Tensor networks and representation theory

三维拓扑场论(更广泛地说是模函子)与表象理论和二维共形场论有许多联系。这特别适用于带边界的三维流形上的拓扑场论。最近的发展表明，这些结构也扩展到不是半简单的范畴。基于这些最新发展，我们建议研究一些独立感兴趣的具体问题，同时有助于形成一般理论发展的方向。我们有两个论文项目的详细目标：1。二维共形场理论中缺陷场与边界场关联函数的弦网描述。1B.对CFT相关器的串网描述的全息理解。三维拓扑场理论中的缺陷与等变Frobenius-Schur指示符。关于三维流形和二维张量网络模型的拓扑场理论。这个项目的潜在和超越的长期目标是为么半群范畴构造模函子，这些范畴仍然遵守某些有限条件，但表现出对偶的一般化，例如Grothendieck-Verdier结构。

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.