1] } and Σ(k) in M R ) is described in the form of a function σ : Well-known formulae [19,41,42] for the Hodge numbers of a Calabi-Yau (n − 1)-fold hy-persurfaceẐ are recorded here in the notation adopted above:

Triangulations, smoothness and projectivity
The polytope ∆ in M R is regarded as the Newton polyhedron of the defining equation of a Calabi-Yau hypersurface Z s in P n Σ . Because Z s is not smooth, in general, we are interested in its projective crepant resolution;Ẑ in (17)(18)(19) stands for such a resolution. Such resolutions may always be constructed from the data of the polytope for n ≤ 4 [19]. For Calabi-Yau hypersurfaces of complex dimensions greater than 4, this is not the case, so that we need to explicitly check in each example.
Constructing a fan Σ over the faces of a polytope ∆, the resulting cones may be nonsimplicial or have (lattice-) volume 6 greater than one. Consequently, the toric variety P n Σ has singularities. We may, however, subdivide the fan Σ to cure such singularities. We denote the corresponding map between fans by φ : Σ −→ Σ. For σ ∈ Σ , φ(σ ) is given by the σ -containing cone σ ∈ Σ with the smallest dimension. The map φ (resp.φ) induces a toric morphism P n Σ −→ P n Σ of (partial) singularity resolution. Such a morphism will preserve the Calabi-Yau condition of a hypersurface if all of the onedimensional cones introduced are generated by points on the polytope ∆ (remember that the origin is the only internal point for a reflexive polytope). A (partial) crepant desingularization of Z s is hence equivalent to finding a triangulation of the polytope ∆ in which every n-simplex contains the origin. This is called a star triangulation and the origin is the star point. A maximal desingularization of P n Σ keeping Z s Calabi-Yau is found by using all points 7 on ∆. A triangulation using all points of a polytope is called a fine triangulation.
When subdividing cones in Σ, we want the ambient space P n Σ to be a projective variety (i.e. it should be a Kähler manifold), which implies projectivity of the Calabi-Yau hypersurface; the Kählerian nature combined with the trivial canonical bundle implies Ricci-flatness. For an n-dimensional toric variety P n Σ given in terms of a fan Σ, a Weil divisor D = i a i D i is also Cartier if we can find a support function ψ D with the following properties: • ψ D is linear on each cone • For a given cone of maximal dimension, σ ∈ Σ(n), ψ D can be described by an element m σ of M satisfying where the sum runs over all one-dimensional cones (determined by the primitive vectors 6 The "lattice volume" of a cone in ∂∆ is defined by multiplying k! to the volume of a k-dimensional cone cut-off at ∂∆. The smallest lattice k-simplex has the lattice volume 1. 7 In fact, it makes sense to relax this requirement, as points which lie in the interior of facets of ∆ do not lead to divisors intersecting a Calabi-Yau hypersurface. This can be seen as follows: for any facet F we can find a normal vector nF such that < nF , νi > = 1 for all vectors νi on F . This means that the intersection ring contains a linear relation of the form with includes some contribution of divisors whose corresponding primitive vectors νj are not in F but lie on other facets. Let us now assume we have refined Σ such that there is a point νp interior to the facet F . The associated divisor Dp can only have a non-zero intersection with divisors D k for which ν k also lies in F , as all others necessarily lie in different cones of the fan Σ. This means that the above relation implies where we sum over all toric divisors coming from points on F . The Calabi-Yau hypersurface is given as the zero-locus of a section of −K P n Σ = j Dj, where we sum over all toric divisors. We now find by using the same argument again. Hence Dp does not meet a generic Calabi-Yau hypersurface. Correspondingly, a refinement of Σ introducing νp does not have any influence. In this language, the cone of ample curves (or, equivalently, the Kähler cone) is described as the set of divisors for which ψ D is strongly convex. This means that ψ D | σ > −a j for all one-dimensional cones not in σ. Note that this implies that m σ = m σ for two different cones σ and σ in Σ(n). If this cone of ample curves is non-empty, we can find a line bundle which is very ample, i.e. it defines an embedding of the toric variety P n Σ into P m for some m. Conversely, if no strongly convex support function exists, the corresponding toric variety cannot be projective.
A strongly convex support function ψ D defines a 'lift' of∆ into R n+1 by assigning the value under ψ D to each point on∆. The triangulation can then be seen as the upper facets of the resulting polyhedron in R n+1 . Note that strong convexity means that no n + 2 points of∆ are mapped to a hyperplane in R n+1 , i.e. faces of∆ which are subdivided into more than one simplex by a triangulation are not coplanar after this lift. Conversely, any triangulation which descends from a triangulation of a lift of∆ (with the property that no n + 2 points are on a common non-vertical plane) to R n+1 can be used to construct a projective toric variety. Triangulations with this property are called regular triangulations.
A maximal projective crepant desingularization (referred to as an MPCP of Z s in [19]) is hence achieved by finding a fine regular star triangulation of ∆. If all cones of such a triangulation have lattice volume unity the ambient space, and hence the Calabi-Yau hypersurface, become completely smooth and the map φ : Σ → Σ can rightfully be called a resolution. Triangulations of this type are called unimodular. We will also call simplices of volume unity and cones over such simplices unimodular.
It turns out that unimodular triangulations do not necessarily exist for polytopes of dimension n ≥ 4. This is only relevant for Calabi-Yau manifolds of dimension ≥ 4 though, as one may always find a triangulation which is 'good enough' in the case of Calabi-Yau threefolds [19]. While any fine triangulation is also unimodular in two dimensions, a simplex can fail to be unimodular in three dimensions or more, even if it does not contain any points besides its vertices. A standard example for this is given in figure 1. For a star triangulation of a reflexive polytope, the lattice volume 8 of any simplex is given by the volume of its 'outward' face, i.e. the face which lies on a facet of∆. For a polytope of dimension three or less, the facets are at most two-dimensional, so that any fine triangulation is automatically unimodular and the ambient space is smooth. For a Calabi-Yau threefold, the facets of∆ are three-dimensional. Hence even for a fine triangulation we are not guaranteed a smooth ambient space, as there might be a simplex S with volume greater than unity. However, the singularities which are induced by such cones are point-like: they are located at the intersection of the four divisors spanning S. These points do not meet a generic hypersurface, so that a fine triangulation is still enough to ensure smoothness for Calabi-Yau threefolds. In the case of Calabi-Yau fourfolds, there can be singularities along curves of P n Σ which are induced by faces which are not unimodular in a fine triangulation. These generically meet a hypersurface in points, so that smoothness is no longer automatic. For Calabi-Yau fourfolds, we thus need to check explicitly if a triangulation giving a resolution exists.
In the following, we will assume that we have found a fine regular star triangulation which makes P n Σ a smooth projective toric variety and leads to a smooth Calabi-Yau fourfold hyper- surfaceẐ. This means we fix a fan Σ and a map φ : Σ → Σ. We will come back to the issues discussed in this section, when we look at examples in section 4.

Computing h 2,2 via decomposition
LetẐ be a non-singular compact Calabi-Yau (n − 1)-fold defined as a hypersurface of a smooth toric ambient space P n Σ . Let {ν i } i=1,··· ,|Σ (1)| be the primitive vectors for all the 1-dimensional cones of the toric fan Σ . For each one of them, there is a toric divisor D i given by {X i = 0} ⊂ P n Σ . The restriction of D i to the hypersurfaceẐ is denoted byŶ i . This set of toric divisors Ŷ i is used to introduce a stratification (15). The primary stratum can be regarded as a hypersurface of T n , and its complement is denoted by Y := ∪ iŶi .
For a compact non-singular irreducible algebraic varietyẐ and a divisor Y with normal crossings inẐ (such that Z =Ẑ\Y is non-singular), we can write a long exact sequence In order to use the exact sequence (25), we first note [38] that H i c (Z) vanishes for i = 0, · · · , n − 2, and More information about the non-vanishing parts of H n−1 c (Z) will be provided later.
The main focus in this article is to study H 4 (Ẑ) for a Calabi-Yau fourfoldẐ in a toric ambient space of dimension n = 5. In this case, only the weight-4 components of H 4 c (Z), H 4 c (Ẑ) and H 4 c (Y ) are relevant to determining H 4 (Ẑ), and we learn from (25) that they satisfy Let us first focus on H 4 c (Y ) . Y is not necessarily non-singular, but it can be written as Y = ∪ iŶi where eachŶ i is a non-singular divisor ofẐ. Note that eachŶ i is not necessarily irreducible. This happens when ν i is in a codimension-2 face of ∆. The (co)homology groups of such geometries can be computed by using the Mayer-Vietoris spectral sequence, and one finds that As eachŶ i is a smooth hypersurface of a toric variety, its h 1,0 = h 3,1 vanishes (see e.g. [38]). Hence H 4 (Y ) contributes only to the (H 4 c (Ẑ)) 2,2 component in (27)(28)(29). Furthermore, keeping in mind that all divisors ofẐ occur as components of toric divisors, it is obvious from (30) that all vertical (2, 2)-forms ofẐ are contained in the H 4 (Y ) subgroup of H 4 (Ẑ).
In order to verify the claim, one needs to note that the kernel of (30) . The cokernel of the same homomorphism therefore corresponds to which gives rise to the [H 5 (Y )] 2,2 component. It can be determined by exploiting other parts of the long exact sequence (25). Focusing on the weight-4 components, we find that the following is exact: Using (26), as stated before.

Stratification and geometry of divisors
For the purpose of capturing H 4 c (Y ) and H 4 (Z) in terms of combinatorial data of the toric ambient space, we take a moment to digress. To start off, we describe stratifications of Calabi-Yau hypersurfacesẐ of a toric ambient space P n Σ . There are two distinct stratifications to which we pay attention: one is associated with the toric fan Σ, and the other with its refinement 10 Σ .
The stratification induced by Σ is easier to describe, it descends from that on P n Σ straightforwardly: each stratum ofẐ is of the form Z σ =Ẑ ∩ T σ for σ ∈ Σ .
The stratification corresponding to Σ iŝ here, the strata are labelled by the faces of ∆, namely, ∆ itself, codimension-1 faces Θ [n−1] 's, and all other faces Θ [n−k] 's of codimension 2 ≤ k ≤ n − 1. It is understood in the expression above, and also in expressions later, that ( is a dual pair of faces. Due to the one-to-one correspondence between faces of ∆, the faces of ∆ and the cones in Σ, the strata in (34) are in one-to-one correspondence with the cones of the fan Σ. The primary stratum, Z ∆ , corresponds to the 0-dimensional cone 0 ∈ Σ, and Z Θ [n−1] to the individual 1-dimensional cones in Σ(1).
has a stratification associated with the maximal simplicial subdivision Σ , where the number of T k−1−p in this decomposition is equal to the number of p-simplices in The geometries of Z ∆ (k = 0) and all the other of Z Θ [n−k] are given by hypersurfaces of T dim C (Θ)=(n−k) . For each one of the Z Θ , only the terms in the Newton (Laurent) polynomial corresponding to Θ [n−k] ∩ M are relevant in determining Z Θ [n−k] ⊂ T n−k (which also means all terms originating from ∆ ∩ M are relevant in determining Z ∆ ⊂ T n ). Obviously the primary stratum Z ∆ is the same as Z :=Ẑ\Y introduce before.
The stratification for the fan Σ ,Ẑ = σ∈Σ Z σ , is therefore obtained by decomposing the stratification (34) under (35). Put differently, where φ : Σ −→ Σ is the map of toric fans (mapping cones to cones) associated with the subdivision refining Σ to Σ and σ( as the exceptional geometry appearing in a resolution of singularities of P n Σ associated with the k-dimensional cone σ( Θ [k−1] ).

Combinatorial formula for
Just like various Hodge numbers of a Calabi-Yau toric hypersurfaceẐ are computed by using the stratification and the Hodge-Deligne numbers of the strata [38,19], Hodge numbers of divisorsŶ i 's of a Calabi-Yau toric hypersurfaceẐ can also be computed essentially with the same technique. Once h 2,2 (Ŷ i )'s are computed, it is almost straightforward (as already explained) to determine h 2,2 [H 4 c (Y )]. The 'Euler characteristics' of Hodge-Deligne numbers of compact support cohomology groups (for some geometry X) are These numbers have the following three nice properties, which were exploited heavily in [38,19], and will also be in the following.
The first two are the additivity, and the multiplicativity, Finally, when an algebraic variety X (of complex dimension n − 1) is compact and smooth, i.e the mixed Hodge structure on cohomology groups is pure, In a slight abuse of language, we will frequently refer to the e p,q , as Hodge-Deligne numbers in the text.
Each divisor componentŶ i (corresponding to ρ i ∈ Σ (1)) of a Calabi-Yau (n − 1)-foldẐ is compact and smooth, and hence h n−3,n−3 (Ŷ i ) = h 1,1 (Ŷ i ) is the same as 11 e n−3,n−3 c (Ŷ i ). The compact geometryŶ i has a stratification associated with Σ , or to be more specific, Because of additivity, e n−3,n−3 c (Ŷ i ) is obtained by summing up e n−3,n−3 c (Z σ ) (σ ≥ ρ i ). Using multiplicativity, this calculation is further boiled down to the computation of e p,q c (Z Θ ) of various faces Θ ≤ ∆, for which the algorithm of [38] (in combination with (26)) can be used.
Here, we record a few crucial formulas from [38] for Hodge-Deligne numbers of the strata Z Θ . For a face of dimension f there is the 'sum rule' where f is the dimension of the face Θ. Here, the functions ϕ k are defined as where jΘ stands for the polytope which is obtained by scaling all vertices of the face Θ by j and then taking the convex hull. We introduce a notation in this article, Obviously the sum rule (42) can be written as Furthermore, the Hodge numbers obey [38] for any p > 0. For a face of dimension f ≥ 4 we also have that This can be regarded, however, as a special case of the more general problem of determining h n−3,n−3 (Ŷ i ) of a divisorŶ i of a Calabi-Yau (n − 1)-fold embedded in a toric ambient space P n Σ . We shall study the more general version of the problem for n ≥ 5.
For any 1-dimensional cone ρ i generated by a primitive vector ν i in the lattice N , there is a divisorŶ i ofẐ. Depending on which face contains ν i in its interior, the computation h n−3,n−3 (Ŷ i ) has to be treated separately.
The cases we have to study are then • k 0 = 1: i.e., ν i is one of vertices of ∆.
• k 0 = n − 1: i.e., ν i is an interior point of a codimension-2 face. (a 3dim face, if n = 5) We work on those five cases one-by-one from now.
The case k 0 = 1: i is a vertex of ∆.Ŷ i is composed of the following strata: only contributes when the corresponding p-simplex in Θ [k−1] contains ν i as one of its faces. This is why, for example, only one point T 0 (one 1-simplex) from E Θ [1] of a given dual pair ( Θ [1] , From the first stratum, From the k-th group of strata (k = 2, · · · , (n − 2)), each pair ( Here, we have introduce the notation ν i * 1 ( Θ) for the number of internal 1-simplices in the face Θ which end on ν i , or in other words have ν i as a face. In the language of cones We also introduce ν i 1 ( ∆ ≤p ) for the number of 1-simplices (again containing ν i as a face) contained in the p-skeleton (faces of dimension p or less) of ∆.
The contribution from the last group of strata, k = n − 1, is the same as above, except that points, which is not necessarily 1. Hence the contribution to e n−3,n−3 c is e 0,0 c (Z Θ [1] ) times larger than that of (50). We thus finally obtain This formula for the divisorsŶ i ofẐ is quite like that ofẐ in (17). The first two terms originate from toric divisors (divisors of the ambient space P n Σ ofẐ and the linear equivalence restricted toŶ i ). The correction term at the end of (53) also looks quite similar to the last term of (17); we will come back in section 3.6 to discuss the geometric interpretation of this term more in detail.
The three cases 1 < k 0 < (n − 3), k 0 = (n − 3) and k 0 = (n − 2): The stratification ofŶ i is described schematically by In all the three cases, the contributions to e n−3,n−3 c from all but the first group of strata remain the same as in the k 0 = 1 case. The first group of strata gives rise to In all the three cases, 1 < k 0 < n − 3, k 0 = n − 3 and k 0 = n − 2, we have that The expression for the factor e n−k 0 −2,n−k 0 −2 For the two other cases, k 0 = (n − 3) and k 0 = (n − 2), however, there is an extra term to Therefore, it turns out that h 1,1 (Ŷ i ) remains the same as in (53) The correction term in the last line of (57) is determined by using the sum rule in [38], (42).
i is a surface given by a Laurent polynomial in T 3 . Therefore, In the case i is a curve given by a Laurent polynomial in T 2 . Thus, = − * (2Θ [2] i ) + 4 * (Θ [2] i ).
Again, the first two lines of (57) are understood as toric divisors ofẐ restricted onŶ i . The (n − 1)-dimensional redundancy among them is simply given by the number of toric linear equivalences. The geometric interpretation of the correction termē n−k 0 −3,n−k 0 −3 The cases k 0 = n − 1: containing ν i as a face. Therefore, Note that the "number of linear equivalences" is different from that in all the other cases where
Here, 1 ( ∆ ≤p ) denotes the number of 1-simplices on the p-skeleton of ∆ and * 1 ( Θ) the number of internal 1-simplices on a face Θ.

Combinatorial formula for
Using the sum rule (45) of [38] we immediately find that, There are six unknowns in the left-hand sides, so that we need three more conditions in order . They come from the fact that all of the Hodge numbers of components where neither p = q, nor p + q = n − 1 are tightly constrained for a compact smooth hypersurface of a toric variety. For p = 0 (or q = 0) such Hodge numbers even vanish. Thus, for the closure of Z Θ [n−1] (satisfying the condition above), it follows that for any facets Θ [n−1] of ∆. Similarly for the closure of Z ∆ =Ẑ, we obtain where the factor 1 is due to e 0,0 (E Θ [2] ) = (−) 2+1 [χ(2d ball) − χ(S 1 )] = 1 (see e.g., [38]).

Vertical, horizontal and the remaining components
We can now compute h 2,2 (Ẑ) by summing (64) and h 2,2 (Ẑ) in (75), due to the exact sequence (29). In fact, this is how Ref. [19] derived the formula (17,18) for h 1,1 and h 3,1 of a Calabi-Yau hypersurface fourfoldẐ, and it is a straightforward generalization to use (29) to determine h 2,2 (Ẑ) in this way. The dimension of the H 2,2 component, however, does not have to be determined in this way: for a smooth fourfoldẐ, the formula [9,12] makes it possible to determine h 2,2 (Ẑ) from the three other Hodge numbers that are already determined through (17)(18)(19).
The exact sequence (29) is still very useful for the purpose of studying the decomposition (2). This is because all the independent divisors of a hypersurface Calabi-Yau fourfoldẐ ⊂ P 5 Σ appear in the form of irreducible components of toric divisorsŶ i ofẐ. We can therefore take a set of generators of the vertical component Homological equivalence among the generators has already been taken care of in the study of [H 4 (Y )] 2,2 in section 3.4. In this section, we start off with identifying which subspace of [H 4 (Y )] 2,2 corresponds to the vertical component H 2,2 V (Ẑ). Mirror symmetry is then used to identify the horizontal component . We will comment on the geometry associated with the remaining component H 2,2 RM (Ẑ; C) at the end.

The vertical component
Let us discuss which subspace of H 1,1 (Ŷ i ) is generated by vertical cycles for the cases with k 0 = 1, n − 3 = 2, n − 2 = 3 and n − 1 = 4. We begin with a divisorŶ i corresponding to a vertex ν i of ∆ (i.e., k 0 = 1). The formula (53) can be rewritten as ) is the number of 1-simplices whose endpoints are ν i and one of the interior points of a face Θ [n −2] , and ν i * • ) is the number of 1-simplices which run through the interior of Θ [n−2] , but have both end points on the boundary of Θ [n −2] . The first two terms account for the dimension of the space of algebraic cycles obtained as the intersection ofŶ i with another toric divisor (i.e., a divisor ofẐ that descends from the toric ambient space P n=5 Σ ). When the intersection of a pair of toric divisors of the formŶ i ∩Ŷ j corresponds to a 1-simplex counted in ν i * • 1 ( Θ [n−2] ), however, the toric divisorŶ j corresponding to an interior point of a codimension-two face of ∆ consists of * (Θ [1] ) + 1 irreducible components, each one of which are independent in H 1,1 (Ẑ). Therefore each one ofŶ i ∩ [Ŷ j ] irr 's can be taken as an independent generator of the vertical algebraic cycles. The term ν i * • 1 ( Θ [n−2] ) * (Θ [1] ) should therefore be counted as a part of the vertical component H 2,2 (Ẑ). ] ), however, correspond to algebraic cycles of the form Y i ∩Ŷ j , withŶ j corresponding to a point in a face of codimension-three or higher. In such cases, Y i ∩Ŷ j consists of * (Θ [1] ) + 1 irreducible components, but only one linear combination of those irreducible components,Ŷ i ∩Ŷ j , should be regarded as a vertical cycle. Thus, there are algebraic but non-vertical cycles left over in

1-simplices counted in
Let us move on to H 1,1 (Ŷ i ) for a toric divisorŶ i corresponding to an interior point of Θ [1] (i.e., k 0 = n − 3 = 2). The first line of (57) is rewritten just like in (77), and the same interpretation applies also in this case. The remaining correction termē 1,1 c (Z Θ [3] ) in (57) in the k 0 = n − 3 case is not regarded as part of the vertical component either. To see this, let us use a long exact sequence like (25), decomposingŶ i into the primary stratum Z ρ i = T k 0 −1 × Z Θ [3] i and the restŶ i \Z ρ i = ρ i <σ∈Σ Z σ .
Note first that in this case,Ŷ i is an (n−2)-fold which can be regarded as a flat family of toric (n − 3) = (k 0 − 1)-dimensional varieties over a curve Σ Θ [2] i , which is the closure of Z Θ [2] i ⊂ T 2 .
The fibre over any point in Z Θ [2] i is given by toric data (ν i -containing simplices within the face i is a Riemann surface (of genus * (Θ [2] i )) with a finite number of punctures. The number of punctures (denoted by k Θ [2] i ) is given by The compact Riemann surface Σ Θ [2] i is obtained by filling these k Θ [2] i punctures in Z Θ [2] i . Any one of those punctures is assigned to one of the 1-dimensional faces Θ [1] ≤ Θ [2] i (see also footnote 14), and the toric k 0 − 1-dimensional fibre geometry is determined by the ν i -containing simplices in the face [1] . From this point of view, the first line of (57), with −(n − 1) replaced by −(k 0 − 1), accounts for the number of irreducible divisors ofŶ i modulo the i punctures. Divisors of a generic fibre correspond to divisors ofŶ i by sweeping them over the whole base.
toric linear equivalence acting in the fibre direction (see Figure 2). The remaining contribution to (57) i . This correction term is now understood as counting only the generic fibre class for an independent divisor inŶ i , while removing the total fibre class at each one of the k Θ [2] i points in Σ Θ [2] i from the formula of h 1,1 (Ŷ i ). This subtraction is necessary because the total fibre class over any point in Σ Θ [2] i is algebraically equivalent (see Figure 3).
The correction termē 0,0 i should always be negative, because this term accounts for the algebraic equivalence relations among the generators of H 2,2 c (Ŷ i ). It is not hard to see this. For a dual pair of faces ( Θ [2] i , Θ [2] i ) and an interior point ν i of Θ [2] i , list up all the dual pairs of faces {( Θ i .
Each one of those pairs leaves (Θ a )] points in Σ Θ [2] i for which the fibre of rather than the generic fibre The last term is obviously negative as the number of edges of a two-dimensional face Θ [2] is always greater than or equal to 3. Thus,ē 0,0 Figure 3: Over specific points, the fibre degenerates and becomes reducible. All fibres are algebraically (and hence homologically) equivalent, however.
The algebraic equivalence relations encoded inē 0,0 c (Z Θ [2] i ) are sometimes among algebraic cycles in the vertical component, and sometimes among cycles in the non-vertical component. Figure 4 is a schematic picture of the singular fibres ofŶ i −→ Σ Θ [2] i at the * (Θ [1] a ) + 1 punctures associated with a given face Θ [1] a ≤ Θ [2] i . There is always at least one combination of the total fibre classes that is in the vertical component; such a combination of the total fibre classes is in fact even in the space of vertical cycles generated by toric divisors. Such a fibre class is algebraically equivalent to the generic fibre class ofŶ i −→ Σ Θ [2] , and is subtracted by the algebraic equivalence on Σ Θ [2] i corresponding to the (3 − #{Θ Suppose that the dual face pair ( Θ [3] a , Θ [1] a ) for an a ∈ A i is such that all the 1-simplices counted in ν i * 1 ( Θ [3] a ) end at some interior points of Θ [3] a on the other boundary of ν i . This means that all the [1 + * (Θ [1] a )] total fibre classes ofŶ i −→ Σ Θ [2] i belong to the space of vertical cycles (see Figure 4). Let Av i be the subset of the labels a ∈ A i satisfying the condition above, and ( Θ [3] v , Θ [1] v ) be such a dual pair of faces. The space of vertical but non-toric cycles originating from a ) whose boundary other than ν i is not in the interior of Θ [3] . Such pairs of faces are denoted by ( Θ [3] nv , Θ [1] nv ). The remaining − Θ [1] nv ≤Θ [2] i * (Θ [1] nv ) independent algebraic equivalences reduce the dimension of the space of algebraic, but non-vertical four-cycles. To v (left hand side) and Θ [3] nv (right hand side). The fibre components drawn in red, blue and green correspond to fibre components which are obtained by intersecting appropriate divisors. These components arise from one-simplices connecting ν i to an interior point of the face Θ [3] a . When there is a one-simplex connecting ν i to a point on the boundary of Θ [3] a , only the sum of several components of different fibres, drawn in grey, arises from an intersection of divisors (so that it should be considered vertical). Again, the two degenerate fibres on the left hand side are algebraic equivalent, as are the two degenerate fibres on the right hand side.
Finally, in the cases of k 0 = n − 1, it is easy to see that all of the generators of the space are vertical cycles. The space of vertical cycles generated by the toric divisors, however, has a dimension given by (62) without being multiplied by e 0,0 Having seen which subspace of H 2,2 c (Ŷ i ) is identified with a part of the vertical component H 2,2 (Ẑ), we are now ready to determine the dimension of the vertical component in H 2,2 V (Ẑ).
Setting aside the components in [H 4 (Y )] 2,2 that have turned out to be non-vertical, we have where Here, * • 1 ( Θ [3] ) is the number of 1-simplices that run through the interior of a face Θ [3] and have at least one boundary point in the interior of Θ [3] . * • 1 ( Θ [3] ) is the number of all other 1-simplices that run through the interior of the face Θ [3] ; It only counts those one-simplices which start and end on the boundary of the face Θ [3] . The Chow group Ch 2 (Ẑ) is obtained by taking a quotient of the space of algebraic (complex)two-cycles by rational equivalence, whereas we have also exploited algebraic equivalence in studying the cohomology group H 2,2 (Ẑ). The Deligne cohomology H 4 D (Ẑ; Z(2)) and the closely related Chow group Ch 2 (Ẑ) not only contains information on the flux field strength G (4) for F-theory, but also on the three-form potential C (3) . Truly of interest in the context of physics application, though, will be H 4 D (Ẑ p ; Z(2)) forẐ p corresponding to a point p ∈ M * in some Noether-Lefschetz locus, rather than that ofẐ p at a generic point p ∈ M * . Fluxes which are in the primary horizontal subspace may also become algebraic in a Noether-Lefschetz locus, where we are expected to end up with under the superpotential W ∝ Ẑ G (4) ∧ Ω, as we have already mentioned in section 2. Ignoring the original context of physics applications, however, let us leave an interesting observation on Ch 2 (Ẑ) forẐ corresponding to a generic point p ∈ M * . The relevance of the refined data contained in the Ch 2 (Ẑ) (compared with homology) to F-theory fluxes has recently been discussed in [45] (see also the literatures therein).
ForẐ p at a generic point in complex structure moduli space p ∈ M * , algebraic cycles generate a subspace of H 2,2 (Ẑ) with a dimension no less than V tor (Ẑ) + V cor (Ẑ) + RM a (Ẑ) + V alg (Ẑ), and possibly larger than this by at most N V 1 (Ẑ). Only the first term of which descends from algebraic cycles of the toric ambient space. Apart from the last term, all the equivalence relations that have been exploited are linear (rational) equivalence. The last term, V alg (Ẑ), introduces algebraic equivalence relations among those cycles as we have already seen. They are associated with divisorsŶ i ofẐ corresponding to a interior points ν i of two-dimensional faces Θ [2] ≤ ∆. The threefoldsŶ i can be seen as flat fibrations of surfaces over curves Σ Θ [2] with genus g = * (Θ [2] ). The fibre classes over any two points in Σ Θ [2] are mutually algebraically equivalent, and they are identified in the cohomology group. Under rational equivalence, however, they form a family of inequivalent classes parametrized by g = * (Θ [2] ) complex parameters. This is analogous to divisors (points) on the curve Σ Θ [2] , which are classified under linear equivalence by Pic(Σ Θ [2] ), which has g = dim C [Pic 0 (Σ [2] Θ )] complex parameters more than the discrete data (the first Chern class) counted in H 1,1 (Σ Θ [2] ). Noting that h 2,1 (Ẑ) = ( Θ [2] ,Θ [2] ) * ( Θ [2] ) * (Θ [2] ) = ( Θ [2] ,Θ [2] ) * ( Θ [2] )g(Σ Θ [2] ), we see that the Ch 2 (Ẑ) group contains h 2,1 (Ẑ) more complex parameters than the cohomology group, and that this difference comes from divisorsŶ i ofẐ that are regarded as flat surface fibration over curves with g > 0. To this end, it is convenient to verify a couple of relations among the combinatorial data first. Let us introduce the following decomposition in order to facilitate the discussion:
We claim that This also means that the following relation holds: Let us verify the relations (93) one by one. As for the first one, note that Obviously one only needs to verify that (97) is equal to 1 (∆ ≤n−2 ) in order to prove that H mon (Ẑ) = V tor (Ẑ m ). Secondly, we see that the first term in (97) counts the number of lattice points in M at the "lattice-distance-2" that are not in the interior of n-dimensional cones, while the second term in (97) counts lattice points at the "lattice distance 1". Now remember that we assume existence of a fine unimodular triangulation of the polytope ∆ (so that bothẐ m and the ambient toric variety P n Σ are smooth). For a cone whose base at the lattice-distance 1 is a minimum volume k-simplex, lattice points at the distance 1 are the k + 1 vertices of the k-simplex, while those at the distance 2 consist of k+1 1 + k+1 2 = k+2 2 =: N k,2 points corresponding to the vertices and 1-simplices on the k-simplex (see figure 5). 13 Because any cone of the fan Σ can be decomposed into such cones of the fan Σ , for any faces Θ < ∆. Using recursion with respect to the faces of Θ, similar relations can be derived for the number of internal one-simplices, * 1 (Θ). This proves the equality between 1 (∆ ≤n−2 ) and (97) and also the first relation H mon (Ẑ) = V tor (Ẑ m ) in (93).
The second one of the relations (93) H red (Ẑ) = V alg (Ẑ m ) can be verified for each one of dual pairs, ( Θ [2] , Θ [2] ). Using the relation (98) for the 2-dimensional face Θ [2] < ∆, we find that 14 13 The slice of such a cone at the distance h becomes a k-dimensional pyramid of height h (see footnote 16). The number of interior points of such a pyramid is N k,h−k−1 = (h − k) k /k!, which becomes positive only when h > k. That is, a lattice point corresponding to a k-simplex is found only at the lattice distance h > k. At the distance 2, ∂(2 ∆), the lattice points correspond only to vertices or 1-simplices on ∂ ∆.
According to mirror symmetry, the subspace of non-vertical components with the dimension also be derived purely combinatorially, without looking at the geometry of the curve and punctures on it. To see this, note first that k Θ [2] = VB, where VB is the number of lattice points appearing on the boundary of a two-dimensional simplicial complex Θ [2] . Now, let T be the number of 2-simplices in Θ [2] , EI and EB the number of 1-simplices in the interior and boundary of Θ [2] , and VI and VB the number of points in the interior and boundary of Θ [2] . The topology of Θ [2] and ∂ Θ [2] indicates that From this, we find that 3VI − EI = 3 − VB = 3 − k Θ [2] .
The space generated by H cor generators must also be a part of the primary horizontal subspace.
The H cor -dimensional subspace of the primary horizontal component must be D 2 ΩẐ that involve at least one deformation that is not represented by a monomial, the last term of (18). It is reasonable that the expression of H cor (Ẑ) above vanishes when there is no pair of dual faces where * ( Θ [1] ) * (Θ [3] ) = 0, because there should be no non-monomial deformation of complex structure in that case.
The correction term H red (Ẑ) in (91) is mirror to the V alg (Ẑ m ) algebraic equivalences, and hence will represent some redundancy in the description of the quadratic deformations of complex structure by H mon (Ẑ) + H cor (Ẑ) generators and RM m a (Ẑ) for the non-horizontal non-vertical components. H red (Ẑ) can be split into three, just like we did for V alg (Ẑ). The dimensions of those three pieces are denoted by H red mon (Ẑ), H red cor (Ẑ) and H red rm (Ẑ). By using the mirror symmetry, we finally arrive at the following formula for the vertical, horizontal and remaining components of H 2,2 (Ẑ) of a Calabi-Yau fourfoldẐ obtained as a hypersurface of a toric variety P 5 Σ .
This result shows under which circumstances the remaining component is present. It is quite reasonable from the perspective of mirror symmetry, though, that the remaining component H 2,2 RM (Ẑ) has a dimension that is symmetric under the exchange of ∆ and ∆. The term RM a (Ẑ) describes a space of algebraic cycles on divisorsŶ i ofẐ that are not obtained by restriction of divisors ofẐ; that is, they come from

Triangulation (resolution) independence
As we have already explained in section 2, formulating F-theory compactification on resolved fourfolds only makes sense if the dimensions of the vertical, horizontal and the remaining com-ponents (i.e., h 2,2 V , h 2,2 H and h 2,2 RM ) of a Calabi-Yau fourfoldẐ are independent of the choice of crepant resolution of singularities in Z s (fine regular unimodular triangulation of ∆). 15 As we have discussed, this follows from the independence of the complex structure moduli space on which resolution is chosen.
In this section 3.6.3 we supplement this general argument with a more specific discussion of triangulation independence for the construction discussed in this section, i.e. for resolutionsẐ of Z s obtained by fine unimodular triangulations of the polytope ∆.
It is easy to see, first, that H mon (Ẑ) and V alg tor (Ẑ) do not depend on the triangulation of ∆ (or ∆), because their expressions only involve the numbers of lattice points in polytopes. Since V tor (Ẑ) and H red mon (Ẑ) are mirror to H mon (Ẑ m ) and V alg tor (Ẑ m ), they are also independent of the triangulation of ∆ (or ∆); although the expression of V tor (Ẑ) involves a number of 1-simplices explicitly, we have seen by using (98) that V tor (Ẑ) is equal to H mon (Ẑ m ), and the number of 1-simplices used in V tor (Ẑ) does not depend on the choice of triangulation. This means that both are independent of triangulations.
The dimensions of other components such as V cor (Ẑ), V alg cor (Ẑ), RM a (Ẑ), etc., however, involve counting the number of 1-simplices with much more specific restrictions, and it is not obvious at first sight how we see triangulation-independence. Let us look at (108), however, where four terms are grouped into two. The first two terms do not depend on triangulation of ∆, but they may depend on triangulation of ∆; the last two terms, on the other hand, do not depend on triangulation of ∆, but they may depend on triangulation of ∆. The dimension of h 2,2 RM (Ẑ), however, has no chance of depending on the choice of triangulation of ∆ by construction. This means that the last two terms of (108) combined-[RM m a (Ẑ) + H red rm (Ẑ)]-should not depend on the choice of triangulation of ∆, not just on triangulation of ∆. Taking its mirror, we see that the combination of the first two terms, does not depend on the triangulation of ∆. This proves that h 2,2 RM (Ẑ) is independent of which (fine, regular, unimodular) triangulation is chosen.
In order to prove that h 2,2 V (Ẑ) is also independent of triangulation of ∆, note that V alg (Ẑ) and N V 1 (Ẑ m ) + N V 3 (Ẑ m ) are independent of triangulation; they depend only on numbers of lattice points, not on 1-simplices. This means that the combination V alg cor (Ẑ) + V alg rm (Ẑ) is also independent of triangulation, because V alg tor (Ẑ) is, and so is the combination [V cor (Ẑ) + RM a (Ẑ)] because of the relation (93). From all above, we see that the combination 15 To be more precise, F-theory formulation suggests this resolution independence only for Calabi-Yau fourfolds where Zs is given by a Weierstrass-model elliptic fibration over B3, andẐ is a crepant resolution of Zs such that Z −→ B3 remains a flat fibration. Thus, the statement here is stretching the "suggestion" a bit too far by not demanding flat elliptic fibration, and also restricting the range of validity by focusing onẐ that are obtained as a hypersurface of a toric fivefold. Thus, an attempt of formulating flux in F-theory using resolved modelsẐ will still survive, even when the dimensions h 2,2 V , h 2,2 H and h 2,2 RM may turn out to depend on resolutions for some Calabi-Yau fourfold Zs which does not admit an elliptic fibration. the second group of terms in (88), is also independent of triangulation. Obviously the independence of [H cor (Ẑ) + H red cor (Ẑ)] also follows from mirror symmetry. We have therefore seen that the six groups of terms in (88, 107, 108) are separately independent of the choice of triangulations of ∆ and ∆ in a toric-hypersurface realization of smoothẐ and singular Z s . This statement is almost the same as a similar statement in section 2, although the argument in section 2 is about any crepant resolutionẐ of Z s , andẐ does not have to be a toric hypersurface. When a Calabi-Yau fourfold hypersurfaceẐ of a toric fivefold is also realized as a complete intersection in an ambient space of higher dimensions, the separation between [V tor + V alg tor ] and [V cor + V alg cor ] may not remain the same, in general. One can also see that the argument for triangulation independence here exploits combinatorics of toric data, (98), but still relies partially on mirror symmetry. This means that there must be some triangulation independent relations involving such numbers as * • 1 (Θ [3] ), * 1 ( Θ [1] ) etc.

Examples
A couple of examples of toric-hypersurface Calabi-Yau fourfolds are presented in this section. We begin with the pair of sextic (6) ⊂ P 5 and its mirror in section 4.1, where the geometry is so simple that we can compute everything by hand. It serves well for the purpose of digesting such notions as stratification and mixed Hodge structure. We will see how things work together nicely so that the long exact sequence (25)  and h 2,2 RM . The results in this section are used as an input in section 6, along with additional results in appendices C and B, to study how the number of flux vacua depends on the number of generations N gen or on the choice of unification group of low-energy effective theories.

The sextic and its mirror
As a first canonical example, let us discuss the sextic fourfoldẐ 6 , a degree-six hypersurface of P 5 , and its mirror manifold denoted byẐ 6,m . With this definition, it is easy to find (using index theorems and the Lefschetz hyperplane theorem) that the Hodge numbers ofẐ 6 are In this presentation of the Hodge numbers, p starts from 0 and increases to the right, while q begins with 0 and increases upward. We will use the same presentation style in this article, when we write down e p,q c of some cohomology groups. For the sextic, h 1,1 (Ẑ 6 ) = 1 is generated by H|Ẑ 6 , restriction of the hyperplane class of P 5  As a toric variety, P 5 can be described by a fan over the faces of a polytopeP 6 , whose six vertices in Z ⊕5 are given by The dual polytope P 6 has six vertices in Z ⊕5 : and one quickly recognizes that the fan Σ over the faces of P 6 gives rise to an orbifold of P 5 .

Geometry of the sextic: mixed Hodge structure of its subvarieties
The sexticẐ 6 has six toric divisorsŶ i , i = 1, · · · , 6, corresponding to the six vertices ofP 6 . These toric divisorsŶ i are all (6) ⊂ P 4 . Similarly,Ŷ i ∩Ŷ j are surfaces (6) ⊂ P 3 , whileŶ i ∩Ŷ j ∩Ŷ k are curves (6) ⊂ P 2 andŶ i ∩Ŷ j ∩Ŷ k ∩Ŷ l six points in P 1 . These facts can be read out from the fact that the k-dimensional faces Θ [k] of the polytope P 6 are k-dimensional pyramids of height-6 (7 points in one edge), 16 which are regarded as the complete linear system of the divisor 6H of P k . Exploiting e.g. index theorems in combination with the Picard-Lefschetz hyperplane theorem one easily finds that h p,q (Ŷ i ) = We begin with the direct computation of H k (Y ) using the explicit results (113) and the Mayer-Vietoris spectral sequence, where Y = ∪ iŶi is the complement of the primary stratum Z P 6 inẐ 6 . The compact support cohomology groups of Z P 6 , on the other hand, are obtained by following the algorithm of [38] (which we have reviewed partially in the previous section). Those computations allow us to see that the long exact sequence (25) nicely reproduces the Hodge diamond (110); see (121).
It is important in the sum rule (42,45) and the algorithm of [38] that they can be applied to polytopes that are not necessarily for the complete linear system defining a family of Calabi-Yau hypersurfaces; the sum rule (45) has been used in (66,67), for example. Thus, we will compute e p,q c (Z Θ [k] ) from the Hodge diamonds of the subvarieties (113), and confirm that the sum rule (42,45) is indeed satisfied at the end of this section.
The cohomology group of the compact (but singular) geometry Y = ∪ iŶi is computed by the Mayer-Vietoris spectral sequence. At the stage of d pq where dim(E p,q 1 ) is shown in the (p + 1)-th column from the left and the (q + 1)-th row from the bottom, in a form maintaining the information of the Hodge filtration. To proceed to the stage E p,q 2 in the spectral sequence calculation, we need to know the morphisms d pq .
We conclude from this that and all other Hodge-Deligne numbers of the cohomology groups H k (Y ) with k = 0, 1, 2, 5, 6 vanish. As for H k (Y ) with k = 3, 4, Let us now move on to the computation the Hodge-Deligne numbers of the cohomology group of the top (primary) stratum Z 6 =Ẑ 6 \ Y = Z P 6 . They are determined by the algorithm of §5 in [38], which is precisely the one we adopted in section 3.5. We only need the value of ϕ's of the polytope P 6 , ϕ 1 (P 6 ) = 1, ϕ 2 (P 6 ) = 456, ϕ 3 (P 6 ) = 3431, ϕ 4 (P 6 ) = 3431, ϕ 5 (P 6 ) = 456, in order to use the algorithm. 17 Using the Hodge-Deligne numbers of other cohomology groups H k c (Z P 6 ) given in (26), we also see that (120) With all the Hodge-Deligne numbers of the cohomology groups H k c (Z P 6 ) and H k (Y ) determined, we are now ready to see that the long exact sequence (25) reproduces the Hodge diamond of the sextic (110):  (117) and (119).
The study above used the Mayer-Vietoris spectral sequence and the Hodge numbers of subvarieties (113) to determine the Hodge-Deligne numbers of H k (Y ). This is doable by hand only for such a simple geometry as the sextic, however. The Hodge-Deligne numbers of H k (Y ) in more complicated geometries are dealt with much more systematically under the approach using the toric stratification (15) and the algorithm and sum rule of [38]. We therefore confirm from here, toward the end of this section, that the sum rule (42) is satisfied indeed by e p,q c (Z Θ [k] )'s of various faces Θ [k] of the polytope P 6 .

Geometry of the mirror-sextic: toric stratification
Let us now use the mirror-to-sexticẐ 6,m fourfold to have a close look at the stratification associated with the toric divisors Ŷ i . The stratification ofẐ 6 was very simple- -because all the faces Θ [k] of the polytopeP 6 are lattice-isomorphic to the minimal simplex of k-dimension, so that any one of E Θ [k] 's consists of a single point. The faces Θ [k] (k < n = 5) of the polytope P 6 are not, however.
and Z Θ [4−k] as well as that of the primary stratum can be computed, and we can use the multiplicativity and additivity of e p,q c 's to compute e p,q (Y ) for the mirror sextic, not just for the case of the sextic. All these details, however, are not recorded here.

RM
Let us now evaluate the dimension of the vertical, horizontal and the remaining components in H 2,2 given by (88, 107, 108), respectively; we do this both for the sexticẐ 6 and its mirrorẐ 6,m .
Let us begin with the vertical component H 2,2 V of the sexticẐ 6 . Because all the faces of the polytope P 6 do not have an interior point (except P 6 itself), both V cor (Ẑ 6 ) and V alg (Ẑ 6 ) vanish. Therefore the vertical component comes purely from intersections of toric divisors, where we have used (89). The vertical component H 2,2 V (Ẑ) is generated by H 2 |Ẑ 6 , as we discussed already at the beginning of this section 4.1.
The same formula (88) can be used also to determine the dimension of the vertical (2,2)forms on the mirror fourfoldẐ 6,m , h 2,2 V (Ẑ 6,m ). We first note that V cor (Ẑ 6,m ) and V alg (Ẑ 6,m ) vanish, as there are no interior points in Θ [1] , Θ [2] and 2 Θ [2] . In order to evaluate V tor (Ẑ 6,m ), we need to count the number of 1-simplices in the polytope P 6 . We find that  6 . It is not difficult to see this directly. First, RM a (Ẑ 6 ) = RM m a (Ẑ 6,m ) vanishes because all the faces Θ [3] of P 6 are minimal three-dimensional simplex, and no 1-simplices are introduced upon triangulation; * • 1 ( Θ [3] ) = 0. Secondly, RM m a (Ẑ 6 ) = RM a (Ẑ 6,m ) also vanishes, because all the one-dimensional faces Θ [1] of P 6 do not have an interior point.
Similarly, one can evaluate the dimension of the horizontal component h 2,2 H (Ẑ 6 ) and h 2,2 H (Ẑ 6,m ) directly from the formulae in the previous section, although we already know their results. As in the discussion for the vertical component above, one can see that both H red and H cor vanish for bothẐ 6 andẐ 6,m . Thus, the horizontal component only comes from the monomial deformation H mon ; the results are and h 2,2 H (Ẑ 6,m ) = H mon (Ẑ 6 ) = 1 − 0 + 0 = 1 (133) by using ϕ 3 ( P 6 ) = 1.

Computations in practice
For practical applications to fourfolds more complicated than the sextic, we clearly do not want to evaluate (88, 107, 108) by hand. What is even worse, it is a non-trivial task to find an appropriate triangulation of ∆. Certainly the value of h 2,2 V , h 2,2 H and h 2,2 RM do not depend on the choice of triangulation, as we have seen at the end of section 3.6, but at least we have to make sure that there is at least one triangulation that is fine, regular and unimodular (see section 3.1.1), or otherwise the formulae in the previous section cannot be applied.
In this section we hence explain how to evaluate (88, 107, 108) in practice using existing computer software. The computation of Hodge numbers is most efficiently carried out using the package PALP [48] (described in some more detail in [49]). In the present context, PALP is also useful to construct a reflexive polytope from a combined weight system.
In order to obtain triangulation we use the package TOPCOM [46]. It is able to perform fine regular triangulations, which are, however, not necessarily star. Furthermore, as we have explained in section 3.1.1, a fine triangulation does not always give rise to a smooth Calabi-Yau hypersurface, let alone a smooth ambient space.
Even though a fine star triangulation is naturally related to a maximally subdivided fan, we may also use a fine regular triangulation of ∆ which is not star for practical purposes. Let us hence assume we have found a regular, fine (non-star) triangulation of ∆. This clearly gives rise to triangulations of all faces of ∆ which are mutually consistent, i.e. the triangulations of any two neighbouring faces induce the same triangulation on their intersection. We may then construct a fan Σ (or, if we like a star triangulation) over all simplices on ∂ ∆ obtained this way. As discussed e.g. in [19] (see also [50]), regularity of the original triangulation implies existence of a strongly convex support function on the simplices on ∂ ∆, which in turn can be lifted to a strongly convex support function on the cones of Σ .
We hence feed the configuration of integral points on ∂ ∆ into TOPCOM to generate a fine regular triangulation, given in terms of five-simplices. As explained above, this can be cast into the data of a fan Σ , or, equivalently, a star triangulation. Such manipulations can be conveniently carried out using the computer algebra system sage [47]. In particular, sage already contains many routines to construct and analyse lattice polytopes.
Having obtained a regular star triangulation, we only need to check unimodularity. Again, this can be easily done by checking that the lattice volumes of all five-dimensional cones are unity.
With a smooth ambient space at hand, we are ready to evaluate the formulae (88, 107, 108). This can again efficiently be done using sage. The computation of the whole procedure outlined above (without any optimization) can be done in a few hours (for polytopes ∆ with O(100) points such as the mirror sextic) to a few days (for polytopes with several 1000 points such as the mirrors of the cases discussed in section 4.4) using an off-the-shelf PC at the time of writing.
Note that a straightforward evaluation of eqs. (88, 107, 108) requires a triangulation both ∆ and ∆, even though we expect the final result to be independent of any triangulation.

Correction terms at work
The sextic and its mirror are clearly very degenerate examples for the evaluation of (88, 107, 108), as the polytope P 6 for P 5 does not have any interior points to its faces and furthermore does not require triangulation. In this section 4.3, we present some examples for which various correction terms in (88, 107, 108) give a non-zero contribution.
Finally, we obtain RM m a (Ẑ B ) = 4 in the following way. It comes from the same four dual pairs of type ( Θ [1] , Θ [3] dia ). As noted already, * ( Θ [1] ) = 1 for all 1-dimensional faces of the polytope P B , and the 1-simplex in figure 8 contributes to * • 1 (Θ [3] dia ), combinatorially; each pair of dual faces ( Θ [1] , Θ [3] dia ) gives rise to 1×1 contribution to RM m a (Ẑ B ), and there are four such pairs. The geometric interpretation of this component, however, is not as directly related to the 1-simplex in Θ [3] dia as the interpretation of RM a (Ẑ B,m ) is. Remember that N V 3 (Ẑ B ) = 0, and note that V cor (Ẑ B,m ) = H cor (Ẑ B ) vanishes in this example. The face Θ [3] dia does not have an interior point or 1-simplex ending on such an interior point. This means that RM m a (Ẑ B ) originates purely from N V 1 (Ẑ B ) in this example. Let us now focus on one of the four pairs of ( Θ [1] , Θ [3] dia ), and let ν i andŶ i be the interior point of Θ [1] and the corresponding divisor ofẐ B , respectively. This divisorŶ i contains a group of strata forming a P 1 fibration over a surface Z Θ [3] dia ; this surface has e 1,1 c (Z Θ [3] dia ) = 1, and this is the origin of RM m a (Ẑ B ) = 0. The Hodge diamond of this divisorŶ i is reconstructed by collecting e p,q c of all the relevant strata (see (41)), and we found that it is (138) h 2,0 (Ŷ i ) = 0, because * (Θ [3] dia ) = 0. All the components in H 1,1 c (Ŷ i ) and H 2,2 c (Ŷ i ) are therefore algebraic, but the exact sequence (79) shows that only a 10-dimensional subspace of the h 2,2 c (Ŷ i ) = 11-dimensional space of algebraic cycles is realized in the form of vertical cycles. The remainingē 1,1 c = 1-dimensional in H 2,2 c (Ŷ i ) is regarded as a part of the RM m a (Ẑ B ) component, which is algebraic (and hence non-horizontal), but not vertical. In this example, we see that the RM m a (Ẑ B )-component is also characterized by (109), not just those the elements in RM a (Ẑ) are.
In this example, the correction terms for H 1,1 (Ẑ B ) and H 1,1 (Ẑ B,m ), i.e. the last term in (17), vanish and all divisor classes are generated by toric divisors for both fourfolds. This means we can easily compute the dimension of H 2,2 V (Ẑ B ) and H 2,2 V (Ẑ B,m ) using the intersection ring of the ambient toric space restricted toẐ B andẐ B,m . This computation reproduces the dimensions of the vertical components in (137), h 2,2 V (Ẑ B ) = 440 and h 2,2 V (Ẑ B,m ) = 32. Using mirror symmetry, we can then also recover the dimensions of the horizontal components. From h 2,2 = 476 it follows again that the remaining component must be 4-dimensional.
A second example is defined by the dual pair of polyhedra where we have displayed the vertices in the form of a matrix. It turns out that these two polyhedra are equivalent, so that they define the same Calabi-Yau fourfold, which is hence selfmirror. We have found fine regular and unimodular triangulations for both P S and P S , which are used for the computations below. The hodge numbers are (141) As expected for a self-mirror fourfold with h 2,2 RM = 0, the result is h 2,2 V = 1 2 h 2,2 . Let us comment on the computation of N V 3 ( P S ), which was zero in the example before. There are three 1-dimensional faces with * ( Θ [1] ) = 5. In P S they have vertices at {(0, 0, −1, 1, −1), (0, 0, 5, 1, 5)}, {(0, 0, −1, 1, −1), (0, 0, 11, 1, 5)} and {(0, 0, 5, 1, 5), (0, 0, 11, 1, 5)}. They are dual to 3-dimensional faces each of which contains 10 points interior to their 2-dimensional faces. As all other 1-dimensional faces have * = 0, we hence find N V 3 ( P S ) = 150. Just like in the example P B , the correction term for H 1,1 (Ẑ S ) vanishes and we can independently verify that h 2,2 V (Ẑ S ) = h 2,2 V (Ẑ S,m ) = 41 by using the intersection ring of the ambient space.
We now discuss an example for which h 2,2 RM = 0 and not all toric divisors are irreducible. The vertices of the polytopes defining this example and its mirror are and We have found a fine, regular and unimodular triangulation for both polytopes. The hodge numbers are Evaluation of (88, 107) and (108) gives In this example, the correction term to h 1,1 (Ẑ 3,m ) is equal to 10 and some of the toric divisors are reducible, giving rise to a non-zero contribution to V cor (Ẑ 3,m ). One can check that the intersection ring of the ambient toric space restricted toẐ 3,m gives rise to 199 four-cycles in this case, as expected from V tor (Ẑ 3,m ) + V alg tor (Ẑ 3,m ) = 199. For V cor (Ẑ 3 ), the correction term to h 1,1 vanishes and all 206 vertical four-cycles are obtained by restriction of intersections of toric divisors.

Elliptic fourfolds
In this section we come to our objects of interest, which are elliptic fourfolds. With the machinery to treat toric hypersurfaces in place, let us discuss elliptic fibrations over toric base manifolds. A smooth elliptic fourfold over a toric base B 3 not supporting any gauge group can be obtained by taking a generic Calabi-Yau hypersurface in an appropriate fibration of P 2 123 over B 3 . The elliptic fibration morphism π then descends from the toric morphism projecting down to B 3 . Torically, we may realize such a situation by setting up a polytope ∆ π B 3 with vertices Here, the v i are the generators of the 1-dimensional cones in the fan of the toric variety B 3 . The above assignment ensures that X 1 (the coordinate associated with (−1, 0, 0, 0, 0)) is a section of −3K B 3 and X 2 (the coordinate associated with (0, −1, 0, 0, 0)) is a section of −2K B 3 .
For appropriate base manifolds B 3 , a generic hypersurfaceẐ π B 3 in the toric variety defined by a fan over faces of the polytope ∆ π B 3 is a smooth elliptic Calabi-Yau manifold with h 1,1 (Ẑ π B 3 ) = h 1,1 (B 3 ) + 1. A compactification of F-theory on such a manifold will not give rise to any gauge symmetry in the low energy effective action.
Another class of examples we will discuss in the following is given by taking a toric base defined by Let us denote the resulting threefolds by B (n) 3 . We can also characterize them as P [O P 2 ⊕ O P 2 (n)], i.e. they are themselves fibrations of P 1 over P 2 . Hence the resulting fourfolds are K3 fibred and for F-theory compactifications there is a heterotic dual. One may think of [z 1 : z 2 ] as coordinates on the P 1 fibre (they also define two sections of the P 1 fibration) and [z 3 : z 4 : z 5 ] as coordinates on the P 2 base.
Using the methods described earlier in this paper, we may now easily compute the Hodge numbers and the dimension of H 2,2 V , H 2,2 H and H 2,2 RM for these examples. The results are given in Table 1.  For a fourfold given by toric data such as (147), we may engineer models with a prescribed gauge group G over a divisor S in the base by choosing the monomials appearing in the defining equation of the hypersurfaceẐ π B 3 appropriately. If S is given by a toric divisor of the ambient space, this is equivalent to deleting points from the dual polytope ∆ π B 3 , G in the M -lattice. If this process results in another reflexive polytope ∆ π B 3 , G , we may construct the dual ∆ π B 3 , G , which contains more points than ∆ π B 3 . If the resulting hypersurfaceẐ G is smooth, these extra points can be interpreted as the exceptional divisors of a resolution.
We may also reverse this process and define our models with an appropriate ∆ π B 3 , which is what we will do in the following. This anticipates the resolution process (which is another way to find the new vertices to add) and only allows those monomials in ∆ leading to the desired fibre structure. This approach is similar to the one used in [51] and the tops of [52,53,54]. Equivalently, the polytopes used may also be constructed from the combined weight systems of appropriately resolved fourfolds.
In the following we will present a few examples for gauge groups SU(5) and SO(10).

SU(5) along P 2
For a fourfold given as a hypersurface in a toric ambient space via a reflexive polytope such as (147), we may engineer fibres of type I 5 in Kodaira's classification along a divisor [z k ] (leading to SU(5) gauge symmetry in a compactification of F-theory) in the base by adding the vertices 18 :   (5) gauge group based on the reflexive polytopes ∆ π P 3 , SU (5) and ∆ π B (n) 3 , SU (5) .
Note that in each case, h 1,1 has increased by four, corresponding to the exceptional divisors. The value of h 2,2 V (Ẑ π B (0) 3 , SU (5) ) = 9 has already been independently computed in [27]. In the last 3 , SU (5) ) = 8 because the 10 matter curve is absent there. 18 To be more precise, we take the convex hull of these points and the polytope (147). In the resulting polytope, ve 1 is not a vertex, but lies on an edge between v k and ve 2 .  .

SO(10) along P 2
In a similar fashion as done in the last section we may construct the polytopes ∆ π B 3 , SO (10) . As (generic) models with gauge group SO(10) can be obtained by a further degeneration of SU(5) models, the polytope ∆ π B 3 , SO(10) contains ∆ π B 3 , SU (5) . The vertices which have to be added to ∆ π B 3 , SU (5) to achieve gauge group SO(10) along z 1 are: For these cases we find the hodge numbers given in Table 3.

Fourfolds which are not hypersurfaces of toric varieties
In the last section, we have shown how H 2,2 (Ẑ, Q) is decomposed into vertical cycles, horizontal cycles and the remaining part H 2,2 RM (Ẑ, Q) for the case of hypersurfaces in toric varieties. As we have seen, the remaining part can be non-zero when divisors ofẐ have algebraic cycles which are not obtained by intersecting with another divisors, cf. (109). In this section we discuss another class of Calabi-Yau fourfolds which can have a non-vanishing remaining part and are motivated in the study of F-theory compactifications.: elliptic fibrations with singular fibres of type I 5 along a divisor S.
Let B 3 be a smooth Fano threefold, and let Z s be a fourfold given by imposing a Weierstrass equation in the ambient space We may choose the Weierstrass equation to be where These conditions leave moduli for s and the a r=5,4,3,2,0 , and we assume that they are chosen generic (in the sense that they are not in a Noether-Lefschetz locus). Fourfolds Z s obtained in this way have an A 4 singularity along the subvariety s = X 2 = X 1 = 0. A compact and non-singular fourfoldẐ can be obtained by carrying out the canonical resolution of the A 4 singularity in Z s first, and then going through a small resolution. The elliptic fibration π :Ẑ −→ B 3 remains a flat morphism [30]. Some more details about this resolution procedure are reviewed in appendices A.1 and A.2 so that there is no ambiguity in the notation used in this article.
This class of geometries generalizes the examples discussed in section 4.4, whereẐ is realized as a hypersurface of a toric variety. As a canonical example, we can think of B 3 = P 3 , and S a quadratic or cubic (or possibly degree d = 4) hypersurface of B 3 . It is known that S = P 1 × P 1 in the d = 2 case, S = dP 6 when d = 3, and S = K3 for d = 4. In these cases, i * : H 1,1 (B 3 ; C) −→ H 1,1 (S; C) is not surjective, which is a welcome feature in F-theory compactifications [20,21].
In this section we study the difference between the algebraic and vertical components in H 2,2 (Ẑ; Q) for this class of geometries. This is done by dividingẐ into a collection of (transversally intersecting) divisors Y = ∪ iŶi ⊂Ẑ and its complement Z :=Ẑ\Y , just like we did in section 3 when studying the case whereẐ is a toric hypersurface. Here, we choose a collection of divisors Y = ∪ iŶi exploiting the fibration structure: the first fiveŶ i , i = 0, 1, 2, 3, 4, are reserved for the five irreducible components of the family of I 5 Kodaira fibres over the divisor S in B 3 , but apart from those, the collection ∪ iŶi consists of the zero section σ of the elliptic fibration π :Ẑ −→ B 3 and π * (D i )'s, where {D i } is some basis of Pic(B 3 ).
We use the long exact sequence (25) with the separationẐ = Y Z described above. The There are two things to be worked out: the first is to determine the (dimension of the) algebraic components in [H 4 c (Y )] 2,2 for cases whereẐ is not necessarily a hypersurface of a toric variety, and the second is to determine . It is known that h 2,2 [H 5 c (Y )] = 0 whenẐ is a hypersurface of a toric variety, but this is not necessarily true for more general cases (note that 2 + 2 ≥ 3, and Z is smooth and non-compact).
Let us now work out [H 4 c (Y )] 2,2 . It is determined by the Mayer-Vietoris spectral sequence, and by (30), in particular. This task consists of determining H 4 (Ŷ i )'s, H 4 (Ŷ i ∩Ŷ j )'s and the morphism between them.
. ForŶ i = π * (D i ) for one of the basis elements of N S(B 3 ), H 1,1 (π * (D)) is generated by restrictions of the divisors σ,Ŷ 0,1,2,3,4 and divisors of B 3 , where we assume that D i ·S is not empty in B 3 for now. To verify this, the Hodge diamond of the threefold π * (D) ⊂Ẑ can be determined by decomposingẐ into strata first, and then by collecting strata that belong to π * (D). It is convenient to start from the following stratification of the three fold π * (D): the surface D ⊂ B 3 is decomposed into D\(D ∩ S), D ∩ (S\(Σ (10) ∪ Σ (5) )) and D ∩ Σ (10) and D ∩ Σ (5) , and the elliptically fibred threefold π * (D) ⊂Ẑ is also decomposed accordingly to the stratification in the base. The Hodge number of the compact and smooth geometry π * (D) is obtained by summing up the Hodge-Deligne numbers of the strata. To h 1,1 (π * (D)) = h 2,2 (π * (D)), only the first two strata contribute.
The claim is now verified 19 .
We assume that a basis {D i } of N S(B 3 ) is chosen such that the number of D i 's with D i · S = φ is the same as the dimensionρ B of the image of In order to determine the Hodge numbers of the five other divisorsŶ 0,1,2,3,4 comprising the I 5 fibre over the the divisor S ⊂ B 3 , we introduce the following stratification of S: Here, Σ (10) is the curve in S given by a 5 | S = 0, and Σ (5) the curve in S given by (a 0 a 2 5 − a 2 a 5 a 3 + a 4 a 2 3 )| S = 0. Those two curves intersect in S, and P E6 and P D6 denote the collection of such intersection points of two different kinds; P E6 are the points where a 5 = a 4 = 0, and P D6 where a 5 = a 3 = 0. Σ • (10) is defined as Σ (10) \(P E6 ∪ P D6 ). The curve Σ (5) forms a double point singularity at each one of the points in P D6 ; Σ • (5) is obtained by resolving the double point singularities (to obtain a curve Σ (5) ), and then removing the lift of the points of P E6 and P D6 . Finally, S • := S\(Σ (10) ∪ Σ (5) ) (see [55,56]).
The Hodge numbers of the five divisors can be calculated by using the stratification of the geometry of S as described above, and the additivity and multiplicativity of the Euler characteristics of the Hodge-Deligne numbers e p,q . The appendix A.3 explains how to carry out the computation in practice, taking the Hodge numbers ofŶ 4 as an example. The result is where the Hodge numbers are given in terms of h 2,0 (S) and h 1,1 (S) and the genera g 10 andg 5 of the (resolved) matter curves; we assume that h 1,0 (S) = 0 here. The appendix A contains more information on the generators of H 1,1 (Ŷ 4 ).
Similar computations can be carried out for four other divisors in the I 5 fibre. The results are:

Determination of H 4 (Y )
Next, H 4 (Y ) is determined as the kernel of (30). Since we have chosen all of theŶ i 's to be irreducible, h 4 (Ŷ i ·Ŷ j ) = 1 for any pair of those divisors (i = j) with a non-empty intersection Y i ·Ŷ j . We switch to the dual (homology group) language in this subsection and the next.
The second group ii) of four-cycles above are mapped into ⊕ i H 4 (π * (D i )), and also leave a cokernel of dimension ρ B . These cycles are represented by π * (curves ⊂ B 3 ).
A given four-cycle of the form π * (D i ) ·Ŷ 1,2,3,4 (with D i · S = φ) among the group iii) of four-cycles appears once in H 4 (Ŷ 1,2,3,4 ) and once in H 4 (π * (D i )). Thus, all of the 4ρ S generators of H 4 (Ŷ 1,2,3,4 ) are in the cokernel in H 4 (Y ). The remaining four-cycles in the third group are of the form π * (D i ) ·Ŷ S . As only aρ B -dimensional subspace of H 1,1 (S) descends from H 1,1 (B 3 ), the images of these cycles in H 4 (π * (D i )) are mapped to aρ B -dimensional subspace of H 4 (Ŷ S ), so that their contribution to the cokernel is ρ S −ρ B -dimensional.
So far, we have identified 2ρ B + 4ρ S + ρ S −ρ B independent generators of the cokernel of (161) in H 4 (Y ). Besides the summands already covered in the discussion above, there are (1 + 1 + 2 + 2 + 3) = 9-dimensions remaining in ⊕ i H 4 (Ŷ i ), and eight dimensions remaining in ⊕ i<j H 4 (Ŷ i ·Ŷ j ). We chooseŶ 2 ·Ŷ 4 as the representative of the 1-dimensional kernel and may conclude that

The vertical components in H 4 (Y )
The (2ρ B + 1)-dimensional subspace of H 4 (Y ) which may be represented by σ · D i and D i · D j is clearly composed of vertical four-cycles. However, not all of H 4 (Y ) arises in this way.
Intersections of the formŶ i ·Ŷ j for i, j ∈ {0, 1, 2, 3, 4} and i = j are all vertical four-cycles as well. They are already contained in the (2ρ B + 1)-dimensional subspace of H 4 (Y ) referred to at the beginning of this subsection, however.
The last remaining group of vertical four-cycles are of the formŶ i ·Ŷ i for i = 0, 1, 2, 3, 4. Such self-intersections can be computed by using such relations aŝ Noting that both c 1 (N S|B 3 ) and c 1 (B 3 )| S are in the image of (155), and using the relations (223-224), one finds that those vertical four-cycles are not independent from those that we have discussed already. Therefore, we conclude that the vertical four-cycles form a subspace of dimension 2ρ B + 4ρ B + 1 in H 4 (Y ).
The remaining 5(ρ S −ρ B )-dimensional subspace of H 4 (Y ) cannot be represented by vertical four-cycles. This subspace has a clear geometric interpretation: for any cycle in S which does not descend from B 3 , we can form five vertical four-cycles by taking the fibre components of the I 5 fibre over S. For every two-cycle Σ in H 2 (S) whose Poincaré dual is in the cokernel of (155), there exists a three-chain γ in B 3 with Σ = ∂γ. This three-chain γ and the boundary morphism

The remaining component
; there are (ρ S −ρ B ) independent choices of Σ and consequently π −1 (γ). This means that only 4(ρ S −ρ B ) out of the 5(ρ S −ρ B ) non-vertical cycles in H 4 (Y ) can become non-trivial four-cycles in H 4 (Ẑ). If there are more cycles in H 5 (Ẑ, Y ) than the ones considered, this number may decrease.
As the five-chains π −1 (γ) constructed above map only to non-vertical cycles in H 4 (Y ) under the boundary map (168), we find that there is at most a (2ρ B + 4ρ B + 1)-dimensional subspace of H 4 (Ẑ) that is represented by vertical four-cycles. If there are more five-chains in H 5 (Ẑ, Y ) than those we discussed above, this subspace may be smaller. Since all the independent divisors are included in the collection ∪ iŶi , all the vertical four-cycles have already been listed up.
The case (ρ S −ρ B ) > 0 is not without phenomenological motivation [20,21]. The non-vertical cycles spanning at most 4(ρ S −ρ B )-dimensions in H 4 (Ẑ), as we discussed above, are precisely the Poincaré dual of four-forms that can break SU(5) unification symmetry down to the gauge group of the Standard Model, while keeping the vector field of U(1) Y massless. In this scenario, those four-cycles of H 4 (Y ) need to represent topologically non-trivial four-cycles in H 4 (Ẑ). If h 2,0 (S) = 0, those four-cycles are automatically algebraic, so they are not horizontal either. The four-forms (four-cycles) of this type in the class of geometries studied in this section are another class of examples of the remaining component H 2,2 RM (Ẑ), and they are also characterized by the property (109). See [57] for an explicit construction of such cycles.
Certainly, all the components of H 2,2 V (Ẑ) are contained in (168), and some of H 2,2 RM (Ẑ) also are, but we cannot say we have exhausted all the components of H 2,2 RM (Ẑ). Therefore we should use the mirror symmetry, h 2,2 H (Ẑ) = h 2,2 V (Ẑ m ), in order to determine h 2,2 H (Ẑ). That will be doable once we can construct a smooth modelẐ as a complete intersection Calabi-Yau of a toric ambient space [58,59]. Such an analysis will be a generalization of what we have done in section 3. A crucial step in constructing such a smooth model is to carry out the resolution of the singularities in the Weierstrass model Z s in toric language. This process was to add the points (150) in section 4.4.1. The essence of this procedure is in the fact that the divisor S, the component of the discriminant locus for I 5 fibre is a toric divisor of B 3 (represented by a vertex in the polytope). This procedure works not just in the case B 3 itself is toric (as in section 4.4.1), but also in cases where B 3 is regarded as a complete intersection in a toric ambient space P k Σ * , as long as this property-S is regarded as a toric divisor of P k Σ * restricted on B 3 -is maintained, as discussed in [51]. 20 It is an extra step to verify that a further small resolution exists, which is equivalent to finding an appropriate triangulation of the polytope; a smooth modelẐ needs to come out as a complete intersection Calabi-Yau, while satisfying the flat-fibration condition [30]. It is beyond the scope of this article, however, to study all of these issues.

Distribution of rank of unification group and number of generations
There is mounting evidence that the decomposition of H 4 (Ẑ; R) of a Calabi-Yau fourfoldẐ in a family π : Z −→ M * has a non-trivial H 2,2 RM (Ẑ; R) component. In addition to the families of Z = K3×K3 in [16] (see also (5)), we have seen in sections 3-5 that the four-forms in H 2,2 RM (Ẑ; R) are often Poincaré dual to topologically non-trivial cycles on divisors. Flux in such a four-cycle violates the condition (8), and hence we are led to take H scan ⊗ R for an ensemble with the same unbroken symmetry in the effective theories to be the primary horizontal subspace The vacuum index density distribution dµ I of such a subensemble of F-theory flux vacua becomes a product of a prefactor and a distribution ρ I (top-dimensional differential form) on M * , as in (13,14). Since the integral of ρ I over the fundamental domain of M * often returns a number of order unity (see [14]; so called "D-limit" regions may have to be removed from M * ), the prefactor therefore gives an estimate of the number of flux vacua with a given set of gauge group, matter representations, multiplicities and choice of (B 3 , [S]). The distribution of the value of coupling constants of such a class of low-energy effective theories is given by the distribution ρ I on M * without being integrated. In this article and Ref. [22], we only discuss physics consequences coming from the prefactor. We learn how the number of vacua 20 In case B3 is a complete intersection of a toric ambient space P k Σ * and S is given by s|B 3 = 0 of some section s ∈ Γ(P k Σ * ; O(DS)) on the toric ambient space, where DS is a Cartier divisor of the fine unimodular Σ * , it is possible to embed the original toric ambient space P k Σ * into another toric variety, the total space of O(DS), by s, and so are B3 and S. B3 is still a complete intersection of the new toric ambient space, and now S is also regarded as a toric divisor, satisfying the condition in the main text. This case, however, lacks generality, in that the divisor class of S needs to be in the image of i * : Pic(P k Σ * ) −→ Pic(B3). Such a situation can be improved by choosing a different embedding in many cases. An alternative approach exists in the case the divisor S of B3 is very ample. One can use the projective embedding of B3 using the complete linear system of S; the divisor S is regarded as the hyperplane divisor of the new ambient space P dim|S|−1 restricted on the image of B3, one of the conditions assumed in the main text. We do not know, however, what kind of conditions to be imposed on B3 so that the image of B3 in P dim|S|−1 is regarded as a complete intersection. depends on the choice of the algebraic information (7-brane gauge group = unification group R) and topological information (number of generations N gen ) of the low-energy effective theories, starting with concrete examples and extracting the essence later on.
Let us begin with the choice R = SU(5) unification and (B 3 , [S]) we studied explicitly in section 4.4.1 (except the one with B 3 = P 3 ). At the beginning, we pay attention to the subensembles for individual N gen , the net chirality in the 10 +5 vs 10 + 5 representations of SU(5). This is done by taking H scan ⊗ R to be that of the real primary horizontal subspace of the families in section 4.4.1, and G (4) fix to be the flux generating chirality. As this class of F-theory compactifications has a dual description in Heterotic string theory, it is well-known how chirality is generated. For the Heterotic string compactified on an elliptically fibred Calabi-Yau threefold with the base B 2 = S = P 2 , π Het : Z Het −→ B 2 = S. Reference [60] introduced the origin of chirality in the form of where γ is a divisor with (possibly half) integral coefficient on a spectral surface C ⊂ Z Het , i C : C → Z Het and j : Σ (10) → C are the embedding maps. It is known from [61,62] that The F-theory dual description of γ F M W has also been known in the literature [25]; necessary details are reviewed briefly in the appendix C; see (235) for the explicit form of the flux. The maximum contribution to the D3-brane charges from G which implies L * K. In such cases, one can even use (76) to derive a relation where the constant term comes from "44" in (76); the relation (76) implies that h 2,2 H does not increase quadratically in h 3,1 , but only linearly.
The reference [14] suggests that the prefactor (2πL * ) K/2 /[(K/2)!] in (13) for L * K be replaced by 21 in the opposite case K L * . It is therefore better, in the cases characterized by (171), to use (174) for the estimate of the number of flux vacua in a given subensemble. L * is always an upper convex quadratic function of N gen , for some c > 0 as in (239), and we can expand the exponent √ 2πKL * in terms of N gen for relatively small number of generations. Retaining only the next-to-leading term, we obtain It is interesting to note that the distribution function (176) has a factorized form. The first factor e K /2 depends on the algebraic information such as R as well as (B 3 , [S]), but it does not depend on N gen . The second factor contains the N gen -dependence. The distribution function on the value of coupling constants of the low-energy effective theories, ρ I , can be multiplied to (176), as it was in (13), if one wishes. One should not expect that the formula (14) remains to be a very good approximation at the quantitative level because the condition K L * does not hold, but still it is not terribly bad to expect that ρ I still retains qualitative aspects of the distribution of the coupling constants after smearing over local regions in the moduli space M * .
The distribution of the number of generations in the low-energy effective theories is given by the second factor, which is the Gaussian distribution with the variance 1/(4πc), for relatively small N gen . This should be regarded as a very robust prediction of landscapes based on F-theory flux compactification, as long as the condition (171) is satisfied in the relevant family of fourfolds π : Z −→ M R * . As for the coefficient c in (175), we can read it out from (239), for the examples we studied in section 4.4.1. Also for many other choices of (B 3 , [S]), the coefficient c is determined by the intersection ring associated with only a few divisors inẐ, and is expected to be more or less of order unity, as in (177). Of course the Gaussian distribution is not a good approximation for N 2 gen large enough to be comparable to the absolute limit χ(Ẑ)/24; the distribution absolutely vanishes, since it is impossible to satisfy the D3-tadpole condition without supersymmetry breaking. Since the value of χ(Ẑ)/24 often comes at the order of hundreds 21 The following argument does not depend too much on the expression exp[ √ 2πKL * ]. Even when this expression were something like K L * /(L * )! ≈ exp[L * ln(K/L * )], the following argument still holds true with little modification. to thousands [9,12,63], however, the distribution of N gen is approximately Gaussian at least for the range N gen 10. The distribution for this range (covering N gen = 3) will be sufficient information for our practical interest. The variance N 2 gen is of order 1/(4πc) ∼ 10 −1 .
Let us now focus on the first factor of the distribution function (176), e K/2 , from which we learn the statistical cost of unified symmetry (7-brane gauge group) R with higher rank. For the purpose of taking ratios of the number of vacua with different unified symmetries R 1 and R 2 , the ratio e (K 1 −K 2 )/2 may be studied by e 3(h 3,1 1 −h 3,1 2 ) instead, when the condition (171) is satisfied. This makes things much easier, since it is much easier to compute/estimate h 3,1 than h 2,2 H . Appendix B exploits this ∆K ≈ 6∆h 3,1 relation and develops the discussion further. In the rest of this section, however, we stick to the dimension K of the primary horizontal space. It is also worth mentioning that e K/2 is regarded as a refinement of the popular "10 500 ", a crude estimate of the number of flux vacua of Type IIB Calabi-Yau orientifold compactifications.
It has been widely accepted at the intuitive level that more general flux leads to geometry stabilized at more general complex structure, and hence with less unbroken symmetry (fewer independent divisors). This intuition has been made quantitative by the factor e K/2 ; Ref. [16] used a family of Calabi-Yau fourfolds that are topologicallyẐ = K3 × K3, and the computation of K for other families in this article adds more examples. The results in the familyẐ = K3×K3 and in the cases in section 4.4 remain similar, though there is difference in detail. In the family of Z = K3 × K3, K is linear in the rank of 7-brane gauge group, and K = 21 × (20 − rank 7 ) (where vanishing cosmological constant is not required; see [16] for more). The value of K becomes smaller for a subensemble with higher-rank unification group R. This remains true in the cases we studied in section 4.4 (see table 4).
At the quantitative level, the difference in the dimension of the flux scanning space K for SU(5) unification and SO(10) unification, K R=A4 − K R=D5 , remains more or less around order O(10) for all the geometries listed in table 4, and the ratio of the number of vacua with a stack of SU(5) 7-branes to that with a stack of SO(10) 7-branes remains of order e O (10) . The fraction of vacua with a stack of SU(5) 7-branes in [S] in all the flux vacua on B 3 is given by e −∆K 0-4 /2 ; the value of ∆K 0-4 = K R=φ − K R=A4 can be 1000-10000 for B 3 = P 3 and B (n) 3 studied in section 4.4, as opposed to the value ∆K 0-4 ∼ 100 in the familyẐ = K3 × K3. As we have discussed, we may estimate ∆K R 1 R 2 by (six times) the number of complex structure moduli which are fixed when enhancing R 1 to R 2 . The result forẐ = K3 × K3 is certainly very special. For more general fourfolds, where the number of moduli is much larger, it seems plausible that the number of complex structure moduli we need to fix to achieve a gauge group of rank 4 or 5 is typically O(100) to O(1000), irrespective of the specific (B 3 , [S]) used. Besides our concrete results (summarized in table 4), such numbers are typical for the dimensions of the middle cohomology of Calabi-Yau fourfolds.
Note that we have assumed a fixed choice of [S] so far. This way of thinking is perfectly reasonable when comparing the statistical cost of enhancing a group of low rank, which we assume is given on [S], to a group of higher rank. When we want to tackle the physically more interesting question of comparing the abundance of models with no gauge group to a model with gauge group R for some [S], we need to be able to sum over various choices of [S]. While we do not expect this to make a relevant contribution in cases where h 1,1 (B 3 ) is small, an exhaustive discussion is beyond the scope of the present work.  Table 4: The value of K R=φ , K R=A4 and K R=D5 in the first three rows are extracted from tables 1, 2 and 3, respectively. In the case B 3 = P 1 × K3, the value of ∆K 0-4 = K R=φ − K R=A4 and ∆K 4-5 = K R=A4 − K R=D5 is determined by using 21 × ∆rank 7 .
(possibly apart from non-perturbative symmetry breaking exponentially suppressed to the level harmless in phenomenology). Eventually this statistical cost needs to be compared against the cost of alternatives. In the application to the dimension-4 proton decay problem, an alternative will be a discrete symmetry, while in the application to the "approximately rank-1" problem of Yukawa matrices (cf. [65,66]), the alternative is to tune a moduli parameter. See also [18].
Obviously one can also exploit the distribution ρ I in (14) to study distribution of the values of the coupling constants in a low-energy effective theory. Certainly the formula 22 (14) is not expected to be very precise, because we expect L * K in many cases. Experience in explicit numerical studies suggests (see [15] and also Fig. 5 of [16]) that qualitative aspects of the actual distribution are still captured by the distribution function ρ I even in cases where L * K.
It should also be noted that the scanning component of the flux, G scan , may play another role in addition to determining the coupling constants of the low-energy effective theories. All the fluxes in the form of G give rise to effective theories with the same N gen , but the number of extra vector-like pairs of matter in the SU(5) GUT5 + 5 representations may vary among such an ensemble of vacua. It has been known widely since the work of [67] that there tend to be many vector-like pairs that do not seem to be present in supersymmetric Standard Models that work phenomenologically well. It took an enormous effort to find a topological choice that leads to small number of vector like pairs. Such study as [67,65], however, was equivalent to only use the flux G As already mentioned in section 3.6.1, specifying fluxes in F-theory (or, more generally Mtheory) backgrounds via a four-form G 4 in (co)homology is not sufficient information to properly characterize all degrees of freedom [68,69]. In particular, we need to specify the three form C 3 .
It is an open problem how to include this data in the vacuum counting problem, i.e. the setup of [14,15].
A Geometry of Elliptic-fibred Calabi-Yau fourfold for general F-theory SU(5) models

A.1 Construction
The geometry considered in section 5, smooth elliptically fibred Calabi-Yau fourfolds for general F-theory SU(5) models, has been studied in [30,25,70], see also [51,71]. We briefly review the construction of the geometry here, so that no ambiguity remains in the notation used.
The construction begins with an ambient space where B 3 is a complex 3-dimensional (Fano) variety. A fourfold Z s is defined as a hypersurface of A 0 by the equation (152). The projection π A 0 : A 0 −→ B 3 defines an elliptic fibration morphism π Zs : Z s −→ B 3 .
Let X 3 , X 2 and X 3 be the homogeneous coordinates corresponding to the W P 2 1:2:3 fibre of the ambient space π A 0 : A 0 −→ B 3 , and σ be the zero section defined by X 3 = 0. D X 1 , D X 2 and D X 3 denotes the zero locus of X 1 , X 2 and X 3 in A 0 , respectively. It follows that Consider a line bundle on B 3 , and let s be a global holomorphic section of this line bundle. The divisor defined by the zero locus of s is denoted by S, i.e. s ∈ Γ(B 3 ; O B 3 (S)). The section s is used in the hypersurface equation of Z s in (152), which implements the condition for an I 5 Kodaira fibre over S [72]. The divisor π * A 0 (S) in A 0 is denoted by D S . First blow-up The fourfold Z s has A 4 singularity along a subvariety of A 0 given by Thus, we blow up at this locus and let A 1 := Bl Y 1 A 0 . The blow-up morphism is denoted by ν 1 : A 1 −→ A 0 , and the exceptional divisor by E 1 . The center of the blowup Y 1 is contained in Z s , and Z s is of multiplicity k = 2 along Y 1 . Thus, where Z Second blow-up The fourfold Z (1) s still has A 2 singularity along the subvariety in A 1 . Thus, let A 2 := Bl Y 2 A 1 . The blow-up morphism is denoted by ν 2 : A 2 −→ A 1 , and the exceptional divisor E 2 . The hypersurface Z (1) s of A 1 contains Y 2 with the multiplicity 2, and hence ν * 2 (Z (1) where Z (2) s is the proper transform of Z (1) s under this blow-up morphism. The proper transforms of E 1 , D (1)

Small resolution The fourfold Z
(2) s still has singularities at loci with codimension higher than two. These singularities can be resolved while the proper transform of Z (2) s remains a flat family of curves over B 3 [30]. We provide a description of two such small resolutions in the following. The two small resolutions correspond, when B 3 and [S] are the ones studied in section 4.4.1, to having v e 1 , v e 3 in the SR ideal, or having v e 2 , v e 4 in the SR ideal. We only describe the first resolution in detail and then add a brief comment concerning the second type.
As a consequence of four successive morphisms, (ν One also finds that A simple way to represent intersections among divisors of the ambient space is by a diagram such as figure 9. It organizes the information in the same way as a fan over faces of a polytope does in the context of toric geometry. In fact, figure 9 is a two-dimensional projection of a A 4 'top' with a triangulation corresponding to the resolution [a]. intersect; if a set of points in this diagram does not share a simplex, the corresponding divisors has empty intersection. This diagram shows the history of successive blow-ups starting from the original ambient space A 0 . This diagram for A [b] It is obvious from this property of the intersection ring thatẐ [a] andẐ [b] are not the same geometries. In fact, we can turnẐ [a] intoẐ [b] by a flop transition, which can already be anticipated from the fact that they only differ by a small resolution. The rich net of phases connected by flop transitions in F-theory compactifications with non-abelian gauge groups and the connection to group-and gauge theory has been explored in [30,31,32,33,34,35]. Any physics consequences in an SU(5) symmetric vacuum should remain the same whether the resolution [a] or [b] is used in formulating the flux background, as we have remarked in section 2. It is quite likely under the argument relying on mirror symmetry (as in section 2) that h 2,2 V , h 2,2 H and h 2,2 RM will indeed turn out to be the same for both [a] and [b]. For this reason, we pay attention only to the small resolution [a] described above, and use in section 5 in the main text, as well as in the rest of this appendix. The superscript [a] will hence be dropped completely in the following.
The degeneration over the matter curve Σ (5) is: Over a generic point in Σ (5) these surfaces become curves that are denoted by C α,β,γ,δ, and C ζ , respectively.
At any one of E 6 points, At any one of D 6 points, Let us first work out the Hodge-Deligne numbers of the various strata of the surface S ⊂ B 3 . Using the divisor η := c 1 (N S|B 3 ) + 6c 1 (S) on S the number of E 6 and D 6 points is given by N E := (5K S + η) · (4K S + η), N D := (5K S + η) · (3K S + η).
Thus the only non-vanishing e p,q c of P E6 and P D6 are e 0,0 c (P E6 ) = N E , e 0,0 c (P D6 ) = N D .
However, there is a double point singularity at each of the points in P D6 . The resolved curve Σ (5) has the genusg The curve Σ • (5) is obtained by removing the points P E6 and the lift of the points P D6 from Σ (5) . Thus we find that e p,q c ( Σ • (5) ) = Finally, the Hodge-Deligne numbers of S • := S\(Σ (10) ∪ Σ (5) ) are obtained by using the additivity of e p,q c . Assuming that h 1,0 (S) = 0, we find that We are now ready to compute the Hodge numbers of the exceptional divisorsŶ 0,1,2,3,4 = E 0,1,2,3,4 |Ẑ. Let us takeŶ 4 as an example; the Hodge numbers of the otherŶ i are determined analogously. The stratification ofŶ 4 is: is generated by π * Let us start from a family for (B 3 , [S], R 1 ), and suppose that the enhancement of the 7brane symmetry R 1 to R 2 occurs when a section f ∈ Γ(B 3 ; O(D)) of some line bundle O B 3 (D) vanishes along S; the section f is used in a defining equation of the Weierstrass model, and the geometry gets more singular for f | S = 0. The line bundle O B 3 (D) here is determined by Tate's algorithm, as we will discuss more explicitly later on. Requiring f to vanish along S, the number of independent complex structure moduli is reduced roughly by The value of ∆K 4-5 in table 4 is estimated reasonably well by 6×h 0 (S; O S (D| S )) = 3(5−n)(4−n) indeed. 23 Let us now use this argument to study how the statistical cost varies for different enhancement of symmetries R 1 → R 2 . Suppose that the Weierstrass model Z s is parametrized by the generalized Weierstrass equation (Tate's form), When b i vanishes along S with the order of vanishing n i , let b i = s n i b i|n i . The dictionary between the order of vanishing n i 's and the 7-brane symmetry R is known [72], and the necessary information is recorded in table 5.
An immediate generalization of the A 4 → D 5 enhancement is the enhancement A m → D m+1 , m ∈ 2N. In these cases, f = b 1|0 and D = −K B 3 for any m ∈ 2N, not just for m = 4. Thus, the same value h 0 (S; O S (−K B 3 | S )) = h 0 (S; O S (S − K S )) provides an approximate upper bound on ∆h 3,1 for any m ∈ 2N. 23 It is a little misleading to use these examples to emphasize that this estimate is good. The P 1 -fibration structure over S in B (n) 3 defines a normal coordinate of S that remains well-defined globally on B (n) 3 . We do not intend to claim that 6h 0 (S; OS(D|S)) is a good estimate of ∆K rather than a good estimate of the upper bound on ∆K.  Table 5: Order of vanishing n i of b i 's required for various types of singular fibre (7-brane gauge group); information relevant to the discussion in this section is extracted from a table in [72]. m ∈ 2N is assumed in this table. The order of vanishing for b 2 is 1 for any one in the I s k and I * s k series.
One can also think of two separate chains of symmetry enhancement, A m → A m+1 → A m+2 and D m+1 → D m+2 → D m+3 . The statistical cost increases at the same pace in these two chains; as the rank increases by two from A m to A m+2 , or from D m+1 to D m+3 , we need to set the sections b 3|m/2 | S , b 4|m/2+1 | S , b 6|m+1 | S and b 6|m+2 | S to zero in any one of those two chains (see table 5). This is consistent with the result above that the statistical cost for the enhancement A m → D m+1 remains much the same for any m ∈ 2N. Noting that b i|n i ∈ Γ(B 3 ; O B 3 (−iK B 3 −n i S)), and hence one finds that the statistical cost for enhancement by rank-one, measured by ∆K/∆m ≈ 6∆h 3,1 /∆m, becomes larger for higher rank m in the case c 1 (N S|B 3 ) < 0; if c 1 (N S|B 3 ) > 0, on the other hand, the cost for one-rank enhancement decreases for higher rank m, because ∆h 3,1 is bounded from above by smaller value. One should be careful in interpreting this phenomenon for c 1 (N S|B 3 ) > 0; it looks as if higher rank gauge group becomes just as "natural" as lower rank gauge group at first sight, but it may also be that the choice of complex structure for such a high rank enhancement has already become impossible.
Let us finally look at the chain of symmetry enhancement E m → E m+1 ; A 4 → D 5 is also regarded as a part of this chain. In this chain, we need to set the section to zero for the enhancement E m → E m+1 . Thus, the statistical cost for one-rank enhancement, ∆K/∆m, becomes increasingly large in higher rank (larger m), if K S < 0. [Note that there are chiral multiplets in the adjoint representation of E m in the spectrum below the Kaluza-Klein scale, if K S > 0 instead.] 24 Interestingly, the behaviour of ∆K/∆m is controlled by the normal bundle N S|B 3 in the (IIB-like) A m type and D m type chains of symmetry enhancement, and by the canonical bundle K S in the E m type chain available in F-theory.

C Flux controlling the net chirality
In order to consider an ensemble of fluxes leading to effective theories with a given number of generations (net chirality) N gen , G fix needs to be chosen so that it generates the net chirality N gen . If we take the scanning space H scan to be the real primary horizontal subspace H 4 H (X; R), then all the flux vacua end up with effective theories with one and the same value of N gen in such an ensemble. This appendix begins with writing down the four-form flux generating the net chirality, which is already a well-understood subject in the literature. We then move on to compute the D3-tadpole bound We implicitly used that G fix ∧ G (4) scan = 0, which follows because the chirality generating flux G (4) fix is chosen within the vertical component H 2,2 V (Ẑ; R) (because the matter surface belongs to the space of vertical cycles), and the primary horizontal subspace is orthogonal to the vertical component.
The choice of (B 3 , [S]) in section 4.4.1 is a simple generalization of [27]; B 3 = P[O P 2 ⊕O P 2 (n)] instead of B 3 = P[O P 2 ⊕ O P 2 ] = P 1 × P 2 . In order to determine the flux G (4) fix generating the chirality of SU(5) unification models, the conditions for Lorentz SO(3,1) symmetry (6) and unbroken SU(5) symmetry (8) were imposed on the space of vertical four-forms H 2,2 V (Ẑ; R). As we can think of the Kähler form as being expanded in a basis consisting of divisors of the base, the section, the generic fibre class and the exceptional fibre components, these constraints also automatically make the D-term (7) vanish. It is legitimate, as we stated above, to search the chirality generating flux G (4) fix from H 2,2 V (Ẑ; R), because the matter surface belongs to the space of vertical four-cycles. We have seen in section 4.4.1 that the space H 2,2 V (Ẑ; R) has nine dimensions, while the conditions of unbroken SO(3,1) Lorentz and SU(5) unified symmetries result in eight independent constraints. This is true for all the cases with −3 ≤ n ≤ 2, not just the case n = 0 in [27]. After a straightforward computation, it turns out that G (4) fix = λ 5Ŷ 4 ·Ŷ 2 + (2S + (3 + n)H P 2 ) · (2Ŷ 1 − 2Ŷ 4 −Ŷ 2 +Ŷ 3 ) , where λ ∈ R and H P 2 is the hyperplane divisor of the base P 2 . We have confirmed that this flux is equivalent to the one given in [25] and, in the case of n = 0, to the choice of [27] with λ = −3β, exploiting rational equivalence. The net chirality for the SU(5) 10-10 representations is given by N gen = χ 10 = −λ(18 − n)(3 − n).
The choice of such chirality generating flux is quantized due to the condition G fix ∈ [H 4 (Ẑ; Z)] shift . It is not an easy task, to say the least, to determine the integral basis of [H 4 (Ẑ; Z)] shift ∩ H 2,2 V (Ẑ; Q), but fortunately this task can be detoured in the cases we are dealing with. A dual description in Heterotic string theory exists for the cases we are dealing with here, and it is known that (169) gives rise to the net chirality (170). We should thus identify λ F M W = λ, and the quantization fix ) 2 /2 and L * for λ = ±1/2, when L * becomes maximal for a given n. The result of K := dim R [H 4 H (Ẑ; R)] is copied from table 2.
follows from that of that of λ F M W [25]. As discussed in [25], G fix with this quantization condition satisfies G It is not obvious from the above equation whether L * becomes integer or not, but L * is indeed; see table 6. The computation here is essentially that of [55], although K3-fibre is used instead of the stable degeneration limit dP 9 ∪ dP 9 , and a specific resolution of singularity is employed. The value of the D3-tadpole upper bound L * depends on the number of generation N gen . This result is used in section 6 (and also [22]) to derive the distribution of N gen in the landscape of F-theory flux vacua.