Tensor networks and representation theory

点击翻译按钮获取中文摘要

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.