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Modular categories as representations of the 3-dimensional bordism 2-category

Bruce Bartlett, Christopher L. Douglas, Christopher J. Schommer-Pries, Jamie Vicary

TL;DR

This work completes a program linking 3D extended TQFTs to modular tensor categories by showing linear representations of structured bordism 2-categories correspond to modular tensor categories equipped with square-root data for the anomaly/global dimension. It develops a robust 2-categorical framework: finite presentations of symmetric monoidal 2-categories, their linear representations in $\mathbf{2Vect}_k$, and a skein-theoretic interior-string-diagram calculus that recovers modular and ribbon structures. The key contributions include establishing a bijection between extended 3D TQFTs and MTCs with square-root anomaly, and outlining how various bordism structures (oriented, signature, $p_1$) correspond to modular data with corresponding root choices. The results pave the way for explicit skein-theoretic computations in extended TQFTs and solidify the role of modular tensor categories as complete algebraic receivers for 3D bordism theories within fully weak higher-categorical settings.

Abstract

We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.

Modular categories as representations of the 3-dimensional bordism 2-category

TL;DR

This work completes a program linking 3D extended TQFTs to modular tensor categories by showing linear representations of structured bordism 2-categories correspond to modular tensor categories equipped with square-root data for the anomaly/global dimension. It develops a robust 2-categorical framework: finite presentations of symmetric monoidal 2-categories, their linear representations in , and a skein-theoretic interior-string-diagram calculus that recovers modular and ribbon structures. The key contributions include establishing a bijection between extended 3D TQFTs and MTCs with square-root anomaly, and outlining how various bordism structures (oriented, signature, ) correspond to modular data with corresponding root choices. The results pave the way for explicit skein-theoretic computations in extended TQFTs and solidify the role of modular tensor categories as complete algebraic receivers for 3D bordism theories within fully weak higher-categorical settings.

Abstract

We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.

Paper Structure

This paper contains 12 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Modular tensor categories and TQFTs
  3. TQFTs and modular functors
  4. Modular functors and modular tensor categories
  5. Universal skein theory for TQFTs
  6. Overview
  7. Conventions
  8. Acknowledgments
  9. Presentations and representations
  10. Presentations of symmetric monoidal 2-categories
  11. Linear categories and bimodules
  12. Extending presentations

Key Result

Theorem 1

Linear representations of the componentwise-signature bordism 2- category ${\mathbf{Bord}^{\mathrm{csig}}_{1,2,3}}\xspace$ are classified by modular tensor categories (MTCs) equipped with a square root of the anomaly of each factor.

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Theorem 2.1: CSPthesis, Theorems 2.96 and 2.78
  • Definition 2
  • Lemma 1: pstragowski-thesis
  • Definition 3: Linear category, enriched tensor product
  • Definition 4
  • ...and 12 more