Vanishing Quantum Dimensions in Low-dimensional Topology

提出的研究的第一个主要主题是量子6J符号和3流形不变量。首席研究员Patureau-Mirand和Turaev已经证明，量子维度为零的范畴自然产生修正的6J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，目的是通过推导这些修改的6J符号的显式公式，并从重新归一化的3-流形不变量构造拓扑量子场理论(TQFT)来进一步理解这些对象。这项工作与体积猜想有关。本文还将致力于构造三维流形的广义Kashaev型量子双曲不变量，目的是定义非紧群的Chern-Simons型不变量。提出的研究的最后一个主题是构造阴影的新的状态和不变量。随着量子场论涌入低维拓扑学，在过去的二十年里，链式和三维流形不变量理论出现了革命性的观点。1984年Jones发现了V.Jones多项式，1989年E.Witten解释了它的三维量子场论，为使用新的代数方法研究拓扑学打开了大门。这些发展导致了一个新的数学分支，被称为“量子拓扑学”。量子维度的消失丰富了量子拓扑学的许多有用和有趣的主题；例如，量子三维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，都与低维拓扑中量子维度的消失有关。

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.