Topology of Orbifolds, Vertex Algebras and Quantum Cohomology

该项目的第一部分是关于orborold的拓扑不变量和弦理论不变量。在物理预测的基础上，Chen和Ruan提出了一种新的上同调理论。这里的研究对象是上同调的代数结构及其与其他上同调理论的关系。手征de Rham复形的应用和推广，是本项目第二部分的重点。手征de Rham复形是与流形相关的顶点代数的非锥束。特别地，我们将研究与一般奥氏同调有关的顶点代数和模，以及它们与奥氏上同调、椭圆亏格和量子上同调的关系。该项目第三部分的目的是研究与自旋型上同调场理论相关的几何和代数结构，以及相应的量子上同调和Gromov-Witten不变量的自旋类似。这一部分涉及E.Witten的一个猜想，即具有r-自旋结构的Riemann曲面的模空间上的交数的母函数是r-KdV族的Atau函数。来自量子场论物理模型的想法是数学中任何令人兴奋的最新发展的原因，特别是在拓扑学和几何学方面。这个项目研究了受弦理论启发的几个几何问题，弦理论是目前统一各种物理力的理论的候选者。由于缺乏实验工具，从事弦理论工作的物理学家正在复杂的数学模型上测试他们的方法，他们的结果往往表明，各种看似遥远的数学对象之间存在猜想联系。当前项目的目标是对其中一些猜想和相互作用进行数学研究。除了产生新的数学结果外，它还可能导致一些物理模型的数学基础，并证明分析它们的物理方法是正确的。

The first part of the project is concerned with topological and string-theoreticinvariants of orbifolds. Based on physical predictions, Chen and Ruan introduceda new cohomology theory for orbifolds. The object of study here will be algebraicstructures on the orbifold cohomology and relations with other cohomology theories. Applications and generalizations of the chiral de Rham complex, acanonical sheaf of vertex algebras associated to a manifold, are in the focus ofthe second part of the project. In particular, vertex algebras and modulesrelated to general orbifolds and their relation with orbifold cohomology,elliptic genus and quantum cohomology will be investigated. The aim of the thirdpart of the project is to study geometric and algebraic structures related tocohomological field theories of spin type and the corresponding spin analogs ofquantum cohomology and Gromov-Witten invariants. This part is related to aconjecture by E.Witten stating that the generating function of intersectionnumbers on the moduli spaces of Riemann surfaces with r-spin structures is atau-function of the r-KdV hierarchy.Ideas coming from physical models of quantum field theory are responsible formany exciting recent developments in mathematics, in particular in topology andgeometry. This project studies several geometric problems inspired by stringtheory, the current candidate for the theory unifying all kinds of physicalforces. Because of lack of experimental tools, physicists working in stringtheory are testing their methods on sophisticated mathematical models, and theirresults often indicate the existence of conjectural ties between variousseemingly remote mathematical objects. The goal of the current project is amathematical study of some of these conjectures and interactions. Besidesproducing new mathematical results, it may lead to a mathematical foundation forsome physical models and a justification of physical ways of analyzing them.