The Topology of Quantum Invariants

这个项目的目标是试图理解三维量子不变量的拓扑结构。目前的知识状态似乎不足以解释不变量携带什么样的拓扑信息，因此在三维拓扑中它们可能有什么样的应用。然而，它揭示了（通过TQFT和Vassiliev理论的框架）大量非常有趣的代数结构，这些结构似乎与三维拓扑特别相关，因此应该被认为是一种三维“新代数拓扑”。Roberts打算尝试弥合这种新拓扑和经典代数拓扑之间的差距。调查主要有三个方面。第一个是k理论，它可以帮助解释量子结不变量和Vassiliev不变量之间的关系（例如，为什么琼斯多项式是一个多项式）。第二个是Rozansky-Witten理论，这是一个非常有趣的TQFT的新例子，它提供了不变量解释的新提示，并且可以在复杂或超凯勒几何中有额外的应用。三是卡沙耶夫、村上和村上的双曲体积猜想。这个猜想似乎符合将量子化（特别是6j符号）与经典几何联系起来的一般模式；可以为这个猜想提供概念背景的想法。对数学家来说，结就是把一根绳子缠在一起，然后把它的两端粘在一起。很明显，做这件事的方式在质量上是不同的；试图描述和理解这些方式形成了拓扑学的一个分支，当定量问题被忽略时，它可能被认为是几何学的剩余部分。结理论的主要工具是“拓扑不变量”，这些数字可以与结相关联，并编码有关其结构的一些信息。在过去的16年里，从量子场论物理学中借鉴的思想导致了许多新的美丽的不变量的发现，这些不变量被称为“量子不变量”，但由于这种不寻常的起源，它们的意义仍然非常神秘，它们的潜力尚未实现。该项目的目标是用一些新的想法来试图解释这个谜团，并弥合几何、拓扑和物理学之间的一些差距。这项工作应该有助于这些学科之间健康的思想交流，并在每个学科中激发新的工作。

DMS-0103922Justin D. Roberts The goal of this project is to try to understand the topology underlying quantum invariants in three dimensions. The current state of knowledge seems inadequate for explaining what kind of topological information the invariants carry, and therefore what kind of applications in three-dimensional topology they might have. However ithas revealed (through the frameworks of TQFT and Vassiliev theory) a wealth of very interesting algebraic structures, which seem to be specifically related to topology in three dimensions, and so should be thought of as kind of ``new algebraic topology'' in three dimensions.Roberts intends to try to bridge the gap between this new topology and classical algebraic topology. There are three main areas of investigation. The first is K-theory, which could help to explain the relationship between quantum knot invariants and Vassiliev invariants (for example, why the Jones polynomial is a polynomial). The second is Rozansky-Witten theory, a very interesting new example of a TQFT which provides new hints about intepretation of the invariants, and could have additional applications in complex or hyperkaehler geometry. The third is the hyperbolic volume conjecture of Kashaev, Murakami and Murakami. This conjecture seems to fit into a general pattern of ideas relating quantization (and specifically, 6j-symbols) to classicalgeometry; ideas which might provide a conceptual context for the conjecture.To a mathematician, a knot is what you get by tangling up a piece of string and then gluing its ends together. It is clear that there are qualitatively different ways of doing this; trying to describe and understand these ways forms a branch of topology, which might be thought of as what remains of geometry when quantitative questions are ignored. The main tools of knot theory are "topological invariants", numbers which can be associated to knots and which encode someinformation about their structure. During the last sixteen years, ideas borrowed from the physics of quantum field theory have led to the discovery of many new and beautiful invariants called "quantum invariants", but because of this unusual origin, their meaning remains very mysterious, and their potential unfulfilled. The goal of the project is to use some new ideas to try to explain this mystery and tobridge some of the gaps between geometry, topology, and physics. The work should contribute to the healthy exchange of ideas between these disciplines, and stimulate new work in each.