Quantum cohomology of homogeneous spaces

齐次空间的量子上同调

• 批准号: 345815019
• 负责人: Dr. Christoph Bärligea
• 金额: --
• 依托单位: Lehrstuhl Analysis (Prof. Abbondandolo)
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/345815019?language=en
• 关键词: Quantum; cohomology; homogeneous; spaces

Gromov-Witten theory has its origins in physical models of string theory. Groundbraking work of Kontsevich and Manin on the moduli space of stable maps gave the theory a rigorous mathematical foundation and contributed to solve outstanding enumerative problems in algebraic geometry. Thereby, they initiated a new branch of research with far-reaching applications in algebraic and symplectic geometry and theoretical physics.Roughly speaking, Gromov-Witten invariants count curves of fixed degree and genus on a smooth projective variety. The considerer curves have to satisfy specific incidences which force the resulting number of curves to be finite. Gromov-Witten invariants can be summarized in the so-called quantum cohomology ring. This is a graded algebra whose quantum product structure is a deformation of the ordinary cup product. The associativity of this product yields non-trivial relations among the enumerative solutions.The present project is about Gromov-Witten theory on homogeneous spaces - a class of smooth projective varieties which carry a transitive group action. In this case, the moduli space of stable maps has particularly desirable properties which allow an intuitive approach to Gromov-Witten theory. In particular, it is possible to make sharper statements about the minimal degrees in quantum products. First of all, we will study in this project the distribution of these minimal degrees and try to compute a minimal upper bound of all minimal degrees.Moreover, the aim of the project is to prove finer properties than irreducibility of the moduli space of stable maps. Properties of the moduli space such as quasi-homogeneity simplify the description of Gromov-Witten invariants and prepare the path for a theoretical understanding of their structure. Thus, the project will be finally about new methods for computing Gromov-Witten invariants.

格罗莫夫-维滕理论起源于弦理论的物理模型。 Kontsevich 和 Manin 在稳定映射模空间方面的开创性工作为该理论提供了严格的数学基础，并有助于解决代数几何中突出的枚举问题。从而，他们开创了一个新的研究分支，在代数和辛几何以及理论物理方面具有深远的应用。粗略地说，格罗莫夫-维滕不变量在光滑射影簇上计算固定次数和亏格的曲线。考虑者曲线必须满足特定的关联，这迫使所得到的曲线数量是有限的。 Gromov-Witten 不变量可以概括为所谓的量子上同调环。这是一个分级代数，其量子乘积结构是普通杯乘积的变形。该乘积的结合性在枚举解之间产生了非平凡的关系。当前的项目是关于齐次空间的 Gromov-Witten 理论 - 一类带有传递群作用的光滑射影簇。在这种情况下，稳定映射的模空间具有特别理想的属性，可以直观地了解 Gromov-Witten 理论。特别是，可以对量子乘积的最小度做出更清晰的陈述。首先，我们将在这个项目中研究这些最小度的分布，并尝试计算所有最小度的最小上限。此外，该项目的目的是证明比稳定映射模空间的不可约性更精细的性质。模空间的性质（例如准齐质性）简化了 Gromov-Witten 不变量的描述，并为其结构的理论理解奠定了基础。因此，该项目最终将涉及计算 Gromov-Witten 不变量的新方法。