Topological Quantum Field Theory in dimensions two and three

本文的研究对象是二维和三维的拓扑量子场论。TQFT生成几何对象的拓扑不变量，如纽结和三维流形。我们将研究TQFT在经典拓扑问题中的应用，例如找出一个曲面上的给定Serre纤维(或更多的，一般地，在一个2维奥氏模型上)是否有一个截面。这就需要对作者在1999年引入的同伦QFT进行仔细的研究。在另一个方向上，我们将给出一个三维态和TQFT的一般代数框架，并将后者与三维外科TQFT联系起来，如下所示：由有限半单么半范畴C导出的状态和TQFT与由C的Drinfeld二重得到的外科TQFT同构。证明将使用Bruguieres-Virelzier-Habiro的底部纠缠技术。量子物理一直是新的数学见解和技术的来源。大约20年前，菲尔兹奖牌获得者E.Witten和M.Atiyah提出了一种这样的见解，称为拓扑量子场论。它允许我们从一个新的物理角度来处理真实世界的几何对象，如实体的表面或空间中的打结圆。我们将使用这个视点来解决与曲面有关的某些几何问题。特别是，我们将回答在量子场论引入之前很久就提出的几个经典问题。我们还将研究源自量子场论和量子引力的数学技巧之间的联系。这项研究将涉及另一位菲尔兹奖牌获得者V.Drinfeld介绍的深度代数技术。这个项目将展示物理学、几何学和代数之间基本联系的新方面。

The proposed research is on Topological Quantum Field Theory (TQFT) in dimensions two and three. TQFTs produce topological invariants of geometric objects such as knots and 3-dimensional manifolds. We shall study applications of TQFTs to classical topological problems such as finding whether a given Serre fibration over a surface (or more, generally, over a 2-dimensional orbifold) has a section. This will require a careful study of Homotopy QFTs introduced by the author in 1999. In another direction, we will give a general algebraic framework for 3-dimensional state-sum TQFTs and connect the latter to 3-dimensional surgery TQFTs as follows: the state sum TQFT derived from a finite semisimple monoidal category C is isomorphic to the surgery TQFT derived from the Drinfeld double of C. The proof will use the technique of bottom tangles due to Bruguieres-Virelizier-Habiro.Quantum physics has invariably served as a source of new mathematical insights and techniques. One such insight introduced about 20 years ago by the Fields medalists E. Witten and M. Atiyah is called Topological Quantum Field Theory. It allows us to approach real-world geometric objects such as surfaces of solid bodies or knotted circles in the space from a new, physical perspective. We will use this viewpoint to address certain geometric problems concerning the surfaces. In particular, we will answer several classical questions raised well before the introduction of Quantum Field Theory. We will also study a connection between mathematical techniques arising from Quantum Field Theory and from Quantum Gravity. This study will involve deep algebraic techniques introduced by another Fields medalist V. Drinfeld. This project will exhibit new facets of the fundamental connection between physics, geometry, and algebra.