Moduli of Spin Curves, Geometric Vertex Algebras and Quantum Cohomology

摘要DMS -0104397.研究者将研究与拓扑弦理论和二维引力的物理模型有关的两组问题，以及它们与几何和其他数学分支的相互作用。基于物理学的思想和方法，E. Witten指出，具有高自旋结构的Riemann曲面的模空间的某些拓扑不变量集合在一个生成函数中，该生成函数可由一个无穷大的微分方程组确定（所谓的Gelfand-Dickey层次）。该提案的第一部分的目标是通过使用Gromov理论中的技术来研究与此猜想相关的几何和代数结构-维滕不变量与量子上同调。除了揭示几何学和分析学之间的紧密联系之外，该项目这一部分的目标是开发工具来分析更一般的自旋类型的上同调场论和相应的量子上同调和Gromov-Witten不变量的自旋类似物。 第二部分的建议是有关的应用和推广的theso-called手征的德拉姆复杂，一个典型的层顶点代数介绍了联合工作与Malikov和Schechtman。本文的主要目标是为二维超共形场论的光滑簇和轨道折叠的物理模型找到严格的几何基础。弦论是一种试图统一各种力的物理理论，特别是将爱因斯坦的引力理论与量子理论结合起来。因为这个理论还不能在实验上得到证实，从事弦理论研究的物理学家们正在用复杂的数学模型来检验它。通常同一个物理量可以用一种以上的方法来计算，而这些方法给出的答案用非常不同的几何结构来表示。这表明了数学中各种看似遥远的部分之间存在着深刻的联系，并导致了连接不同数学对象（通常是几何和分析性质）的非平凡结构的形成。当前项目的目标是对其中一些结构和相互作用进行数学研究。它除了产生新的数学结果外，还应对某些物理模型进行数学验证，并使分析这些模型的数学方法和物理方法之间有更直接的联系。

Abstract for DMS - 0104397.The investigator will study two groups of problems related tophysical models of topological string theory and two-dimensionalgravity and their interaction with geometry and other branches ofmathematics. Based on physical ideas and methods, E.Wittenconjectured that certain topological invariants of the moduli spacesof Riemann surfaces with higher spin structures assemble in agenerating function that can be determined by an infinite system ofdifferential equations (the so-called Gelfand-Dickey hierarchy).The goal of the first part of the proposal is to study geometric andalgebraic structures related to this conjecture by using techniquesfrom the theory of Gromov-Witten invariants and quantumcohomology. Besides exposing deep connections between seeminglyunrelated areas of geometry and analysis, a goal of this part of theproject is to develop tools for analyzing more general cohomologicalfield theories of spin type and the corresponding spin analogs ofquantum cohomology and Gromov-Witten invariants. The second part ofthe proposal is related to applications and generalizations of theso-called chiral de Rham complex, a canonical sheaf of vertexalgebras introduced in joint work with Malikov and Schechtman. Themain goal here is to find a rigorous geometric foundation for thephysical models of two-dimensional superconformal field theory onsmooth varieties and orbifolds.String theory is a proposed physical theory which attempts to unifyall kinds of forces and, in particular, combines Einstein's theoryof gravity with the quantum theory. Because this theory cannot yetbe verified on experiments, physicists working in string theory aretesting it on sophisticated mathematical models. Often the samephysical quantity can be computed by more than one method whichgives answers expressed in terms of very different mathematicalstructures. This suggests the existence of deep ties between variousseemingly remote parts of mathematics and leads to formulation ofnon-trivial conjectures connecting different mathematical objects,usually geometric and analytical in nature. The goal of the currentproject is a mathematical study of some of these conjectures andinteractions. Besides producing new mathematical results, it shouldgive mathematical verification of some physical models and providemore direct links between mathematical and physical methods ofanalysis of these models.