No Arabic abstract
In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we will also prove their conjectures in low genus cases.
We propose Picard-Fuchs equations for periods of nonabelian mirrors in this paper. The number of parameters in our Picard-Fuchs equations is the rank of the gauge group of the nonabelian GLSM, which is eventually reduced to the actual number of K{a}hler parameters. These Picard-Fuchs equations are concise and novel. We justify our proposal by reproducing existing mathematical results, namely Picard-Fuchs equations of Grassmannians and Calabi-Yau manifolds as complete intersections in Grassmannians. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds, including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi-Yau threefold and Gulliksen-Neg{aa}rd Calabi-Yau threefold, and find agreement with existing results in the literature. The generating functions we propose for non-complete intersection Calabi-Yau manifolds are genuinely new.
In this paper, we study holomorphic discs in K3 surfaces and defined the open Gromov-Witten invariants. Using this new invariant, we can establish a version of correspondence between tropical discs and holomorphic discs with non-trivial invariants. We give an example of wall-crossing phenomenon of the invariant and expect it satisfies Kontsevich-Soibelman wall-crossing formula.
In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. This exten
In the paper, we study the wall-crossing phenomenon of reduced open Gromov-Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open Gromov-Witten invariants.
We give a pedagogical review of the computation of Gromov-Witten invariants via localization in 2D gauged linear sigma models. We explain the relationship between the two-sphere partition function of the theory and the Kahler potential on the conformal manifold. We show how the Kahler potential can be assembled from classical, perturbative, and non-perturbative contributions, and explain how the non-perturbative contributions are related to the Gromov-Witten invariants of the corresponding Calabi-Yau manifold. We then explain how localization enables efficient calculation of the two-sphere partition function and, ultimately, the Gromov-Witten invariants themselves.