Superpotentials and Geometric Invariants of Parallel/Complete Coincident/Part Coincident D-brane System on Compact Calabi-Yau Manifold

TL;DR

This paper investigates the superpotentials and geometric invariants of various D-brane configurations on compact Calabi-Yau threefolds using duality and mirror symmetry, revealing phase transitions and decoupling phenomena.

Contribution

It constructs dual F-theory fourfolds for different D-brane configurations and computes superpotentials and invariants, highlighting differences and phase transitions.

Findings

Superpotentials for parallel D-branes show decoupling behavior.

Differences in superpotentials indicate phase transitions.

Evidence of gauge symmetry enhancement and geometric singularities.

Abstract

For D-brane system with three D-branes on compact Calabi-Yau threefolds, the dual F-theory fourfolds for parallel/complete coincident/part coincident D-brane system is constructed by the type II/F-theory duality. Complete coincident means that the three D-branes coincide and part coincident represents the coincident of two of the three D branes. The low energy effective superpotentials are calculated by mirror symmetry, GKZ-system method and the type II/F-theory duality on the B-model side, respectively. Using the mirror symmetry, A-model superpotentials and the Ooguri-Vafa invariants are obtained from the B-model side. These results indicate that the superpotential contributed by one of three parallel D-branes is identical with the D-brane system with only one D-brane, which is a signal of decoupling of the parallel topological D-branes. However, the superpotential and Oogrui-Vafa invariants are different among parallel, complete coincident and part coincident D-brane system, which show the evidence of the phase transition due to the enhanced gauge symmetry in the low energy theories and the geometrical singularity.

Equations

$${\mathcal{W}}_{\mathcal{N}=1}(z,\hat{z})={\Pi }_{\gamma }(z,\hat{z})={\int }_{\gamma }{\Omega }^{(3,0)}(z,\hat{z}),\gamma \in H_3(M_3,\mathcal{D})$$

$$\begin{array}{rl}{\mathcal{W}}_{\mathcal{N}=1}(z,\hat{z})& =\sum N_{\alpha }{\Pi }_{\alpha }(z,\hat{z})\\ & ={\mathcal{W}}_{open}(z,\hat{z})+{\mathcal{W}}_{closed}(z)\\ & ={\mathcal{W}}_{brane}(z,\hat{z})+{\mathcal{W}}_{flux}(z),\\ {\Pi }_{\alpha }(z,\hat{z})& ={\int }_{\alpha }{\Omega }^{(3,0)}(z,\hat{z}).\end{array}$$

$$\begin{array}{rl}{\mathcal{W}}_{\mathcal{N}=1}=& {\mathcal{W}}_{classical}(t,\hat{t})+{\mathcal{W}}_{instanton}(q,\hat{q})\\ =& (\frac{1}{2}{\mathcal{K}}_{jk}{t}^j{\hat{t}}^k+b)+\sum _{k,m}{G}_{k,m}{q}^k{\hat{q}}^m\\ =& (\frac{1}{2}{\mathcal{K}}_{jk}{t}^j{\hat{t}}^k+b)+\sum _n\sum _{k,m}\frac{N_{k,m}}{n^2{q}^{nk}{\hat{q}}^{nm}},\end{array}$$

$$\begin{array}{rl}{\mathcal{W}}_{GVW}(M_4)=& {\int }_{M_4}{G}_4\wedge {\Omega }^{(4,0)}\\ =& \sum _{\Sigma }{N}_{\Sigma }(G_4){\Pi }_{\Sigma }(z,\hat{z})+\mathcal{O}(g_s)+\mathcal{O}(e^{-1/g_s}),\end{array}$$

Full text

3.65.Vf; 02.40.Tt; 04.65.+e; 11.25.Uv

1 Introduction

In the $N=2$ supersymmetric theories, the closed-string mirror symmetry gives an equivalence between A-model parameterized by Kähler moduli in the terms of the quantum geometry and B-model parameterized by complex structure moduli in terms of the classical geometry. Mirror symmetry provides many techniques for the variation of physical structures over their moduli spaces[CQ.14, Ba.18], and in the closed-string sector[LVW.89, BCOV.94] has a relatively perfect solution in the early works. With the appearance of D-branes, we take more attention to the open-string sector in recent years[SU.15]. The supersymmetry breaks into $N=1$ when D-brane is included and that leads the application of open-closed mirror symmetry[MK.94, MP.01, JW.06], e.g., the quantum corrected domain wall tensions on the Calabi-Yau threefolds can be calculated with open-closed mirror for compact Calabi-Yau manifolds.

The D-brane superpotential is a section of a special holomorphic line bundles of the moduli space from the mathematical perspective. It is defined as the F-term of low-energy effective theory and determines the string vacuum structure from the physical perspective. The superpotential is the generating function of all disk instantons from the worldsheet point of view, and it can be calculated by relative period. The D-branes on A-model, which are called A-branes, wrap on the special Lgarangian submanifolds of Calabi-Yau manifold $W_{3}$. Correspondingly, B-branes wrap on holomorphic submanifolds of Calabi-Yau manifolds $M_{3}$. The expression of instanton expansion of superpotential on the A-model side encodes the number of BPS states which corresponds to the Ooguri-Vafa invariants mathematically[AV.00]. These invariants give an important mathematical language which has not been studied systematically in theory so far. It is hard to calculate in the A-model because of the non-perturbed quantum corrections. Nevertheless, we can figure it out in B-model, and then mirror it to A-model. For non-compact Calabi-Yau manifolds, several methods that can be used to calculate the D-brane effective superpotential are localization[AV.00, AKV.02, A.05], topological vertex and direct integration related to $N=1$ special geometry[LMW.02.1, LMW.02.2]. Further, for compact Calabi-Yau manifolds, some techniques have been evolved, e.g., mixed Hodge structure variation, Gauss-Manin connection[JS.08, JS.09], the blow-up method[GHKK.09] and the GKZ-generalized hypergeometric system for open-closed sectors. This paper we focus on is to calculate D-brane superpotentials for compact Calabi-Yau manifolds with open-closed mirror symmetry and generalized GZK system[JXT.17, XFJ.14.01, XFJ.14.02, CS.14, ZSS.15, ZH.16, AHMM.09, LS.09].

The dual description of the superpotential in Type-II string theory can be found in F-theory. To be precise, the D-brane superpotentials in the string theory of Type-II is dual to the background flow in F-theory[AHJMMS.09, JMW.09]. This duality provides us with a method to calculate the D-brane superpotentials in Type-II string theory. Some works which are related to the calculation of the superpotentials near the limit point and the invariants for the system with two D-branes have been done[JXT.17]. And it is mentioned that the parallel and the coincident D-branes phases correspond to the Coulomb branch and the Higgs branch of the non-Abelian gauge theory on the worldvolume of D-brane system respectively. Motivated and guided by the work, we calculate and compare the system with three D-branes. The system with three branes are more generalized than two branes, and it complicates the research. The coincident D-branes phases are divided into complete coincident phase and part coincident phase. Complete coincident means that the three D-branes coincide and part coincident represents the coincident of two of the three D branes.

The organization is as follow. In section $2$, we review the background related to physics and mathematics. This section gives an overview of toric brane geometry and GKZ system. In section $3$, we concentrate on two models, D-brane system on the mirror quintic and on hypersurface $X_{8}(1,1,2,2,2)$. The superpotentials of parallel, complete coincident and part coincident D-brane phase are calculated and discussed from each model. The Ooguri-Vafa invariants are extracted in different D-branes models as well. First, we calculate superpotentials and invariants of the parallel D-brane phase (Coulomb phase) with three parallel branes. Second, we calculate the superpotentials and invariants of complete coincident D-brane phase (Higgs phase) in which three parallel D-branes coincide. Third, we calculate the superpotential of part coincident D-brane phase (Coulomb phase-Higgs phase) in which two of the three parallel D-branes coincide. The last section is a brief summary.

2 Toric brane geometry and GKZ system

2.1 Mirror symmetry

Mirror symmetry is the duality between Type II A and Type II B topological string theory. The corresponding D-branes are A-brane and B-brane, respectively. The supersymmetry breaks from $N=2$ to $N=1$ when D-branes are included. Sometimes we also call $N=1$ mirror symmetry as open-closed mirror symmetry since this involves the open and closed part.

Closed mirror symmetry is more fundamental than open-closed mirror symmetry. Taking the brane to infinity and we just suppose there is an infinity, then the open-closed mirror symmetry degenerates to the closed mirror symmetry. It is necessary to note is that there are two ways to explain the open-closed. One way to understand is that an open string falls on the brane and another one on the anti-brane (the brane and anti-brane can be regards as two branes that carry opposite charges), then two discs was glued to be a sphere, which is closed string. Another way to understand is that closed string does not needs brane since brane in string theory is just a boundary condition. Closed string is a Rieman surface without boundary, which can be projected into Calabi-Yau manifold, while open string is Rieman surface with boundary.

The A-brane is wrapped on the special Lagrangian submanifold $L$ of the Calabi-Yau manifold $W$, while the B-brane is wrapped on the holomorphic submanifold $\mathcal{D}$ of the Calabi-Yau manifold $M$. Mirrored symmetry with D-brane has a more precise description in mathematics: The equivalence between the derived category of coherent sheaves on a Calabi-Yau manifold and the Fukaya category of its mirror. $N=1$ Mirror symmetry contacted two completely different D-brane geometries $(W,L)$ and $(M,D)$.

2.2 The effective superpotentials in Type-II string theory and F-theory

On the B-model side, the space-filling D5-branes wrap on an reducible curve $C=\sum_{i}\mathcal{C}_{i}$ and C embedded in a divisor $\mathcal{D}$ of Calabi-Yau 3-fold $M_{3}$. The effective superpotential is:

[TABLE]

The superpotential can be written as a linear combination of relative period[LMW.02.1, LMW.02.2]:

[TABLE]

On the A-model side, a general form of D-brane superpotential is:

[TABLE]

where $t^{j}$ and $\hat{t}^{k}$ are the closed and open Kähler moduli respectively, $q=exp(2\pi it)$, $\hat{q}=exp(2\pi i\hat{t})$. $\{G_{k,m}\}$ are the open Gromov-Witten invariants labeled by relative homology class. $\vec{m}$ represent the elements of $H_{1}(L)$, $\vec{k}$ represent the elements of $H_{2}(W_{3})$, and $\mathcal{K}_{jk}$ is the combination coefficient. $\{N_{\vec{k},\vec{m}}\}$ are the Ooguri-Vafa invariants.

There is a duality between the type II string theory with D-brane systems on complex Calabi-Yau threefold $M_{3}$ and the F-theory compactified on the Calabi-Yau fourfold $M_{4}$ without any branes but with fluxes.

The superpotential of 4-form flux $G_{4}$ in F-theory compactified on the Calabi-Yau 4-fold $M_{4}$ is a section of the Hodge line bundle in the complex structure moduli space $\mathcal{M}_{CS}(M_{4})$. This superpotnetial is called Gukov-Vafa-Witten superpotential[GVW.99, AHJMMS.09]:

[TABLE]

The leading term of above equation on the right-hand side is the D-brane superpotential $\mathcal{W}_{N=1}$ and $g_{s}$ is the string coupling strength.

There is a duality between the periods of holomorphic $(4,0)$ form on the non-compact 4-fold $\tilde{M}_{4}$ and the relative periods of the brane geometry ($M_{3}$, $\mathcal{D}$). From mirror symmetry, one obtains the relation between the different compactifications:

[TABLE]