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3d TQFTs and 3-manifold invariants

Kursat Sozer, Alexis Virelizier

Abstract

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We give an overview of 3-dimensional topological quantum field theories (TQFTs) and the corresponding quantum invariants of 3-manifolds. We recall the main algebraic concepts and constructions, such as modular and spherical fusion categories, the Witten-Reshetikhin-Turaev and Turaev-Viro theories, and the relation between these two TQFTs. We also briefly discuss generalizations of these constructions by providing a (non-exhaustive) review of some recent works on 3-dimensional extended TQFTs, defect TQFTs, homotopy QFTs, and non-semisimple TQFTs.

3d TQFTs and 3-manifold invariants

Abstract

This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics. We give an overview of 3-dimensional topological quantum field theories (TQFTs) and the corresponding quantum invariants of 3-manifolds. We recall the main algebraic concepts and constructions, such as modular and spherical fusion categories, the Witten-Reshetikhin-Turaev and Turaev-Viro theories, and the relation between these two TQFTs. We also briefly discuss generalizations of these constructions by providing a (non-exhaustive) review of some recent works on 3-dimensional extended TQFTs, defect TQFTs, homotopy QFTs, and non-semisimple TQFTs.
Paper Structure (27 sections, 3 theorems, 64 equations)

This paper contains 27 sections, 3 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. Pivotal categories
  4. Penrose graphical calculus
  5. Spherical categories
  6. Braided categories
  7. Ribbon categories
  8. Fusion categories
  9. $6j$-symbols
  10. Modular categories
  11. Three dimensional TQFTs
  12. The surgery approach
  13. Surgery on framed links
  14. The WRT invariants of closed 3-manifolds
  15. The surgery TQFT
  16. ...and 12 more sections

Key Result

Theorem 4.1

The expression is invariant under the Kirby moves on $L$. This expression yields, therefore, a well-defined topological invariant $\mathrm{WRT}_{\mathcal{B}}$ of closed connected oriented 3-manifolds.

Theorems & Definitions (3)

  • Theorem 4.1
  • Theorem 5.1
  • Theorem 6.1