Mirror Symmetry, D-brane Superpotential and Ooguri-Vafa Invariants of Compact Calabi-Yau Manifolds

The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi-Yau manifolds. By using the GKZ-generalized hypergeometric system, we will calculate the B-brane superpotentials of two non-fermat type compact Calabi-Yau hypersurfaces in toric varieties,respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri-Vafa invariants for the mirror Calabi-Yau manifolds.


Introduction
The theory of topological string which is derived from the two dimensional (N,N) = (2, 2) superconformal field theory has gained a great development over the past years and it has influenced deeply on the mathematics. The D-brane superpotential, the generating functional of correlation function, is particularly vital physical quantity which is a section of a special holomorphic line bundles of the moduli space from the mathematical perspective. Through the superpotential we can derive a series of important properties for the CY manifolds such as Yukuwa couplings, Ooguri-Vafa invariants and so on. Therefore the calculation of the superpotential is very meaningful.
Some important properts of the moduli spaces for various Calabi-Yau manifolds [1][2][3] has been well studied via the mirror symmetry which was first mentioned in the local operator algebra of the N = 2 string theory [4]. It is well known that mirror symmetry connects two different moduli spaces which are respectively parameterized by kahler geometry deformation and complex geometry deformation in A-and B-model. In A-model there exist contributions from the instantons while there is none in Bmodel. So calculating superpotential directly in A-model is considerably difficult. In fact only in several special cases we know the corresponding brane configuration on mirror A-model side for a given brane configuration in the compact CY manifold [6] derived from the GKZ system in B-model. In GKZ system the superpotential is related to period integral. And the Hodge theoretic approach [5] provides a useful insight on studying the period integrals for CY manifolds period integral satisfies Picard-Fuchs differential equation which is closely related to the GKZ system..
Recently, for compact CY manifolds, there are some great development in calculating the quantum corrected domain wall tensions on the CY threefolds via open-closed mirror symmetry [7,8]. The properties of some compact Calabi-Yau manifolds has been studied in refs. [25,26,28]. In this note, we compute the D-brane superpotentials for two non-fermat CY threefolds in detail via mirror maps and GKZ hypergeometric system.
The structure of this paper is as follows. In section 2 we describe the generalized GKZ hypergeometic system. The solution of GKZ hypergeometic system is just the integral period. We also outline the approach to constructing the corresponding polyhedron ∆ and its mirror polyhedron ∆ * for the Calabi-Yau manifold. Then we review how to calculate superpotential. In section 3 we analyze two non-fermat type compact CY manifolds in toric varietis, respectively. and compute their superpotentials as well as some disk invariants with the method referred previously. The last section is the conclusion.

Toric Geometry, Relative Period Integrals and GKZ System
We divide this section into two parts to review some related background.

Superpotential on D-brane
In the presence of some background fluxes and space-filling D-branes, the type II string theory compactification on Calabi-Yau manifold gives rise to the N = 1 low effective energy theories [25], whose effective superpotential is captured by the relative period integral of the holomorphic three form Ω(z) around the integral relative cycle with boundaries in the D-branes [14]. As is listed in the refs. [15][16][17][18] that the above integral is derived from the action of a holomorphic Chern-Simons theory on the brane which wraps the holomorphic curves.
For a D-brane wrapping internal cycles of Calabi-Yau manifold X, the corresponding effective superpotential is [16] where if there are N branes, A is a holomorphic U(N) gauge connection on X and Ω is the holomorphic three form on X. For type IIB string, the effective superpotential is a linear combination of the relative period integrals [18,19,21].
whereN a stands for the homology class, which is wrapped with the D-brane.Π a (ϕ, ξ) represents the period integralΠ Here, ξ and ϕ stands for the open-and closed-string moduli respectively. The internal background fluxes H = H RR + H N S lead to an effective superpotential [22][23][24][25][26][27][28] which is defined by where Ω denote the holomorphic three-form on the Calabi-Yau manifold, and τ denote the complex couplings for the type II string of B-model. In this note, wo only consider the RR flux, the induced superpotential becomes Therefore the combined superpotential generated by D-brane and flux is [29,30] here the coefficient N Σ denotes both the D-brane topological charge and the RR flux quantum data and Π Σ (ϕ, ξ) denotes the integral of the three-form Ω(ϕ) over the threechains in the relative integer homology group, which is defined by The relative period integral referred previouslyΠ a (ϕ, ξ) is equal to the domain wall tension T (ϕ, ξ) [6,17,18,29,30]. T (ϕ, ξ) is defined as . As is depicted in the refs. [3,8,31,32], at the critical points, the domain wall tensions are considered as normal function from which the Abel-Jacobi invariants can be derived .
For the D-brane in A-model, the superpotential which is expressed in terms of the flat closed/open coordinates can be calculated as the generating functional of the correlation functions [18,29,[33][34][35].It is defined by Here, q = e 2πit ,q = e 2πit and n k, m is the Ooguri-Vafa invariant. Mirror symmetry, which indicates that the two superpotentials for D-branes in A-and B-model, respectively, are related to each other by the mirroe map, gives us a method to computing the

Ooguri-Vafa invariant which is closely related to the open Gromov-Witten invariant
G k, m [6]. The superpotentials and the Ooguri-Vafa invariants are as follows [7,36] : where q a = e ta (a = 1, 2) and Here the mirror map t a is is defined as t a = ∂aω 0 ω 0 .

Toric Geometry and GKZ System
The generalized hypergeometric system was first introduced in refs. [37], and soon after gained a fast development in mirror symmetry [5,[38][39][40][41]. Define a mirror pair of hypersurfaces (X, X * ) in two toric ambient spaces (V, V * ), respectively. The toric varieties (V, V * ) are related to the fans (Σ(∆), Σ(∆ * )) induced by two dual polyhedron (∆, ∆ * ). The defining polynomial for the hypersurface is defined as: Or we can write the above equation in another way Here a i is complex parameter and X k are inhomogeneous coordinates on the open torus, x i is the homogeneous coordinates.
The general integral period is expressed as It is referred in [38,39] that the period can be annihilated by a GKZ hypergeometric differential system the operators D l and Z j are expressed as The torus invariant algebraic coordinates z a in the large complex structure limit [3] where l a is the set of basic vectors which denote the generators of the Mori cone. Then, by θ a = z a ∂ za , (2.16) changes into where l is the linear combination of l a . One can refer [31,42,43] for more details. The result to the GKZ system is described as 3 Study of two compact non-fermat type Calabi-Yau manifolds In this section we will calculate the superpotentials and disk invariants for two compact CY in the weighted projective space, respectively, with the method mentioned in the section 2.
according to (2.16) the GKZ system for the two-parameters family become The solution to this GKZ system are written as Similarly, in this model the on-shell superpotentials satisfy W + C = W − C according to the Z 2 symmetry. So the superpotentials is described as At the critical point z 3 = 1, on-shell superpotentials are expressed as the mirror maps are ) and the corresponding inverse mirror maps are Analogous to computing the disk invariants of the X 7 (1, 1, 1, 1, 3), we have the results listed in the following tables:   and are impossibly obtained from the perturbative or localization way which is important methods to compute the D-brane superpotential in non-compact Calabi-Yau manifold in A-model. A effective approach to obtain the D-brane superpotential is by using the blown-up geometry of target space along the submanifold wrapped by the D-branes [10,47]. The alternative approach to compute the superpotential of the D-brane in compact Calabi-Yau manifold in A-model is via the algebraic geometric method and mirror symmetry.
In this paper, we extend the generalized GKZ system in a Fermat Calabi-Yau threefolds to the compact non-Fermat Calabi-Yau threefolds which are less study so far in contrast to the Fermat type Calabi-Yau threefolds. we first construct the generalized GKZ system for the compact non-fermat type Calabi-Yau manifolds, then work out the corresponding D-brane superpotential in the mirror B-model by the algebraic geometric method. The superpotential in the A-model is obtained accroding to mirror symmetry.
Finally the Ooguri-Vafa invariants are extracted from the A-model superpotential.
These superpotential have potential phenomenological applications. Furthermore, acoording to the type II string/M-theory/F-theory duality, in the weak decoupling limit g s → 0, these superpotentials of Type II string give the Gukov-Vafa-Witten superpotentials W GV W of F-theory compactified on the dual fourfold. On the other hand, since there is not a systematic mathematical method to compute them by now, after all, it is difficult to get, from other approach, those Ooguri-Vafa Invariants predicted in this paper. Those Ooguri-Vafa Invariants provide some concrete data which could potentially be checked by an independent mathematical calculation.