@article {
author = {Liu, C.C. Melissa},
title = {Moduli of $J$-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an $S^1$-equivariant pair},
journal = {Journal of the Iranian Mathematical Society},
volume = {1},
number = {1},
pages = {5-95},
year = {2020},
publisher = {Iranian Mathematical Society},
issn = {2717-1612},
eissn = {2717-1612},
doi = {10.30504/jims.2020.104185},
abstract = {Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^\infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$-action on the pair $(X,L)$ which preserves $J$ and has no fixed points on $L$, we define the Euler number for this $S^1$-equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given $S^1$-action.},
keywords = {Moduli of $J$-Holomorphic Curves,Lagrangian boundary conditions,Open Gromov-Witten Invariants},
url = {https://jims.ims.ir/article_104185.html},
eprint = {https://jims.ims.ir/article_104185_c86f22dd64c9c7ae78d34f83e9d829f4.pdf}
}