Mirror symmetry gives the equivalence between A-model determined by Kähler moduli of quantum geometry and B-model determined by complex structure moduli of the classical geometry in the topological string theory. Supersymmetry breaks from N=2 to N=1 when D-brane is included. Correspondingly, open-closed mirror symmetry^[1-2] between A and B models is developed. On the analogy of the prepotential in N=2 supersymmetric topological string theory, the counterpart is called the non-perturbed holomorphic superpotential in N=1 supersymmetric topological string theory^[3]. In this work, the D-brane superpotential is calculated by mirror symmetry, GKZ-system method, and the type Ⅱ/F-theory duality^[4-8].

Since there is no quantum correction and it is relatively easy to deal with, we calculate the effective superpotential on the side of B-model. Considering that the space-filling D5-branes wrap on a reducible curve C=∑_iC_i and C is embedded in a divisor D of Calabi-Yau 3-fold M₃, the effective superpotential is

$ \begin{array}{*{20}{c}} {{W_{N = 1}}(z,\hat z) = {\Pi _\gamma }(z,\hat z) = \int_\gamma {{\Omega ^{(3,0)}}} (z,\hat z),}\\ {\gamma \in {H_3}\left( {{M_3},D} \right).} \end{array} $

(1)

The effective superpotential^[9] can be expressed as a linear combination of relative period^[10]:

$ \begin{array}{l} {W_{N = 1}}(z,\hat z) = \sum {{N_\alpha }} {\Pi _\alpha }(z,\hat z)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {W_{{\rm{open }}}}(z,\hat z) + {W_{{\rm{closed }}}}(z,\hat z), \end{array} $

(2)

where z and $\hat{z}$ are closed and open parameters, respectively.

On the other hand, the type Ⅱ string theory is dual to F-theory^[11-12]. The D-brane superpotential in type Ⅱ string theory is dual to the background flux superpotential in F-theory^[11]. The D-brane superpotential of type Ⅱ on a compact Calabi-Yau threefold M₃ can be obtained in F-theory which is on a compact Calabi-Yau fourfold M₄. The superpotential of 4-form flux G₄ in F-theory compactified on the Calabi-Yau 4-fold M₄ is a section of the Hodge line bundle in the complex structure moduli space M_cs(M₄). This superpotential is called Gukov-Vafa-Witten superpotential,

$ \begin{array}{l} {W_{{\rm{GVW}}}}\left( {{M_4}} \right) = \int_{{M_4}} {{G_4}} \wedge {\Omega ^{(4,0)}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum {\left( {{N_\Sigma }\left( {{G_4}} \right){\Pi _\Sigma }(z,\hat z)} \right)} + O\left( {{g_s}} \right) + O\left( {{{\rm{e}}^{\left( { - \frac{1}{{{g_s}}}} \right)}}} \right). \end{array} $

(3)

The leading term on the right-hand side of the above equation is the D-brane superpotential W_N=1, and g_s is the string coupling strength.

In the weak coupling limit g_s→0, the D-brane superpotential W_N can be obtained from the GVW superpotential W_GVW of F-theory:

$ \begin{array}{l} {W_{N = 1}}\left( {{M_3},D} \right) = \sum {{N_\Sigma }} \left( {{G_4}} \right){\Pi _\Sigma }(z,\hat z)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \mathop {\lim }\limits_{{g_s} \to 0} {W_{{\rm{GVW}}}}\left( {{M_4}} \right). \end{array} $

(4)

The system with three D-branes is more complex than the system with single or two D-branes. Complicated D-brane systems play the important role in the phenomenological applications such as superstring/ M/F-theory for MSSM. In this work, for the system with three D-branes on the hypersurface P(1, 1, 1, 1, 2), i.e., Sextic, the type Ⅱ string theory and F-theory duality^[12] are used to calculate the D-brane superpotential and Oogrui-Vafa invariants. The phase transitions, the parallel phase→the partial coincident phase→the complete coincident phase, correspond to the enhancement of gauge group U(1)×U(1)×U(1)→U(1)×U(2)→U(3), in the low energy effective theory.

1 Toric geometry of D-branes system and the generalized hyper-geometric system of Gel'fand, Kapranov, and Zelevinski (GKZ-hypergeomtric system)

The Calabi-Yau manifolds in this paper are the hypersurfaces in ambient toric variety^[13-14]. For each pair of reflexive polyhedrons ($\nabla $₄, Δ₄) with a pair of complete fans (Σ($\nabla $₄), Σ(Δ₄)), there is a pair of toric varieties (P_{Σ($\nabla $4)}, P_Σ(Δ4)). The defining polynomial P for the hypersurface M₃ in the homogeneous coordinates x_j of toric ambient space is

$ P = \sum\nolimits_{i = 0}^{p - 1} {{a_i}} \prod\nolimits_{v \in {\Delta _4}} {x_j^{ < v,v_i^* > + 1}} . $

(5)

The n parallel D-branes are defined by reducible divisor:

$ \begin{array}{l} Q(D) = \prod\nolimits_{m = 0}^n {\left( {{\phi _m}{a_0} + {a_i}\sum\nolimits_{v \in {\Delta _4}} {x_j^{k < v,v_i^* > + 1}} } \right)} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \sum\nolimits_{k = 0}^n {{b_k}} \sum\nolimits_{v \in {\Delta _4}} {x_j^{k < v,v_i^* > + n}} . \end{array} $

(6)

The parallel D-brane geometry corresponds to the Coulomb phase of the gauge theory and the corresponding group is U(1)×U(1)…U(1)^{[11, 15]}.

The vertices of the parallel D-branes phase shaping the $\tilde{\nabla }$₅ are

$ \tilde v_j^* = \left\{ {\begin{array}{*{20}{l}} {\left( {v_j^*,0} \right)}&{j = 0, \cdots ,p - 1,}\\ {\left( {mv_i^*,1} \right)}&{j = p + m,0 \le m \le n.} \end{array}} \right. $

(7)

The gauge group U(1)×U(1)…U(1) is promoted to the U(N) group when parallel D-branes coincide, while the phase is translated to the Higgs branch.

We have got the enhanced polyhedron which defines the non-compact four-fold $\tilde{X}$₄^*. Then the P₁ compactification can be obtained by adding the extra vertex. Combined with the compactifying point $\tilde{v}$_c^*, all the points define the enhanced polyhedron $\nabla $₅. This means the real F-theory compact four-manifold is obtained.

The bases for choosing compactifying point are given as follows. 1) The origin is to be included in the enhanced polyhedron. 2) The polyhedron and dual polyhedron are convex polyhedron.

The relative period satisfies the differential equations of Picard-Fuchs^[16], and its differential operator derived by GKZ-system^[4] is

$ \begin{array}{*{20}{l}} {L\left( {{l^a}} \right) = \prod\nolimits_{k = 1}^{l_0^a} {\left( {{\vartheta _0} - k} \right)} \prod\nolimits_{l_j^a > 0} {\prod\nolimits_{k = 0}^{l_j^a - 1} {\left( {{\vartheta _j} - k} \right)} } - }\\ {{{( - 1)}^{l_0^a}}{z_a}\prod\nolimits_{k = 1}^{ - l_0^a} {\left( {{\vartheta _0} - k} \right)} \prod\nolimits_{l_j^a} {\prod\nolimits_{k = 0}^{ - l_j^a - 1} {\left( {{\vartheta _j} - k} \right)} } ,} \end{array} $

(8)

where ${{\vartheta }_{j}}={{a}_{j}}\frac{\partial }{\partial {{a}_{j}}}$ are logarithmic derivative operators in the parameter a_j, and a_j defines the coefficients in the polynomial for the hypersurface. l_a are Mori cone generators of toric variety P_{Σ($\nabla $5}). These generators are also known as the charge vectors of the gauged linear sigma model (GLSM), a∈{1, …, k}.

According to the generators of Mori cone, the local coordinates near the limit point of large complex structure in M₄ moduli space have the forms

$ {z_a} = {( - 1)^{l_0^a}}\prod\nolimits_j {a_j^{l_0^a}} . $

(9)

Mori cone and Kähler cone are dual to each other. The selection of Mori cone generator l_a also gives a set of dual bases for Kähler cone, denoted as J_a∈H^{(1, 1)}(W₄). The corresponding local coordinate is marked as k_a. Since the large complex structure limit point is dual to the large radius limit point, the k_a of Kähler moduli space are also known as flat coordinates.

Local solution of GKZ-system can be derived from basic cycle w₀,

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {w_0}(z;\rho ) = \sum {_{{m_1}, \cdots ,{m_a} \ge 0}} \\ \frac{{\Gamma \left( { - \sum {\left( {{m_a} + {\rho _a}} \right)} l_0^a + 1} \right)}}{{\prod\nolimits_{1 \le i \le p} \Gamma \left( {\sum {\left( {{m_a} + {\rho _a}} \right)} l_i^a + 1} \right)}}{z^{m + \rho }}. \end{array} $

(10)

By using the Frobenius method, the complete cycle vector has the form,

$ \mathit{\boldsymbol{ \boldsymbol{\varPi} }}(z) = \left( {\begin{array}{*{20}{c}} {{\Pi _0} = {{\left. {{w_0}(z;\rho )} \right|}_{\rho = 0}}}\\ {{\Pi _{1,i}} = {{\left. {{\partial _{\rho i}}\left( {{w_0};\rho } \right)} \right|}_{\rho = 0}}}\\ {{\Pi _{2,n}} = \sum\nolimits_{{i_1},{i_2}} {{{\left. {{K_{{i_1}{i_2};n}}{\partial _{{\rho _{{i_1}}}}}{\partial _{{\rho _{{i_2}}}}}{w_0}(z;\rho )} \right|}_{\rho = 0}}} }\\ \ldots \end{array}} \right), $

(11)

n⊂1, …, h, where h equals the dimension of H₄($\tilde{M}$₄). The K_{i1, i2; n} are the combinatoric coefficients of the second derivative of w₀.

The mirror conjecture gives the dual periodic vectors on the side of A-model as follows:

$ {\mathit{\boldsymbol{ \boldsymbol{\varPi} }}^*}(k) = \left( {\begin{array}{*{20}{c}} {\Pi _0^* = 1}\\ {\Pi _{1,i}^* = {k_i}}\\ {\Pi _{2,n}^* = \sum\nolimits_{{i_1},{i_2}} {K_{{i_1}{i_2};n}^*} {k_{{i_1}}}{k_{{i_2}}} + {b_n} + F_n^{{\rm{inst}}}}\\ \ldots \end{array}} \right), $

(12)

where k_i=Π_{1, i}/Π₀ are the flat coordinates. The coefficients K_i1i2;n^* of the leading terms are related to the classical sector in superpotential (K_{i1i2; n}^*=K_{i1i2; n}) and b_n are constants. The F_n^inst stands for the instanton correction sector of the solutions.

Mirror map is given by

$ {k_i} = \frac{{\prod\nolimits_{1,i} {(z)} }}{{{\Pi _0}}}. $

(13)

The open and closed string parameters in Kähler moduli can be separated by linear combination of k_i. The instanton corrections are encoded as a power series expansion in q_i=exp(2πit_I) and $\hat{q}$_i=exp(2πi$\hat{t}$_i)

$ {F^{{\rm{inst }}}}(t,\hat t) = \sum\nolimits_{\mathit{\boldsymbol{r,m}}} {{G_{\mathit{\boldsymbol{r,m}}}}} {q^\mathit{\boldsymbol{r}}}{q^\mathit{\boldsymbol{m}}} = \sum\nolimits_n {\sum\nolimits_{\mathit{\boldsymbol{r,m}}} {\frac{{{N_{\mathit{\boldsymbol{r,m}}}}}}{{{n^2}}}} } {q^{n\mathit{\boldsymbol{r}}}}{\hat q^{n\mathit{\boldsymbol{m}}}}, $

(14)

where {G_{r, m}} are open Gromov-Witten invariants and {N_{r, m}} are Ooguri-Vafa invariants. m represent the elements of H₁(L) and r represent the elements of H₂(W₃).

2 Model: D-brane system on the Sextic

The hypersurface P(1, 1, 1, 1, 2) is given by a polynomial

$ \begin{array}{l} P = {a_1}x_1^6 + {a_2}x_2^6 + {a_3}x_3^6 + {a_4}x_4^6 + {a_5}x_5^3 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_0}{x_1}{x_2}{x_3}{x_4}{x_5}. \end{array} $

(15)

The degree 6 hypersurface P is in the ambient toric variety P_Σ(Δ4). The toric variety is determined by the vertices of the polyhedron Δ₄ as follows:

$ {{v_1} = (2, - 1, - 1, - 1),{v_2} = ( - 1,5, - 1, - 1),} $

$ {{v_3} = ( - 1, - 1,5, - 1),{v_4} = ( - 1, - 1, - 1,5),} $

$ {{v_5} = ( - 1, - 1, - 1, - 1).} $

The vertices of its dual polyhedron $\nabla $₄ are

$ v_0^* = (0,0,0,0),v_1^* = (1,0,0,0), $

$ v_2^* = (0,1,0,0),v_4^* = (0,0,0,1), $

$ v_5^* = ( - 2, - 1, - 1, - 1),v_3^* = (0,0,1,0). $

2.1 Parallel phase of three D-branes:

$\tilde{v}$_c^*=(-1, 1, 1, 1, -1) is selected as the compactifying point.

We consider the parallel D-branes defined by the reducible divisor D=D₁+D₂+D₃, which can be written as the degree 15 homogeneous equations,

$ \begin{array}{l} Q = {b_0}{\left( {{x_1}{x_2}{x_3}{x_4}{x_5}} \right)^3} + {b_1}x_1^2x_2^2x_3^2x_4^2x_5^5 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_2}{x_1}{x_2}{x_3}{x_4}x_5^7 + {b_3}x_5^9\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \backsim \prod\nolimits_{i = 1}^3 {\left( {{\phi _i}{a_0}{x_1}{x_2}{x_3}{x_4}{x_5} + {a_1}x_5^3} \right)} . \end{array} $

(16)

The open-closed system is encoded in the enhanced polyhedron $\tilde{\nabla }$₅ whose vertices are

$ {\tilde v_0^* = (0,0,0,0,0),\tilde v_1^* = (1,0,0,0,0),} $

$ {\tilde v_2^* = (0,1,0,0,0),\tilde v_3^* = (0,0,1,0,0),} $

$ {\tilde v_4^* = (0,0,0,1,0),\tilde v_5^* = ( - 2, - 1, - 1, - 1,0),} $

$ {\tilde v_6^* = (0,0,0,0,1),\tilde v_7^* = (1,0,0,0,0),} $

$ {\tilde v_8^* = (2,0,0,0,1),\tilde v_9^* = (3,0,0,0,1).} $

The generators of Mori cone determined by $\nabla $₅ are given as follows:

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} \ \ \ 0&1&2&3&4&5&\ \ \ 6&\ \ \ 7&\ \ \ 8&\ \ \ 9&c \end{array}\\ \ {{l^1} = \left( {\begin{array}{*{20}{l}} { - 4}&0&1&1&1&1&{ - 2}&\ \ \ 2&\ \ \ 0&\ \ \ 0&0 \end{array}} \right)}\\ \begin{array}{*{20}{l}} {{l^2} = \left( {\begin{array}{*{20}{l}} \ \ \ 0&0&0&0&0&0&\ \ \ 1&{ - 2}&\ \ \ 1&\ \ \ 0&0 \end{array}} \right)}\\ {{l^3} = \left( {\begin{array}{*{20}{l}} \ \ \ 0&0&0&0&0&0&\ \ \ 0&\ \ \ 1&{ - 2}&\ \ \ 1&0 \end{array}} \right).} \end{array}\\ \begin{array}{*{20}{l}} {{l^4} = \left( {\begin{array}{*{20}{l}} { - 1}&1&0&0&0&0&\ \ \ 0&\ \ \ 0&\ \ \ 1&{ - 1}&0 \end{array}} \right)}\\ {{l^5} = \left( {\begin{array}{*{20}{l}} { - 3}&0&0&0&0&1&{ - 2}&\ \ \ 3&\ \ \ 0&\ \ \ 0&1 \end{array}} \right)} \end{array} \end{array} $

(17)

A suitable set of bases is selected to visualize the closed and open moduli,

$ \begin{array}{*{20}{c}} {t = {k_1} + 2{k_2} + 2{k_3} + 2{k_4},}\\ {{{\hat t}_1} = {k_2} + {k_3} + {k_4},{{\hat t}_2} = {k_3} + {k_4},{{\hat t}_3} = {k_4}.} \end{array} $

(18)

The leading terms of the relative periods are

$ \begin{array}{*{20}{c}} {\tilde \Pi _{2,1}^* = 3{t^2},\tilde \Pi _{2,2}^* = {{\left( {t - 2{{\hat t}_1}} \right)}^2},}\\ {\tilde \Pi _{2,3}^* = \left( {t - 2{{\hat t}_2}} \right),\tilde \Pi _{2,4}^* = \left( {t - 2{{\hat t}_3}} \right).} \end{array} $

(19)

The $\tilde{\Pi }$_{2, 1}^* only depends on the close moduli t and is supposed to be the leading term of the bulk potential function F_t(t), while $\tilde{\Pi }$_{2, 2}^*, $\tilde{\Pi }$_{2, 3}^*, and $\tilde{\Pi }$_{2, 4}^* are supposed to lead the D-brane superpotential W(t, $\hat{t}$) which depends on both open $\hat{t}$ and closed t parameters.

Using algebraic coordinates as follows:

$ {z_1} = \frac{{{a_2}{a_3}{a_4}{a_5}b_1^2}}{{b_0^2}},{z_2} = \frac{{{b_0}{b_2}}}{{b_1^2}},{z_3} = \frac{{{b_1}{b_3}}}{{b_2^2}},{z_4} = \frac{{{a_1}{b_2}}}{{{a_0}{b_3}}}. $

(20)

the fundamental period and the logarithmic period are

$ \begin{array}{l} {\Pi _0}(z) = {w_0}(z;0),{\Pi _{1,i}} = {\left. {{\partial _{\rho i}}(z;\rho )} \right|_{{\rho _i} = 0}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\Pi _{2,n}} = \sum\limits_{i,j} {{{\left. {{K_{i,j;n}}{\partial _{{\rho _i}}}{\partial _{{\rho _j}}}{w_0}(z;\rho )} \right|}_{\rho = 0}}} . \end{array} $

(21)

The flat coordinates are given by

$ {k_i} = \frac{{{\Pi _{1,i}}(z)}}{{{\Pi _0}}} = \frac{1}{{{\rm{2 \mathsf{ π} i}}}}\log {z_i} + \cdots . $

(22)

Then the mixed inverse mirror maps in terms of q_i=exp(2πik_i) for {i=1, 2, 3, 4} are

$ \begin{array}{l} {z_1} = {q_1} - 24q_1^2 + 2{q_1}{q_2} - 24q_1^2{q_2} + {q_1}q_2^2 + 2{q_1}{q_2}{q_3} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 24q_1^2{q_2}{q_3} + 2{q_1}q_2^2{q_3} + 288q_1^2q_2^2q_3^2{q_4} + \cdots ,\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_2} = {q_2} + 12{q_1}{q_2} + 414q_1^2{q_2} - 2q_2^2 - 48{q_1}q_2^2 - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 3q_2^2{q_3} + {q_2}{q_3} + 12{q_1}{q_2}{q_3} + 414q_1^2{q_2}{q_3} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1944q_1^2q_2^2 + \cdots ,\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{z_3} = {q_3} + {q_2}{q_3} + 12{q_1}{q_2}{q_3} + 414q_1^2{q_2}{q_3} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 12{q_1}q_2^2{q_3} - 2q_3^2 - 3{q_2}q_3^2 - 36{q_1}{q_2}q_3^2}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1242q_1^2{q_2}q_3^2 - 2{q_2}q_3^2 + \cdots ,} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_4} = {q_4} + {q_3}{q_4} + {q_2}{q_3}{q_4} + 12{q_1}{q_2}{q_3}{q_4} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 414q_1^2{q_2}{q_3}{q_4} + 12{q_1}q_2^2{q_3}{q_4} + q_4^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 12{q_1}q_1^2q_3^2{q_4} + q_3^2q_4^2 + \cdots . \end{array} $

(23)

According to the leading terms, we find the relative periods which correspond to the closed-string period and the D-brane superpotentials in the A-model as follows:

$ \begin{array}{l} {F_t}(t) = 3{t^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}\left( {15768q + 24117750{q^2} + \cdots } \right),\\ {W_1}\left( {t,{{\hat t}_1}} \right) = {\left( {t - 2{{\hat t}_1}} \right)^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}\left( {4416q + 6465456{q^2} + } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 24{{\hat q}_1} + 6\hat q_1^2 + \frac{8}{3}\hat q_1^3 + \frac{3}{2}\hat q_1^4 + 160q\hat q_1^{ - 2} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1632q\hat q_1^{ - 1} + 3952q{{\hat q}_1} - 2184q\hat q_1^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {6600{q^2}\hat q_1^{ - 4} + \cdots } \right),\\ {W_2}\left( {t,{{\hat t}_2}} \right) = {\left( {t - 2{{\hat t}_2}} \right)^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}\left( {4416q + 6465456{q^2} + } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {24{{\hat q}_2} + 6\hat q_2^2 + \frac{8}{3}\hat q_2^3 + \frac{3}{2}\hat q_2^4 + }\\ {160q\hat q_2^{ - 2} - 1632q\hat q_2^{ - 1} + 3952{{\hat q}_2} - }\\ {\left. {2184q\hat q_2^2 + 6600{q^2}\hat q_2^{ - 4} + \cdots } \right)} \end{array}\\ {W_3}\left( {t,{{\hat t}_3}} \right) = {\left( {t - 2{{\hat t}_3}} \right)^2} + \frac{1}{{4{\pi ^2}}}\left( {4416q + 6465456{q^2} + } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {24{{\hat q}_3} + 6\hat q_3^2 + \frac{8}{3}\hat q_3^3 + \frac{3}{2}\hat q_3^4 + }\\ {160q\hat q_3^{ - 2} - 1632q\hat q_3^{ - 1} + 3952q{{\hat q}_3} - }\\ {\left. {2184q\hat q_3^2 + 6600{q^2}\hat q_3^{ - 4} + \cdots } \right),} \end{array} \end{array} $

(24)

where q_i=exp(2πit_i), $\hat{q}$_i=exp(2πi$\hat{t}$_i), {i=0, 1, 2, 3}.

The disk invariants are shown in Table 1.

2.2 Partial coincident phase of three D-branes

The part coincident represents the coincidence of two of the three D branes. "$\tilde{v}$_c^*=(-1, 1, 1, 1, -1)" is selected as the compactifying point.

First, we ignore the interior point $\tilde{v}$₇^* on the one-dimensional edge with $\tilde{v}$₆^*, $\tilde{v}$₈^* and $\tilde{v}$₉^* to obtain the new charge vectors.

The generators of Mori cone determined by $\nabla $₅ are given:

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} \ \ \ 0&1&2&3&4&5&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 6}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 7}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 8}&c \end{array}\\ \ {l^1} = \left( {\begin{array}{*{20}{l}} { - 8}&0&2&2&2&2&{ - 2}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 2}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 0}&0 \end{array}} \right)\\ \ {l^2} = \left( {\begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}&0&0&0&0&0&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 1}&{\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt}{-3}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ \ 2}&0 \end{array}} \right).\\ \begin{array}{*{20}{l}} {{l^3} = \left( {\begin{array}{*{20}{l}} {-1}&1&0&0&0&0&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 1}&{ - 1}&0 \end{array}{\kern 1pt} } \right)}\\ {{l^4} = \left( {\begin{array}{*{20}{l}} { - 6}&0&0&0&0&2&{ - 1}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 3}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 0}&2 \end{array}} \right)} \end{array} \end{array} $

(25)

A suitable set of bases is selected to visualize the closed and open moduli.

$ t = {k_1} + 2{k_2} + 4{k_3},{\hat t_1} = {k_2} + {k_3},{\hat t_2} = {k_3}. $

(26)

The leading terms of the relative periods are:

$ \tilde \Pi _{2,1}^* = 3{t^2},\tilde \Pi _{2,2}^* = {\left( {t - 2{{\hat t}_1}} \right)^2},\tilde \Pi _{2,3}^* = \left( {t - 2{{\hat t}_2}} \right). $

(27)

Using algebraic coordinate:

$ {z_1} = \frac{{a_2^2a_3^2a_4^2a_5^2b_1^2}}{{a_0^8b_0^2}},{z_2} = \frac{{{b_0}b_2^2}}{{b_1^3}},{z_3} = \frac{{{a_1}{b_1}}}{{{a_0}{b_2}}}. $

(28)

Then the mixed inverse mirror maps in terms of q_i=exp(2πik_i) for {i=1, 2, 3} are:

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_1} = {q_1} - 2520q_1^2 - 2{q_1}{q_2} + 7560q_1^2{q_2} + {q_1}q_2^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 6300q_1^2q_2^2 + 6{q_1}{q_2}{q_3} + 22680q_1^2{q_2}{q_3} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 6{q_1}{q_2}q_3^2 - 20160q_1^2q_2^2{q_3} + \cdots .\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_2} = {q_2} + 1260{q_1}{q_2} + 13384350q_1^2{q_2} + 3q_2^2 - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2{q_2}{q_3} + 5040{q_1}q_2^2 - 34706700q_1^2q_2^2 - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2520{q_1}{q_2}{q_3} + \cdots .\\ {z_3} = {q_3} - 2{q_2}{q_3} - 1260{q_1}{q_2}{q_3} + 3150{q_1}q_2^2{q_3} + q_3^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {q_2}q_3^2 - 13384350q_1^2{q_2}{q_3} + 1260{q_1}{q_2}q_3^2 + \cdots . \end{array} $

(29)

D-brane superpotentials in the A-model as follows:

$ \begin{array}{l} \begin{array}{*{20}{c}} {{F_t}(t) = 3{t^2} + \frac{1}{{4{\pi ^2}}}(498991500q + \cdots ),}\\ {{W_1}\left( {t,{{\hat t}_1}} \right) = {{\left( {t - 2{{\hat t}_1}} \right)}^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}\left( {84453030q - 16{{\hat q}_1} - } \right.} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {20\hat q_1^2 + \frac{{368}}{9}\hat q_1^3 - 77\hat q_1^4 + 12600q\hat q_1^{ - 2} + }\\ {\left. {1212960q\hat q_1^{ - 1} + \cdots } \right),} \end{array}\\ {W_2}\left( {t,{{\hat t}_2}} \right) = {\left( {t - 2{{\hat t}_2}} \right)^2} + \frac{1}{{4{\pi ^2}}}\left( {1578722769q + 88{{\hat q}_2} + } \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 22\hat q_2^2 + \frac{{88}}{9}\hat q_2^3 + \frac{5}{2}\hat q_2^4 + 5671296q\hat q_2^{ - 2} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {89251824q\hat q_2^{ - 1} + \cdots } \right). \end{array} $

(30)

Second, we ignore the interior point $\tilde{v}$₈^* on the one dimensional edge with $\tilde{v}$₆^*, $\tilde{v}$₇^* and $\tilde{v}$₉^* to obtain the new charge vectors.

The generators of Mori cone determined by $\nabla $₅ are given:

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} \ \ \ 0&1&2&3&4&5&\ \ \ 6&\ \ \ 7&\ \ \ 8&c \end{array}\\ \begin{array}{*{20}{l}} {{l^1} = \left( {\begin{array}{*{20}{l}} { - 4}&0&1&1&1&1&{ - 2}&\ \ \ 2&\ \ \ 0&0 \end{array}} \right)}\\ {{l^2} = \left( {\begin{array}{*{20}{r}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 0}&0&0&0&0&0&\ \ \ 2&{ - 3}&\ \ \ 1&0 \end{array}} \right).} \end{array}\\ \begin{array}{*{20}{l}} {{l^3} = \left( {\begin{array}{*{20}{l}} { - 2}&2&0&0&0&0&\ \ \ 0&\ \ \ 1&{ - 1}&0 \end{array}} \right)}\\ {{l^4} = \left( {\begin{array}{*{20}{l}} { - 3}&0&0&0&0&1&{ - 2}&\ \ \ 3&\ \ \ 0&1 \end{array}} \right)} \end{array} \end{array} $

(31)

A suitable set of bases is selected to visualize the closed and open moduli.

$ t = {k_1} + {k_2} + {k_3},{\hat t_1} = {k_2} + {k_3},{\hat t_2} = {k_3}. $

(32)

The leading terms of the relative periods are:

$ \tilde \Pi _{2,1}^* = 3{t^2},\tilde \Pi _{2,2}^* = {\left( {t - {{\hat t}_1}} \right)^2},\tilde \Pi _{2,3}^* = \left( {t - {{\hat t}_2}} \right). $

(33)

Using algebraic coordinate:

$ {z_1} = \frac{{{a_2}{a_3}{a_4}{a_5}b_1^2}}{{a_0^4b_0^2}},{z_2} = \frac{{b_0^2{b_2}}}{{b_1^3}},{z_3} = \frac{{a_1^2{b_1}}}{{a_0^2{b_2}}}. $

(34)

Then the mixed inverse mirror maps in terms of q_i=exp(2πik_i) for {i=1, 2, 3} are:

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_1} = {q_1} - 24q_1^2 - 2{q_1}{q_2} + {q_1}q_2^2 - 24q_1^2q_2^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 3{q_1}{q_2}{q_3} - 2112q_1^2{q_2}{q_3} - 3{q_1}q_2^2{q_3} + \cdots .\\ {z_2} = {q_2} + 24{q_1}{q_2} + 972q_1^2{q_2} + 3q_2^2 + 168{q_1}q_2^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 8136q_1^2q_2^2 - {q_2}{q_3} - 24{q_1}{q_2}{q_3} + \cdots .\\ {\kern 1pt} {\kern 1pt} {z_3} = {q_3} - {q_2}{q_3} - 48{q_1}{q_2}{q_3} - 972q_1^2{q_2}{q_3} + q_3^2 + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 12{q_1}q_2^2{q_3} + 1260q_1^2q_2^2{q_3} - 720{q_1}{q_2}q_3^2 + \cdots . \end{array} $

(35)

D-brane superpotentials in the A-model as follows:

$ \begin{array}{l} \begin{array}{*{20}{c}} {{F_t}(t) = 3{t^2} + \frac{1}{{4{\pi ^2}}}\left( {15768q + 24117750{q^2} + \cdots } \right),}\\ {{W_1}\left( {t,{{\hat t}_1}} \right) = {{\left( {t - 2{{\hat t}_1}} \right)}^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}(4832q + } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {6267312{q^2} - 14{{\hat q}_1} + \frac{{25}}{6}\hat q_1^2 - }\\ {\left. {\frac{{94}}{{45}}\hat q_1^3 + \frac{{361}}{{280}}\hat q_1^4 + 160q\hat q_1^{ - 1} + \cdots } \right),} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {W_2}\left( {t,{{\hat t}_2}} \right) = {\left( {t - 2{{\hat t}_2}} \right)^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}(4964q + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {6215169{q^2} + 10{{\hat q}_2} + \frac{{17}}{6}\hat q_2^2 + }\\ {\left. {\frac{{62}}{{45}}\hat q_2^3 + \frac{{233}}{{280}}\hat q_2^4 - 8q\hat q_2^{ - 1} + \cdots } \right).} \end{array} \end{array} $

(36)

2.3 Complete coincident phase of three D-brane

The complete coincident means that the three D-branes coincide. We ignore the interior point $\tilde{v}$₇^* and $\tilde{v}$₈^*, and select $\tilde{v}$_c^*=(-1, 1, 1, 1, -1) as the compactifying point.

The generators of Mori cone determined by $\nabla $₅ are given:

$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} \ \ \ 0&1&2&3&4&5&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 6}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 7}&c \end{array}\\ {l^1} = \left( {\begin{array}{*{20}{l}} { - 12}&0&3&3&3&3&{ - 2}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \ 2}&0 \end{array}} \right)\\ {l^2} = \left( {\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} - 3}&3&0&0&0&0&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1}&{ - 1}&0 \end{array}} \right)\\ {l^3} = \left( {\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} - 3}&0&0&0&0&1&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0}&{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1}&1 \end{array}} \right) \end{array} $

(37)

A suitable set of bases is selected to visualize the closed and open moduli.

$ t = {k_1} + 2{k_2},\hat t = {k_2}. $

(38)

The leading terms of the relative periods are

$ \tilde \Pi _{2,1}^* = 3{t^2},\tilde \Pi _{2,2}^* = {(t - 2\hat t)^2}. $

(39)

Using algebraic coordinate:

$ {z_1} = \frac{{a_2^3a_3^3a_4^3a_5^3b_1^2}}{{a_0^{12}b_0^2}},{z_2} = \frac{{a_1^3{b_0}}}{{a_0^3{b_1}}}. $

(40)

Then the mixed inverse mirror maps in terms of q_i=exp(2πik_i) for {i=1, 2} are

$ \begin{array}{l} {z_1} = {q_1} - 369600q_1^2 - 2{q_1}{q_2} + 337444800q_1^2{q_2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {q_1}q_2^2 - 137496092160q_1^2q_2^2 + \cdots ,\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_2} = {q_2} + 184800{q_1}{q_2} + 525959313264q_1^2{q_2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} q_2^2 - 168168000{q_1}q_2^2 + \cdots . \end{array} $

(41)

D-brane superpotentials in the A-model as follows:

$ \begin{array}{l} \begin{array}{*{20}{c}} {{F_t}(t) = 3{t^2} + \frac{1}{{4{\pi ^2}}}(4465834118784q + \cdots ),}\\ {{W_1}(t,\hat t) = {{(t - 2\hat t)}^2} + \frac{1}{{4{{\rm{ \mathsf{ π} }}^2}}}(1488611372928q + } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {96\hat q + \frac{{156}}{5}{{\hat q}^2} + \frac{{1738}}{{105}}{{\hat q}^3} + \frac{{4101}}{{385}}{{\hat q}^4} - }\\ {\left. {106456166409q{{\hat q}^{ - 1}} + \cdots } \right).} \end{array} \end{array} $

(42)

The disk invariants are shown in Table 2.

3 Summary

The superpotential and Ooguri-Vafa invariants of D-brane system with three D-branes on hypersurface P(1, 1, 1, 1, 2), i.e., Sextic are calculated. Different from previous works on the systems with single and two D-branes, the system with three D-branes complicates the research.

The results show that there are three phases: parallel phase, partial coincident phase and complete coincident phase. The parallel phase of the D-brane system corresponds to the Coulomb branch of gauge theory, and the superpotential contributed by one of the three D-branes is identical to the one contributed by the D-brane system with single brane on the same Calabi-Yau manifold. The partial coincident phase of the D-brane system corresponds to the Coulomb-Higgs branch. The D-brane system changes from parallel D-brane phase to partial coincident D-brane phase, and that corresponds to the transition from the U(1)×U(1)×U(1) to U(1)×U(2) in terms of gauge theory. The complete coincident phase of the D-brane system corresponds to the Higgs branch. The D-brane system changes from parallel D-brane phase to complete coincident D-brane phase, and that shows the feature of the phase transition from U(1)×U(1)×U(1) to U(3) in terms of gauge theory.

Therefore, in the low effective theory of the system with three D-branes on compact Calabi-Yau manifolds, the superpotentials have rich phase structures. On the other hand, these superpotentials have potential phenomenological applications. We also try to calculate the D-brane superpotential with the method of A_∞ structure and derived category approach and path algebras of quivers.