Braided tensor categories, higher Picard groups, and classification of topological phases of matter

辫状张量类别、高皮卡德群以及物质拓扑相的分类

• 批准号: 2302267
• 负责人: Dmitri Nikshych
• 金额: $ 18.26万
• 依托单位: University of New Hampshire
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302267&HistoricalAwards=false
• 关键词: Braided; tensor; categories; higher; Picard

This project concerns the study of braided tensor categories and their higher-dimensional analogs. These are abstract algebraic structures consisting of objects that can be fused using certain natural rules. Such categories are indispensable in the study of classical and quantum symmetries. They are also used as mathematical models for a potential hardware design of a quantum computer. Namely, they correspond to topological phases of matter relevant to quantum computation and are used to predict the existence of new types of such phases, their behavior, and their physical realization. This project is motivated by these connections and deals with algebraic aspects of the theory of (higher) tensor categories: their structure, classification, and arithmetic properties. The emphasis is on categories that are most widely used in applications. Such categories admit an additional symmetry constraint called braiding that is used to model the interaction of quantum particles. Student recruitment and training are essential components of the proposed research activity.This project will address fundamental questions concerning the structure and classification of braided tensor categories and 2-categories. Categorical techniques will be employed to interpret and solve algebraic problems. Previously developed tools such as categorical Witt and Picard groups will be generalized and combined into higher-categorical groups, providing a useful homotopy-theoretic machinery. The concrete research problems include the following: (1) classification of braided fusion 2-categories and description of their Witt invariants, (2) computation of groups of minimal extensions corresponding to symmetry-protected topological phases of matter, (3) study of Hecke algebras associated with fusion categories and their integral forms, (4) classification of fiber functors on non-degenerate braided fusion categories using a categorical analog of Belavin-Drinfeld triples, and (5) classification of non-semisimple pointed braided tensor categories.This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目涉及编织张量类别及其较高维的类似物的研究。这些是由可以使用某些自然规则融合的对象组成的抽象代数结构。在经典和量子对称性的研究中，此类类别是必不可少的。 它们还用作量子计算机潜在硬件设计的数学模型。也就是说，它们对应于与量子计算相关的物质的拓扑阶段，并用于预测这种新型阶段的存在，其行为和它们的物理实现。这些项目是由这些联系的动机，并涉及（高）张量类别理论的代数方面：它们的结构，分类和算术属性。重点是在应用中最广泛使用的类别。此类类别承认了一种称为编织的其他对称约束，用于对量子颗粒的相互作用进行建模。学生招聘和培训是拟议的研究活动的重要组成部分。该项目将解决有关编织张量类别和两类的结构和分类的基本问题。将采用分类技术来解释和解决代数问题。先前开发的工具（例如分类WITT和PICARD组）将被概括并合并为更高分类组，从而提供有用的同质理论机制。具体的研究问题包括以下内容：（1）编织融合2类别的分类及其不变性的描述，（2）计算与物质的对称性保护拓扑相对应的最小扩展的组，（3）使用与Hecke代数相关的融合类别及其集成形式的分类（4）fiber formitification foe foi foffiper的拓扑阶段， Belavin-Drinfeld三元组的类似物以及（5）非偏鼻尖的编织张量类别的分类。该项目由代数和数字理论计划和既定的计划共同资助，刺激竞争研究（EPSCOR）的既定计划（EPSCOR）。这一奖项反映了NSF的法定任务和审查的范围。