Groups and Algebraic Structures in Topological Quantum Field Theory

AbstractAward：DMS-1007255首席研究员：Constantin Teleman拟议的研究包括拓扑学，二维量子场论和范畴论接口的几个主题相关项目。第一个项目，规范镜像对称，结合了射影流形上的群作用，等变K-理论和范畴论的思想（开闭TQFT和等变K-理论的一个代数Brauer群），一个应用是从辛流形的规范福谷范畴确定GIT等价物的Gromov-Witten理论。基于Kontsevich、Costello、霍普金斯和Lurie在“扩展”拓扑场论方面的最新进展，提出了二维规范拓扑量子场论的具体描述。PI有一个具体的建议，耦合一个二维TQFT到一个紧凑的量子群和量化的规范理论。 这结合了物理学的思想（朗道-金斯堡超势）与PI和合作者在等变（扭曲）K理论和黎曼曲面上主丛模的一般指数公式上的早期工作。第二个项目探讨了由PI和合作者引入的“higheralgebras”作为更高类别的简化模型，以及有限同伦类型的同伦TQFT的一个玩具例子。希望这能为外来同源理论与QFT思想的相互作用提供一个良好的工作基础。第三个密切相关的项目是利用拓扑方法，沿着PI和合作者已经完成的环面群的路线，将Chern-Simons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论支配的，一个复杂的数学框架，它是从上个世纪的量子力学发展而来的。 量子场论从未与广义相对论（另一个得到充分支持的物理理论）相调和，过去60年来的许多理论研究都围绕着重新调和这两者。拓扑量子场论是一种玩具式的尝试，试图在避免分析困难的同时解决问题：距离和大小（例如质量）的概念被抛弃，时空的几何与量子场论的代数结构直接相关。由于Kontsevich、霍普金斯和Lurie的工作，在过去的十年里，对代数结构的理解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思想结合起来-以规范理论的形式-这在现实的物理理论中是不可或缺的。(One应该记得，粒子物理学的所谓“标准模型”，包括电磁、弱和强相互作用，伊萨规范理论。

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.)