Gauge Theory and Categorification

拟议研究的主要目标是研究几何表示理论中各种结构的物理实现，特别是结同调的物理框架和几何朗兰兹程序。物理实现是基于拓扑规范理论和拓扑弦理论，它们描述了物理规范理论/弦理论的超对称部分。最近，“范畴化”的思想在数学的各个分支中导致了许多引人注目的发展，这些发展很可能具有物理解释。这些问题包括表示论中的各种问题，以及低维拓扑中多项式结不变量的“分类”。最近在物理学中出现了相关的结构，在四维规范理论中，这项研究与几何朗兰兹纲领产生了新的有趣的联系。在对偶弦理论中，纽结和链环的陈-西蒙斯不变量在BPS态中的实现导致了三阶纽结同调的形成。这些发展指出了本提案中描述的物理学和数学中各种问题之间的许多新联系。更广泛的影响：这个提议，一方面，应该推进对规范理论动力学和弦理论的理解，另一方面，在数学的相关领域，包括枚举几何，同调代数，低维拓扑学和表示论，导致重要的教训和新的结果。PI积极参与组织与拟议研究有关的各种主题的研讨会，讲习班和会议，并指导各级学生。

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.