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3-dimensional defect TQFTs and their tricategories

Nils Carqueville, Catherine Meusburger, Gregor Schaumann

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.

3-dimensional defect TQFTs and their tricategories

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.

Paper Structure

This paper contains 22 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction and summary
  2. Bordisms
  3. Stratified bordisms
  4. Standard stratified bordisms
  5. Decorated bordisms
  6. Tricategories from defect TQFTs
  7. Free 2-categories and Gray categories
  8. Bicategories
  9. Tricategories
  10. Duals
  11. Bicategories
  12. Tricategories
  13. Gray categories from defect TQFTs
  14. Gray categories with duals from defect TQFTs
  15. Examples
  16. ...and 7 more sections

Key Result

Theorem 1.1

Every 3-dimensional defect TQFT $\mathcal{Z}: \operatorname{Bord}_{3}^{\mathrm{def}}(\mathds{D}) \to \operatorname{Vect}_\Bbbk$ gives rise to a $\Bbbk$-linear Gray category with duals $\mathcal{T}_\mathcal{Z}$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 32 more