Mathematical Sciences: Holomorphic Invariants of 3-Manifolds

9626544劳伦斯三维流形的Witten-Reshetikhin-Turaev(Wrt)量子不变量族被定义为依赖于一个参数的离散不变量族，该参数可以是任何单位根。本课题旨在研究这种不变量对参数的全纯函数的推广。在村上、大树、罗赞斯基和首席调查者的工作之后，目前只有可能为特定的流形定义这样的扩展。这个项目的目标是更好地理解这个函数、它所取的特定值、它的渐近展开以及原始流形的几何和拓扑的组合描述之间的关系。这将导致有限类型3-流形不变量的组合公式，Casson不变量的推广，以及Chern-Simons-Witten Feynman积分的离散化以及与Kontsevich积分的比较。尽管人们对所谓的三维节点和链节不变量的量子家族已经知道很多，这些量子家族通常是复杂参数的多项式函数，但目前对它们的三维流形来说还不是这样。从为数不多的关于这些函数的已知结果中，可以看出它们具有有趣的数论性质，对这些性质的研究是本项目的主题。这些不变量也有一个作为费曼积分的公式，希望这个项目将有助于更好地理解一般的费曼积分，目前费曼积分的基础并不完全严格。***

9626544 Lawrence The Witten-Reshetikhin-Turaev (WRT) family of quantum invariants of 3-manifolds was defined only as a family of discrete invariants dependent on a parameter that can be any root of unity. This project aims to study the extension of this invariant to a holomorphic function of the parameter. Following work of Murakami, Ohtsuki, Rozansky and the Principal Investigator, it is currently possible to define such an extension only for specific manifolds. The goal of the project is to understand better the relationship between this function, specific values that it takes, its asymptotic expansion, and a combinatorial description of the geometry and topology of the original manifold. This should lead to combinatorial formulae for finite type 3-manifold invariants, generalizations of the Casson invariant, as well as a discretization of the Chern-Simons-Witten Feynman integral and comparisons with the Kontsevich integrals. Although much is known about the so-called quantum family of invariants of knots and links in three dimensions, which are generally polynomial functions of a complex parameter, the same is currently not true for their 3-manifold counterparts. From the few known results on these functions, they are seen to have intriguing number theoretic properties, the investigation of which is the subject of this project. These invariants also have a formulation as a Feynman integral, and it is hoped that this project will contribute to the better understanding of Feynman integrals in general, which currently sit on a not entirely rigorous foundation. ***