Homotopy Quantum Field Theory

同伦量子场论

• 批准号: 1202335
• 负责人: Vladimir Touraev
• 金额: $ 18.05万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1202335&HistoricalAwards=false
• 关键词: Homotopy; Quantum; Field; Theory

Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory dealing with manifolds and cobordisms endowed with maps to a given target space. The aim of the project is to establish foundations of 3-dimensional HQFT in the case where the target space is the Eilenberg-MacLane space of type K(G,1) where G is a discrete group (finite or infinite). This should generalize the known results of the PI with N. Reshetikhin, O. Viro, and A. Virelizier. The project will specifically address the following problems:(1) produce a state sum construction of 3-dimensional HQFTs from spherical G-fusion G-categories; (2) introduce an appropriate notion of a G-modular G-category and give a surgery construction of a 3-dimensional HQFT from such a category. Solutions to these problems will include definitions of the required classes of G-categories and of the corresponding HQFTs. Finally, the PI will establish a fundamental relation between the constructions (1) and (2) via a G-version of the Drinfeld-Joyal-Street center of categories: the HQFT obtained by construction (1) from a spherical G-fusion G-category C is equivalent to the HQFT obtained by construction (2) from the appropriately defined G-center of C. The project develops new techniques in geometry arising from ideas of quantum physics. Specifically, the project will focus on the notion of Topological Field Theory introduced by the Fields medalist Edward Witten in the 1980's. Topological Field Theory allows topologists to analyze the shape of geometric objects from the microscopic viewpoint decomposing them into small elementary pieces. Mathematical formulation of Topological Field Theory given by the PI and co-authors has successfully led to creating bridges between previously unrelated areas of pure mathematics like the geometric theory of knotted strings in space on one hand, and the algebraic theory of representations and categories on the other hand. The project considerably enlarges the class of geometric objects which can be analyzed using Topological Field Theory. We include in consideration not only the objects themselves but also relations between them known in mathematics as maps or mappings. The principal aim of the project is to analyze the maps from the above mentioned microscopic viewpoint and to derive corresponding notions in algebra and theory of categories. The project will introduce and develop two different solutions to this problem and establish a subtle theorem relating these two solutions. This project will lead to a better understanding of geometric objects and their maps. It will also produce new powerful algebraic notions and techniques. Potential areas of applications outside of pure mathematics include theoretical physics and quantum computations.

同伦量子场论是拓扑量子场论的一个分支，它涉及到给定目标空间上的映射的流形和上边界。该项目的目的是在目标空间是K(G，1)型Eilenberg-MacLane空间的情况下建立三维HQFT的基础，其中G是离散群(有限或无限)。这应该推广了N.Reshetikhin、O.Viro和A.Virelzier的PI的已知结果。该项目将具体解决以下问题：(1)从球面G-融合G-范畴产生三维HQFT的状态和构造；(2)引入G-模G-范畴的适当概念，并由这样的范畴给出三维HQFT的外科构造。这些问题的解决办法将包括G-类别和相应的HQFT所需类别的定义。最后，PI将通过范畴的Drinfeld-JoYal-Street中心的G-版本建立构造(1)和(2)之间的基本关系：通过从球面G-融合G-范畴C构造(1)获得的HQFT等价于通过构造(2)从适当定义的C的G-中心获得的HQFT。该项目在几何方面开发了源于量子物理思想的新技术。具体地说，该项目将重点介绍由菲尔兹奖牌获得者爱德华·维顿在20世纪80年代提出的拓扑场论的概念。拓扑场论允许拓扑学家从微观角度分析几何对象的形状，将它们分解成小的基本碎片。PI和合著者给出的拓扑场论的数学表述成功地在以前不相关的纯数学领域之间建立了桥梁，一方面是空间中结弦的几何理论，另一方面是表示和范畴的代数理论。该项目极大地扩大了可以用拓扑场理论分析的几何对象的类别。我们不仅考虑对象本身，还考虑它们之间的关系，在数学上称为映射或映射。该项目的主要目的是从上述微观角度分析地图，并推导出相应的代数和范畴论概念。该项目将为这个问题引入和开发两种不同的解决方案，并建立一个将这两种解决方案联系在一起的微妙定理。这个项目将导致对几何对象及其地图的更好的理解。它还将产生新的强大的代数概念和技术。纯数学之外的潜在应用领域包括理论物理和量子计算。