Tensor networks and representation theory

张量网络和表示论

• 批准号: 449480360
• 负责人: Professor Dr. Christoph Schweigert
• 金额: --
• 依托单位: Bereich Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/449480360?language=en
• 关键词: Tensor; networks; representation; theory

Topological field theories in three-dimensions (and more generally modular functors) have many connections to representation theory and two-dimensional conformal field theory. This applies in particular to topological field theory on three-manifolds with boundaries. Recent developments have shown that these structures also extends to categories that are not semisimple. Based on these recent developments, we propose to study some concrete problems that are of independent interest and, at the same time, help to shape directions for the developments of the general theory.We have detailed goals for two dissertation projects:1a. String-net descriptions for correlation functions of defect and boundary fields in two-dimensional conformal field theory. 1b. A holographic understanding of the string-net description of CFT correlators.2a. Defects in three-dimensional topological field theories and equivariant Frobenius-Schur indicators.2b. Topological field theory on 3-manifolds with boundaries and tensor network models in two dimensions.The long term goal underlying and transcending this project is the construction of modular functors for monoidal categories that still obey certain finiteness conditions, but exhibit generalizations of dualities, e.g. Grothendieck-Verdier structures.

三维拓扑场论（以及更一般的模函子）与表示理论和二维共形场论有许多联系。这尤其适用于具有边界的三流形的拓扑场论。最近的发展表明，这些结构也扩展到非半简单的类别。基于这些最新的发展，我们建议研究一些具有独立意义的具体问题，同时有助于为一般理论的发展塑造方向。我们有两个论文项目的详细目标：二维共形场理论中缺陷场与边界场相关函数的弦网描述。1 b。CFT相关器的弦网描述的全息理解。三维拓扑场理论和等变Frobenius-Schur指标的缺陷。三维流形的拓扑场理论及二维张量网络模型。长期目标的基础和超越这个项目是一元范畴的模函子的构造，仍然服从一定的有限条件，但表现出对偶性的推广，如Grothendieck-Verdier结构。