Open mirror symmetry for higher dimensional Calabi-Yau hypersurfaces

Compactifications with fluxes and branes motivate us to study various enumerative invariants of Calabi-Yau manifolds. In this paper, we study non-perturbative corrections depending on both open and closed string moduli for a class of compact Calabi-Yau manifolds in general dimensions. Our analysis is based on the methods using relative cohomology and generalized hypergeometric system. For the simplest example of compact Calabi-Yau fivefold, we explicitly derive the associated Picard-Fuchs differential equations and compute the quantum corrections in terms of the open and closed flat coordinates. Implications for a kind of open-closed duality are also discussed.


Introduction
Mirror symmetry is an efficient tool to compute the enumerative invariants of Calabi-Yau manifolds such as Gromov-Witten invariants as first demonstrated in [1] to the quintic threefold. The genus zero Gromov-Witten invariants are equivalent to the number of holomorphic spheres in the manifold, and it is quite natural to ask whether one can consider JHEP03(2016)160 more general enumerating problem in the framework of mirror symmetry. One of these attempts has been undertaken in [2] for the case of holomorphic disks in the compact Calabi-Yau manifold, and the disk partition function for the quintic threefold was clarified (see also [3]). This open mirror symmetry prescription was subsequently developed in [4][5][6] for other one-and two-parameter Calabi-Yau threefolds. By extending the holomorphic anomaly equation for the closed topological strings [7], the higher genus invariants for open topological strings have been also studied [8][9][10][11]. More mathematical treatment of disk enumeration can be found in [12,13] for the A-model and [14] for the mirror B-model.
As described in [2], in order to derive the disk partition function of the open topological string, it is sufficient to introduce the branes from the involution on the Calabi-Yau manifold. However, this involution brane does not have an explicit open string moduli dependence and therefore the structure of open string deformation cannot be extracted from this "on-shell" formalism. Based on the earlier works of [15,16], the incorporation of open string moduli was carried out in [17] for the case of compact Calabi-Yau threefolds. By studying the associated Hodge structure, they derived the Picard-Fuchs differential equations depending on both open and closed string moduli parameters.
Thereafter, using the toric geometry prescription, an alternative and more efficient method to derive the equivalent differential system was constructed in [18]. Interestingly, this formulation implies that there exists a duality between the off-shell open topological strings on a Calabi-Yau threefold with branes and the closed topological string on a Calabi-Yau fourfold without branes [19][20][21][22][23].
Therefore the natural question is whether one can construct the off-shell formalism and recognize signs of a similar duality in higher dimensions. On-shell mirror symmetry prescription and the direct calculation using the localization formula have been argued in a certain class of higher dimensional Calabi-Yau manifolds in several contexts. However, to the best of our knowledge, an explicit generalization including the open string deformation has not been elucidated in dimensions greater than three. The aim of this paper is to provide a first step to clarify this fascinating issue, which may have a wide range of applications. For instance, in the case of Calabi-Yau fivefolds, our prescription might be useful to compute the non-perturbative corrections in the effective superpotential of the M-theory compactification studied in [24]. This paper is organized as follows. First we take a look at the disk enumeration problem from the aspect of mirror symmetry in section 2. It gives a brief introduction to the closed and on-shell open mirror symmetry in general dimensions. In section 3, we describe how to incorporate the open string deformation into the present framework and construct the necessary ingredients via Hodge theoretical approach. Furthermore, we explicitly derive a Picard-Fuchs system associated with the open and closed string deformations for an example of compact Calabi-Yau fivefold. In section 4, we present another efficient formulation in terms of the toric geometry and show that the resulting differential system is equivalent to the one obtained in section 3. Finally, in section 5, we will also refer to the implications of the off-shell formalism for a certain kind of open-closed duality. Section 6 is devoted to the conclusion and discussion. We summarize the localization formulas for the related A-model computations in appendix A. Monodromies under the analytic continuation in the moduli JHEP03(2016)160 space is also discussed in appendix B. In appendix C, we discuss about disk two-point functions, and the differential equations derived in section 4 is solved in appendix D.

Disk enumeration and open mirror symmetry
In this section, we describe the disk enumeration problem on complex odd dimensional Calabi-Yau hypersurfaces from the viewpoint of mirror B-model.
Let us consider a Calabi-Yau m-fold X m+2 ⊂ CP m+1 defined by a degree m + 2 Fermat hypersurface m+2 i=1 x m+2 i = 0 in a projective space CP m+1 . Its mirror manifold Y m+2 ⊂ CP m+1 is described by a resolution of a (Z m+2 ) m orbifold of the hypersurface where ρ = e 2πi/(m+2) . Note that there are only m independent identifications due to the existence of overall rescaling induced by Z (1) m+2 × Z (2) m+2 × · · · × Z (m+1) m+2 . Next we consider the inclusion of branes. On the A-model side, we further introduce an involution brane on the Lagrangian submanifold L m+2 = X R m+2 in X m+2 as the real locus which is defined by the fixed points of an anti-holomorphic involution This means that there are two choices for specifying the involution brane. In the A-model on X m+2 , let us consider the worldsheet n-point function on the disk D as Instead of (2.3), by using the automorphism group of CP m+1 , the Lagrangian submanifold L m+2 can be also defined from fixed points under the identification The efficient way to compute the open Gromov-Witten invariants is to make use of the open mirror symmetry. In [14], by using the matrix factorization [25], a set of holomorphic two-cycles in X 3 wrapped by the B-brane mirror to the involution brane on L 3 was clarified. Analogously, the mirror geometry of the Lagrangian submanifold L m+2 ⊂ Y m+2 can be described by 2m-dimensional two isolated holomorphic curves It would be interesting to derive this expression from the matrix factorization. Let us consider a chain integral where Ω is the holomorphic m-form on the mirror Calabi-Yau Y m+2 depending on the complex structure modulus ψ. Γ is an m-chain with the homologically trivial boundary Clearly the cycle C + − C − defined from (2.8) satisfies this condition for all ψ. This quantity is a higher dimensional analog of the tension of a BPS domainwall [26] realized by a brane wrapped on Γ ⊂ Y 3 . In the case of Calabi-Yau threefold, it is well understood that this chain integral is related to the D-brane superpotential for the four dimensional N = 1 supersymmetric compactifications (see, for example [27,28] and references therein). Therefore, our result for Calabi-Yau fivefold is naturally expected to provide the "M-brane superpotential" in the super-mechanics theory considered in [24]. The integral (2.9) is regarded as the on-shell disk partition function whose expression in the vicinity of the large complex structure point is given by [29] where c is a normalization constant, 2 and z = 1 In appendix B, we see that this constant can be fixed by monodromy analysis as performed in [2].

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is a local coordinate around the large complex structure point. The disk partition function (2.10) satisfies the inhomogeneous Picard-Fuchs equation associated with C + − C − given by [2,12,14] (see also [30][31][32][33]) where θ z = z∂/∂z. The solutions to the homogeneous Picard-Fuchs equation give the periods of the holomorphic m-form Ω on the mirror manifold Y m+2 , which can be expressed by integration over the integer homology group 3 as where p = 0, . . . , m. Solving the Picard-Fuchs equation (2.13), we obtain the closed string periods as Two of these solutions are expressed as where Ψ(x) = d dx log Γ(x) is the digamma function. The complexified Kähler modulus t of X m+2 is determined by the mirror map It has been conjectured in [29] (and later proved in [35]) that the disk partition function (2.10) is related to the instanton part of the disk one-point function O h m X m+2 JHEP03(2016)160 which satisfy the relations 4 With the use of the mirror map (2.17), by expanding the right hand side of (2.18) around the large radius point q = e 2πit = 0, one obtains the degree d open Gromov- For example, the degree one open Gromov-Witten invariant of X m+2 is given by It is worth noting that the disk two-point functions are obtained from the disk one-point functions and the three-point functions on CP 1 by where θ q = q∂/∂q. By using the results listed in table 2-7 of appendix A, we can explicitly check this formula. We will revisit this formula in appendix C. In [29], a multiple covering formula for the disk one-point function in general dimension was found to be It has been shown in [36] that the genus zero Gromov-Witten invariants can be obtained from the formula for the three-point function on CP 1 as where q = e 2πit and z(q) is the inverse mirror map derived from (2.17). Utilizing this formula and a prescription of [37], it is possible to find exact Kähler potential of the Kähler moduli space for Calabi-Yau fivefolds with one Kähler parameter. It would be interesting to obtain general expression for manifolds with multiple Kähler parameters as performed in [38] for Calabi-Yau fourfolds. 5 The open Gromov-Witten invariants with even degree are zero. This can be justified by identifying the disk partition function as a normal function whose expression is generically constrained under the shift t → t + 1. See [2,14] for more details.

Off-shell formalism via relative periods
The geometric structure of open/closed B-model and the associated off-shell superpotential depending on the open and closed string moduli has been studied from the viewpoint of N = 1 special geometry [19,20] for the case of non-compact Calabi-Yau threefolds [15,16]. In [17] (see also [41]), the study of off-shell superpotential was extended to compact Calabi-Yau threefolds and the corresponding open/closed Picard-Fuchs equation was clarified.
In this section, we utilize the method of [17] in general dimensions and explicitly compute the open/closed Picard-Fuchs equation for septic Calabi-Yau fivefold X 7 . Then we will solve the resulting differential system in the vicinity of the Landau-Ginzburg point, and also check the consistency with the on-shell result.

Open string moduli and relative cohomology
Let us introduce a 2m-dimensional curve C u depending on an open string modulus u in the mirror Calabi-Yau Y m+2 . This curve is not necessarily holomorphic, except at the value of u where the on-shell quantity is realized. Consider a chain integral where Γ u (u) is an m-chain whose boundary is defined by ∂Γ u = C u . This is a higher dimensional analog of the off-shell superpotential defined in [15][16][17] for Calabi-Yau threefolds. Here the curve C u (u) is defined to yield the two holomorphic curves C ± in (2.8) at the critical points of (3.1) with respect to the variation of open string modulus. In order to study the moduli dependence of the chain integral (3.1) systematically, it is convenient to introduce the notion of the relative cohomology group and the variation of the mixed Hodge structure [15][16][17]. In what follows, we will briefly look at mathematical preliminaries.
Let us consider the divisor V embedded in the mirror Calabi-Yau manifold Y m+2 through the map i : V ֒→ Y m+2 . 6 The space of relative forms Ω * (Y m+2 , V ) is subspace of the forms Ω * (Y m+2 ) induced by the morphism i * : Ω * (Y m+2 ) → Ω * (V ). Then we can define the JHEP03(2016)160 Since H m (V ) is trivial, the first factor is equal to H m (Y m+2 ) and thus describes the closed string sector. The second factor is called variable cohomology H m−1 var (V ) of V [16], which varies with the embedding i : V ֒→ Y m+2 .
Correspondingly, one can consider the relative cohomology group H m (Y m+2 , V ) and construct a relative m-form Ξ as a pair of an m-form Ξ ∈ H m (Y m+2 ) and an (m − 1)-form ξ ∈ H m−1 (V ), such that i * Ξ − dξ = 0. The equivalence relation is given by where α is an (m − 1)-form on the mirror Calabi-Yau Y m+2 and β is an (m − 2)-form on the divisor V . The pairing between a relative m-cycle Γ a in the relative homology group As a result, we can define the relative period integrals which can be regarded as a combination of the integral (3.1) and (2.14).
Here Ω(z, u) is the relative holomorphic m-form which will be later constructed as a residue integral. The relative homology basis Γ a ∈ H m (Y m+2 , V ) captures the m-cycles Γ p in (2.14) and the m-chain Γ u in (3.1). The convenient method to study the moduli dependence of relative periods is the variation of mixed Hodge structure. Since the standard Hodge decomposition does not vary holomorphically under the complex structure deformation, it is convenient to use the Hodge filtration 8) in order to analyze the moduli space in the B-model. Note that

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Another ingredient which is required to define the mixed Hodge structure is the finite increasing weight filtration satisfying the relation Thus the filtration F p ∩W m describes the Hodge structure associated with the closed string sector. The (mixed) Hodge structure can be naturally equipped with the Gauss-Manin connection ∇ satisfying the Griffiths transversality such as ∇F p ⊂ F p−1 . As a consequence of this transversality, by acting ∇ on the relative holomorphic m-form, we can derive the open/closed Picard-Fuchs equations governing the relative period integrals.
A schematical picture of the variation of mixed Hodge structure is given by (3.12) where ∂ z and ∂ u denote the infinitesimal closed/open string deformations on the space of the relative middle cohomology H m (Y m+2 , V ), respectively. As we will see later, each constituent of the above structure can be expressed as a residue integral.

The extended Griffith-Dwork algorithm
The Griffiths-Dwork algorithm [42,43] is to represent a holomorphic m-form on a hypersurface as a residue of a meromorphic (m + 1)-form on a projective space. In [17], this algorithm was extended to derive the Picard-Fuchs equation governing the relative period integrals of compact Calabi-Yau threefolds. Here we consider the case of higher dimensional compact Calabi-Yau manifolds Y m+2 .

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Due to the Griffiths transversality, a basis of the relative cohomology group H m (Y m+2 , V ) can be constructed as π ≡ (π m,m , π m−1,m , . . . , π 0,m , π m−1,m+1 , π m−2,m+1 , . . . , π 0,m+1 ) in terms of the polynomials (2.1) and (3.13). Here dy i denotes the exclusion of a component dy i and the integration is taken over a curve γ in the projective space CP m+1 surrounding the hypersurface Y m+2 . The indices of the relative cohomology basis (3.15) are labeled so that each element corresponds to the constituent of the mixed Hodge structure described in the diagram (3.12). By taking the derivatives of Ω with respect to the closed string modulus ψ and open string modulus φ, we obtain the residue integral representations for the closed relative m-forms corresponding to the closed string sector and the closed relative (m − 1)-forms whereγ is a tube in CP m+1 surrounding the intersection {P = 0} ∩ {Q = 0}. We have introduced the abbreviations for the monomials as and

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The next step to obtain the Gauss-Manin connection associated with the open/closed Picard-Fuchs operators is to find the derivatives of the basis elements in π with respect to the open/closed string moduli. Some of the derivatives are, by definition, given by In order to obtain other relations at the level of cohomology, we need to know the residue integral expression of the exact relative forms as well as the closed relative forms. The residue integral expression of an (m − 2)-form on the divisor V ⊂ Y m+2 is given by where q i (y) is a homogeneous polynomial of degree (k − 1)(m + 2) + ℓ(m + 1) + 1. By acting the de Rham differential, one can obtain the exact relative (m − 1)-form as where ∂ i = ∂/∂y i . Similarly one can also construct the exact relative m-form on Y m+2 . Since the basis elements are now represented by derivatives of the relative holomorphic m-form Ω, from the linear differential system of π, one can construct the open/closed Picard-Fuchs operators L i satisfying which also annihilate the relative periods (3.6). Here the symbol "≃" means that the equation holds at the level of cohomology. As we will demonstrate explicitly, in order to construct the open/closed Picard-Fuchs operators for Y m+2 , it is sufficient to find the expansion of ∂ φ π k,m+1 and ∂ ψ π 0,m+1 in terms of the linear combination of the basis elements π k,m+1 , where k = 0, 1, 2, . . . , m − 1.

Example: septic Calabi-Yau fivefold
As an explicit example, let us consider the septic Calabi-Yau fivefold X 7 in the projective space CP 6 . First we look at the on-shell open Gromov-Witten invariants, which should be encoded in the off-shell formulation described above. Then we will explicitly derive the open/closed Picard-Fuchs equations and solve the system.

On-shell open Gromov-Witten invariants
The mirror geometry (2.1) of the septic Calabi-Yau fivefold with a closed string modulus ψ is defined by the homogeneous degree seven polynomial Y 7 : P (ψ) = y 7 1 + y 7 2 + y 7 3 + y 7 4 + y 7 5 + y 7 6 + y 7 7 − 7ψy 1 y 2 y 3 y 4 y 5 y 6 y 7 = 0, As a mirror of the Lagrangian submanifold L 7 ∼ = RP 5 in (2.3), we consider two isolated holomorphic 4-dimensional curves (2.8) in the mirror septic Y 7 as As described in (2.10), in the vicinity of large complex structure point z = 1/(7ψ) 7 = 0, the chain integral (2.9) associated with the curves C ± is given by (3.30) The closed string periods Π 0 (z) and Π 1 (z) are obtained by taking m = 5 in (2.16), and the inverse of the mirror map (2.17) is where q = e 2πit . Note that the log 2 solution in (2.15), can be regarded as a generating function of the closed genus zero Gromov-Witten invariants as indicated in (2.20), where the summation in the last equality is referred to as the multiple covering formula [34,44]. (3.35) where θ q = q∂/∂q and we have used a relation θ z = I 1 (z)θ q . Plugging into the multiple covering formula (2.25), we obtain the positive integer invariants n open d (h 2 ) as listed in the right of table 1.

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First, by using the aforementioned formula (3.20) and the expression for the exact relative forms (3.25) with m = 5, we can easily find that Similarly, with the use of (3.20) and (3.25), we obtain the relation among higher order derivatives as for fourth order, for fifth order and for sixth order. The above relations are not sufficient for the closure of the relevant linear differential system. To fix the relations completely, we require an additional relation, which can be obtained by (3.21). After a very long calculation, we obtain a relation for the sixth order derivatives of the relative holomorphic five-form as where we have used (3.20) and (3.25) repeatedly.
As a result, we obtain closed relations for the expansion of ∂ φ π * ,6 and ∂ ψ π 0,6 in terms of the linear combination of the basis elements π * ,6 as and where where L 1 and L 2 are given by

Solutions to the off-shell open/closed Picard-Fuchs equations
Now that we have derived the Picard-Fuchs equations capturing the open string deformation, in the remaining part of this section, we will solve the off-shell Picard-Fuchs equations (3.46) in the vicinity of the Landau-Ginzburg point ψ = 0. As performed in [17], we also check that the inhomogeneous Picard-Fuchs equation (2.12) can be reproduced at the on-shell point (3.14). In order to find the relative period integrals annihilated by L 1 and L 2 , we first specify the solutions determined by the factorized operators L 1 and L 2 . Let us consider a function where u ≡ φ(φ − 7ψ) 6 . This function indeed satisfies L 1 χ = 0. By acting the operator L 2 on this ansatz, we obtain a differential equation for the undetermined function λ(u) as We can easily find the solutions of this equation as where the underbrace means that the unity is to be omitted. Their power series expansions are given by (3.52) Next step to obtain the relative periods determined by L 1 and L 2 is to perform an integration of the solutions (3.51) with respect to φ as represented by Evaluating these integrals, we finally obtain the relative periods involved with open string deformation as five independent solutions to the equations (3.46) as (3.54) Here we have labeled the relative periods associated with the closed string sector around the Landau-Ginzburg point as

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As well as the closed string mirror map t(ψ) ≡ Π 1 (ψ)/Π 0 (ψ) around the Landau-Ginzburg point, now we can also extract the open mirror map bŷ t(ψ, φ) ≡ Π 6 (ψ) Π 0 (ψ) , (3.56) up to an overall constant. The inverse of these mirror maps are (3.58) Then, the higher dimensional analogue of the orbifold disk partition function W (t,t) in [17] is given by whose coefficients can be regarded as the orbifold disk invariants up to an overall normalization constant. It would be interesting to check whether these numbers can be reproduced from other independent calculations. Moreover, from the solution Π 8 (ψ, φ), we can extract the on-shell disk partition function around the Landau-Ginzburg point ψ = 0. By taking a difference of the period Π 8 (ψ, φ) at the two critical points, we obtain which satisfies the inhomogeneous Picard-Fuchs equation associated with C + − C − as (3.61) Here we have replaced the standard Picard-Fuchs operator L in (2.12) into L ′ ≡ −7 6 Lψ by taking into account the normalization of the relative holomorphic five-form as described in [1,14]. Therefore we conclude that the off-shell Picard-Fuchs equations (3.46) certainly reproduce the on-shell properties.

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4 Off-shell formalism via generalized hypergeometric system So far we have seen that the open/closed Picard-Fuchs equations for relative periods can be derived from the Griffith-Dwork method. In this section, we will describe more efficient method to obtain the same Picard-Fuchs system utilizing the generalized hypergeometric system and the gauged linear sigma model (GLSM). In the context of off-shell open mirror symmetry, this approach was initiated in [16,19,20] and has been applied for non-compact Calabi-Yau threefolds with toric branes.
Analogously, the open/closed Picard-Fuchs equations for compact Calabi-Yau threefolds can be also obtained from similar framework of the toric geometry [18,21,45,46]. For mathematical details, we refer the reader to [47]. Here we will apply this prescription to the higher dimensional Calabi-Yau hypersurfaces X m+2 . For the septic Calabi-Yau fivefold X 7 , we explicitly demonstrate that a system of differential equations obtained from this approach is indeed equivalent to the Picard-Fuchs system derived in the previous section.

Generalized hypergeometric system for period integrals
As pioneered in [48], various Calabi-Yau manifolds can be realized as the IR fixed points of two dimensional N = (2, 2) GLSM. For example, the abelian GLSM corresponding to the Calabi-Yau hypersurfaces X m+2 with one Kähler modulus contains a chiral superfield P = Φ 0 with U(1) charge −m − 2 and m + 2 chiral superfields Φ i with U(1) charge 1. These chiral superfields interact with each other through a gauge invariant superpotential which satisfy the Calabi-Yau conditions n i=0 l α i = 0. As shown in [49], a Calabi-Yau manifold Y mirror to X can be described by a Landau-Ginzburg theory. In the viewpoint of two dimensional N = (2, 2) supersymmetry algebra, the mirror symmetry exchanges the left-moving supersymmetry generators as where parameters z α represent the complex structure moduli of the mirror manifold Y . The mirror Landau-Ginzburg theory also has a twisted superpotential W subject to the constraints (4.5), from which the mirror geometry of Y can be specified. It can be easily shown that the mirror geometry (2.1) is derived from the Landau-Ginzburg superpotential (4.6) with (4.5) by using (2.11) and the change of variables Y i = y m+2 i . As a consequence, the period of the holomorphic m-form on Y can be constructed from (4.6) and the resulting function is known to be annihilated by a set of differential operators of the generalized hypergeometric system of the Gel'fand-Kapranov-Zelevinsky (GKZ) type [50][51][52] Here we have introduced the parameters a i by scaling the coordinates as Y i → a i Y i such that n i=0 Y In general, the standard GKZ system defined from D α contains redundant solutions as well as the periods [51]. One common method for finding the full set of periods is to normalize D α into D α as defined in (4.7) and factorize the resulting expression. For example, the differential operator (4.7) for the Calabi-Yau hypersurfaces X m+2 specified by the charge vector (4.1) takes the form where we ignored an overall constant 1/a 0 a 1 · · · a m+2 . Then one finds the Picard-Fuchs operator defined in (2.12) as an irreducible lower order component of the operator (4.9).

Extended GKZ system for relative period integrals
Let us turn to the inclusion of open string deformation in higher dimensional Calabi-Yau manifolds from the viewpoint of toric geometry. This can be achieved by introducing the

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toric branes as first demonstrated for non-compact Calabi-Yau threefolds in [53,54]. Recall that the m-dimensional Calabi-Yau hypersurfaces X m+2 can be defined by a single U(1) charge vector in (4.1). In this setup, the off-shell open mirror symmetry can be formulated as follows. where the last two entries represent the U(1) charges of newly introduced chiral superfields of the GLSM on X m+2 , say Φ m+3 and Φ m+4 . 8 Note that the sum of the charges in each vector must be zero due to the Calabi-Yau condition. Let z and z be closed and open string moduli associated with charge vectors l and l. According to the formula (4.5), we obtain as an additional constraint. Here the new coordinates Y m+3 and Y m+4 are mirror dual to the lowest components of the superfields Φ m+3 and Φ m+4 , respectively. Taking into account the contributions from Y m+3 and Y m+4 in the twisted superpotential (4.6), we obtain a defining equation of a submanifold Comparing with (3.14), we find that the on-shell behaviour can be realized by taking z = −1. for later convenience. Correspondingly, the associated moduli parameters are transformed into the form

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to satisfy the constraint (4.5). Substituting the set of U(1) charge vectors (4.16) into the general formula of GKZ operators (4.7), we obtain differential equations of the extended GKZ system of Calabi-Yau hypersurfaces X m+2 with a toric brane as As already mentioned above, this kind of GKZ system can be factorizable. Indeed, we see that (m + 1) Note that in order to include the closed string sector and obtain a complete system of differential equations, we also need to consider one more operator D 3 associated with the charge vector l = l 1 + l 2 given by Again we can factorize the operator as (m + 2) Eventually we find that a complete set of differential equations from the extended GKZ system is determined by { D 1 , D 2 , D 3 }. In the following section, we will investigate several aspects of this system and verify that the toric geometry approach for higher dimensional Calabi-Yau hypersurfaces indeed satisfy nontrivial checks.

On-shell disk partition function
Here we will first explicitly show that from the solutions to the extended GKZ system, the on-shell disk partition function (2.10) is precisely reproduced at the specific point of moduli space.
Recall that the expression (2.10) is expanded around the large complex structure point z = 0. According to (4.15) and (4.17), the on-shell solution should be realized from the expansion around (z 1 , z 2 ) = (0, −1). Therefore, as performed in [18] for the quintic Calabi-Yau threefold, it is convenient to reparametrize z 1 and z 2 as

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so that the on-shell critical point is represented by u = 0. In these variables, we consider a kernel of the extended GKZ operators { D 1 , D 2 , D 3 } satisfying Solving these equations, we find that the function τ off takes the form where the coefficients a i,j are subject to the following recurrence relations: To analyze the on-shell behavior, it is sufficient to look at the terms depending only on the variable v. With the use of the parametrization m = 2m + 1 (m ∈ N), we further obtain the expression for odd dimensional Calabi-Yau hypersurfaces as where d ∈ N. Taking a normalization such that a 0,m+1 = 2c(m + 2)!!, we see that the above solution leads to (2.10) as (4.27)

Off-shell Picard-Fuchs equations
Next, we will compare the system { D 1 , D 2 } of differential equations determined by extended GKZ operators with our previously obtained open/closed Picard-Fuchs equations in (3.46) for the septic Calabi-Yau fivefold. From (2.11), (4.14), and (4.17), we can rewrite the variables z 1 , z 2 in terms of ψ and φ as Then the GKZ operators can be expressed up to overall constant factors as

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where θ ψ = ψ∂ ψ and θ φ = φ∂ φ . Using (4.30), the operator D 1 can be further evaluated as Obviously, in the case of m = 5, the differential operator P 1 in (4.30) agrees with the factorized Picard-Fuchs operator L 1 in (3.47). Meanwhile, using the operator L 1 in (3.46), the remaining Picard-Fuchs operator L 2 can be expressed as up to an overall constant. Comparing with the operator P 2 in (4.31), we find that Therefore it turns out that the kernel of the differential operators { D 1 ψ, D 2 ψ} constructed from the toric geometry prescription indeed coincides with that of the open/closed Picard-Fuchs operators (3.46).

Open-closed duality in higher dimensions
In the previous section, we discussed the extended GKZ system associated with the charge vectors (4.16) which describes the open topological strings on the m-dimensional Calabi-Yau hypersurface X m+2 . Alternatively, we can also consider that these charge vectors describe a non-compact Calabi-Yau (m + 1)-fold X m+2 embedded in the non-compact space without brane. To discuss this description more precisely, as considered in [21] for m = 3, let us compactify the embedding space by further adding a chiral superfield Φ m+5 and an extra charge vector l 3 as where the last entry represents the U(1) charges of Φ m+5 . Then one obtains the compact elliptically fibered Calabi-Yau (m + 1)-fold X c m+2 whose base CP 1 has the Kähler form associated with the extra charge vector l 3 .
Let t i and h i , i = 1, 2, 3 be the Kähler moduli and Kähler forms associated with the charge vectors l i . The generating function of the classical intersection numbers κ i 1 i 2 ...i m+1 = JHEP03(2016)160 X c m+2 h i 1 ∧ h i 2 ∧ · · · ∧ h i m+1 are determined by the charge vectors as where we have defined and correspondingly Here we consider the odd dimensional case with m = 2m+1. Taking the volume of the base CP 1 to infinity (Im s → ∞) corresponds to the decompactification limit X c m+2 → X m+2 . In this limit, the following 2m independent two-point functions on CP 1 are relevant: 9 These are the generating functions of the degree d genus zero Gromov-Witten invariants . We see that in the limit Im s → ∞, the first two-point functions in (5.5) coincide with the two-point functions for the Calabi-Yau m-fold X m+2 : 10 The remaining two-point functions in (5.5) are naturally expected to give the following off-shell disk partition functions in the vicinity of the large radius point: The last equality means that the marked point on CP 1 for the operator O h 2m−p+1 1 transformed into a boundary (or hole) on which the disk D p can end. In other words, the boundary ∂D p of the disk D p should be mapped to a real (2p + 1)-dimensional subspace of the off-shell Lagrangian submanifold in X m+2 . This relation is a higher dimensional generalization of the open-closed string duality for Calabi-Yau threefolds discussed in [19][20][21][22]. It would be interesting to check this relation by explicitly defining and computing the off-shell disk partition functions in the A-model. Note that even for Calabi-Yau threefolds, this problem has not been fully elucidated. 9 In the case of the quintic Calabi-Yau threefold, CP 1 correspond to Π2,2 and Π2,3 in the appendix A of [21], respectively. 10 For m = 5, we will check this coincidence in appendix D.

JHEP03(2016)160 6 Conclusions and discussions
We studied the open and closed string deformations for a class of Calabi-Yau hypersurfaces in general dimensions.
To formulate the open mirror symmetry and extract the generating functions for topological invariants, we employed two methods using 1) the Gauss-Manin system for the relative cohomology group and 2) the GKZ system associated with the generalized hypergeometric functions. We have shown that these independent approaches yield the same differential system even for higher dimensions. In particular, for the simplest case of the compact Calabi-Yau fivefold, we have explicitly derived the Picard- Throughout this paper, we have paid attention to the compact Calabi-Yau hypersurfaces. Meanwhile, as explored in [16,19,20], it can be possible to formulate the off-shell mirror symmetry for the noncompact Calabi-Yau in a similar way. To generalize our higher dimensional construction into the local Calabi-Yau m-folds may shed light on novel aspects of the local mirror symmetry [55] and the geometric engineering [56].
From the viewpoint of flux compactifications, the closed string periods are related to the Gukov-Vafa-Witten type flux superpotential [57], and the relative periods are correspond to the superpotential arising from branes wrapping the internal cycles of the manifold. Although it is not obvious whether our result for m > 5 can be useful for the space-time compactification, there exists a possible application for the case of m = 5, namely the Calabi-Yau fivefolds. Indeed, in [24], the flux compactification of M-theory on Calabi-Yau fivefolds and its relation to the one-dimensional super-mechanics theories has been intensely studied. Naively we expect that holomorphic disks in Calabi-Yau fivefolds considered in this work are related to the superpotential in the 2a sector of this theory. It would be quite interesting to contemplate whether one can apply higher dimensional mirror prescription into this fascinating setup.
Further study on the open-closed duality is also important. Although we have shown that the charge vectors for brane geometry defined in the off-shell mirror formalism are naturally interpreted to describe another manifold without branes, a more detailed analysis is required to understand the correspondence completely. It would be necessary to reinterpret the off-shell mirror symmetry in the A-model viewpoint for the brane geometry and find the whole structure of the closed Gromov-Witten partition functions for higher dimensional Calabi-Yau manifolds with multiple Kähler parameters for the closed dual geometry.
Finally, we comment that there is another interesting direction for the future research about higher dimensional Calabi-Yau manifolds. It has been clarified that the Type IIB superstring on AdS 5 ×S 5 admits an A-model formulation through the pure spinor formalism and it was shown in [58] that the Coulomb branch of this model can be captured by a GLSM on a superprojective space. Thus one can expect that to analyze the A-model on the specific Calabi-Yau supermanifolds may provide some insights into the AdS/CFT correspondence JHEP03(2016)160 from the worldsheet perspective. Since this particular type of supermanifolds are supposed to be related to the (typically higher dimensional) local Calabi-Yau varieties by means of reversion of Grassmann parity of the fibers [59], it would be quite useful to establish the higher dimensional local mirror symmetry by modifying our present construction and the works in [16,19,20].

A Direct calculation in the A-model via localization
In this appendix, we describe how to compute the closed and open Gromov-Witten invariants of higher dimensional Calabi-Yau hypersurfaces in the A-model via the localization [60]. The direct enumeration of the holomorphic spheres and disks can be done by using the localization.

A.1 Counting holomorphic spheres
Here we consider the Calabi-Yau m-fold X m+2 ⊂ CP m+1 defined by a degree m+2 hypersurface in the A-model. Let O h i be observables defined by the elements h i ∈ H 2i (CP m+1 , C), where h ∈ H 2 (CP m+1 , C) is the hyperplane class of CP m+1 . The genus 0 and degree d

Gromov-Witten invariant is defined by
where M 0,n (CP m+1 , d) is the compactified moduli space of degree d and n-pointed stable maps from genus 0 stable curves to CP m+1 . The map ev i is the evaluation map at the i-th marked point from the moduli space Note that, in order to obtain the nontrivial quantity in (A.1), the constraint 11 should be satisfied. The Gromov-Witten invariants (A.1) can be computed by using the Atiyah-Bott fixed point theorem for the T m+2 -equivariant action on the embedding space CP m+1 . This torus action by T m+2 = U(1) m+2 is equipped with the equivariant parameters λ 1 , . . . , λ m+2 . The T m+2 -equivariant action induces an action on M 0,n (CP m+1 , d). The connected components of the fixed points of them are labeled by the tree graphs Γ which consist of the vertices v ∈ V ert(Γ) and the edges e ∈ Edge(Γ) with degrees d e ∈ N whose summation is fixed by The complex dimension of the moduli space is dim C M0,n(CP m+1 , d) = (m + 2)d + m − 2 + n, and the rank of coherent sheaf E d is (m + 2)d + 1. The integration in (A.5) is decomposed as where E i,j (λ 1 , . . . , λ m+2 ) = (m+2)de a=0 is the contribution from an edge e ∈ Edge(Γ) which connects two vertices labeled by i, j, and It was conjectured in [61] that the multiple covering formula for this kind of n-point function is given by where n d (h p 1 , . . . , h pn ) take integer values. The results for some one-and two-point functions for m-dimensional Calabi-Yau manifolds are listed in table 2 (for m = 7), table 3 (for m = 9), and table 4 (for m = 11).

A.2 Counting holomorphic disks
In the following, we consider the odd dimensional Calabi-Yau hypersurface X m+2 with m = 2m + 1. (A.13) Let L m+2 = X R m+2 be a Lagrangian submanifold in X m+2 which is the real locus defined by the fixed points of an anti-holomorphic involution on CP m+1 : (A.14)

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where i = 1, . . . , m + 1. The degree d disk Gromov-Witten invariant (see [62,63] for a mathematical detail) of X m+2 is defined by Here M D,n (CP m+1 /CP m+1 R , d) is the moduli space of degree d and n-pointed stable disk maps to CP m+1 , where the boundary of the disk ∂D is mapped to the Lagrangian CP m+1 R obtained as the set of fixed points of (A.14). e(E R d ) is the Euler class of a real vector bundle E R d on M D,n (CP m+1 /CP m+1 R , d) with real rank (m + 2)d + 1. We do not give a mathematically rigorous definition of (A.15). Instead, as in [29], we follow the definition in [12] for the quintic Calabi-Yau threefold, and assume that their analysis can be straightforwardly extended into higher dimensions. As in (A.2), then one finds the following condition for obtaining the nontrivial quantity: 16) As was described in appendix A.1, let us apply the Atiyah-Bott fixed point theorem to compute the disk Gromov-Witten invariant (A.15). As performed in [12], we consider a subtorus T m+1 ⊂ T m+2 which preserves the Lagrangian L m+2 as where i = 1, . . . , m+1. The T m+1 -equivariant parameters λ 1 , . . . , λ m+1 on L m+2 are defined by [12] ρ * (λ 2i−1 ) = −ρ * (λ 2i ) = λ i , ρ * (λ m+2 ) = 0, The T m+1 -equivariant action induces an action on M D,n (CP m+1 /CP m+1 R , d), and the connected components of the fixed points are labeled by open tree graphs Γ. Each tree graph Γ consists of vertices v ∈ V ert( Γ), edges e ∈ Edge( Γ) with degrees d e ∈ N, and also a half-edge e with an odd degree d 0 . The total degree of Γ is given by (A.20) The each vertex v ∈ V ert( Γ) is labeled by i v ∈ {1, . . . , m + 2} and also colored by a set S v ⊂ {1, . . . , n} in (A.4). The each edge e ∈ Edge( Γ) connects two different vertices v, u with different labels i v = i u , whereas the half-edge e have only one vertex v. Some of the open tree graphs with and without marked points are described in figure 4, 5, and 6. Then as discussed in [29,35] (see also [64]), one expects ϕ ≡ e(E R d )ev * 1 (h p 1 ) · · · ev * n (h pn ),  where E i,j (λ 1 , . . . , λ m+2 ) and V i (λ 1 , . . . , λ m+2 ) are defined in (A.8) and (A.9), respectively, and

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Suppose that the normal function associated with C + − C − (see [14] for details) is defined by Since the Landau-Ginzburg monodromy transforms the classical terms as Π 1 2 + Π 0 4 → Π 1 2 − Π 0 4 + Π 5 4 , we find that the natural choice for the normalization factor is c = π 3 . This result is consistent with the observation in the appendix of [29].

C On-shell disk two-point function
In this appendix, we will verify the formula (2.24) for the m = 2m + 1 dimensional Calabi-Yau hypersurface X m+2 : (C.1) Note that to obtain the expression (2.24), we need to use the divisor equation.
Let O p be the observable in the mirror B-model on Y m+2 corresponding to the observable O h p in the A-model on X m+2 . These operators obey the fusion rule [34,66] where η is a constant. Two different factorizations of a four-point function by the fusion rule yield a relation [34,66] From the fusion rule we have Then we can expect a relation for disk two-point functions in the B-model For q = m − p + 1, this relation is nothing but the mirror dual of (C.1).

D Solutions to the extended GKZ system
For the mirror Calabi-Yau hypersurfaces Y m+2 , the solutions of the extended GKZ operators { D 1 , D 2 , D 3 } constructed in section 4.2 can be obtained by the Frobenius method.