Groups and Algebraic Structures in Topological Quantum Field Theory

摘要奖：DMS-1007255首席研究员：康斯坦丁·泰尔曼建议的研究包括几个主题相关的项目，涉及拓扑学、二维量子场论和范畴理论。第一个项目，GauedMirror对称性，结合了射影流形上的群作用，等变K-理论和范畴理论(开闭的TQFT和等变K-理论的猜想Brauer群)的思想。一个应用是从不对称流形的规范Fukaya范畴确定Git商的Gromov-Wittenn理论。基于Kontsevich、Costello、Hopkins和Lurie在“扩展”拓扑场理论方面的最新进展，提出了规范拓扑量子场论在二维中的具体描述。PI提出了将二维TQFT耦合到紧致对称群并将规范理论量子化的具体方案。这结合了物理学(Landau-Ginzburg超势)的思想和PI及其合作者在等变(扭曲)K理论和Riemann曲面上主丛的模数的一般指数公式方面的早期工作。第二个项目探索了PI和合作者作为更高范畴的简化模型而引入的“更高代数”，以及有限同伦类型的同伦TQFT的一个玩具例子。这有望成为外来同源理论与QFT思想相互作用的良好工作场所。第三个密切相关的项目是通过拓扑学方法，沿着PI和环面群合作者已经完成的路线，将陈-西蒙斯规范理论构建为0-1-2-3理论。为了解释这项研究的背景，我们必须回忆起，支配宇宙中能量和物质的基本相互作用被认为是由量子场理论支配的，这是从上个世纪量子力学开始演变而来的一种理性化的数学框架。量子场理论从来没有与广义相对论--另一个得到充分支持的物理理论--调和在一起，而在过去的60年里，许多数学研究都集中在对两者的理解上。拓扑量子场论是在避免分析困难的同时解决这个问题的一种尝试：放弃了距离和量级(例如质量)的概念，时空的几何与量子场论的代数结构直接相关。在过去的十年里，由于康采维奇、霍普金斯和卢里的工作，在理解代数结构方面取得了实质性的进展。PI的项目围绕着将这些最新的发展与对称性的想法相结合--以规范理论的形式--众所周知，对称性在现实的物理理论中是不可或缺的。(人们应该回忆一下，粒子物理的所谓“标准模型”，包括电磁、弱和强相互作用，是一种规范理论。)

AbstractAward: DMS-1007255Principal Investigator: Constantin TelemanThe proposed research comprises several, thematically relatedprojects at the interface of topology, 2-dimensional quantumfield theory and category theory. The first project, gaugedmirror symmetry, combines group actions on projective manifolds,equivariant K-theory and ideas from category theory (open-closedTQFTs and a conjectural Brauer group of equivariant K-theory).One application would be the determination of Gromov-Wittentheory of GIT quotients from the gauged Fukaya category of asymplectic manifold. A concrete description of gauged topologicalquantum field theories in two dimensions is proposed, based onrecent progress by Kontsevich, Costello, Hopkins and Lurie on'extended' topological field theories. The PI has a concreteproposal for coupling a 2-dimensional TQFT to a compact symmetrygroup and quantizing the gauged theory. This combines ideas fromphysics (Landau-Ginzburg super-potentials) with earlier work bythe PI and collaborators on equivariant (twisted) K-theory andthe general index formula on moduli of principal bundles overRiemann surfaces. The second project explores the 'higheralgebras' introduced by the PI and collaborators as simplifiedmodels of higher categories, and the a toy example of ahomotopical TQFT for finite homotopy types. This is hoped to be agood working ground for the interaction of exotic homologytheories with ideas from QFT. A third, closely related project isthe construction of Chern-Simons gauge theory as a 0-1-2-3 theoryby topological methods, along the lines already accomplished bythe PI and collaborators for torus groups.To explain the context of this research, one must recall that thefundamental interactions governing energy and matter in theuniverse are believed to be governed by quantum field theory, asophisticated mathematical framework that has evolved from thebeginnings of quantum mechanics over the last century. Quantumfield theory has never been reconciled with general relativity --another well-supported physical theory -- and much mathematicalresearch over the last six decades has centered aroundreconciling the two. Topological quantum field theory is a toyattempt to come to grips with the problem while avoiding theanalytical difficulties: the notions of distance and magnitude(for instance, mass) are abandoned, and the geometry ofspace-time is directly related to the algebraic structure of thequantum field theory. Substantial progress in understanding thealgebraic structure has been made over the last decade thanks towork by Kontsevich, Hopkins and Lurie. The PI's projects revolvearound integrating these recent developments with the idea ofsymmetry -- in the form of gauge theory -- which is known to beindispensable in realistic physical theories. (One should recallthat the so-called 'standard model' of particle physics,comprising the electromagnetic, weak and strong interactions, isa gauge theory.)