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

Jurgen Fuchs, Christoph Schweigert, Alessandro Valentino

TL;DR

The paper develops a rigorous, category-theoretic framework for topological boundary conditions and surface defects in 3d Reshetikhin–Turaev TFTs based on modular tensor categories. It shows that consistent boundaries correspond to Witt-trivializations $\mathcal{C}\to\mathcal{Z}(\mathcal{A})$ and that the associated boundary (resp. defect) theories are captured by the bicategory of $\mathcal{A}$-modules (resp. $\mathcal{W}_{d}$-modules) with Wilson lines realized as module functors. Central functors and Drinfeld centers encode the admissible interfaces, while Lagrangian algebras and special symmetric Frobenius algebras provide concrete invariants and a bridge to abelian Chern–Simons theories and to string-diagram techniques. The results generalize known abelian CS and Turaev–Viro analyses and establish a folding-trick viewpoint relating boundaries and defects within a unified higher-categorical setting. This establishes a robust, scalable framework for extending RT/Turaev–Viro TFTs to manifolds with codimension-two and higher structures.

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.

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

TL;DR

The paper develops a rigorous, category-theoretic framework for topological boundary conditions and surface defects in 3d Reshetikhin–Turaev TFTs based on modular tensor categories. It shows that consistent boundaries correspond to Witt-trivializations and that the associated boundary (resp. defect) theories are captured by the bicategory of -modules (resp. -modules) with Wilson lines realized as module functors. Central functors and Drinfeld centers encode the admissible interfaces, while Lagrangian algebras and special symmetric Frobenius algebras provide concrete invariants and a bridge to abelian Chern–Simons theories and to string-diagram techniques. The results generalize known abelian CS and Turaev–Viro analyses and establish a folding-trick viewpoint relating boundaries and defects within a unified higher-categorical setting. This establishes a robust, scalable framework for extending RT/Turaev–Viro TFTs to manifolds with codimension-two and higher structures.

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.

Paper Structure

This paper contains 14 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Module categories
  4. Bicategories and Deligne products
  5. Drinfeld center and enveloping category
  6. Central functors
  7. Lagrangian algebras
  8. The Witt group
  9. Bicategories for boundary conditions
  10. Bicategories for surface defects
  11. Lagrangian algebras and abelian Chern-Simons theories
  12. Relation with special symmetric Frobenius algebras
  13. String diagrams
  14. Frobenius algebras from string diagrams

Key Result

Proposition 2.4

Let $\mathcal{A}$ be a $\Bbbk$-linear monoidal category. For any object $U\,{\in}\,{\mathcal{A}}$ denote by $F_U\colon {\mathcal{A}_{\mathcal{A}}}\,{\to}\,{\mathcal{A}_{\mathcal{A}}}$ the module endofunctor that acts on objects by tensoring with $U$ from the left, $F_U(V) \,{:=}\, U\,{\otimes}\, V$. is an equivalence of monoidal categories.

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 23 more