Gromov-Witten Theory of Deligne-Mumford Stacks

Gromov-Witten理论涉及从节点黎曼曲面到目标空间的稳定映射的模空间上的积分。这是一个发展迅速的学科，与数学和弦理论的许多领域都有联系。在过去几十年的许多工作中，已经揭示了轨道折叠/堆叠在几何和拓扑学中的关键作用，例如群作用和模量问题的研究。这两个主题在很长一段时间内都没有被放在一起考虑，直到最近Gromov-Witten理论被扩展到堆栈/orbifold目标空间。PI建议研究Gromov-Witten orbifolds/stacks理论的几个基本方面。PI将追求对复曲面目标的轨道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。Gromov-Witten理论与双有理几何的关系，特别是所谓的crepant归结猜想，也将被研究。Gromov-Witten理论的orbifolds/stacks到曲线的模空间的应用，特别是Hurwitz-Hodge积分的性质和计算，也将被研究。Gromov-Witten理论的主题在于数学和弦理论的几个领域的边界。拟议的研究将进一步推进有关这些领域的知识，特别是镜像对称的orbifolds/堆栈，枚举几何的Deligne-Mumford堆栈，可积系统，几何的模空间的曲线。这也将进一步促进代数几何、辛几何、组合数学、数学物理和弦理论之间的相互作用。

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.