Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the K\"ahler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order K\"ahler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.


Introduction
At present F-theory compactifications on elliptic Calabi-Yau fourfolds provide the richest class of explicit N = 1 effective theories starting from string theory. The reason is that the construction of Calabi-Yau fourfolds as algebraic varieties in a projective ambient space is very simple and toric, or more generally non-abelian gauged linear σ-model descriptions provide immediately trillions of geometries [1].
In fact, geometric classifications of certain compactifications with restricted physical features seem possible even though this has been achieved mostly for elliptic Calabi-Yau threefolds, where it has been argued that there exists only a finite number of topological types in this class [2].
Most of the generic compact toric examples allow for elliptic fibrations and in addition for each of them there is a huge degeneracy of possible flux choices, which together with non-perturbative effects have been argued to solve the moduli stabilization problem by driving the theory to a particular vacuum. Ignoring the details of how this happens for the concrete geometry under consideration it has been shown that by degenerating the fourfold in a controlled way viable phenomenological low energy particle spectra will emerge in four dimensions as was worked out in the F-theory revival starting with the papers of [3][4][5][6].
An additional nice feature of F-theory is a largely unified description of gauge-or brane moduli in terms of the complex structure moduli space of the fourfold. Together with mirror symmetry this results in a large variety of geometrical tools that can be used to study the physically relevant structures on these moduli spaces. In this paper we want to improve on these tools following the line of the papers [7][8][9][10][11][12].
Of particular interest when studying the F-theory effective action associated to a given Calabi-Yau fourfold are the admissible fluxes. There are two different types, namely horizontal and vertical fluxes, and in general both are necessary to construct phenomenologically viable models. While determining a basis of fluxes over C is relatively straightforward, it has been shown that the fluxes are quantized [13] and finding the proper sublattice -in particular for the horizontal part -is more involved. However, horizontal fluxes on a Calabi-Yau fourfold W can be identified with the charges of topological B-branes on a mirror manifold M . In this work we use the derived category description of the latter and the asymptotic charge formula in terms of the Gamma class [22][23][24] to determine properly quantized fluxes on W . We provide formulas that allow to write down the integral fluxes -and in many cases an integral basis -in terms of the intersection data on M .
We then restrict to the case of non-singular elliptic Calabi-Yau fourfolds and find explicit expressions for several elements of the monodromy group Γ M . We show that a generic subgroup of the monodromy generates the SL(2, Z) action on the Kähler modulus of the fiber. This explains certain modular properties of the topological string amplitudes on M that we also analyze in detail. We find that the genus zero amplitudes in the type II language that determine the Kähler potential and the superpotential are SL(2, Z) quasimodular forms, extending results of [14]. We also show that similar features hold for the genus one amplitude, which is conjectured to be related to the gauge kinetic terms. As in the Calabi-Yau threefold case we find that these amplitudes are related via certain holomorphic anomaly equations, from which they can be reconstructed in simple situations [15][16][17][18][19].
Finally, we study the global structure of the properly quantized horizontal flux superpotential for a particular example. To this end we analytically continue the integral periods to the generic conifold locus, the generic orbifold and the Gepner point. We find that aligned flux stabilizes the theory at the conifold where the scalar potential vanishes. Somewhat surprisingly the complex 8 × 8 continuation matrix can be expressed analytically up to five real constants.
In the rest of the introduction we describe the principle structures associated to the moduli space of Calabi-Yau fourfolds. This will set our notation and guide the reader in later chapters, where we add to this discussion.

Mathematical and physical structures on the moduli space
Let us give a very short account of the complex structure moduli space of Calabi-Yau fourfolds W , its algebraic and differential structures and their physical interpretation.
As far as the differential structure and some aspects of mirror symmetry are concerned this is based on the analysis of [7][8][9]. The analysis can be viewed as a generalization of the ones that lead to special geometry for Calabi-Yau threefolds [20] and was discussed with emphasis on mirror symmetry in [21].
Calabi-Yau manifolds are equipped with a Kähler (1, 1) form ω and a no-where vanishing holomophic (4,0) form Ω with the relation ω 4 /12 = Ω ∧Ω. The complex structure moduli space M is unobstructed and of complex dimension h 3,1 (W ). Further key structures are the bilinear intersection form on the horizontal cohomology α pq , β rs ∈ H 4 hor (W ) = H 40 ⊕ H 31 ⊕ H 22 hor ⊕ H 13 ⊕ H 04 α pq , β rs = W α pq ∧ β rs = 0 unless p = s and q = r , (1.1) which is even as the dimension is even and transversal with respect to the Hodge type as indicated.
Moreover there is a positive real structure where α is a primitive form in H p,q with p + q = n. In particular e −K(z) = R(Ω(z)) , (1.3) defines the real Kähler potential K for the Weil-Petersson metric G i = ∂ j∂ K, which is closely related to kinetic terms of the moduli fields in the N = 1 4d effective action. Here ∂ j = ∂ ∂z i or∂ are the derivatives with respect to the generic coordinates z i on M and their complex conjugates.
Because the intersection (1.1) is even on fourfolds one gets a mixture of algebraic and differential conditions on the periods and if we consider the cohomology over Z we get lattice structures somewhat similar to that of K3 surfaces. In particular the relations lead to non-trivial constraints on the periods. In [12] these relations have been used to fix an integral basis for particular one parameter Calabi-Yau fourfolds. Moreover, the authors used the Gamma class formula for the 8-brane charge as a non-trivial check of their results. We verified that the algebraic constraints can be used to fix an integral basis for the mirror of the two-parameter elliptic Calabi-Yau fourfold X 24 but found that this method quickly becomes unpractical if the number of moduli increases. Our approach is somewhat complementary in that we use the Gamma class formula to fix integral periods and the constraints (1.4) can be used to supplement our technique and as a non-trivial check. In particular, this approach scales well with the number of moduli.
Other immediate data are the 4-point couplings By the usual relation of the horizontal and vertical cohomology rings of W to the (chiral,chiral) and (chiral,anti-chiral) rings of the N = (2, 2) superconformal theory on the worldsheet -with their U (1) l × U (1) r charge bigrading corresponding to the Hodge type grading 1 -and the axioms of the CFT one sees however that these 4-point couplings are not fundamental, but factorize into three-point couplings with the independent associativity condition Here the latin indices run over the moduli fields associated to either the complex structure moduli on W whose tangent space is associated to harmonic forms in H 3,1 (W ) (dual to H 1,3 (W )) or Kähler moduli on M whose tangent space is associated to harmonic forms in H 1,1 (M ) (dual to H 3,3 (M )). The greek indices are associated to elements in H 2,2 hor (W ) and H 2,2 vert (M ), respectively. Theη's define a constant intersection form with respect to a fixed basis of H hor 4 (W ) or a suitable K-theory basis extending H vert * , * (M ). More specifically we can identifyη (2) in a reference complex structure near large radius with the inverse of the pairing on H 2,2 hor (W ) andη (1) with the inverse pairing on H 3,1 (W ) ⊕ H 1,3 (W ), which by (1.1) is block diagonal. This property is maintained throughout the moduli space due to the charge grading.
The basic idea of mirror symmetry is to calculate these couplings, which are nontrivial sections of tensor bundles over M, from the periods of Ω. The latter can be obtained as the solutions of the Picard-Fuchs differential equations. We denote an integral basis of periods by Π κ (z) = Γ κ Ω, where κ = 1, . . . , dim H 4 hor and Γ κ is a fixed 4-cycle basis in H hor 4 (W, Z). This is physically relevant as the flux superpotential is given with respect to this basis by (half) 2 integer flux quanta n κ ∈ Z, quantized due to a Dirac-Zwanziger quantization condition and additional constraints discussed in [13]. The analysis of attractor points and cosmologically suitable minima of the associated scalar potential relies therefore crucially on this basis. Interpreted in the A-model the triple couplings C α ij (t) in the flat coordinates given by the mirror map t k (z) ∝ [C k ] (ω + iB), where [C k ] is an integral curve class on M and B is the Neveu-Schwarz B-field, encode the quantum cohomology of M . In particular each coefficient of the Fourier expansion C α ij (e 2πit k ) counts the contribution of a holomorphic worldsheet instanton in a given topological class. These contributions are directly related to Gromov-Witten invariants at genus zero. Gromov-Witten invariants at genus one can be calculated from the Ray-Singer Torsion, starting with the genus zero data. Both genus zero and genus one worldsheet instanton series give rise to a remarkable integrality structure in terms of additional geometric invariants of embedded curves [25].
An interesting aspect of these generating functions is that they are modular forms of the monodromy group Γ preserving the intersection form in the integral basis. For generic Calabi-Yau fourfolds this aspect is too difficult to appreciate in the sense that not much is known about the corresponding automorphic forms, but for elliptically fibered Calabi-Yau spaces, there is a subgroup of Γ which acts as the modular group on the Kähler modulus τ of the elliptic fiber in M . The precise way this subgroup is embedded in Γ can be inferred using specific auto-equivalences of the derived category of B-branes, as we will see in section 3.2.
It turns out that there is a clash between holomorphicity and modularity in the τ dependence of the triple couplings and the Ray-Singer torsion, which leads for Calabi-Yau threefolds to the holomorphic anomaly equations. We will discuss analogous holomorphic anomaly equations for fourfolds in section 4.

The period geometry of Calabi-Yau fourfolds
In this section we show how to determine integral horizontal fluxes on a Calabi-Yau fourfold W . To this end we interpret the flux lattice as the charge lattice of A-branes on W . This in turn is related via homological mirror symmetry to the charge lattice of B-branes on a mirror manifold M . B-branes on M form the bounded derived category of coherent 2 As pointed out in [13] the combination G4 − c 2 (M ) 2 ∈ H4(M, Z) has to be integral. However, in the concrete examples discussed below c2(M ) is even.
sheaves D b (M ). Given a brane E • ∈ D b (M ) the asymptotic behaviour of the charge can be calculated using the Γ-class. Moreover, an C-basis of fluxes on W can be obtained as the solution to a set of differential equations, the Picard-Fuchs system. Integral generators are then linear combinations of solutions with the correct asymptotic behaviour.
A similar calculation has been used in [26] to obtain the quantum corrected A-model cohomology ring for certain non-complete intersection Calabi-Yau fourfolds. In some cases the asymptotic behaviour was not sufficient to uniquely determine integral elements. As was pointed out in [26], the Jurkiewicz-Danilov theorem and the Lefschetz hyperplane theorem prevent this behaviour for the induced cohomology on complete intersections in toric ambient spaces. In general algebraic constraints on the periods can be used to supplement the above procedure.

The structure of H 4 (W, Z)
The structure of H 4 (W, Z) for a Calabi-Yau fourfold is surprisingly subtle and in this paper we will only be interested in finding an integral basis for the period lattice. However, even this notion demands justification.
We first discuss the structure of H 4 (W, C). By the definition of a Calabi-Yau manifold, H 4,0 (W, C) is generated by a unique, holomorphic 4-form that we call Ω. Then H 3,1 (W, C) is generated by first-order derivatives ∂ z i Ω -modulo a part in H 4,0 (W, C) -where z i are complex structure coordinates. Due to the existence of the the harmonic (4, 0) form, H 3,1 (W, C) can be identified with the first order deformations of the complex structures and by the Tian-Todorov theorem the latter are unobstructed. H 1,3 (W, C) and H 0,4 (W, C) are obtained from these spaces by complex conjugation.
The interesting part is thus H 2,2 (W, C). By Lefschetz decomposition the cohomology splits into Here the subgroup of primitive classes is given by where ω is the Kähler form. On the other hand the so-called primary vertical cohomology is generated by the SL(2, Z) Lefschetz action from the primitive classes in H 1,1 (W, C), i.e.
We now denote the subspace of cohomology generated by derivatives ∂ z i 1 · · · ∂ z in Ω of the holomorphic 4-form as the primary horizontal cohomology H 4 H (W, C). Since the Kähler class is independent of the complex structure, it follows from where H 2,2 RM (W, C) is the subgroup of primitive classes that are neither horizontal nor vertical.
The naive expectation that mirror symmetry maps vertical into horizontal classes and vice versa while the remaining component maps into itself can not hold. It would lead to a contradiction when applied to the geometry studied in [26], where additional "vertical" cycles appear in the quantum deformed A-model intersections. A true statement about the relation under mirror symmetry would therefore require a more refined notion of verticality. This subtlety is avoided when phrasing the problem in terms of branes and homological mirror symmetry.

Fixing an integral basis
Re e iθ Ω Σ = 0 , (2.6) and the Kähler class restricts to zero ω| Σ = 0. In other words, Σ is a special lagrangian cycle that can be wrapped by a topological A-brane L. The central charge of this brane is then given by the period Note that this is equal to the superpotential generated by a flux quantum along Σ. By homological mirror symmetry [28,29], the topological A-branes on W are related to B-branes on the mirror M . The latter correspond to elements in D b (M ), the bounded derived category of coherent sheaves on M . Given a B-brane that corresponds to a complex E • ∈ D b (M ), the asymptotic behaviour of the central charge is where J is the Kähler class on M . The details of this formula will be discussed in the next section. The crucial fact is that the central charges of A-and B-branes are identified via the mirror map. While a construction for all objects in D b (M ) is in general not available, the central charge only depends on the K-theory charge of a complex of sheaves. Our approach to fix an integral basis for the period lattice will be to construct elements E • in D b (M ) that generate the algebraic K-theory group K 0 alg (M ) and calculate the asymptotic behaviour of the central charges. Using the mirror map, these can be interpreted as the leading logarithmic terms of generators of the period lattice. The subleading terms are given by the corresponding solutions to the Picard-Fuchs equations.

B-branes and the asymptotic behaviour of the central charge
A set of maps f i : E i → F i , such that the f i commute with the coboundary maps, corresponds to an element f ∈ Hom(E • , F • ). Objects as well as morphisms are identified under certain equivalence relations but a more detailed discussion of topological branes and D b (M ) is outside the scope of this paper and can be found e.g. in [30]. However, we note that if there is an exact sequence where F is a coherent sheaf and E i are locally free sheaves, i.e. equivalent to vector bundles, then the complex . Now given the Kähler class J, the asymptotic charge of a B-brane that corresponds to the complex E • is given by (2.12) The characteristic class Γ C (M ) can be expressed in terms of the Chern classes of M and for a Calabi-Yau manifold the expansion reads The Chern character of the complex is given by where E i is the vector bundle corresponding to the locally free sheaf E i and the involution (. . . ) ∨ acts on an element β ∈ H 2k (M ) as β ∨ = (−1) k β.
A general basis of 0-, 2-, 6-and 8-branes has been constructed in [26]. The 8-brane corresponds to the structure sheaf O M and the 6-branes are generated by locally free resolutions of sheaves O J i , where the divisors J i generate the Kähler cone. The 0-brane is represented by the skyscraper sheaf O pt. . A basis of 2-branes was constructed as where ι is the inclusion of the curve C a that is part of a basis for the Mori cone and K 1/2 C a is a spin structure on C a . The asymptotic charges have been calculated in [26] and for the readers convenience they are reproduced below. We now describe a construction of 4-branes which in many cases leads to an integral basis. Given effective divisors D i , i ∈ I that correspond to codimension one subvarieties of M and S = i∈I D i , the Koszul sequence is exact and provides a locally free resolution of the coherent sheaf O S . When I contains only one element, this is just the familiar short exact sequence The latter implies the equivalence . This is the locally free resolution employed in [26] to calculate the central charges for a basis of 6-branes. More generally, we can use the Koszul sequence to describe branes wrapped on arbitrary cycles that are intersections of subvarieties of codimension one. If a basis of H 2,2 V (M, C) ∩ H 4 (M, Z) can be constructed this way, then, as we described above, this leads to an integral basis of the period lattice in the mirror. In particular the asymptotic behaviour then uniquely singles out a solution to the Picard-Fuchs system. For a Calabi-Yau hypersurface M in a toric variety P ∆ , the cohomology of the ambient spaces is generated by elements in H 1,1 (P ∆ ). As was pointed out by the authors of [26], the quantum Lefschetz hyperplane theorem then guarantees that H 2,2 V (M, C) is generated by restrictions of elements in H 2,2 (P ∆ , C).
The formula for the asymptotic central charge gives the following results: • 8-brane: • 6-brane wrapped on J a : The charge of the 0-brane is universally Z asy (O pt. ) = 1. We denoted the generators of the Kähler cone by J i and the Kähler form is given by J = t i J i . Finally we need the intersection matrix of the 4-cycles mirror dual to the B-branes. They are not given by the classical intersection numbers in the A-model but rather by the open string index (2.23) The Todd class Td(M ) is for a Calabi-Yau fourfold given by and introduce the intersection matrix η ij = χ(v i , v j ), the inverse matrix η −1 will act on the period vector Π corresponding to the mirror dual cycles. For example

Elliptically fibered Calabi-Yau fourfolds
Although the methods to find integral generators of the period lattice are applicable to general Calabi-Yau manifolds we now restrict to elliptic fibrations such that for a general choice of complex structure on M the fiber exhibits at most I 1 singularities over loci of codimension 1 in the base B. In particular we require the presence of a section. Fourfolds of this type have been previously studied in [14]. It turns out that the intersection ring and the relevant topological invariants are completely determined by the base. Note that this geometric setup is completely analogous to the threefolds studied in [16,18].

Geometry of non-singular elliptic Calabi-Yau fourfolds
As far as it carries over to the fourfold case, we follow the notation in [18] which we now quickly review. The generators of the Mori cone of the base B are given by In particular we assume that the Mori cone is simplicial. Let E be the section so that its divisor class is given by [E].
We now obtain curvesC In the following we will mostly drop the square brackets and assume that the distinction between subvarieties and corresponding classes is clear from the context. The intersection ring of M is determined in terms of intersections on B via where P is any polynomial in h 11 (B) + 1 variables. We denote the complexified areas of the curves in the base bỹ where ω is the Kähler class and B is the Neveu-Schwarz B-field. The complexified area of the fiber will be calledτ The generators of the Mori cone and the dual generators of the Kähler cone provide a natural choice of basis for divisors and curves from the geometric perspective. However, as was already observed for elliptically fibered threefolds, the SL(2, Z) subgroup of the monodromy acts more naturally in a different choice of basis. We introduce with 8) and the dual basis The complexified areas corresponding to C k and C e are now given by respectively. Finally we introduce the exponentiated complexified areas with similar definitions for Q k and q e . We also define the topological invariants of the base and denote the k-th degree component of ch(F • ) by ch k (F • ). The definitions above are straightforward extensions of the corresponding threefold expressions introduced in [18]. For Calabi-Yau fourfolds a basis of middle-dimensional cycles has to be specified as well. It turns out that for elliptically fibered fourfolds with at most I 1 singularities in the fibers such a basis is given by We call the 4-cycles H k = π −1C k , k = 1, . . . , h 11 (B) that result from lifting a curve in the base to a 4-cycle in M the π-vertical 4-cycles. As we will see in section 4.5.1 the genus zero amplitudes that correspond to π-vertical 4-cycles have particularly simple modular properties. Using the Koszul sequence (2.16) we calculate with the volume form V and where we assume that for effective divisorsD a . The Chern characters of the 6-branes are given by

Fourier-Mukai transforms and the SL(2, Z) monodromy
The B-model periods are multi-valued and experience monodromies along paths encircling special divisors in the complex structure moduli space. Homological mirror symmetry [28] implies that the corresponding monodromies in the A-model lift to auto-equivalences of the derived category [28,31,32]. Furthermore, an important theorem by Orlov states that every equivalence of derived categories of coherent sheaves of smooth projective varieties is a Fourier-Mukai transform.
where π 1 and π 2 are the projections from X × Y to Y and X respectively. The object E is called the kernel and R and L indicate that one has to take the left-or right derived functor in place of π * , π * or ⊗. For our purpose the nice property of this picture is that certain general monodromies correspond to generic Fourier-Mukai kernels. This allows us to write down closed forms not only for the large complex structure monodromies but also for a certain generic conifold monodromy and a third type that is special to elliptically fibered Calabi-Yau.
Let D be one of the generators of the Kähler cone and C the dual curve. The limit in which C becomes large corresponds to a divisor in the Kähler moduli space. It is well known [31] that the Fourier-Mukai transform corresponding to the monodromy around this large radius divisor acts as We choose a basis of branes and calculate the monodromy for the large radius divisor corresponding to D j , acting on the vector of charges. One can obtain a similar expression for the monodromỹ T e , corresponding toD e . Another auto-equivalence, the Seidel-Thomas twist, corresponds to the locus where, given a suitable loop based on the point of large radius, the D8-brane becomes massless. Its action on the brane charges is given by (3.23) As was explained in [12], for a Calabi-Yau fourfold and this monodromy is of order two. Elliptically fibered Calabi-Yau manifolds with at most I 1 singularities exhibit yet another type of auto-equivalence. Physically it corresponds to T-duality along both circles of the fiber torus. The corresponding action Φ on the derived category was first studied by Bridgeland [34] in the context of elliptic surfaces. Calculations for Calabi-Yau threefolds can be found in [35] and were elaborated on in the subsequent review [36]. In full generality the auto-equivalences and their implications for the modularity of the amplitudes on elliptic Calabi-Yau threefolds with I 1 singularities [37] have been presented in [38].
We can decompose the Chern character of a general brane E • as (3.24) Here we introduced n, n E , n e , s ∈ Q, and F i , B i are pullbacks of forms in H i,i (B, C). The volume form on M is denoted by V . Adapting the calculation in [36] to Calabi-Yau fourfolds, we find that the Chern character of the transformed brane is given by with c 1 = π * c 1 (B). Using the formulae for the Chern characters of the basis of branes introduced above, this translates into the matrix for the corresponding monodromy.
We can now explicitly calculate that and another careful calculation reveals It follows that generate a group isomorphic to P SL(2, Z), the modular group. In particular, Q k , q are invariant under T , while some of theQ k obtain a sign under T -transformations if the canonical class of the base is not even. As was already noted by [18], this makes Q k and q the correct expansion parameters for the topological string amplitudes to exhibit modular properties.

Toric construction of mirror pairs
To fix conventions we will briefly review the Batyrev construction of Calabi-Yau n-fold mirror pairs (M, W ) as hypersurfaces in toric ambient spaces [39]. The data of the mirror pair is encoded in an n + 1-dimensional reflexive lattice polytope ∆ ⊂ Γ and the choice of a regular star triangulation of ∆ and the polar polytope that is embedded in the dual lattice Γ * . We denoted the real extensions of the lattices by Γ R and Γ * R respectively. The triangulation of ∆ * leads to a fan by taking the cones over the facets that in turn is associated to a toric variety P ∆ . he family M of Calabi-Yau n-folds is given by the vanishing loci of sections P ∆ ∈ O(K ∆ * ) The mirror family W is obtained by exchanging ∆ ↔ ∆ * . Even for a generic choice of section the Calabi-Yau varieties thus constructed might be singular. For n ≤ 3 the singularities can be resolved by blowing up the ambient space. This is not always possible for fourfolds. However, all models studied in this paper can be fully resolved by toric divisors.

Toric geometry of elliptic fibrations
For F-theory we need Calabi-Yau manifolds that are elliptically fibered. One way to construct these is by taking a torically fibered ambient space such that the hypersurface constraint cuts out a genus one curve from the fiber [40]. Toric fibrations can be understood in terms of toric morphisms. toric morphism φ : P ∆ → P ∆ B in turn is encoded in a lattice morphisms such that the image of every cone in Σ is completely contained inside a cone of Σ B . We obtain a fibration with the fan of the generic fiber given by is exact.
We can now obtain elliptically fibered mirror pairs (M, W ) from the following construction [12]. First we combine a base polytope ∆ B and a reflexive fiber polytope ∆ F and embed them into a n + 1-dimensional polytope ∆ as follows: This describes a reflexive pair of polytopes (∆, ∆ * ) given by the convex hulls of the points appearing in (3.34). Using the Batyrev construction one gets an n-fold M from the locus given by (3.31) on the ambient space P ∆ . As mentioned above, M inherits a fibration structure from the ambient spaces P ∆ → P ∆ B and we can identify a map In the following we will consider fibers constructed as E 8 hypersurfaces One can obtain a fibration using the E 8 fiber and a base B from the following toric data: In particular, fibrations of this type have a section and at most I 1 singularities in the fiber.

Picard-Fuchs operators
The periods of the holomorphic n-form on a Calabi-Yau n-fold are annihilated by a set of differential operators, the Picard-Fuchs system. For Calabi-Yau varieties constructed as hypersurfaces in a toric ambient space it is easy to write down differential equations for which the solution set is in general larger than that spanned by the periods. However, in many cases the solution sets are equal and it is sufficient to study the so-called GKZ-system. How to derive the GKZ-system from the toric data and the relation to the Picard-Fuchs system is explained e.g. in [41].

Amplitudes, geometric invariants and modular forms
The topological string A-model encodes Gromov-Witten invariants, counting holomorphic maps On the other hand, intersecting with the pull-back of the fundamental class [M ] leads to vanishing invariants. The latter property of Gromov-Witten invariants is called the Fundamental class axiom. It follows that for fourfolds the invariants with g ≥ 2 vanish. We will now review the Calabi-Yau fourfold invariants for g = 0, 1 and how they are encoded in various observables of the topological A-model.

Review of genus zero invariants
From the general virtual dimension formula we find vir dimM 0,1 = 2 , and given γ ∈ H 2,2 (M, Z) we obtain well-defined invariants In particular, the classical terms corresponding to F (0) γ are determined by Z asy (O γ ). While the Gromov-Witten invariants are in general rational numbers, they are conjecturally related to integral instanton numbers n 0,β via (4.6) The Gromov-Witten invariants can also be related to meeting invariants m β 1 ,β 2 [25], which for β 1 , β 2 ∈ H 2 (M, Z) virtually enumerate rational curves of class β 1 meeting rational curves of class β 2 . They are recursively defined via the following rules.

Genus one invariants
At genus one, the virtual dimension vanishes for Calabi-Yau manifolds of any dimension. The corresponding invariants are encoded in the holomorphic limit of the genus one free energy Assuming h 2,1 = 0 it has the general form In this expression χ is the Euler characteristic of M , ∆ is the discriminant and z(t) is the mirror map in terms of the algebraic coordinates z and the flat coordinates t. The coefficients b i can be fixed by the limiting behaviour of F (1) in the moduli space.
Assuming that the coordinates z are chosen such that z i (t) = t i + O(t 2 ), the large radius limit (4.14) At genus one, the conjectured relation of the Gromow-Witten numbers to integral invariants n 1,β is more involved and has been worked out in [25]. It involves the meeting invariants as well as the genus zero Gromov-Witten invariants and is given by In Appendix A.1 we provide genus one invariants of the one parameter fourfold geometries discussed in [12]. In the following, apart from studying the modular properties of the amplitudes, we calculate the integral invariants for E 8 fibrations with bases P 3 and P 1 × P 2 . We provide some of the invariants in Appendix A.3. To our knowledge the latter case has not been studied in the literature before and provides further evidence supporting the conjectured relations.

Quasi modular forms and holomorphic anomaly equations
In this section we further explore aspects of modularity on elliptically fibered fourfolds (with at most I 1 singular fibers) that has previously been observed by [14]. The latter authors have proven modularity of the 4-point function with all legs in the base and found modular expansions for the genus zero string amplitudes discussed in section 4.1. We aim here to derive corresponding modular anomaly equations. For the K3 case this was done in [42] and for elliptic threefolds in [16,17]. Our strategy will be the following: We study the the generic degree 24 hypersurface X 24 in P(1, 1, 1, 1, 8, 12) that has been used by [14] to illustrate the modular structure and we find the modular anomaly equations. We borrow the differential operators of [37,43] and find their corresponding version for CY fourfolds by comparing with the observed modular anomaly equations. Then, we conjecture the general form of such differential quations for multiparameter families of elliptically fibered toric CY fourfolds with at most I 1 singularities in the fiber. This leads to a generalized version of the modular anomaly equations that we observed for X 24 . At the end of the day we provide data for another CY fourfold supporting our conjecture.
For X 24 it was found in [14] that the instanton parts of the genus zero free energies F (0) γ admit an expansion where P  It turns out that corresponding 4-cycles can be directly related to the basis introduced above (3.13).
First consider the π-vertical 4-cycles H i . Note that we can express them as which satisfies the intersection relations (3.14), where a ij is the inverse of a ij . Now the asymptotic part of the corresponding genus zero amplitudes follows from the computations of sections 2.3 and 3.1, and reads Notice that the double logarithmic part is proportional to τ and only the i-th (twisted) base Kähler parameter T i appears in the classical part of the amplitude. The first property is analogous to the behaviour of the periods ∂ T i F 0 for CY threefolds, where F 0 is the prepotential. The second property can be satisfied by choosing a special basis of the threefold periods. Following the same lines of the analysis that has been carried out for ∂ T i F 0 in [37,43], we find that the corresponding polynomials P On the other hand, the leading behaviour of the periods over the cycles H i is of the form F where the constant s i can be determined as We find that P H k belongs to the case (b). This can be seen from the factorization of the Yukawa coupling C T i T j T k T l in (1.6), which has modular weight -2. In the next section, to illustrate the reason for the different modular behaviour of the F H k amplitudes, we study the fourfold X 24 . As a special remark, recall the monodromy transformationsT :=T −1 e andS introduced in section 3.2. By introducing the factors in (3.29), we find that they generate the modular group. However, T and S do not belong to the monodromy group of the geometry. On the other hand,T andS act on the fiber parameter as the modular group, sincẽ We note that the coordinates introduced (3.10) transform underT andS as This has been explained already in [37,38] for CY threefolds. We find that the same argument holds for CY fourfolds. On the one hand,T acts onT k trivially. On the other hand, straightforward calculations show that up to exponentially small termsS acts as T k → T k + a k 2 . This leads to (4.22). Moreover, note that the Dedekind eta function transforms as (4.23) It follows that Q k and η 12a k are modular objects with the same multiplier system. In particular, we can rewrite the instanton part of the string amplitudes in the coordinates (3.10) as where each factor in the parenthesis transforms as a modular form of weight -6c 1 (B) · β. A similar expansion can be obtained for the genus one string amplitudes F (1) .

Modularity on the fourfold X 24 (1, 1, 1, 1, 8, 12)
We now study the fourfold X 24 which has been introduced in [9]. Its integrality and modular properties have been further discussed in [14,25].  Here ∆ * B is the polytope for the base P 3 and ∆ * F the fiber polytope for the E 8 fiber with special inner point ν * F 3 = (2, 3). We summarize the toric data in the following table which provides the points of the polytope ∆ * of P ∆ * together with the corresponding toric divisors We use the Sage -Mathematics Software System [45] to calculate toric intersection numbers and the Mori cone. We also provide a worksheet to illustrate the use of Sage for determining the topological invariants and the asymptotic expansions of the integral periods. It can be downloaded from the page The Polytope ∆ * describes a degree 24 hypersurface X 24 given by the locus P ∆ in P ∆ * = P (1, 1, 1, 1, 8, 12). Let X * 24 be the mirror manifold of X 24 defined by the locus P ∆ * = 0 in P ∆ , where P ∆ * = x 0 z 6 (α 1 u 24 1 + α 1 u 24 2 + α 3 u 24 3 + α 4 u 24 4 ) + α 0 (u 1 u 2 u 3 u 4 )xyz + α 6 x 3 + α 7 y 2 . (4.28) Here the α i parametrize the complex structure of X * 24 . By considering the torus action on the homogeneous coordinates of P ∆ , x i → λ In particular the large complex structure limit is defined to be the point at z = 0, this is the maximal degeneration point which corresponds to a large radius limit for the mirror manifold M [39]. For the case of X * 24 we have the following two large complex structure variables .

(4.30)
Essential for the B-model description is the nowhere vanishing holomorphic (4,0) form. This can be written as the residuum where X i are inhomogeneous coordinates on P ∆ . Using the methods of [41], one can obtain the GKZ differential operators from the Mori cone vectors of M . From the GKZ operators one can extract the Picard-Fuchs operators L i Π(z) = 0. Solving the latter differential equations, we obtain the periods Π κ (z) = Γ κ Ω(z) in (1.8). In our present case we find the Picard-Fuchs operators , where θ a = z a ∂ z a . The components of the discriminant of these Picard-Fuchs operators are (4.33) Using (3.13), we can determine a basis {H b , H b } of 4-cycles on X 24 given by (4.34) For later convenience we introduce a special basis {H b , H • b } and refer to this as a 'pure modular basis', where H • b is given by The respective genus zero string amplitudes in the basis {H b , H b } given by (4.18) and (4.19) are (4.36) Here τ and t are the Kähler moduli corresponding to the flat coordinates, which appear in the leading order of the mirror map of z e and z b respectively. For H • b the associated amplitude is given by F In [14] it has been observed that F has modular weight k Ctttt = −2. The intersection matrix of 4-cycles η (2) in the pure modular basis takes the form Moreover, in [14], there have appeared signs of a modular anomaly equation for F (0) H b . We find that the periods F For the case of F we find that it does not follow the relation in (4.37), but another kind of recursive relation given by On the other hand, the genus one string amplitude can be easily computed from (4.11) and (4.14). For X 24 this reads  (1) transforms with modular weight k 1 = 0 as expected. Moreover, we observe a recursive relation for F (1) in terms of the amplitude F As a special remark, the π-vertical period F H b in X 24 closely resembles the quadratic logarithmic solution of the Picard-Fuchs operators in the elliptically fibered Calabi-Yau threefold given by an E 8 fibration over P 2 . In the following section we make use of this similarity and extend it to the language of differential operators introduced in [37,43]. We find that (4.37), (4.38) and (4.40) can be derived from such special differential relations. Then we give a conjectural, generalized version of the modular anomaly equations for fourfolds. In Appendix A.4 we provide data supporting the modular anomaly equations (4.37), (4.38) and (4.40).

Derivation of modular anomaly equations
In this section we use the approach of [37,38,43] to derive modular anomaly equations for general, non-singular elliptic Calabi-Yau fourfolds with E 8 fibers as described by the toric data in (3.37). In particular, we find recursive relations satisfied by the periods over π-vertical cycles and the genus one free energies. On the other hand, we argue that the relation (4.38) is special to X 24 and stems from a holomorphic anomaly equation satisfied by the 4-point couplings that we derive in 4.5.3.
Recall that z e , z b are complex structure parameters that can be expressed in terms of the mirror map as and When taking derivatives with respect to the Eisenstein series E 2 (q), we can keep either z b fixed or t fixed. In the first case one has to account for the q dependence of z b . To distinguish between these operations L E 2 (q) is defined in [37,43] to be the derivative with One immediately obtains the relations In [18], the following non-trivial results for the elliptic threefold X 18 → P 2 have been derived As we noted above, the asymptotic behavior of the periods over π-vertical cycles closely resembles that of the double logarithmic periods for elliptic Calabi-Yau threefolds. Indeed we verified that for X 24 → P 3 the relations hold. Moreover, for any rational or logarithmic functions f (z e , z b ) and g(X 0 ) one finds We can relate the two differential operators in (4.42) by making use of (4.45) and the chain rule to obtain H b in the analogous threefold relation and find Together these relations immediately imply the recursive relation observed in (4.37), We are now ready to generalize the discussion,

Modular anomaly equations for periods over π-vertical 4-cycles
We now consider a general non-singular elliptic Calabi-Yau fourfold M with E 8 fiber as described by the toric data in (3.37). The definition of the differential operators (4.42) can be extended to multiparameter families as Furthermore the relations (4.43) now read L E 2 (q) z i = L E 2 (q) τ = 0. We conjecture the generalization of (4.45) to be given by Note that F (0),inst H i on the right hand side of the last equation is singled out as the unique π-vertical period which only involves t i and τ . Using the chain rule and (4.51), L E 2 can be expressed as Another useful relation we borrow from [37,43] by replacing a linear combination of ∂T i F (0) that matches the leading asymptotic behaviour of F We also assume L E 2 F (0),inst H k = 0 for the instanton contributions to (4.18). This determines the multiparameter version of the recursive relation (4.37) for the amplitudes F (0) H k associated to the π-vertical 4-cycles H k We provide evidence of this relation for the geometry with base P 1 × P 2 in Appendix A.5.

Genus one modular anomaly equation
For the same Calabi-Yau fourfold M described in section 4.5.1, we discuss now the modular anomaly equation for the genus one string amplitude. Recall the form of F (1) given in (4.11). Due to (4.50), we find that L E 2 acts non-trivially only on the determinant contribution, However, we acted on both the classical and the instanton contributions. Denote the classical part by P (b a + 1)t a , which is the linear polynomial appearing in (4.13). This gives a non-trivial contribution when acting with the differential operator L E 2 on F (1) where Using both results (4.55) and (4.56) together with the expressions (4.52) and (4.45), we find the genus one modular anomaly equation for elliptically fibered Calabi-Yau fourfolds following the construction in section 3.4, (4.58) Again we provide the corresponnding data for the case that B = P 1 × P 2 in Appendix A.5 which provides a non-trivial check.

4-point coupling modular anomaly equation
From the B-model perspective the 4-point couplings C pqrs are rational functions in the complex structure variables z e , z i . The A-model 4-point couplings can be expressed in the mirror coordinates t and are related to these via As we reviewed in the introduction, the 4-point coupling can be factorized in terms of the 3-point couplings C γ ab . On the A-side the latter are derivatives of the string amplitudes γ . The factorization of the 4-point function is given by (4.60) Now we act with L E 2 on the A-model 4-point coupling with all legs in the base, i.e. C ijkl (τ,T ), with i, j, k, l = 1, . . . h 11 (B). This leads to the relation where we have used (4.53). We can now insert (4.52) to get a recursive relation of C ijkl with respect to the Eisenstein series E 2 , i.e. a modular anomay equation for the 4-point coupling.
As an example we go back to the E 8 fibration over P 3 . We apply the modular anomaly equation (4.61) to C tttt ≡ C It turns out that this implies the recursive relation (4.38). To see this we insert the factorization of C given in (4.60). We choose the basis {H b , H • b } introduced in (4.35). In such a basis the equation we found in (4.62) can be brought into the form We can now cancel on both sides of the equation and integrate with respect to t. The result is the modular anomaly equation (4.38) satisfied by F It is immediately clear that this does not generalize to multiparameter families and periods F (0)

19). We can always obtain a basis
has modular weight 0. This basis only has to satisfy that η (2) is anti-block-diagonal. Then the 4-point coupling with all legs in the base is given by Acting with L E 2 leads to the relation which cannot be factorized as was possible in the case of X 24 .
5 Horizontal flux vacua for X *

24
We will now use the integral period basis for the mirror X * 24 of X 24 (1, 1, 1, 1, 8, 12) to study the admissible horizontal fluxes and the corresponding vacua. To this end we analytically continue the basis to various special loci in the complex structure moduli space. Note that the structure of the moduli space is similar to that of the mirror of the threefold X 18 (1, 1, 1, 6, 9) which has been studied in [47].
Recall the defining equation of X * 24 , The two components of the discriminant are given by the vanishing loci of First we introduce a new set of complex structure variables by rescaling the homogeneous coordinates on P (1, 1, 1, 1, 8, 12). The defining equation (5.1) becomes In these variables the components of the conifold become where we introduced φ = φ − ψ 6 . The general structure of the moduli space is sketched in figure 1. Note that z e and z b are the Batyrev variables and the large complex structure divisors LR 1 and LR 2 correspond to z e = 0 and z b = 0 respectively. On the other hand, using φ and ψ as variables, both ∆ 1 = 0 and ∆ 2 = 0 have a forth-order tangency with LR 2 . Only after resolving LR 2 ∩ {∆ 1 = 0} we get LR 1 as one of the exceptional divisors. This is reflected in the fact that the point {z e = 0} ∩ {z b = 0} corresponds to a double-scaling limit in φ and ψ. The two divisors that correspond to the components of the conifold are labelled with C 1 and C 2 respectively. Furthermore, we will analyze solutions around the orbifold divisor O 1 that is given by ψ = 0.
Finally note that ∆ 1 and ∆ 2 as well as LR 1 and LR 2 are exchanged under the involution First we study the possible fluxes around C 1 ∩ LR 1 . To this end we choose local coordinates and z e . We transform and solve the Picard-Fuchs equations to obtain a vector of eight solutions with asymptotic behaviour given by We demand that the leading monomial of each period is absent from the other solutions to specify the vector uniquely. This is related to the integral basis at large complex structure via The matrix T c can be obtained by numerical analytic continuation and is given by Using the algebraic constraint 10) and the integral monodromies corresponding to LR 1 and C 1 6 we reduced the numerical uncertainty to five real values r i , i = 1, ..., 5. Due to the size of the expressions we relegated the elements f * , * and the numerical values into appendix A.5.
To further simplify the analysis we will move away from z e = 0 and introduce .
The corresponding vector of solutions is given by This is related to the integral basis at large complex structure via The numerical value of T c as well as those of the other continuation matrices in this section are provided in a Mathematica worksheet that can be downloaded from [46].
We now obtain the monodromy action on Π LR when transported along a lasso wrapping C 1 . Using the algebraic constraints we find the analytic expression for (T c T c ) T η −1 T c T c . Unfortunately we are unable to solve the resulting equation for T c . However, note that where v = ± (0, 10, 0, 6, 1, 4, −1, 0). In other words, the monodromy M c corresponds to a Seidel-Thomas twist, where the charge of the shrinking brane is given by whereT 2 is the monodromy corresponding to LR 2 . The involution exchanges C 1 and C 2 and transforms the brane vanishing at C 1 , up to large complex structure monodromies, into a D 8 -brane.
To obtain a vanishing superpotential at c 1 = 0, we can turn on n ∈ Z units of flux along the cycle with period π c . For this to be a supersymmetric minimum we also have to check that D i W = 0. In flat coordinates t i c this condition reads where K i = ∂ i K and K is the Kähler potential As flat coordinates we can use the normalized periods In terms of these, the vanishing period reads Using the numerical result for T c · T c we find that ∂ i K are regular at c 1 = 0 and therefore D i π c ∼ (t 1 c ) i−1/2 . The scalar potential is given by where G ij is the inverse of the metric G ij = ∂ i ∂jK. We restrict to t 2 c = 0 and introduce Re(t 2 c ) = x, Im(t 2 c ) = y. Then the leading terms of the scalar potential are v =0.020174 x 2 + y 2 + 0.31715x 2 + 0.31715y 2 − 2.8019x x 2 + y 2 + O(x 3 , y 3 ) . (5.25) A plot is shown in figure 2. We checked that this is the dominant contribution at least up to order seven, where we calculated the coefficients to a precision of twenty digits. Deep inside the radius of convergence |t 1 c | ≈ |c 1 | < 1/256 the potential is well approximated by the leading order v ≈ 0.020174 · |c 1 |. Our findings are in agreement with [12] where it was argued that for Calabi-Yau fourfolds the Conifold is generically stabilized by aligned flux.

Orbifold O 1
To expand around O 1 ∩ LR 2 we use the variables z b and We find a vector of solutions to the transformed Picard-Fuchs system with leading terms It is related to the integral basis at large complex structure via However, in contrast to the analytic continuation matrix to the conifold, T o can be determined exactly with the help of the Barnes integral method. The latter has been discussed for one-parameter models in [48] and can be adapted to this two-parameter model. We give the analytic expression in the Mathematica worksheet that can be found online [46]. The monodromy acting on Π LR when transported along a lasso wrapping O 1 is of order six and given by To analyze the possible fluxes we will again move away from the large complex structure divisor and introduce the variable We demand that the leading monomial of each period is absent from the other solutions to specify the vector uniquely. It is related to the previous basis via where the numerical expression for T o has been calculated with a precision of around fifty digits. From the solution vector it follows that every choice of flux leads to a vanishing superpotential at o 1 = 0. Moreover, our numerical analysis shows that D σ i W = 0, i = 1, 2 is generically satisfied at O 1 . If one chooses the flux superpotential in terms of o 1 = x + Iy at o 2 = 0. A plot of the potential, expanded to order eleven, is shown in figure 3. Note that the radius of convergence is o 1 < 216 · (2 − 2 3/4 ) ≈ 69. We did a Monte Carlo scan over non-vanishing flux vectors and found that the scalar potential was always positive at x = y = 0. Moreover, the behaviour close to the origin was qualitatively the same in that the gradient vanished at x = y = 0 but the Hessian was undefined.
We also performed an analytic continuation to the special locus P where the Calabi-Yau becomes a Gepner model. However, the behaviour of the scalar potential was qualitatively the same as for a generic point on O 1 .

Conclusions and Outlook
We described a very efficient method to obtain the integral flux superpotential using the central charge formula defined in terms of theΓ class . This method is simple enough to be applied to multi moduli cases. In particular if the Calabi-Yau fourfold is embedded in a toric ambient space it is in general straightforward to find a basis by toric intersection calculus and the Frobenius method for constructing the periods at the points of maximal unipotent monodromy. Example calculations in Sage can be found on our homepage [46].
We then restrict to non-singular elliptic Calabi-Yau fourfolds and study universal monodromies in the integral basis of the horizontal cohomology and the dual homology. Using this basis we provide general expressions for the monodromies corresponding to T i -shifts, that act as t i → t i + 1 on the Kähler moduli. In the derived category these correspond to the auto-equivalences induced by tensoring with the line bundles of the dual divisor. Physically this is the integral Neveu-Schwarz B-field shift and the action on the periods follows directly from their leading logarithms which are determined again by A-model intersection numbers. In particular, the T e -shift acts as the parabolic operator T in SL(2, Z) on the fiber parameter.
More non-trivially we extend Bridgelands construction of an auto-equivalence of elliptic surfaces to the class of elliptic Calabi-Yau fourfolds with at most I 1 singularities in the fibers. This provides an action of the order two element S in SL(2, Z) on the fiber parameter. Apart from being a non-trivial check of the integrality of our periods, these auto-equivalences generate the full PSL(2, Z) action on the elliptic parameter. This gives rise to modular properties of the genus zero and holomorphic genus one amplitudes as well as a holomorphic anomaly that we analyze in detail.
Let us summarize the types of the amplitudes and the results. The virtual dimension formula (4.2) is positive for genus zero. Therefore we need a meeting condition for rational curves with γ ∈ H 4 (M, Z) (mod torsion) and get different amplitudes F (0) γ (q) for each γ, whose geometry with respect to the fibration structure plays an important role. In genus one the virtual dimension is zero and we get a universal amplitude F (1) . For g > 1 the dimension is negative and hence all higher genus amplitudes vanish. Finally one can also consider the modular properties of the 4-point functions. The clearest situation arises for the genus zero amplitudes associated to π-vertical 4-cycles H k and for the genus one amplitude as well as for the 4-point functions with all legs in the base. In each case we get a complete and universal answer for the holomorphic anomaly equations which can be derived using the methods in [18,37].
For genus zero amplitudes over 4-cycles that are not π-vertical we observe a modular anomaly equation only for the E 8 fibration over P 3 . However, we argue that this is a consequence of the modular anomaly equation of the 4-point function which factorizes for two-parameter families. We also check the integrality of the curve counting invariants of [25] at genus one for various new cases.
In order to study the global properties of the horizontal flux superpotential relevant for F-theory compactifications, we analytically continued the periods of the mirror X * 24 to the following critical divisors displayed in figure 1, whose symmetry implies that we only need to consider the left half of it. We first studied the conifold divisor C 1 . Here we could determine an analytic expression for the 8 × 8 continuation matrix T c in (5.9) up to five numerical coefficients 7 . We also generalized the result of [12] that flux along the vanishing cycle stabilizes the theory at this divisor. We further analyzed the possible flux superpotentials at the generic orbifold divisor O 1 and its special locus P .
The most obvious generalization of this work is to include singular elliptic fibrations. Our formalism for fixing the integral periods explained in section 2.3 will work essentially unchanged and with the same technical tools as long as we have Calabi-Yau spaces embedded into toric varieties and the resolutions of the singularities can be described torically. This will be essential to probe in a quantitative way the flux stabilization mechanism of realistic F-theory vacua. The generalization of the construction of the Bridgeland auto-equivalence should also be possible in principle. In fact at least in the Calabi-Yau threefold case the results for the all genus amplitudes which can be expressed in terms of Weyl-invariant Jacobi-Forms [49,50] indicate that the affine Weyl-group of the singularity will appear as part of the auto-equivalences of the derived category of the A-model.
Using the choice of basis for 4-cycles in (3.13) we obtain Hence the (3.14) intersections follow as q 2 e η 48 Q + P