Gromov-Witten Theory of Deligne-Mumford Stacks

Gromov-Witten理论研究从节点黎曼曲面到目标空间的稳定映射在模空间上的积分。这是一门发展迅速的学科，与数学和弦理论的许多领域都有联系。在过去的几十年里，许多研究都揭示了轨道/堆叠在几何和拓扑中的关键作用，例如群作用和模问题的研究。直到最近Gromov-Witten理论被推广到堆叠/轨道目标空间之后，这两个主题才被考虑在一起。PI提出研究轨道/堆的Gromov-Witten理论的几个基本方面。PI将追求轨道目标的轨道Gromov-Witten不变量的显式计算，并研究Gromov-Witten不变量的生成函数的结构。格罗莫夫-威滕理论与双几何的关系，特别是所谓的蠕变分辨猜想，也将被研究。本文还将研究轨道堆的Gromov-Witten理论在曲线模空间中的应用，特别是Hurwitz-Hodge积分的性质和计算。格罗莫夫-维滕理论的主题处于数学和弦理论几个领域的边界上。本文所提出的研究将进一步推进这些领域的知识，特别是轨道/堆栈的镜像对称，Deligne-Mumford堆栈的枚举几何，可积系统，曲线模空间的几何。这也将进一步促进代数几何、辛几何、组合学、数学物理和弦理论之间现有的相互作用。

Gromov-Witten theory concerns integrals over moduli spaces of stable maps from nodal Riemann surfaces to a target space. It had been a rapidly developing subject, with connections to many areas of mathematics and string theory. The crucial role of orbifolds/stacks in geometry and topology, for example the study of group actions and moduli problems, had been revealed in many works over the past decades. The two subjects were not considered together for a long time, until only very recently after Gromov-Witten theory was extended to stack/orbifold target spaces. The PI proposes to study several fundamental aspects of Gromov-Witten theory of orbifolds/stacks. The PI will pursue explicit calculations of orbifold Gromov-Witten invariants of toric targets and study structures of generating functions of Gromov-Witten invariants. Relations between Gromov-Witten theory and birational geometry, especially the so-called crepant resolution conjecture, will also be studied. Applications of Gromov-Witten theory of orbifolds/stacks to moduli spaces of curves, in particular properties and calculations of Hurwitz-Hodge integrals, will also be studied.The subject of Gromov-Witten theory lies on the boundary of several fields of mathematics and string theory. The proposed research will further advance the knowledge about these fields, in particular mirror symmetry for orbifolds/stacks, enumerative geometry of Deligne-Mumford stacks, integrable systems, the geometry of moduli spaces of curves. This will also further promote the existing interactions between algebraic geometry, symplectic geometry, combinatorics, mathematical physics, and string theory.