Orbifold Gromov-Witten理论及相关问题研究

Gromov-Witten理论是近三十年来几何物理中最重要的理论之一。它的物理背景超弦理论为该理论提供了很多重要现象，如镜像对称猜想、Virasoro猜想等。在这些猜想的推动下，辛流形上的Gromov-Witten理论及其相关领域得到飞速发展。辛orbifold Gromov-Witten理论由Chen-Ruan率先建立并引起广泛的研究，但相比于辛流形的研究，发展明显缓慢，包括许多基础性的问题尚未解决。其难度主要源于没有足够有效的方式控制orbifold。人们逐渐意识到groupoid是一个描述orbifold的有效语言。本项目的核心目标即是将orbifold Gromov-Witten理论放入groupoid的框架中。其中包括在groupoid框架下orbifold的结构、稳定映射模空间的光滑性、（相对）orbifold Gromov-Witten理论与辛双有理几何等问题。

Gromov-Witten theory is one of the most important theories in geometry and physics over the last thirty years. Its physical background, superstring theory, provides many important phenomena for this theory, like Mirror Symmetry Conjecture, Virasoro Conjecture, etc. Driven by these conjectures, Gromov-Witten theory of symplectic manifolds and its related fields have developed rapidly. The Gromov-Witten theory for symplectic orbifolds was first constructed by Chen-Ruan, and caused extensive research. However, unlike the research on Gromov-Witten theory of symplectic manifolds, orbifold Gromov-Witten theory develops slowly, and there are still many basic problems have not been solved. The difficulty is mainly due to the lack of an effective way to control orbifolds. People gradually realize that the language of groupoids is effective to describe orbifolds. The core goal of this project is to put the orbifold Gromov-Witten theory into the framework of groupoids. These include the structure of orbifolds in the framework of groupoids, the smoothness of the moduli spaces of stable maps, (relative) orbifold Gromov-Witten theory and symplectic birational geometry, etc.