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. ***
