Quantum Invariants, Enhanced Moduli, and Integrable Systems

量子不变量、增强模和可积系统

• 批准号: 1601438
• 负责人: Tony Pantev
• 金额: $ 12.14万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601438&HistoricalAwards=false
• 关键词: Quantum; Invariants; Enhanced; Moduli; Integrable

This award supports research in the field of algebraic geometry. The research sets the stage for understanding the basic structure of fundamental problems in a way suitable for pragmatic use in a broad spectrum of applications. The results of the work are intended to be immediately relevant to questions in the theory of integrable systems, string theory, gauge theory, and the study of topological insulators. In addition, the research project aims to have concrete applications to field theory and quantization. The project also aims to build a new arsenal of geometric techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by providing research opportunities on the interface of geometry and string theory for graduate students and postdoctoral associates in mathematics and physics, and by development of courses on formal localization and shifted quantization and on Calabi-Yau integrable systems and higher Donaldson-Thomas theory. The project integrates ideas from derived geometry and quantum field theory to unravel the hidden complexity of moduli problems, to extract new enumerative invariants of varieties, and to study completely integrable systems. The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. Three directions will be studied. The first aims to characterize those moduli spaces that admit a realization as the critical locus of a potential. The characterization requires the development of the formalism of shifted symplectic and Poisson structures in derived geometry and the theory of isotropic foliations. In the second project a new method will be investigated for constructing motivic orientation data on non-abelian cohomology by building explicit Lagrangian foliations and computing the associated potential functions. The project aims to construct higher Chern-Simons functionals, and sets the stage for extracting enumerative invariants from higher dimensional quantum field theory. The final project searches for Calabi-Yau integrable systems that realize the tame or wild meromorphic Hitchin fibrations for ADE structure groups.

该奖项支持代数几何学领域的研究。该研究为理解基本问题的基本结构奠定了基础，以适合在广泛应用中务实使用的方式。该作品的结果旨在与可集成系统，字符串理论，规程理论和拓扑绝缘体研究理论中的问题立即相关。 此外，该研究项目旨在在现场理论和量化上提供具体的应用。 该项目还旨在建立一种适用于代数周期，符号拓扑和高能量物理学理论的几何技术的新库。这将通过为研究生和数学和物理学的博士后伙伴提供有关几何学和弦理论界面的研究机会，并通过开发有关正式本地化和转移量化的课程以及Calabi-yau的可整合系统以及更高的Donaldson-Thomas-Thomas-Thomas-Thomas-Thomas-thomas理论。该项目整合了从衍生的几何形状和量子场理论中，以揭示模量问题的隐藏复杂性，提取新的列出品种的新列出，并研究完全可以集成的系统。这些问题的解决将巩固和揭开几何，象征性拓扑和田间理论中现有的几种量化方案。将研究三个方向。第一个旨在表征那些承认认识为潜力关键基因源的模量空间。该特征需要在衍生几何形状和各向同性叶子理论中发展的符号和泊松结构的形式主义的发展。在第二个项目中，将研究一种新方法，用于通过建立明确的拉格朗日叶子并计算相关的潜在功能来构建非亚伯同谋的动机取向数据。该项目旨在构建更高的Chern-Simons功能，并为从高维量子场理论中提取枚举不变性奠定了基础。最终项目搜索了Calabi-Yau可集成的系统，这些系统实现了ADE结构组的驯服或野生Meromorormorphic Hitchin纤维。