Gromov-Witten Theory of Deligne-Mumford Stacks

Gromov-Witten理论涉及从节点Riemann曲面到目标空间的稳定映射的模空间上的积分。这是一门发展迅速的学科，与数学和弦论的许多领域都有联系。在过去的几十年里，许多工作已经揭示了orbillold/Stack在几何和拓扑学中的关键作用，例如研究群作用和模问题。这两个主题很长一段时间都没有被考虑在一起，直到最近Gromov-Witten理论才被扩展到堆叠/或双目标空间。国际和平研究所建议研究格罗莫夫-威腾的奥比诺德/堆栈理论的几个基本方面。PI将显式计算环形目标的奥比霍尔德Gromov-Witten不变量，并研究Gromov-Witten不变量的生成函数的结构。我们还将研究Gromov-Witten理论与双曲几何之间的关系，特别是所谓的Crepant分解猜想。本文还将研究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。Gromov-Witten理论的主题位于数学和弦理论的几个领域的边界上。本文的研究将进一步加深人们对这些领域的认识，特别是关于orbilold/Stack的镜像对称性、Deligne-Mumford栈的计数几何、可积系、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.