Gauge Theory and Categorification

规范理论与分类

• 批准号: 0757647
• 负责人: Sergei Gukov
• 金额: $ 51万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757647&HistoricalAwards=false
• 关键词: Gauge; Theory; Categorification

The main goal of the proposed research is to study the physical realization of various constructions in geometric representation theory, notably the physical framework for knot homologies and the geometric Langlands program. The physical realization is based on topological gauge theory and topological string theory which describe the supersymmetric sector of the physical gauge theory/string theory. Recently the idea of "categorification" led to a number of remarkable developments in various branches of mathematics which may well have a physical interpretation. These include a variety of problems in representation theory, as well as "categorification" of polynomial knot invariants in low-dimensional topology. Related structures recently emerged in physics.In four-dimensional gauge theory the study has led to new interesting connections with the geometric Langlands program. The realization of Chern-Simons invariants of knots and links in terms of BPS states in the dual string theory led to formulation of triply-graded knot homologies. These developments point to many new connections between various problems in physics and mathematics that are described in this proposal. Broader impact: This proposal, on the one hand, should advance the understanding of gauge theory dynamics and string theory and, on the other hand, lead to important lessons and new results in related areas of mathematics, including enumerative geometry, homological algebra, low-dimensional topology, and representation theory. The PI is actively involved in organizing seminars, workshops and conferences on various topics related to the proposed research, and in mentoring students at all levels.

拟议研究的主要目的是研究几何表示理论中各种结构的物理实现，尤其是结的物理框架和几何兰兰兹计划。物理实现是基于拓扑规理论和拓扑弦理论，描述了物理量规理论/字符串理论的超对称部门。最近，“分类”的想法导致了数学各个分支的许多显着发展，这些发展很可能具有物理解释。这些包括在表示理论中的各种问题，以及低维拓扑中多项式结的“分类”。最近在物理学中出现的相关结构。在四维规程理论中，该研究导致与几何兰兰兹计划建立了新的有趣联系。双字符串理论中BPS状态的结与Chern-Simons不变性和链接的实现导致了三位级的结同源性的表述。这些事态发展指出了本提案中描述的物理学和数学各种问题之间的许多新联系。更广泛的影响：一方面，该提议应提高对量规理论动力学和弦理论的理解，另一方面，该提议在数学相关领域（包括列举几何学，同源代数，低维拓扑结构和代表理论）的相关领域的重要课程和新结果。 PI积极参与组织研讨会，研讨会和会议，讨论与拟议的研究有关的各种主题，以及在各级指导学生。