Mathematical Sciences: Holomorphic Invariants of 3-Manifolds

3流形的wittenreshetikhin - turaev （WRT）族仅被定义为依赖于一个参数的离散不变量族，该参数可以是任意单位根。本课题旨在研究将该不变量推广到参数的全纯函数。根据Murakami， Ohtsuki， Rozansky和主要研究者的工作，目前可以只对特定流形定义这样的扩展。该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。这将导致有限型三流形不变量的组合公式，Casson不变量的推广，以及chen - simons - witten Feynman积分的离散化以及与Kontsevich积分的比较。虽然我们对三维结点和连杆的所谓量子不变量族（通常是复数参数的多项式函数）了解很多，但对于它们的3流形对立物来说，目前还不是这样。从这些函数的少数已知结果来看，它们具有有趣的数论性质，这是本项目的研究主题。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，目前它还没有完全严格的基础。＊＊＊

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. ***