Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

Journal of High Energy Physics, Jan 2018

Cesar Fierro Cota, Albrecht Klemm, Thorsten Schimannek

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Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

HJE Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds Cesar Fierro Cota 0 1 3 Albrecht Klemm 0 1 2 3 Thorsten Schimannek 0 1 3 0 Endenicher Allee 62 , D-53115 Bonn , Germany 1 Nussallee 12 , D-53115 Bonn , Germany 2 Hausdorff Center for Mathematics, Universita ̈t Bonn 3 Bethe Center for Theoretical Physics, Universita ̈t Bonn 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. Flux compactifications; Topological Strings; F-Theory 1 Introduction 1.1 Mathematical and physical structures on the moduli space 2 The period geometry of Calabi-Yau fourfolds 3 Elliptically fibered Calabi-Yau fourfolds The structure of H4(W, Z) Fixing an integral basis B-branes and the asymptotic behaviour of the central charge Geometry of non-singular elliptic Calabi-Yau fourfolds Fourier-Mukai transforms and the SL(2, Z) monodromy Toric construction of mirror pairs Toric geometry of elliptic fibrations Picard-Fuchs operators 4 Amplitudes, geometric invariants and modular forms Review of genus zero invariants Genus one invariants Quasi modular forms and holomorphic anomaly equations Modularity on the fourfold X24(1, 1, 1, 1, 8, 12) Derivation of modular anomaly equations 4.5.1 4.5.2 4.5.3 Modular anomaly equations for periods over π-vertical 4-cycles Genus one modular anomaly equation 4-point coupling modular anomaly equation 5 6 Horizontal flux vacua for X2∗4 5.1 5.2 Conifold C1 Orbifold O1 Conclusions and outlook A Supplementary data A.1 Curve counting invariants for one parameter fourfolds A.2 Toric data for MPE18×P2 A.3 Geometric invariants for MPE18×P2 A.4 Modular expressions for X24(1, 1, 1, 1, 8, 12) A.5 Modular expressions for MPE18×P2 A.6 Analytic continuation data for X24(1, 1, 1, 1, 8, 12) 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 HJEP01(28)6 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 [3–6]. 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 [7–12]. 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 [14–16] 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 ΓM . We show that a generic subgroup of the monodromy generates the SL(2, Z) action on the K¨ahler 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 – 2 – language that determine the K¨ahler potential and the superpotential are SL(2, Z) quasimodular forms, extending results of [17]. 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 [18–22]. 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 later sections, where we add to this discussion. Note added: after this article appeared on the arxiv, Georg Oberdieck pointed out that our results in section 4.5 match with his and Aaron Pixton’s conjectured holomorphic anomaly equation on Calabi-Yau n-folds appearing in [23, 24]. Moreover, he explained to us the explicit form of the generalized holomorphic anomaly equation for the Gromov-Witten potentials on Calabi-Yau fourfolds, which we include now in appendix B. We performed further non-trivial checks of his conjecture with our data beyond the material that appeared already in appendix A.5. 1.1 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 [7–9]. The analysis can be viewed as a generalization of the ones that lead to special geometry for Calabi-Yau threefolds [25] and was discussed with emphasis on mirror symmetry in [26]. moduli space M is unobstructed and of complex dimension h3,1(W ). Further key structures are the bilinear intersection form on the horizontal cohomology αpq, βrs ∈ Hh4or(W ) = Z W hαpq, βrsi = αpq ∧ βrs = 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 R(α) = ip−qhα, α¯i > 0 , – 3 – (1.2) where α is a primitive form in Hp,q with p + q = n. In particular e−K(z) = R(Ω(z)) , (1.3) Z W Z W defines the real K¨ahler potential K for the Weil-Petersson metric Gi¯ = ∂j ∂¯¯K, which is closely related to kinetic terms of the moduli fields in the N = 1 4d effective action. Here ∂j = ∂zi or ∂¯¯ are the derivatives with respect to the generic coordinates zi 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 Ω ∧ Ω = 0 , Ω ∧ ∂i1 . . . ∂in Ω = 0 , for n ≤ 3 , (1.4) 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 X24 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 Cijkl(z) = hΩ, ∂i∂j ∂k∂lΩ(z)i . 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 grading1 — and the axioms of the CFT one sees however that these 4-point couplings are not fundamental, but factorize into three-point couplings Cijkl(z) = Ciαj (z)ηˆα(2β)Cβkl(z) = Ciαj (z)Cαpk(z)ηˆp(1l) , with the independent associativity condition Ciαj (z)ηˆα(2β)Ckβl(z) = Ciαk(z)ηˆα(2β)Cjβl(z) . (1.5) (1.6) (1.7) 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 H3,1(W ) (dual to H1,3(W )) or K¨ahler moduli on M whose tangent space is associated to harmonic forms in H1,1(M ) (dual to H3,3(M )). The greek indices are associated to elements in Hh2o,2r(W ) and 1The exchange of this identification is the essence of mirror symmetry between W and M . – 4 – Hv2e,2rt(M ), respectively. The ηˆ’s define a constant intersection form with respect to a fixed basis of H4hor(W ) or a suitable K-theory basis extending Hv,ert(M ). ∗ ∗ More specifically we can identify ηˆ( 2 ) in a reference complex structure near large radius with the inverse of the pairing on Hh2o,2r(W ) and ηˆ(1) with the inverse pairing on H3,1(W ) ⊕ H1,3(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 This is physically relevant as the flux superpotential Z W W (z) = G4 ∧ Ω(z) = nκΠκ(z) , (1.8) is given with respect to this basis by (half)2 integer flux quanta nκ Dirac-Zwanziger quantization condition and additional constraints discussed in [13]. The analysis of attractor points and cosmologically suitable minima of the associated scalar ∈ Z, quantized due to a potential relies therefore crucially on this basis. R Interpreted in the A-model the triple couplings Ciαj (t) in the flat coordinates given by the mirror map tk(z) ∝ [Ck](ω + iB), where [Ck] 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 Ciαj (e2πitk ) 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 [27]. 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 K¨ahler 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. 2As pointed out in [13] the combination hG4 − c2(2M) i ∈ H4(M, Z) has to be integral. However, in the concrete examples discussed below c2(M ) is even. – 5 – 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 sheaves Db(M ). Given a brane E • ∈ Db(M ) the asymptotic behaviour of the charge can be calculated using the Γ-class. Moreover, a 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 [28] 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 [28], 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. 2.1 The structure of H4(W, Z) The structure of H4(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 H4(W, C). By the definition of a Calabi-Yau manifold, H4,0(W, C) is generated by a unique, holomorphic 4-form that we call Ω. Then H3,1(W, C) is generated by first-order derivatives ∂zi Ω — modulo a part in H4,0(W, C) — where zi are complex structure coordinates. Due to the existence of the harmonic (4, 0) form, H3,1(W, C) can be identified with the first order deformations of the complex structures and by the Tian-Todorov theorem the latter are unobstructed. H1,3(W, C) and H0,4(W, C) are obtained from these spaces by complex conjugation. The interesting part is thus H2,2(W, C). By Lefschetz decomposition the cohomology splits into Here the subgroup of primitive classes is given by H2,2(W, C) = Hp2r,2im(W, C) ⊕ HV2,2(W, C) . Hp2r,2im(W, C) = {α ∈ H2,2(W, C) | ω ∧ α = 0} , where ω is the K¨ahler 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 H1,1(W, C), i.e. HV2,2(W, C) = {ω ∧ β | β ∈ H1,1(W, C) , ω3 ∧ β = 0} . We now denote the subspace of cohomology generated by derivatives ∂zi1 · · · ∂zin Ω of the holomorphic 4-form as the primary horizontal cohomology HH4 (W, C). Since the K¨ahler class is independent of the complex structure, it follows from ω ∧ Ω = 0 , – 6 – (2.1) (2.2) (2.3) (2.4) in [29], there can be additional primitive classes in Hp2r,2im(W, C)\HH2,2(W, C). The structure is thus vertical. H2,2(W, C) = HH2,2(W, C) ⊕ HR2,M2 (W, C) ⊕ HV2,2(W, C) , (2.5) where HR2,M2 (W, C) is the subgroup of primitive classes that are neither horizontal nor 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 [28], 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. 2.2 Fixing an integral basis A 4-cycle Σ dual to an element in HH4 (W, C) ∩ H4(W, Z) is calibrated symplectically, i.e. (2.6) (2.7) (2.8) and the K¨ahler 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 [30, 31], the topological A-branes on W are related to B-branes on the mirror M . The latter correspond to elements in Db(M ), the bounded derived category of coherent sheaves on M . Given a B-brane that corresponds to a complex E • ∈ Db(M ), the asymptotic behaviour of the central charge is Z M ZBasy(E •) = eJ ΓC(M ) (ch E •)∨ , where J is the K¨ahler 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 Db(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 Db(M ) that generate the algebraic K-theory group Ka0lg(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. Re eiθΩ Σ = 0 , ZA(L) = Ω . Z Σ – 7 – For a Calabi-Yau manifold M , the topological B-branes and the open string states stretched between them are encoded in the bounded derived category of coherent sheaves Db(M ). The objects of this category are equivalence classes of bounded complexes of coherent sheaves E • = . . . E E E d−2 // E −1 d−1 // E 0 d0 // E 1 E A set of maps fi : E i → F i, such that the fi 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 Db(M ) is outside the scope of this paper and can be found e.g. in [32]. HJEP01(28)6 However, we note that if there is an exact sequence Now given the K¨ahler class J , the asymptotic charge of a B-brane that corresponds to Z M Zasy(E •) = eJ ΓC(M ) (ch E •)∨ . The characteristic class ΓC(M ) can be expressed in terms of the Chern classes of M and for a Calabi-Yau manifold the expansion reads ΓC(M ) = 1 + 1 iζ(3) 24 c2 − 8π3 c3 + 1 5760 (7c22 − 4c4) + . . . . The Chern character of the complex is given by . . . // E −1 // E 0 E • = . . . // E −1 / / F // E 0 where F is a coherent sheaf and E i are locally free sheaves, i.e. equivalent to vector bundles, then the complex (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) is equivalent to F inside Db(M ). the complex E • is given by Ca• = ι!OCa (K1a/2) , C – 8 – ch(E •) = . . . − ch(E−1) + ch(E0) − ch(E1) + ch(E2) − . . . , (. . . )∨ acts on an element β ∈ H2k(M ) as β∨ = (−1)kβ. where Ei is the vector bundle corresponding to the locally free sheaf E i and the involution A general basis of 0-, 2-, 6- and 8-branes has been constructed in [28]. The 8-brane corresponds to the structure sheaf OM and the 6-branes are generated by locally free resolutions of sheaves OJi , where the divisors Ji generate the K¨ahler cone. The 0-brane is represented by the skyscraper sheaf Opt.. A basis of 2-branes was constructed as where ι is the inclusion of the curve Ca that is part of a basis for the Mori cone and K1a/2 C is a spin structure on Ca. The asymptotic charges have been calculated in [28] and for the readers convenience they are reproduced below. i∈I 0 ❛❛❛❛00 ⊕ OM (−Di) / / OM − P Di i∈I / / OM 0 / / OM (−D) We now describe a construction of 4-branes which in many cases leads to an integral M and S = T Di, the Koszul sequence basis. Given effective divisors Di, i ∈ I that correspond to codimension one subvarieties of is exact and provides a locally free resolution of the coherent sheaf OS. When I contains only one element, this is just the familiar short exact sequence of complexes in Db(M ). This is the locally free resolution employed in [28] to calculate the central charges for a basis of 6-branes. of elements in H2,2(PΔ, C). 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 HV2,2(M, C) ∩ H4(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 PΔ, the cohomology of the ambient spaces is generated by elements in H1,1(PΔ). As was pointed out by the authors of [28], the quantum Lefschetz hyperplane theorem then guarantees that HV2,2(M, C) is generated by restrictions The formula for the asymptotic central charge gives the following results: // . . . / • 6-brane wrapped on Ja: Zasy (OJa ) = eJ ΓC(M ) [1 − ch (OM (Ja))] cia = 1 24 Ca0aaa + c0a c2(M )JaJi , , c0a = 1 4! Ci0jkltitj tktl + 2 cij titj + citi + c0 , c3(M )Ji , c0 = 7c2(M )2 − 4c4(M ) Z Z M M Z M − Z M cij = c2(M )JiJj , (2.19) • 4-brane wrapped on H = Da ∩ Db: Zasy (ODa∩Db ) = hij titj + hiti + h , hij = h = M • 2-brane wrapped on Ca dual to Ja: DaDbJiJj , hi = DaDb(Da + Db)Ji , (2.21) DaDb(2Da2 + 3DaDb + 2Db2) + c2(M )DaDb the K¨ahler cone by Ji and the K¨ahler form is given by J = tiJi. The charge of the 0-brane is universally Zasy(Opt.) = 1. We denoted the generators of 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 χ(E •, F •) = Td(M ) (ch E •)∨ ch F • . The Todd class Td(M ) is for a Calabi-Yau fourfold given by Z M E where V is the volume form. Note that if we construct a basis of B-branes and introduce the intersection matrix ηij = χ(vi, vj ), the inverse matrix η−1 will act on the period vector Π corresponding to the mirror dual cycles. For example Td(M ) = 1 + + 2V , c2(M ) 12 ~v = (E1•, . . . , En•), Z W Ω ∧ Ω = 0 → ΠT η−1Π = 0 . 3 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 I1 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 [17]. 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 [19, 21]. 3.1 As far as it carries over to the fourfold case, we follow the notation in [21] which we now quickly review. The generators of the Mori cone of the base B are given by {[C˜′k]}, k = 1, . . . , h11(B) = h11(M ) − 1 and the dual basis of the K¨ahler cone is {[Dk′]}. 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 curves ˜k = E · π−1C˜′k , k = 1, . . . , h11(B) , C by the dual basis {[D˜ e], [D˜ k]}, where on M for some representatives C˜′k of [C˜′k]. A basis for the Mori cone on M is given by {[C˜k], [C˜e]}, where [C˜e] is the class of the generic fiber. The K¨ahler cone of M is generated [D˜ k] = π∗[Dk′] , [D˜ e] = [E] + π∗c1(B) . 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 Z M Z M D˜ e · P (D˜ e, D˜ 1, . . . , D˜ h11(B)) = P (c1(B), D1′, . . . , Dh′11(B)) , Z B P (1, D˜ 1, . . . , D˜ h11(B)) = 0 , where P is any polynomial in h11(B) + 1 variables. We denote the complexified areas of the curves in the base by Z Z T˜k = C ˜k B + iω , τ˜ = C ˜e B + iω . where ω is the K¨ahler class and B is the Neveu-Schwarz B-field. The complexified area of the fiber will be called The generators of the Mori cone and the dual generators of the K¨ahler 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 and the dual basis [Ce] = [C˜e] , [Ck] = [C˜k] + a k 2 [C˜e] , ak = Z C ˜k c1(B) , De = D˜ e − 2 2 1 π∗c1(B) = E + 1 π∗c1(B) , Dk = D˜ k . (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) HJEP01(28)6 with similar definitions for Qk and qe. We also define the topological invariants of the base a = c1(B)3 , ai = c1(B)2 · Di′ , aij = c1(B) · Di′ · Dj′ , cijk = Di′ · Dj′ · Dk′ , (3.12) and denote the k-th degree component of ch(F •) by chk(F •). The definitions above are straightforward extensions of the corresponding threefold expressions introduced in [21]. 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 I1 singularities in the fibers such a basis is given by HJEP01(28)6 with Hk = E · π−1Dk′ = E · D˜ k , Hk = π−1C˜′k , Hi · Hj = −aij , Hi · Hj = δij , Hi · Hj = 0 . We call the 4-cycles Hk = π−1C˜′k, k = 1, . . . , h11(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 The complexified areas corresponding to Ck and Ce are now given by respectively. Finally we introduce the exponentiated complexified areas τ = τ˜ and T k = T˜k + a k 2 τ˜ , Q ˜k = exp(2πiT˜k) , q˜e = exp(2πiτ˜) , (3.10) (3.11) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) ch(OHi ) = Hi − 2 ch(OHi ) = Hi − C˜e · hi , 1 C˜k(ckii − aki) + 1 12 V (2ai − 3aii + 2ciii) , hi = Z M E ch3(OHi ) = 2 λa,bE · (Da · Db) · (Da + Db) , Hi = X λa,bD¯ a · D¯ b , ch(OD˜i ) = D˜ i − 2 ch(OE ) = E + 1 1 2 Hkckii + Hi · ai + 6 1 C˜eciii , 6 1 C˜iai + 1 24 V · a . with the volume form V and where we assume that X 1 a,b a,b for effective divisors D¯ a. The Chern characters of the 6-branes are given by Moreover, ch(OM ) = 1, ch(C˜e•) = C˜e and ch(C˜k•) = C˜k. is a Fourier-Mukai transform. Db(X × Y ) and acts as [33, 34]3 The B-model periods are multi-valued and experience monodromies along paths encircling special divisors in the complex structure moduli space. Homological mirror symmetry [30] implies that the corresponding monodromies in the A-model lift to auto-equivalences of the derived category [30, 33, 34]. Furthermore, an important theorem by Orlov states that every equivalence of derived categories of coherent sheaves of smooth projective varieties A Fourier-Mukai transform ΦE : Db(X) → Db(Y ) is determined by an object E ∈ F • 7→ Rπ1∗(E ⊗L Lπ2∗F •) , where π1 and π2 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 K¨ahler cone and C the dual curve. The limit in which C becomes large corresponds to a divisor in the K¨ahler moduli space. It is well known [33] that the Fourier-Mukai transform corresponding to the monodromy around this large radius divisor acts as We choose a basis of branes E • 7→ O(D) ⊗ E • . OM , OE, ODi , OHi , OHi , C˜i, C˜e, Opt. , and calculate the monodromy for the large radius divisor corresponding to Dj , T˜j =  00 00 00    1 0 −δjk acting on the vector of charges. One can obtain a similar expression for the monodromy T˜e, corresponding to D˜e. 3An accessible explanation for physicists of how these calculations are performed can be found in [35]. 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 Z(E•) 7→ Z(E•) − χ(E•, OM )Z(OM ) . (3.23) As was explained in [12], for a Calabi-Yau fourfold χ(OM , OM ) = 2. This implies that Z(OM ) transforms into −Z(OM ) and this monodromy is of order two. Elliptically fibered Calabi-Yau manifolds with at most I1 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 [36] in the context of elliptic surfaces. Calculations for Calabi-Yau threefolds can be found in [37] and were elaborated on in the subsequent review [38]. In full generality the auto-equivalences and their implications for the modularity of the amplitudes on elliptic Calabi-Yau threefolds with I1 singularities [39] have been presented in [40]. We can decompose the Chern character of a general brane E• as Here we introduced n, nE, ne, s ∈ Q, and Fi, Bi are pullbacks of forms in Hi,i(B, C). The volume form on M is denoted by V . Adapting the calculation in [38] to Calabi-Yau fourfolds, we find that the Chern character of the transformed brane is given by ch0(Φ(E•)) = nE , 1 1 ch1(Φ(E•)) = B1 − 2 nE c1 − n · E , ch2(Φ(E•)) = B2 − 2 B1 · c1 + ch3(Φ(E•)) = − 2 B2 · c1 + 1 ch4(Φ(E•)) = −ne V − 61 c12 · F1 · E + 2 c1 · F2 · E + 1 1 24 n c13 · E , 1 12 nE c12 − F1 · E + 2 n c1 · E , 1 1 12 B1 · c12 + s C˜e + 1 2 c1 · F1 · E − F2 · E − 6 1 n c12 · E , with c1 = π∗c1(B). Using the formulae for the Chern characters of the basis of branes introduced above, this translates into the matrix 0 12 ckiiai − ak 112 (3aiiai − 2ciiiai − a)  for the corresponding monodromy. We can now explicitly calculate that and another careful calculation reveals It follows that (S˜ · T˜e−1)3 = −I . S =  h11(B) Y i=1 i T˜−ai/2   S˜ , T =  h11(B) Y i=1  T ˜ai/2 i  T˜e−1 , generate a group isomorphic to PSL(2, Z), the modular group. In particular, Qk, q are invariant under T , while some of the Q˜k obtain a sign under T -transformations if the canonical class of the base is not even. As was already noted by [21], this makes Qk and q the correct expansion parameters for the topological string amplitudes to exhibit modular properties. 3.3 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 [41]. The data of the mirror pair is encoded in an n+1-dimensional reflexive lattice polytope Δ ⊂ Γ and the choice of a regular star triangulation of Δ and the polar polytope Δ∗ = {p ∈ Γ∗R | hq, pi ≥ −1, ∀q ∈ Δ} , (3.30) that is embedded in the dual lattice Γ∗. We denoted the real extensions of the lattices by ΓR and Γ∗R respectively. The triangulation of Δ∗ leads to a fan by taking the cones over the facets that in turn is associated to a toric variety PΔ. The family M of Calabi-Yau n-folds is given by the vanishing loci of sections PΔ ∈ O(KΔ∗ ) PΔ = X Y ν∈Δ∩Γ ν∗∈Δ∗∩Γ∗ aν xhνν∗,ν∗i+1 = 0 . The mirror family W is obtained by exchanging Δ ↔ Δ∗. 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 [42]. Toric fibrations can be understood in terms of toric morphisms. A toric morphism φ : PΔ → PΔB in turn is encoded in a lattice morphisms φ : Γ → ΓB , such that the image of every cone in Σ is completely contained inside a cone of ΣB. We obtain a fibration with the fan of the generic fiber given by ΣF ∈ ΓF if the morphism φ : Γ → ΓB is surjective and the sequence 0 → ΓF ֒−→ Γ −φ→B ΓB → 0 , In the following we will consider fibers constructed as E8 hypersurfaces EE8 : X6( 1, 2, 3 ) = {(x, y, z) ⊂ P2( 1, 2, 3 ) : x6 + y3 + z2 − sxyz = 0}. (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) We can now obtain elliptically fibered mirror pairs (M, W ) from the following construction [12]. First we combine a base polytope ΔB and a reflexive fiber polytope ΔF and embed them into a n + 1-dimensional polytope Δ as follows: ν∗ ∈ Δ∗ νiF ∗ 0 ΔF ∗ 0 ΔB∗ sij ΔB . . νjF . νjF ΔF ν ∈ Δ For a fixed νiF ∗ ∈ ΔF ∗ and νjF ∈ ΔF we introduced sij = hνjF , νiF ∗i + 1 ∈ Z>0. This describes a reflexive pair of polytopes (Δ, Δ∗) 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 PΔ. As mentioned above, M inherits a fibration structure from the ambient spaces PΔ → PΔB and we can identify a map M = {x ⊂ PΔ|PΔ (x) = 0} −→ B = PΔB π One can obtain a fibration using the E8 fiber and a base B from the following toric data: div. ν¯∗ i l(e) 0 −6 D1 In particular, fibrations of this type have a section and at most I1 singularities in the fiber. We will denote as MBE8 the elliptically fibered n-fold given by the fibration EE8 ֒→ MBE8 → 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 [43]. B. maps (3.37) Amplitudes, geometric invariants and modular forms The topological string A-model encodes Gromov-Witten invariants, counting holomorphic f : Σg,p¯ → M , from pointed curves Σg,p¯ of genus g into M . The general formula for the virtual dimension of the moduli stack of stable maps4 into a Calabi-Yau M is given by vir dim M¯ g,n(M, β) = (dim M − 3)(1 − g) + n , M¯ g,n(M, β) and Σ the domain of f . where n is the number of marked points and we require f∗[Σ] = β ∈ H2(M ) for f ∈ While for Calabi-Yau threefolds the virtual dimension is zero at all genera with n = 0, in the case of fourfolds it is non-negative only when g = 0, 1. A positive virtual dimension 4A map is stable if it has at most a finite number of non-trivial automorphisms that preserve marked and nodal points. N0,β(γ) = ev∗1(γ) , Z ξ β≥0 with ξ = [M¯ 0,1(M, β)]virt.. From the topological string theory perspective they are encoded in the instanton part of the normalized double-logarithmic quantum periods F (0) = classical + X N0,β(γ)qβ . γ In particular, the classical terms corresponding to Fγ(0) are determined by Zasy(Oγ ). While the Gromov-Witten invariants are in general rational numbers, they are conjecturally related to integral instanton numbers n0,β via X N0,β(γ)qβ = β≥0 X n0,β(γ) X∞ qdβ β≥0 d=1 The Gromov-Witten invariants can also be related to meeting invariants mβ1,β2 [27], which for β1, β2 ∈ H2(M, Z) virtually enumerate rational curves of class β1 meeting rational curves of class β2. They are recursively defined via the following rules. can be compensated by intersecting with classes on M pulled back along the evaluation maps evi = f (pi) : M¯ g,n → M , i ∈ 0, . . . , n . 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 H2,2(M, Z) we obtain well-defined invariants From the general virtual dimension formula we find vir dim M¯ 0,1 = 2 , and given γ ∈ mβ1,β2 = mβ2,β1 . Z γi ∪ γj . (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) 1. The invariants are symmetric, 2. If either deg(β1) ≤ 0 or deg(β2) ≤ 0, then mβ1,β2 = 0. 3. If β1 6= β2, then mβ1,β2 = X n0,β1 (γi)η( 2 ),ij n0,β2 (γj ) + mβ1,β2−β1 + mβ1−β2,β2 , i,j where γi ∈ HV4 (M, Z) form a basis mod torsion and Genus one invariants for Calabi-Yau fourfolds haven been calculated for example in [27, 28]. 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 F (1) = classical + X N1,βqβ . β≥0 Assuming h2,1 = 0 it has the general form In this expression χ is the Euler characteristic of M , Δ is the discriminant and z(t) is the mirror map in terms of the algebraic coordinates z and the flat coordinates t. The coefficients bi can be fixed by the limiting behaviour of F (1) in the moduli space. Assuming that the coordinates z are chosen such that zi(t) = ti + O(t2), the large t→∞ lim F (1) = − 24 1  c3(M ) ∪ Ji ti + regular , bi = − 24 c3(M ) ∪ Ji − 1 . At genus one, the conjectured relation of the Gromow-Witten numbers to integral invariants n1,β is more involved and has been worked out in [27]. It involves the meeting invariants as well as the genus zero Gromov-Witten invariants and is given by energy radius limit implies X i   Z M 1 Z M β>0 + 1 24 1 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 E8 fibrations with bases P3 and P1 × P2. 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. where Gi¯j is the inverse of the metric Gi¯j = ∂i∂¯jK. We restrict to tc2 = 0 and introduce Re(tc2) = x, Im(tc2) = y. Then the leading terms of the scalar potential are v = 0.020174px2 + y2 + 0.31715x2 + 0.31715y2 − 2.8019xpx2 + y2 + O(x3, y3) . (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 1 inside the radius of convergence |tc | ≈ |c1| < 1/256 the potential is well approximated by the leading order v ≈ 0.020174 · |c1|. Our findings are in agreement with [12] where it was argued that for Calabi-Yau fourfolds the Conifold is generically stabilized by aligned flux. 5.2 To expand around O1 ∩ LR2 we use the variables zb and We find a vector of solutions to the transformed Picard-Fuchs system with leading terms Πo = o51, o51 log (zb) , o15 log2 (zb) , o15 log3 (zb) , o1, o1 log (zb) , o1 log2 (zb) , o1 log3 (zb) + O(o17, z) . It is related to the integral basis at large complex structure via However, in contrast to the analytic continuation matrix to the conifold, To can be determined exactly with the help of the Barnes integral method. The latter has been discussed for one-parameter models in [50] and can be adapted to this two-parameter model. We give the analytic expression in the Mathematica worksheet that can be found online [48]. ΠLR = To · Πo . (5.26) (5.27) (5.28) the orbifold O1 in coordinates o1 = x + Iy, o2 = 0. The monodromy acting on ΠLR when transported along a lasso wrapping O1 is of order six and given by Πo = To′ · Πo′ , To analyze the possible fluxes we will again move away from the large complex structure divisor and introduce the variable Solutions in the new variables are Πo′ = o1, o17, o112, o119, o15, o111, o117, o123 + O(o12, o2) , 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 To′ has been calculated with a precision of around fifty digits. (5.29) (5.30) (5.31) (5.32) (5.33) (5.34) From the solution vector it follows that every choice of flux leads to a vanishing superpotential at o1 = 0. Moreover, our numerical analysis shows that Dσi W = 0, i = 1, 2 is generically satisfied at O1. If one chooses the flux superpotential W = To−′ 1To−1ΠLR,0 = −(0.237201 − 0.907908i)o1 + (97.5605 − 9.49343i)o1o2 − (24181.7 + 1211.32i)o1o22 + O(o4) , this leads to the scalar potential v = 0.011139161558549787439 + O(x2, y2) . in terms of o1 = x + Iy at o2 = 0. A plot of the potential, expanded to order eleven, is shown in figure 3. Note that the radius of convergence is o1 < 216 · (2 − 23/4) ≈ 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 O1. For some recent discussion of moduli stabilization at the point of large complex structure in a particular example see also [51]. 6 Conclusions and outlook We described a very efficient method to obtain the integral flux superpotential using the central charge formula defined in terms of the Γˆ 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 [48]. 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 Ti-shifts, that act as ti → ti + 1 on the K¨ahler 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 Te-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 I1 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 ∈ H4(M, Z) (mod torsion) and get different amplitudes Fγ(0)(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 Hk 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 [21, 39]. For genus zero amplitudes over 4-cycles that are not π-vertical we observe a modular anomaly equation only for the E8 fibration over P3. 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 [27] 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 X2∗4 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 C1. Here we could determine an analytic expression for the 8 × 8 continuation matrix Tc 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 O1 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 autoequivalence 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 Weylinvariant Jacobi-Forms [52, 53] 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. 7Further details about this highly non-trivial analytic continuation can be found at [48]. M/n0,d(J 2) Acknowledgments 1582400 791944986400 783617464399966400 031333248042176116592000 17510976 48263250 71161472 7677952 4347594 1919808 852480 6255156277440 117715791990353760 36449586432 181688069500 354153540352 9408504320 3794687028 988602624 259476480 100346754888576 905026660335000 2336902632563200 15215566524416 4368985908840 669909315456 103646279680 HJEP01(28)6 We would like to thank Andreas Gerhardus, Thomas Grimm, Babak Haghighat, Min-Xin Huang, Hans Jockers, Amir-Kian Kashani-Poor, Sheldon Katz and Georg Oberdieck for discussions and comments. In particular we would like to thank Georg Oberdieck for notes he provided shortly after this paper appeared on the arxiv. AK would like to thank Min-Xin Huang and Sheldon Katz for comments on the draft and the IHES for hospitality during the time when this work was initiated. CFC would like to thank the financial support from the fellowship “Regierungsstipendiaten CONACYT-DAAD mit Mexiko” under the grant number 2014 (50015952) and the BCGS for their generous support. TS would like to thank the BCGS for their generous support as well as Amir-Kian Kashani-Poor and the ENS for hospitality during part of the work on this project. A Supplementary data A.1 Curve counting invariants for one parameter fourfolds Here we report the genus zero and genus one curve counting invariants for nine one parameter fourfolds in toric ambient spaces with generalized hypergeometric type Picard-Fuchs equations. The genus zero invariants agree with the ones calculated in [12]. The genus one invariants provide a new test for the multi covering formula derived in [27]. Similar checks for one parameter Calabi-Yau spaces in Grassmannian ambient spaces with Apery type Picard-Fuchs operators were provided in [28]. 2734099200 2813440 9058000 146150912 47104 53928 1024 3779200 26873294164654597632 1006848150400512 20201716419250520 132534541018149888 42843921424384 7776816583356 338199639552 27474707200000 HJEP01(28)6 Here we consider the hypersurface MPE18×P2 . This arises from an E8 fibration over the base Hence the polytope Δ∗ corresponding to the fibration over Δ∗E8 is given by div. D0′ D2′ D2′ D1′ D1′ D2′ −1 1 −1 0 −1 ν¯∗B i 0 1 0 0 0 −1 0 1 0 0 0 2 2 2 2 2 2 0 −1 ν¯∗ i 0 0 0 1 0 0 0 0 1 0 0 0 0 3 3 3 3 3 div. coord. KM 2D˜ e 3D˜ e E x0 x y z ˜ ˜ ˜ ˜ ˜ D2 u1 D2 u2 D2 u3 D1 u4 D1 u5 1 1 1 1 1 1 1 1 3 −1 0 −1 (A.1) (A.2) 387176346729900 81906297984 845495712250 5670808217856 4277292544 1203128235 65526084 15090827264 l′(1) l′( 2 ) 0 −2 −3 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 −1 The intersections among divisors lead to the constants in (3.12), (  2 3  cijk = c122 = c212 = c221 = 1 , 0 otherwise , a = 54, ai =   , ai = 9 12 , aij =   The genus zero amplitudes in the basis (A.5) read The Picard-Fuchs equations read L1 = θe(θe − 2θ1 − 3θ2) − 12ze(6θe + 5)(6θe + 1) , L2 = θ12 − z1(θe − 2θ1 − 3θ2)(θe − 2θ1 − 3θ2 − 1) , L3 = θ23 − z2(θe − 2θ1 − 3θ2)(θe − 2θ1 − 3θ2 − 1)(θe − 2θ1 − 3θ2 − 2) . Here the coordinates za are determined by (4.29). The discriminants of the Picard-Fuchs equations are given by HJEP01(28)6 Δ1 = (−1 + 4z1)3 − 54(1 + 12z1)z2 − 729z22 , Δ2 = −h − 1 + 864ze 1 + 216ze(−1 + 4z1) i3 + 4738381338321616896ze6z22 + 4353564672ze3(−1 + 432ze)h1 + 864ze − 1 + 216ze(1 + 12z1) iz2 . Using the choice of basis for 4-cycles in (3.13) we obtain H1 = E · D˜1, H2 = E · D˜2, H1 = D˜22, H2 = D˜1D˜2 . Hence the (3.14) intersections follow as (A.3) (A.4) (A.5) (A.6) (A.7) η( 2 ) =  −3 −2 0 1  .  1 0 0 0  F H(01) = τ 2 + τ t1 + 1 + F H(01),inst(qe, Qe1, Qe2) , 2 F H(02) = 3 τ 2 + τ t2 + 1 τ + + F H(02),inst(qe, Qe1, Qe2) .    2 5 4 3 2 In appendix A.5 we show some of the instanton expansions of the above expressions in terms of quasi-modular forms. Note that we make use of a ‘pure modular’ basis — as in the case of X24 — to compute the modular weight zero components of the F H(0i) periods. We define such a basis as follows F H(01◦) ≡ F H(01) + 32 F H(02) , F H(02◦) ≡ F H(02) + F H(02) + 23 F H(01) . (A.8) (A.10) (A.11) constants related to the Chern classes are 1 1 b1 = − 24 c3 TMPE18×P2 · D1 − 1 = be = − 24 c3 TMPE18×P2 · De − 1 = , 539 4 This leads to the genus one amplitude 1 b2 = − 24 c3 TMPE18×P2 · D2 − 1 = 29 , ˜ , χ = 19728 . where we give part of the expansion of F (1),inst in terms of quasi-modular forms in appendix A.5. The second Chern class of MPE18×P2 can be written in terms of the basis (A.5) as We compute the genus zero Gromov-Witten invariants of (A.9) in appendix A.3. Further c2 TMPE18×P2 = 24H1 + 36H2 + 102H1 + 138H2 . (A.9) Geometric invariants for MPE18×P2 -731942730 605426932980 -139049122837500 17515925402297760 3 -243 -49734 -1719756 -27555633 -277450434 3 3 138240 34833240 1396005840 25003580040 274895998560 -35363790 -12410449830 -583905569940 4 2304 904500 56117574 1515365226 24502800744 4 4 -1698840 -786936060 -55422152100 -1654348658580 -29038118214600 578799000 342447273720 27847911802680 HJEP01(28)6 n0,(0,d1,d2)(H2) d2 = 0 n0,(1,d1,d2)(H2) d2 = 0 n0,(2,d1,d2)(H2) d2 = 0 6 0 0 0 0 0 0 0 0 0 720 -720 1440 -1440 1 -24 -192 -744 -2040 -4560 1 6120 67680 314280 961920 2313000 1 -1036800 -13718160 -69796080 -230700960 2 114 4440 55050 390744 1973472 2 -43920 -2349360 -34350480 -274751280 -1517061600 2 8217540 660289320 11223041760 99434663640 3 -864 -93744 -2528040 -34977312 -318919680 3 3 495360 65718720 2043688320 31523616000 313418304000 -131045040 -23552058960 -851198459760 4 8808 1898622 92087760 2139264666 31152820512 4 4 -6528960 -1654942320 -90803818800 -2326758388560 -36732061356480 2264001480 724510733760 45595230845400 -14568373007280 1290110994869760 -588578400 593689222980 -157013407044000 22030115559925320 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 30 84 180 330 1 -1800 -12240 -39960 -93600 -181800 1 377460 2668140 9566100 2 -30 -870 -8682 -51600 -224112 2 12240 493920 5793120 38578320 182164320 2 -2483820 -147669480 -1999807560 3 234 20196 460512 5535630 44650908 3 3 d2 = 0 d1 = 0 720 0 -3 0 0 0 0 0 0 0 0 0 0 3 81 16578 573252 9185211 92483478 3 -51840 -12547080 -492862320 -8722465560 -95139102240 3 15155910 4798354950 4 -768 -301500 -18705858 -505121742 -8167600248 4 4 617760 279335520 19402918200 574019167320 10012773524400 -231519600 -129069932760 2 3 4 1 2 3 4 1 2 3 4 0 0 0 0 0 0 0 -18 -18 0 0 0 0 0 1 36 288 972 2304 4500 1 -11772 -123228 -498420 -1313172 -2743308 0 0 1161 15174 110151 2 -90 -5166 -681210 -10098990 -80360496 2 47052 2995704 208221228 3364230240 70 -990 -183402 -4442538 -56477430 3 -30744 698400 138997944 3786456528 52885199952 3 6547608 -257783256 -54043640640 -1643238648792 -1602 12496941 487139904 199225723852 4 4 706572 -43754310 -11571378390 -504941463486 -218756626565280 -221005710 18293975928 5483304374166 265936355246088 965359376676 -273025875142044 36253634952195918 (0, 1, 0) (1, 0, 0) ( 0, 0, 2 ) (0, 1, 1) ( 0, 2, 0 ) (1, 0, 1) (1, 1, 0) ( 2, 0, 0 ) β2 -180 (0, 0, 1) (0, 1, 0) (1, 0, 0) ( 0, 0, 2 ) (0, 1, 1) ( 0, 2, 0 ) 378 -2016 72 -342 0 -1422 6894 648 -15012 0 0 0 0 0 5400 -24840 -2160 52920 0 38800 (1, 0, 1) 36720 -18360 376920 0 (1, 1, 0) -11880 57240 -85320 0 0 38800 ( 2, 0, 0 ) 10800 -49680 -4320 105840 0 38800 -1744200 516240 -7069680 3656800 -3499200 FHb 2 Q η48 e Q !2 (0) +P70 (Hb) η48 e Q !3 +··· (A.12) (0) P22 (Hb) = − 18 (0) P46 (Hb) = − 12 5 1 E4E6(35E43 +37E62) (0) E2 P22 (Hb) 2 5E4E6 − 2985984 +208392741E43E64 +27245569E66 (0) P70 (Hb) = 1 1671768834048 29908007E49 +207234483E46E62 q 2 Q +P48 (Hb ) q 2 Q +P72 (Hb ) q 2 Q !3 +··· (A.13) FHb◦ 0 (Hb )+P24 (Hb ) P 0 (Hb ) = 960 P24 (Hb ) = P48 (Hb ) = 5 10368 5 13759414272 ∞ d=1 X σ3(d) d qe , 10321E46 +1680E2E44E6 +59182E43E62 +1776E2E4E63 +9985E64 34974695189E412 +955855257580E49E62 +2375228903358E46E64 +576E2E4E6 19602269E49 +134498081E46E62 +137176487E43E64 +18933723E66 2 ! P72 (Hb ) = 5 12999674453557248 169868512046891311E418 +10991441298020921814E415E62 +154580302495588543E612 +16124313600E23E43E63 35E43 +37E62 3 +8398080E22E42E62 35E43 +37E62 44357407E49 +305364363E46E62 +309961101E43E64 +42023369E66 +432E2E4E6 188980289153801E415 +4041754304722571E412E62 +171728558335663E610 – 46 – d=1 qe , −10321E46 +34440E2E44E6 −59182E43E62 +36408E2E4E63 −9985E64 η48 e Q η48 e Q η48 e Q +··· (A.14) −8718461011E412 −238460285300E49E62 −592848334770E46E64 −239525096180E43E66 −8301513619E68 +7649280E22E42E62(35E43 +37E62)2 ! 0 P 0 P24 P48 = −2 A.5 Modular expressions for MP1×P2 P72 5 2166612408926208 −54494943725199823E418 −3526301098569327294E415E62 26341691595249846705E46E68 −3475678553808910878E43E610 −50493219640852471E612 +337714790400E23E43E63 35E43 +37E62 +1399680E22E42E62(35E43 +37E62)(3711620489E49 +25730000061E46E62 3 +25856467947E43E64 +3371520463E66)+108E2E4E6 5231073695092861E415 FHi P (0) 6(a1d1+a2d2)−2(Hi) P16(H1) = E4 + E4E62 , P16(H2) = P10(H1) = 0, P22(H1) = 0, P28(H1) = P28(H2) = P34(H1) = − 9216 P34(H2) = − 110592 1 1 9 2 3 8 X d1,d2 3 2 85 48 1 192 X d1,d2 FHi◦ P 6(a1d1+a2d2)(Hi ) E8 η12 12 !a d1+a2d2 1 Qe1d1 Qe2d2 P10(H2) = −3E4E6 , P22(H2) = − 32 E42E6 − 32 E4E63 , 16 E4E64 qe η12 21 !a d1+a2d2 1 Qe1d1 Qe2d2 E4 48359E46E6 +161426E43E63 +39047E65 +24E2E4 E43 +3E62 31E43 +113E62 E4 208991E46E6 +755906E43E63 +196319E65 +4E2E4 31E43 +113E62 ! (A.15) 2! (A.16) ∞ d=1 d2 e P0(H1 )=−24(b1+1)X σ3(d) d ◦ q , P0(H2 )=−24(b2+1)X σ3(d) d ◦ q , ∞ d=1 d2 e 49E43+ 94E62, P12(H2 )= 23E43+ 1E2E4E6+E62, 2 −31E2E44−926E43E6−113E2E4E62−226E63 , −137E43E6−47E63−2E2 E44+3E4E62 , 51E46+17E2E44E6+199E43E62+7E2E4E63+14E64 , −648E2E47−E45 31E22+52747E4 E6−4444E2E44E62 −386E2E47−E45 72E22+32849E4 E6−5002E2E44E62 P36(H1 )= 628895E49+438639E2E47E6+9743040E46E62+1649058E2E44E63 P36(H2 )= 333303E49+54875E2E47E6+5411350E46E62+220490E2E44E63 P12(H1 )= P18(H1 )= P18(H2 )= P24(H1 )= P24(H2 )= P30(H1 )= 1 1 16 128 1 1 384 1 2304 1 1152 1 1327104 1 221184 (1) P0 =−2 (1) 3 (1) P18 = (1) P24 = (1) P30 = X d1,d2 F (1),inst = (1) P6(a1d1+a2d2) η12 21 !a1d1+a2d2 Qe1d1Qe2d2 d ∞ d=1 qe , 24 −h11 −6E43+10E2E4E6−5E62 , 33E46−24E2E44E6+2E42 −2E22+71E4 E62−16E2E4E63+13E64 , – 48 – (A.17) 108E22E42 31E43+113E62 577E43+1871E62 +36E2 3099607E47E6 +10537042E44E63+2578903E4E65 . A.6 Analytic continuation data for X24(1, 1, 1, 1, 8, 12) We provide the numerical and — as far as we know them — analytic expressions for the continuation matrices Tc, Tc′, To, To′ in a Mathematica worksheet on the webpage [48]. Due to their special importance we reproduce here the intersection matrix at c1 = c2 = 0 as well as the entries of the continuation matrix to the point ze = c1 = 0: (TcTc′)T η−1TcTc′ = κ 204448 −30916905725128 2836622426104377556352 135 −466556 0 33392 204448 −46656 5 −3083774976 25 0 0 0 0 0 2519424 −308372754976 32651573504 0 0 5 0 0 where 0 1 1 −604656176 24428335104 −15768933728256 0  125 0  0  0  0  32  (A.18) 1 κ= 1572864π4 . f1,1 = −2916π12r44+10260π6r42−27π4r5r4+144r12−32r22−7129 1152 f2,1 = 3 1(6r1+4ir2+3) f3,1 = 3184 216π6r42+48r1+32ir2+3iπr5−404 f7,1 =−12i(2r2−9π3r4) f5,1 = 48 −54π6r42−36iπ3r4−12r1+8ir2+101 f1,2 = 214 3r1r3−√2r2r3+384π2r4−24r5 16π f5,2 =−−i√2πr3+πr3+768π3r4+256i 8π – 49 – −5760π4r12r3 +1280π4r22r3 +285160π4r3 +349440π2r3 +4464π4r3r5 /(768π4r3) 32π4r3 128π4r3 f3,3 = 15i√2π4r32 +15π4r32 +17280π10r42r3 −428544iπ9r42r3 +3840π4r1r3 +2560iπ4r2r3 +142848π6r4r3 −92160iπ5r4r3 +240iπ5r5r3 −32320π4r3 +738048iπ3r3 +15360π2r3 15 3i√2π4r32 +2π4r32 +512π4r1r3 +512iπ4r2r3 +512π4r3 +786432i√2−524288 f5,3 = − −15i√2π4r32 +15π4r32 +17280π10r42r3 +3840π4r1r3 −2560iπ4r2r3 +11520iπ7r4r3 +142848π6r4r3 −32320π4r3 +47616iπ3r3 +15360π2r3 −3932160i√2 −3932160)/(256π4r3) 5i 3√2π4r32 +512π4r2r3 +786432√2 64π4r3 256π4r3 f2,3 = f4,3 = f6,3 = f7,3 = f1,4 = 1 8π r2 = −1. 29219644630091977480074761037 r4 = −0.00948778220735050311547607017424 B General genus zero modular anomaly equation After releasing a pre-print of this paper it was brought to our attention by Georg Oberdieck that there is a conjectured modular anomaly equation for elliptic Calabi-Yau n-folds in [23, 24]. For n = 4 the conjecture implies the modular anomaly equations for the GromovWitten potentials associated to π-vertical cycles and for genus one free energies that we derived in this paper. For the non π-vertical cycles the conjectured anomaly equation agrees with our results for MPE38 and we also checked it for the Gromov-Witten potentials of MPE18×P2. Our results on the modular structure can therefore be seen as a partial derivation and non-trivial check of the holomorphic anomaly equation conjectured in [23, 24] for Calabi-Yau fourfolds. We will now briefly describe the general form of the holomorphic anomaly equations for genus zero Gromov-Witten potentials. Let Fγ(1g,)...,γm be the string amplitude associated to the Gromov-Witten invariants Ng,κ(γ1, . . . , γm) = Z [Mg,n(M,κ)]vir i Y evi∗(γi) , as these fulfill the intersection relations Hi · Hj◦ = δji , Hi · Hj = 0 , Hi◦ · Hj◦ = 0 . where κ ∈ H2(M, Z) and γi ∈ H∗(M, Z). On the one hand, given β ∈ H2(B, Z) conjecture A in [23, 24] implies that Coeff(Fγ(1g,)...,γm , Qβ) ∈ η12c1(B)·β C[E2, E4, E6] , 1 which matches with our Ansatz given in expression (4.24). On the other hand, conjecture B of [23, 24] implies a general modular anomaly equation for Fγ(1g,)...,γm . Following the discussion of sections 3.1 and 4.4, we can make a generalization of the pure modular basis by taking the 4-cycles, Hi = aij akD˜ j D˜ k , Hi◦ = DeD˜ i , i = 1, . . . , h1,1(B), Note that Hi ∈ H4(M, Z) while in general Hi◦ ∈/ H4(M, Z). Let ℓ ∈ H2(B) such that hβ, ℓi 6= 0. Then for a given γ ∈ H2,2(M, C) Georg Oberdieck pointed out to us that conjecture B of [23, 24] implies a modular anomaly equation for Fγ(0), which in the modular basis (B.3) reads ∂Fγ(0,β) = − 12 ∂E2 1 " X βi′Fγ(0,β)′ F Hi,β′′ − hβ′,ℓi2hβ′′,π∗γi+hβ′′,ℓi2hβ′,π∗γi F H(0i),β′ F H(0i◦),β′′ hβ,ℓi2 hβ,ℓi Fπ(0∗)(π∗(γ)∪ℓ),β − hhπβ∗,γℓ,iβ2i Fπ(0∗)ℓ2,β From the properties of the Gysin morphisms it follows that π∗Hi◦ = Di′ and π∗Hi = 0. Hence for a π-vertical 4-cycle Hi, the modular anomaly equation (B.5) of its corresponding string amplitude F H(0i) reduces to equation (4.54). Now we consider the 4-cycles Hi◦ where equation (B.5.4) becomes more involved. It is easy to verify for MPE38 that (B.5) reduces to (4.38). Moreover, when h1,1(B) ≥ 2 equation (B.5) in general implies multiple relations, since it depends on the choice of ℓ. We checked equation (B.5) for MPE18×P2 of which we include the toric data in appendix A.2. 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Cesar Fierro Cota, Albrecht Klemm, Thorsten Schimannek. Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds, Journal of High Energy Physics, 2018, 86, DOI: 10.1007/JHEP01(2018)086