Bicategories for boundary conditions and for surface defects in 3-d TFT

TL;DR

This paper develops a bicategorical framework to describe boundary conditions and surface defects in 3D topological field theories, connecting algebraic structures with topological features.

Contribution

It introduces a bicategory-based approach to characterize boundary conditions and surface defects in 3D TFTs using modular tensor categories and trivializations in the Witt group.

Findings

Boundary conditions described by central functors and trivializations in the Witt group.

Bicategory of boundary conditions linked to module categories over trivializations.

Relation established between Frobenius algebras, Wilson lines, and special defects.

Abstract

We analyze topological boundary conditions and topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type based on arbitrary modular tensor categories. Boundary conditions are described by central functors that lift to trivializations in the Witt group of modular tensor categories. The bicategory of boundary conditions can be described through the bicategory of module categories over any such trivialization. A similar description is obtained for topological surface defects. Using string diagrams for bicategories we also establish a precise relation between special symmetric Frobenius algebras and Wilson lines involving special defects. We compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong on boundary conditions and surface defects in abelian Chern-Simons theories and in Turaev-Viro type TFTs, respectively.