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

这个项目涉及编织张量范畴及其高维类似物的研究。这些是抽象的代数结构，由可以使用某些自然规则融合的对象组成。这些范畴在研究经典对称性和量子对称性中是不可或缺的。它们也被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相，并用于预测这种相的新类型的存在、它们的行为和它们的物理实现。这个项目的动机是这些联系，并处理（高）张量范畴理论的代数方面：它们的结构、分类和算术性质。重点是在应用程序中使用最广泛的类别。这样的分类承认一个额外的对称约束，称为编织，用于模拟量子粒子的相互作用。学生招募和培训是拟议研究活动的重要组成部分。本项目将解决编织张量类和2类的结构和分类的基本问题。分类技术将被用来解释和解决代数问题。以前开发的工具，如范畴Witt群和Picard群，将被推广并结合到更高范畴群中，提供一个有用的同伦理论机制。具体研究问题包括：(1)编织融合2类的分类及其Witt不变量的描述；(2)物质对称保护拓扑相对应的最小扩展群的计算；(3)与融合类相关的Hecke代数及其积分形式的研究；(4)利用范畴类比Belavin-Drinfeld三元组对非简并编织融合类上的纤维函子进行分类。(5)非半简单点编织张量范畴的分类。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.