Vanishing Quantum Dimensions in Low-dimensional Topology

低维拓扑中消失的量子维度

• 批准号: 1007197
• 负责人: Nathan Geer
• 金额: $ 13.55万
• 依托单位: Utah State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007197&HistoricalAwards=false
• 关键词: Vanishing; Quantum; Dimensions; Low; dimensional

The first main theme of the proposed research is quantum 6j-symbols and 3-manifold invariants. The Principal Investigator, Patureau-Mirand and Turaev have shown that categories with vanishing quantum dimensions naturally give rise to modified 6j-symbols and re-normalized link and 3-manifold invariants. The project continues this work with a goal of further understanding these objects by deriving explicit formulas for these modified 6j-symbols and constructing Topological Quantum Field Theories (TQFTs) from the re-normalized 3-manifold invariants. This work is related to the Volume Conjecture. The proposed research will also focus on constructing generalized Kashaev-type quantum hyperbolic invariants of 3-manifolds with an aim of defining Chern-Simons-type invariants for non-compact groups. A final theme of the proposed research is the construction of new state-sum invariants of shadows. With the influx of quantum field theory into low-dimensional topology, the past two decades have seen the emergence of a revolutionary point of view in the theory of link and 3-manifold invariants. The discovery of the V. Jones polynomial by Jones in 1984 and its 3-dimensional quantum field theory interpretation by E. Witten in 1989 have opened the door for the use of new algebraic techniques to study topology. These developments led to a new branch of mathematic known as "quantum topology." Many useful and interesting topics of quantum topology are enriched by vanishing quantum dimensions; for example the construction of quantum 3-manifold invariants and the Volume Conjecture both depend on the subtleties of such vanishing. This project has three components, all concerning the vanishing of quantum dimensions in low-dimensional topology.

第一个主题是量子6 j-符号和3-流形不变量。 主要研究者Patureau-Mirand和Turaev已经证明，具有消失量子维度的范畴自然会产生修改的6 j符号和重新归一化的链接和3流形不变量。 该项目继续这项工作，目标是通过推导这些修改的6 j符号的显式公式并从重新归一化的3流形不变量构建拓扑量子场论（TQFT）来进一步理解这些对象。 这个工作与体积猜想有关。 拟议的研究还将集中在3流形的广义Kashaev型量子双曲不变量的构造，目的是定义非紧群的Chern-Simons型不变量。 建议的研究的最后一个主题是新的状态和不变量的阴影的建设。 随着量子场论在低维拓扑学中的应用，在过去的二十年里，人们看到了链路和三维流形不变量理论中革命性观点的出现。 1984年Jones发现了V. Jones多项式，并由E.维滕在1989年已经打开了大门，使用新的代数技术来研究拓扑结构。 这些发展导致了一个新的数学分支，称为“量子拓扑学”。量子拓扑学的许多有用和有趣的主题都通过消失的量子维度而丰富;例如量子3流形不变量的构造和体积猜想都依赖于这种消失的微妙之处。 这个项目有三个组成部分，都是关于在低维拓扑中量子维的消失。