arXiv subjects
Publications and source records attributed to Fu-Zhong Yang.
In this paper, we study the open topological string amplitudes on Calabi-Yau threefolds by the extended holomorphic anomaly equation. The disk two-point function determined by the domainwall tension, together with the Yukawa couplings, solves the amplitudes with high genus and boundaries recursively. The BPS invariants encoded in the amplitudes are extracted by mirror symmetry.
Open topological string partition function on compact Calabi-Yau threefolds satisfies the extended holomorphic anomaly equation. By direct integration, we solve these equations and obtain partition functions for first several genus and boundaries on complete intersection Calabi-Yau threefolds. Complemented by the unoriented worldsheet contribution, the annulus functions encode the genus one BPS invariants.
By blowing up the ambient space along the curve wrapped by B-branes, we study the brane superpotentials and Ooguri-Vafa invariants on complete intersections Calabi-Yau threefolds. On the topological B-model side, B-brane superpotentials are expressed in terms of the period integral of the blow-up manifolds. By mirror maps, the superpotentials are generating functions of Ooguri-Vafa invariants counting holomorphic disks on the topological A-model side.
The number of rational points in toric data are given for two-parameter Calabi-Yau $n$-folds as toric hypersurfaces over finite fields $\mathbb F_p$ . We find that the fundamental period is equal to the number of rational points of the Calabi-Yau $n$-folds in zeroth order $p$-adic expansion. By analyzing the solution set of the GKZ-system given by the enhanced polyhedron, we deduce that under type II/F-theory duality the 3D and 4D Calabi-Yau manifolds have the same number of rational points in zeroth order. Taking the quintic and its duality as an example, the number of rational points in some specific complex moduli are given by numerical calculation to support our results.
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.
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.
The phase transitions are studied for the D-brane systems with multiple open-string moduli in terms of toric geometry: between the parallel D-brane phase corresponding to the Coulomb branch and the coincident phase corresponding to the Higgs branch. The two separated D-branes on compact Calabi-Yau 3-fold coincide developing the geometric singularity in the corresponding F-theory Calabi-Yau 4-fold in terms of TypeII/F theory duality, and the enhancement of gauge group $U(1)\times U(1)\rightarrow SU(2)$ in terms of gauge theory. For several D-brane system with various closed-string moduli, using the mirror symmetry and the typeII/F theory duality the A-model superpotentials are obtained from the B-model side for the two phases, and the Ooguri-Vafa invariants are extracted from the A-model superpotential. We find the discrete $Z_2$ symmetry of superpotentials in the two parallel D-branes phase of all the models which is a signal of decoupling of the parallel topological D-branes. Furthermore, the \emph{$U(1)$ Ooguri-Vafa invariants} for one of the two parallel D-branes are the same as the invariants for the D-brane system with only one D-brane. However they are different from the \emph{$SU(2)$ Ooguri-Vafa invariants} corresponding to the two D-branes coincident phase. We present these Ooguri-Vafa invariants for two phases with figures to observe the difference, and find that in two phases the points for invariants form a wave-packet respectively. The wave-packets for coincident phase are higher and wider than ones for parallel phase which means more complicate spectrum structure. This is an evidence of the phase transition between the Coulomb branch and the Higgs branch.
In this paper, we extend the GKZ-system method to the more general case: compact Complete Intersection Calabi-Yau manifolds (CICY). For several one-deformation modulus compact CICYs with D-branes, the on-shell superpotentials in this paper from the extended GKZ-system method are exactly consistent with published results obtained from other methods. We further compute the off-shell superpotentials of these models. Then we obtain both the on-shell and off-shell superpotentials for sev- eral two-deformation moduli compact CICYs with D-branes by using the extended GKZ-system method. The discrete symmetrical groups, Z2, Z3 and Z4, of the holo- morphic curves wrapped by D-branes play the important roles in computing the superpotentials, in some sense, they are the quantum symmetries of these models. Furthermore, through the mirror symmetry, the Ooguri-Vafa invariants are extracted from the A-model instanton expansion.
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.
We calculate the D-brane superpotentials for two non-Fermat type compact Calabi-Yau manifolds which are the hypersurfaces of the weighed projective spaces in type II string theory. By constructing the open-closed mirror maps, we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.
In this paper, we make a further step of \cite{Feng} and calculate off-shell superpotential of two Calabi-Yau manifold with three parameters by integrating the period of subsystem. We also obtain the Ooguri-Vafa invariants with open mirror symmetry.
According to the generalized GKZ system for CY manifold with brane and the open-closed Mirror map, We calculate the off-shell D-Brane/F-Theory effective superpotentials and Ooguri-Vafa invariants of several compact Calabi-Yau manifolds .
We calculate the D-brane superpotentials for two Calabi-Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials $\mathcal{W}_{GVW}$ of the dual F-theory compactification on the relevant Calabi-Yau fourfolds in the weak decoupling limit. We also compute the Ooguri-Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov-Witten invariants.
The geometrical description of BPS 3-string junction in the F-theory background is given by lifting a string junction in IIB into F-theory and constructing a holomorphic curve in K3 with respect to a special complex structure of K3. The holomorphic curve is fibration of 1-cycles of the elliptic fiber over the geodesic string junction. The F-theory picture in this paper provides an unifying description of both string and string junction, and is advantageous over the M-theory picture of them.