3-dimensional defect TQFTs and their tricategories

TL;DR

This paper develops a comprehensive mathematical framework for 3-dimensional defect topological quantum field theories (TQFTs), constructing associated tricategories that encode defect fusion and duality, and illustrating how existing theories fit into this structure.

Contribution

It introduces a systematic approach to 3D defect TQFTs via symmetric monoidal functors and constructs a natural tricategory with duals, advancing the algebraic understanding of defect fusion and duality.

Findings

Constructed a tricategory with duals from defect TQFTs

Embedded Reshetikhin-Turaev and Turaev-Viro theories into the framework

Extended existing TQFTs to include defect structures

Abstract

We initiate a systematic study of 3-dimensional `defect' topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.