Superpotentials of D-branes in Calabi-Yau manifolds with several moduli by mirror symmetry and blown-up

We study B-brane superpotentials depending on several closed- and open-moduli on Calabi-Yau hypersurfaces and complete intersections. By blowing up the ambient space along a curve wrapped by B-branes in a Calabi-Yau manifold, we obtain a blow-up new manifold and the period integral satisfying the GKZ-system. Via mirror symmetry to A-model, we calculate the superpotentials and extract Ooguri-Vafa invariants for concrete examples of several open-closed moduli in Calabi-Yau manifolds.


Introduction
The type IIB compactification with branes can be described by an effective N = 1 supergravity theory with a non-trivial superpotential on the open-closed moduli space, because the D-branes wrapping supersymmetric cycles reduce the N=2 supersymmetry to N=1 supersymmetry. In topological string theory, there are two type D-branes. A-branes wrap special lagrangian cycles and B-branes wrap holomorphic cycles that can be even dimensional in a Calabi-Yau threefold.
On the A-model side, the superpotential is related to the topological string amplitudes, which counts disk instantons [1,2]. On the B-model side, the topological string is related to holomorphic Chern-Simons theory [3]. The B-brane superpotential is given by a integral over 3-chain with boundary consisting of 2-cycles γ around the B-branes, which is the sections of a line bundle over the open-closed moduli space described by the holomorphic N=1 special geometry [4][5][6]. The B-brane superpotential can be expressed as a linear combination of the integral of the basis of relative period. When considering B-brane wrapping two curves within the same homology class, the superpotential changes on the two sides of the domain wall whose tension is in terms of the Able-Jacobi map.
In physics, the D-brane superpotential is topological sector of the d=4, N=1 superpymmetric spacetime effective Lagrangian, as well as the generating function of open string topological field theory correlators. It encode the instanton correction and its derivatives determine structure constants of 2d chiral ring and Gauss-Manin connection of the vacuum bundle on the moduli space. The flatness of this connection determines the Picard-Fuchs equations satisfied by period vector. The expansion of superpotentials at large volume phase underlies the Ooguri-Vafa invariants counting the holomorphic disks ending on a lagrangian submanifold on the A-model side. These invariants closely relate to space of the states, non-pertubative effects, and geometric properties of the moduli space.
From the perspective of the deformation theory, the deformations of a curve in Calabi-Yau threefold are given by the sections of the normal bundle. The holomorphic sections lead to massless or light fields in the effective theory and the non-holomorphic sections lead to massive fields whose masses are given by the volume change under infinitesimal deformations. The superpotentials are related to the deformations with masses vanishing at some point in the closed moduli space. In order words, determining the B-brane superpotentials JHEP02(2022)203 is equivalent to solve the deformation theory of a pair (X, S) with curve S and Calabi-Yau manifold X. When a D5 brane wrapping a rational curve, the non-trivial superpotential is defined on a family of curves S, whose members are in general non-holomorphic except at some critical points where S is holomorphic curve. The critical locus corresponds to the supersymmetric vacus, other points in moduli space correspond to the obstructed deformation of the rational curve and excitation about the supersymmetric minimum.
The computation of off-shell superpotential for a toric brane, has been presented by local case in [4,5,7], and extended to compact Calabi-Yaus in [8][9][10]. For this brane, the onshell superpotentials and flat coordinates are the solutions to a system of open-closed Picard-Fuchs equations, which arise as a consequence of the N = 1 special geometry. These equation can be obtained by Griffish-Dwork reduction method or GKZ system. When a B-brane wrapping a curve S in X, the blowing up X along S lead to a new manifold X with an exceptional divisor E [11,12]. Meanwhile, the deformation theory of (X, S) is equivalent to that of (X , E). 1 The B-brane superpotential on the Calabi-Yau threefold X can be calculated in terms of period vector of manifold X . In this note, we first calculate Bbrane superpotentials in Calabi-Yau manifolds with several moduli via blowing up method, then extract Ooguri-Vafa invariants from A-model side at large volume phase by mirror symmetry.
The organization of this paper is as follows. In section 2, we introduce the background and formalism. To begin with, we review D-brane superpotential in the Type II string theory and relative cohomology description, recall the basic toric geometry about constructing Calabi-Yau manifold, generalized GKZ system and its local solutions, and outline the procedure to blow up a curve on a Calabi-Yau manifold. In section 3-5, for degree −9, −8, −12 Calabi-Yau hypersurface and degree-(3, 3) complete intersection Calabi-Yau manifold, we apply the blow up method to the mainfold with a curve on it and obtain a new Kahler manifold an exceptional divisor. The Picard-Fuchs equations and their solutions are derived by GKZ hypergeometric system from toric data of the enhanced polyhedron. The superpotential are identified as double-logarithmic solutions of the Picard-Fuchs equations and Ooguri-Vafa invariants are extracted at large volume phase. The last section is a brief summary and further discussions. In appendix A, we summarize the GKZ-system for two complete intersections Calabi-Yau manifolds X (112|112) [4,4] and X (123|123) [6,6] . In appendix B, we present the compact instanton invariants of above models for first several orders.
For a mirror pair of compact hypersurfaces (X * , X), one may associate a pair of integral polyhedra (∆ * , ∆) in a four-dimensional integral lattice and its dual. The n integral points JHEP02(2022)203 of the polyhedron correspond to homogeneous coordinates x i on the toric ambient space and satisfy linear relations i l j i v i = 0, a = 1, . . . , h 2,1 where l j i is the ith component of the charge vector l j . The integral entries of the vectors l j define the weights l j i of the coordinates x i under the C * action and l j i 's are the U(1) charges of the fields in the gauged linear sigma model (GLSM) [21]. In above description, the mirror Calabi-Yau threefold is determined as a hypersurface in the dual toric ambient space with constraints Here z j denotes the complex structure moduli of X. In terms of vertices v * j ∈ ∆ * , v i ∈ ∆ The complete intersection Calabi-Yau threefolds can be constructed similarly and we omit the details. Equivalently, Calabi-Yau hypersurfaces is also given by the zero loci of certain sections of the anticanonical bundle. The toric variety contains a canonical Zariski open torus C 4 with coordinates X = (X 1 , X 2 , X 3 , X 4 ). The sections are After homogenization, above equation is the same as (2.2). On these manifolds, a mirror pairs of branes, defined in [7] by another N charge vectorsl j . The special Lagrangian submanifold wrapped by the A-brane is described in terms of the vectorsl j satisfying where c j parametrize the brane position. And the holomorphic submanifold wrapped by mirror B-brane is defined by the following equation The N = 2 case, a toric curve, is more interesting to us and it is the geometry setting we are studying in this notes. To handle the toric curve case, we consider the enhanced polyhedron method proposed in [8]. It is possible in a simple manner to construct the enhanced polyhedron from the original polyhedra and the toric curve specified by two charge vectors. We denote the vertices of ∆ by v i , i = 1, . . . , n, with v 0 the origin, its JHEP02(2022)203 charge vectors by l i , and two brane vectors byl 1 andl 2 . We add 4 points to ∆ * to define a new polyhedron ∆ with vertices where we use the abbreviation The first line of (2.4) simply embeds the original toric data associated to ∆ into ∆ , whereas the second and third line translate the brane data into geometric data of ∆ . Given the toric data, the GKZ-system on the complex structure moduli space of X is given by the standard formula Here β = (−1, 0, 0, 0, 0) is the so-called exponent of GKZ system, ϑ j = a j ∂ ∂a j are the logarithmic derivative andv j = (1, v j ). The operators L i 's express the trivial algebraic relations among monomials entering hypersurface constraints, Z 0 expresses the infinitesimal generators of overall rescaling,Z i , i = 0's eare the infinitesimal generators of rescalings of coordinates x j . All GKZ operators can annihilate the period matrix, thus determine the mirror maps and superpotentials.
This immediately yields a natural choice of complex coordinates given by And from the operators L i , it is easy to obtain a complete set of Picard-Fuchs operators D i . Using monodromy information and knowledge of the classical terms, their solution can be associated to integrals over an integral basis of cycles in H 3 (X, Z) and given the flux quanta explicit superpotentials can be written down. For appropriate choice of basis vector l j , solutions to the GKZ system can be written in terms of the generating functions in these variables then we have a natural basis for the period vector For a maximal triangulation corresponding to a large complex structure point centered at z = 0, ∀a, ω 0 (z) = 1 + O(z) and ω 1,i (z) ∼ log(z i ) that define the open-closed mirror maps where S(z) is a series in the coordinates z. In addition, the special solution Π = W open (z) has further property that its instanton expansion near a large volume/large complex structure point encodes the Ooguri-Vafa invariants of the brane geometry.

Blowing up and Hodge structure
Blowing up in algebraic geometry is an important tool in this work. Now, we review the construction and properties of blowing up a manifold along its submanifold. Given S be a curve in a Calabi-Yau threefold X ∈ Z with Z an ambient toric variety, we can blow up along S to obtain a new manifold. According to section 2.2 (d) in [22], for this case that X ⊂ Z is a closed irreducible non-singular subvariety of Z and X is transversal to S at every point S ∩ X, π : Z → Z be the blowup of S. Then the subvariety π −1 (X) consist of two irreducible components, π −1 (X) = π −1 (S ∩ X) ∪ X and π : X → X defines the blow-up of X with center in S ∩ X = S, i.e. X is the manifold obtained from blowing up X along S. The subvariety X ⊂ Z is called the birational transform of X ⊂ Z under the blowup.
First, by the local construction, we consider an three dimensional multidisk in X, ∆ with holomorphic coordinates x i , i = 1, 2, 3, and V is specified by x 1 = x 2 = 0 on each ∆. Then we define the smooth variety∆ Here y 1 , y 2 are the homogeneous coordinates on P 1 . The projection map π :∆ → ∆ on the first factor is clearly an isomorphism away from V , while the inverse image of a point

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z ∈ V is a projective space P 1 . The manifold∆, together with the projection map π is the blow-up of ∆ along V ; the inverse image E = π −1 (V ) is an exceptional divisor of the blow-up. For two coordinates patches U i = (y i = 0), i = 1, 2, they have holomorphic coordinates respectively Next we consider the global construction of the blow-up manifold. Let X be a complex manifold of dimension three and S ⊂ X be a curve. Let {U α } be a collection of disks in X covering S such that in each disk ∆ α the subvariety S ∩ ∆ α may be given as the locus (x 1 = x 2 = 0), and let π α :∆ α → ∆ α be the blow-up of ∆ α along S ∩ ∆ α . We then have and using them, we can patch together the local blow-ups∆ α to form a manifold Finally, sinve π is an isomorphism away from X ∩ (∪∆ α ), we can take X =∆ ∪ π X − S X , together with the projection map π : X → X extending π on∆ and the identity on X − S, is called the blow-up of X along S, and the inverse image π −1 (S) is an exceptional divisor.
From the excision theorem of cohomology in algebraic topology [23], which means that the variation of the mixed Hodge structures of H 3 (X, S) and H 3 (X , E) over the corresponding moduli space are equivalent. The mixed Hodge structure as follow, where Ω denotes the holomorphic p-forms on X . The filtrations have the form and Additionally, the mixed Hodge structure has graded weights

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that take the following form for the divisor E The reason to consider these (graded) weights is the following: the mixed Hodge structure is defined such that the Hodge filtration F m H 3 induces a pure Hodge structure on each graded weight, i.e. on Gr W 2 H 3 and Gr W 3 H 3 . Thus, the following two induced filtrations on 14) The flatness of the Gauss-Manin connection leads to N=1 special geometry and a Picard-Fuchs system of differential equations that govern the mirror maps and superpotentials. The geometric setting we are interested in is a hypersurface X : P = 0 with a curve S on it, S : P = 0, h 1 = h 2 = 0. After blowing up along S, the blow-up manifold X is given globally as the complete intersection in the total space of the projective bundle where (y 1 , y 2 ) ∼ λ(y 1 , y 2 ) is the projective coordinates on the P 1 -fiber of the blow-up X .
We have to emphasize that X is not Calabi-Yau since the first Chern class is nonzero. In addition, the blow-up procedure do not introduce new degrees of freedom associated to deformations of E. Under blowing up map, the open-closed moduli space of (X, S) is mapped into the complex structure deformation of X . This enable us to calculate the superpotential W brane for B-branes wrapping rational curves via the periods on the complex structure moduli space of X determined by Picard-Fuchs equations.

Five branes wrapping lines
The Calabi-Yau threefold X is defined as the mirror of the Calabi-Yau hypersurface X * in P 4 (1,1,1,3,3) with h 2,1 = 2 complex structure moduli and the charge vectors of the GLSM for the A model manifold are given by:

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The hypersurface constraint for the mirror manifold, written in homogeneous coordinates of P (1,1,1,3,3) , is The Greene-Plesser orbifold group G acts as x i → λ g k,i k x i with λ 9 1 = λ 9 2 = 1, λ 3 3 = 1 and weights Z 9 : g 1 = (1, −1, 0, 0, 0), Z 9 : g 2 = (1, 0, −1, 0, 0), Z 3 : g 3 = (0, 0, 0, 1, −1) Next, we add a five-brane wrapping a rational curve on a toric curve S S :  For generic values of the moduli in (3.2), S is an irreducible high genus Riemann surface. But we can make a linearization by following steps: to begin with, we inserted h 1 and h 2 into P , Here η 3 1 = η 9 2 = η 9 3 = 1 and m(x 3 , x 4 , x 5 ) = α 9 + β 9 + γ 9 α 9 + α 3 β 3 γ 3 α 9 φ x 9 3 + Under the action of G = Z 2 9 × Z 3 , (3.5) describes a single line, In other words, these lines in P 4 have a parametrization by homogeneous coordinates U, V on P 1 as the Veronese mapping Thus all obstructed deformations locate at M(S) − M P 1 (S), inducing a non-trivial superpotential, which plays an important role in research on obstruction deformation, especially for a manifold with a submanifold on it. As we know, blowing up is very effective method to handle such case. According to (2.15), we construct the blow-up manifold X given by the complete intersection in projective bundle It is obvious from above defining equations that the moduli of S described by the coefficients of the monomials in h i , i = 1, 2 turn into complex structure moduli of X . We obtain the embedding of the obstructed deformation space of (X,P 1 ) into the complex structure moduli space of X , which is crucial for the following superpotential calculations.

Toric branes and blowing up geometry
Now, we study the A-model manifold, whose toric polyhedron is denoted by ∆ * and charge vectors are denoted by l 1 and l 2 . The integral vertices of polyhedron ∆ * and the charge vectors l 1 , l 2 for A-model manifold,l 1 ,l 2 for A-branes is as table 1.
From above toric data of ∆ * and its dual polyhedron ∆, In B-model, the defining equations of the mirror manifold X and the curve S as follow in torus coordinates where a i 's are free complex-valued coefficients. With the abbreviation of logarithmic deriva-JHEP02(2022)203 Here Z 0 represent the invariance of P under overall rescaling and other Z i 's relate to the invariance of P under the rescaling of torus coordinates X i 's combined with the rescaling of coefficients a i 's.
Operators L i 's relate to the symmetries among the Laurent monomials in P (3.7), By blowing up X along S, the blow-up manifold X is obtained.
After careful observation on the torus symmetry of X , we can obtain the infinitesimal generators which are belong to GKZ system associated to X , (3.10) Here Z 0 , Z 1 are associated with the overall rescaling with respect to P = 0, Q = 0 respectively. Z i , i = 2, . . . 5 are related to the torus symmetry as before.
In addition,Z 6 is related to the torus symmetry (y 1 , y 2 ) → (λy 1 , λ −1 y 2 ). The new L 3 , L 4 incorporate the parameter a 7 , . . . , a 10 that are associated with the open-closed moduli of JHEP02(2022)203 the curve S. All these GKZ operators annihilate the holomorphic three form Ω on X that is the pull back of the homomorphic three form Ω on X, i.e. Ω = π * Ω. Now, we formulate the GKZ-system (3.10) on an enhanced polyhedron ∆ , by adding additional vertices on the original polyhedron ∆ * .
where v i 's are the integral vertices of ∆ and their corresponding monomials in homogeneous coordinates of P 4 are w i . The A-model closed string charge vectors and A-branes charge vectors relate to the maximal triangulation of ∆ and satisfy the relations, The coordinates z j by (2.7) on the complex structure moduli space of X .
, z 2 = a 1 a 2 a 9 a 2 6 a 10 , z 3 = a 6 a 7 a 2 a 8 , z 4 = a 3 a 10 a 6 a 9 (3.12) Next, we convert the L i operators to Picard-Fuchs operators D i , from differential equations about a j (j = 0, . . . , 10) to those about z j (j = 1, . . . , 4) of X b . From table (3.11), we obtain the identity Inserting above relations between the logarithmic derivatives ϑ j w.r.t. a j and the logarithmic derivatives θ j w.r.t. z j into L operators in (3.10), the full set of Picard-Fuchs operators are obtained By the definition of the flat coordinates and mirror maps from Kahler moduli space to complex structure moduli space we obtain the z j as a series of q j upon inversion of the mirror maps In addition, we abbreviate the double logarithmic solutions by their leading terms

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where log(z i )'s are abbreviated as i 's. According to above, a specific linear combination of double logarithmic solutions is constructed, is the dilogarithm function. We extract the disk instantons N d 1 ,d 2 ,d 3 ,d 4 from W brane and present first a few invariants of the form N k,m,k,m+n in table 8. When focusing only on the invariants of the form N k,m,k,m+n , where . . . are terms independent of invariants, andq 1 = q 1 q 3 ,q 2 = q 2 q 4 ,q 3 = q 4 . The superpotential (3.17) can be written as which are essentially the superpotential of the model in [8]. Therefore, our invariants at first several order exactly match with the data of table 5 in [8] and these invariants are marked by blue color in table 8. In addition, we also calculate the invariants at higher order and put them into table 8.

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As before, we find single logarithmic solutions The superpotentials is constructed as follow as linear combination of double logarithmic solutions. We extract and summarize the Ooguri-Vafa invariants of the form N k,m,k,m+n and N k,m+n,k,n in table 9 and table 10. At first several order, our result marked by blue color exactly agree with table 6 in [8] and we also present higher order result in the table.

Branes wrapping rational curves and blowing up geometry
Now, we study the A-model manifold, whose toric polyhedron is denoted by ∆ * and charge vectors are denoted by l 1 and l 2 . A-model manifold and A-brane are specified by the following toric data. The hypersurface equation for X, written in homogeneous coordinates of P (1,1,2,2,2) , is The toric curve S on X is defined as the complete intersection The Greene-Plesser orbifold group G acts as Insert h 1 and h 2 into P = 0 P 1 is a non-holomorphic family due to fourth roots of unity. At critical loci of the parameter space α, β, γ M P 1 (S) : m(x 1 , x 4 ) vanishes identically and S degenerates to Modulo the action of G, (3.23) can be solved holomorphically. Thus the anholomorphic deformation of P 1 (3.25) can be used to described holomorphic deformation of S. From the toric data in table 2, we obtain the GKZ-system of X by (2.6), where Z 0 represents the invariant of P under overall rescaling, Z i 's relate to the torus symmetry, and L i 's relate to the symmetries among monomials consisting of P . As before, all GKZ operators annihilate the period integrals and determine the mirror maps and superpotentials.
After blowing up X along S, we obtain the blow-up manifold defined by, As before, the corresponding infinitesimal generators are obtained, which are belong to the GKZ system of X the GKZ system of X by observation on the torus symmetry, (3.28) where Z 0 , Z 1 are associated with the overall rescaling with respect to P = 0, Q = 0 respectively. Z i , i = 2, . . . , 6 are related to the torus symmetry and The L 3 , L 4 incorporate the parameter a 7 , . . . , a 10 that are associated with the moduli of the curve S.
Then we formulate above GKZ-system in an enhanced polyhedron ∆ (table 3). Here we present the integral points v i of the enhanced polyhedron ∆ and their the corresponding monomials w i 's. l i 's are the generators of Mori cone l i satisfying l 1 = l 1 + l 3 , l 2 = l 2 + l 4 and they are the maximal triangulation of ∆ .

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By definition (2.7), the local coordinates of complex structure moduli space of X are z 1 = a 4 a 5 a 6 a 8 a 3 0 a 7 , z 2 = a 2 1 a 10 a 2 6 a 9 , z 3 = a 3 a 7 a 0 a 8 , z 4 = a 2 a 9 a 1 a 10 (3.29) Next, we convert the L i operators in (3.28) to Picard-Fuchs operators D i , from differential equations about a j (j = 0, . . . , 10) to those about z j (j = 1, . . . , 4), where θ i 's are the logarithmic derivatives with respect to z i 's and each D i corresponds to a specific linear combination among l 1 , l 2 , l 3 , l 4 .

Brane superpotential and disk instantons
Now, we solve the Picard-Fuchs equations (3.30) at z i → 0 and identified the mirror maps and superpotentials. By the techniques we introduced in section 2.1. The unique power series solution, as well as the fundamental period of X, is The single logarithmic solutions are

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such that single logarithmic period of X can be reproduced by Π 1 1 = ω 1,1 +ω 1,3 , Π 1 2 ω 1,2 +ω 1,4 , and open-closed mirror maps are inverse series of flat coordinates, The double logarithmic solutions are denoted by their leading term 3 4 with abbreviations i = log(z i ). The brane superpotential is constructed as linear combination of double logarithmic solutions, This has the expected integrality properties of the Ooguri-Vafa Li 2 multicover formula. The Ooguri-Vafa invariants of the form N (m,k,m+n,k) are exactly match the data in table 5 in [24]. In addition, we also extract the invariants of the form N (m+n,k,n,k) and summarize them into table 11, where the rows and columns are labelled by m and n, respectively.

Five branes wrapping rational curves and blowing up geometry
In this section, we study the A-model manifold, whose toric polyhedron is denoted by ∆ * and charge vectors are denoted by l 1 and l 2 . A-model manifold and A-brane are specified by the following toric data. The mirror hypersurface X is determined by the constraint, where x i 's are homogeneous coordinates in P (1,1,2,2,6) and ψ = z The toric curve S is described by the complete intersection Insert h 1 and h 2 into P = 0 HereP 1 is evidently non-holomorphic because of the sixth roots of unity and hence, a nonholomorphic family of rational curves on. However, at special loci we see that S degenerates as follows, (3.35) where Z 0 represents the invariance of P under overall rescaling, Z i 's relate to the torus symmetry, and L i 's relate to the symmetries among monomials consisting of P . And all GKZ operators above annihilate the period matrix and determine the mirror maps and superpotential.

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where a i 's are free complex-valued coefficients. By simple observation, we can obtain the GKZ system of X as complement to GKZ system of X.
(3.37) where Z 0 , Z 1 are associated with the overall rescaling with respect to P = 0, Q = 0, Z i , i = 2, . . . , 6 are related to the torus symmetry, and L 3 , L 4 incorporate the parameter a 7 , . . . , a 10 that are associated with the moduli of the curve S.
Then we formulate above GKZ-system on an enhanced polyhedron ∆ , z 4 = a 2 a 9 a 1 a 10 (3.38) Next, we convert the L i in (3.37)operators to Picard-Fuchs operators D i , from differential equations about a j (j = 0, . . . , 10) to those about z j (j = 1, . . . , 4), where θ i = z i ∂ ∂z i 's are the logarithmic derivatives and each D i corresponds to a specific linear combination among l i 's.

Brane superpotential and disk instantons
Along the line of 2.1, we solve the differential equations (3.38) at z i → 0 and identify the mirror maps and superpotential. The fundamental period of X as power series solution is The single logarithmic solutions are such that the single logarithmic periods of X are reproduced by Π 1 1 = ω 1,1 + ω 1,2 , Π 1 2 = ω 1,3 + ω 1,4 . With single logarithmic solutions, open-closed mirror maps are inverse series of flat coordinates

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The double logarithmic solutions are denoted by their leading term with abbreviations i = log(z i ). Then we construct two linear combination of double logarithmic solutions and insert the inverse mirror maps to match the disk instantons in [10].
Invariants of the form N m,n,k,k are summarized in table 12, 13, 14, 15, where the rows and columns are labelled by m and n, respectively.

Five branes wrapping lines and blowing up geometry
The underlying manifold X * we are considering in the A-model is the intersection of two cubics in P 5 , whose mirror manifold X can be represented a one-parameter family of bicubics, with group G = Z 2 3 × Z 9 acting on them, where ψ is the complex structure modulus. Turning to the specification of D-brane configurations, we consider the curve S on X P 1 = P 2 = 0, An equivalent and convenient form is easy to obtained, For generic values of the moduli, the S is an irreducible higher genus Riemann surface. But we can always make a linearization by inserting h 1 and h 2 into P 1 , P 2 , This is an one dimensional family of cubic plane elliptic curves in P 2 , called the Hesse pencil. For special value of Ψ, it degenerate into 12 lines [25], Upon the action of group G, they are identified as a single line. Thus the deformation space of (4.4) is embedded in the deformation space of S. And away from that special locus, the obstructed deformation is identified with the unobstructed deformation of S, which means that we can use the obstructed deformation of that line to describe the unobstructed deformation of S.

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For L i 's, they represents the relations among Laurent monomials in P 1 and P 2 in (4.1).
And all GKZ operators annihilate the period matrix and determine the mirror maps and superpotentials.
After blowing up X along S, the blow-up manifold is By careful observations on the defining equations (4.6), the GKZ system of X is obtained as follow where Z 0 , Z 1 are associated with the overall rescaling with respect to P = 0, Q = 0 respectively. Z i , i = 2, . . . , 5 are related to the torus symmetry.

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where v i 's are the integral vertices and w i , their corresponding monomials. The A-model closed string charge vectors and A-branes charge vectors satisfy the relations l 1 = l 1 + 2l 2 + l 3 ,l 1 = l 3 ,l 2 = l 4 . The coordinates z i by (2.7)on the complex structure moduli space of X , z 1 = a 1 a 4 a 5 a 6 a 8 a 10 a 3 0,2 a 2 a 2 7 a 9 , z 2 = a 2 a 7 a 0,1 a 8 , z 3 = a 3 a 9 a 0,1 a 10 (4.8) Next, we convert the L i operators in (4.7) to Picard-Fuchs operators D i , from differential equations about a j to those about z j , where θ i = z i ∂ ∂z i 's are the logarithmic derivatives and each operator D a corresponds to a linear combination among the charge vectors l 1 , l 2 , l 3 .

Brane superpotential and disk instantons
Now, we solve the Picard-Fuchs equations (4.9)derived in the last section and identified the mirror maps and superpotentials. By the methods introduced in 2.1, at z i → 0, the fundamental period of X as series expansion is There are four the single logarithmic solutions

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we obtain the z j as a series of q j upon inversion of the mirror maps In addition, there are also double logarithmic solutions with leading terms where log(z i )'s are abbreviated as i 's. According to above, a specific linear combination of double logarithmic solutions is constructed and its disk instantons expansions are extracted.
We present first a few invariants of the form N m,m+n,m in table 16.

Branes on complete intersections P P P
(112|112) [4,4] and P P P (123|123) [6,6] Similar to the last section, we also calculate the superpotential and extract the Ooguri-Vafa invariants at large volume phase for complete intersections Calabi-Yau manifolds of P (112|112) [4,4] and P (123|123) [6,6] . We summarize the main formulas and tables in appendix A and Ooguri-Vafa invariants for first several orders in appendix C.

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where Z 0 represents the invariance of P under overall rescaling, Z i 's relate to the torus symmetry, and L i 's relate to the symmetries among monomials consisting of P . And all GKZ operators above annihilate the period matrix and determine the mirror maps and superpotential.
After blowing up X along S, the blow-up manifold is obtained as the complete intersection in the projective bundle where a i 's are free complex-valued coefficients. By observation on the symmetry of obove defining equations, we can obtain the GKZ system of X as complement to GKZ system of X.
∂ ∂a 10 , (5.4) where Z 0 , Z 1 are associated with the overall rescaling with respect to P = 0, Q = 0, Z i , i = 2, . . . , 6 are related to the torus symmetry, and L 3 , L 4 incorporate the parameter a 6 , . . . , a 10 that are associated with the moduli of the curve S. The new manifold X is describe by the following charge vectors , z 3 = a 5 a 7 a 0 a 8 , z 4 = a 1 a 10 a 5 a 9 (5.5) Next, we convert the L i in (5.4)operators to Picard-Fuchs operators D i , from differential equations about a j (j = 0, . . . , 10) to those about z j (j = 1, . . . , 4),

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When a 2 7 = a 6 a 8 , the two individual branes coincide. We obtain a new set of charge vectors,

Branes wrapping rational curves and blowing up geometry
The mirror octic hypersurface X arises as the Calabi-Yau hypersurfaces in P 4 (1,1,1,1,4) on which we consider parallel branes which are described by intersections of divisors The GKZ operators are derived by (2.6) where Z 0 represents the invariance of P under overall rescaling, Z i 's relate to the torus symmetry, and L i 's relate to the symmetries among monomials consisting of P . And all GKZ operators above annihilate the period matrix and determine the mirror maps and superpotential.
After blowing up X along S, the blow-up manifold X is obtained as the complete intersection

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where a i 's are free complex-valued coefficients. By observation on above defining equations, GKZ system of X is obtained as complement to GKZ system of X. 10 (5.11) where Z 0 , Z 1 are associated with the overall rescaling with respect to P = 0, Q = 0, Z i , i = 2, . . . , 5 are related to the torus symmetry, and L 3 , L 4 incorporate the parameter a 6 , . . . , a 10 that are associated with the moduli of the curve S.The new manifold X is describe by the following charge vectors , z 3 = a 5 a 7 a 0 a 8 , z 4 = a 1 a 9 a 0 a 10 (5.12) Next, we convert the L i in (3.37)operators to Picard-Fuchs operators D i , from differential equations about a j (j = 0, . . . , 10) to those about z j (j = 1, . . . , 4), · · · (5.13) where θ i = z i ∂ ∂z i 's are the logarithmic derivatives.

Brane superpotential and disk instantons
Along the line of 2.1, we solve the differential equations (5.13) at z i → 0 and identify the mirror maps and superpotential. The fundamental period of X is

JHEP02(2022)203 6 Summary and conclusions
In this work, we calculate the superpotentials in d=4 N=1 supersymmetric field theories arising from IIA D6-branes wrapping special Lagrangian three cycles of Calabi-Yau threefold. The special Lagrangian three-cycles with non-trivial topology are mirror to obstructed rational curves, which correspond to the brane excitation about the supersymmetric minimum. We consider a five brane wrapping a rational curve that coincides with a toric curve S at certain locus of the deformation space M(S). S is described by the intersection of two divisors D 1 ∩ D 2 and its unobstructed deformation space match with the obstructed deformation space of the rational curve wrapped by the five brane. After blowing up, the toric curve S is replaced by a exceptional divisor E without introducing new degree of freedom. All the complex structure moduli and brane moduli are embedded into the complex moduli space of the blow-up new manifold, given as the complete intersection in the projective bundle W = P(O(D 1 ) ⊕ O(D 2 )). From observation on the defining equation of X , we obtain the Picard-Fuchs equations that annihilate the period matrix defined by the natural pairing between the elements of relative homology H * (X, S) and cohomology H * (X, S). Via GKZ system of X , the system of Picard-Fuchs equations are solved at z i → 0. The single logarithmic solutions are interpreted as mirror maps and specific linear combinations of double logarithmic solutions are the B-brane superpotentials. Using multi-cover formula and inverse mirror maps, the Ooguri-Vafa invariants are extracted from A-model side and interpreted as counting disk instantons, i.e. holomorphic disks with boundary in a nontrivial homology class on a special Lagrangian submanifold. It would be interesting to directly extract these invariants on the A-model side directly by adequate localization techniques.

Acknowledgments
This work is supported by NSFC(11475178).
After blowing up X along the curve specified byl 1 ,l 2 , the GKZ system of blow-up manifold X : Toric data for the enhanced polyhedron associated to the X : 0 0 Polyhedron vertices and charge vectors A-model manifold X * and A-branes on it (table 7): GKZ system of X as follows After blowing up along the curve specified byl 1 ,l 2 , the GKZ system of blow-up manifold X :

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The brane superpotential is  with brane II at large volume. k and m label the class t 1 , t 2 of X 9 and n labels the brane winding. Blue entries agree with   Table 11. Ooguri-Vafa Invariants N k,m+n,k,n for X           [3,3] , P [112|112] [4,4] and P [123|123] [6,6] at large volume. The entries * in this table exceed the order of our calculation.