nLab 3d TQFT

Idea

TQFT of dimension 3.

Constructions

From 3-modules realized as sesquialgebra, bialgebras, Hopf algebra, quantum groups, fusion categories:

• Reshetikhin-Turaev model

• Turaev-Viro model

Examples

• 3d Chern-Simons theory

  • Dijkgraaf-Witten theory

  • analytically continued Chern-Simons theory

  • Courant sigma-model

• Levin-Wen model

Related concepts

• TQFT

  • 2d TQFT

    • 2d Chern-Simons theory

  • 3d TQFT

    • 3d Chern-Simons theory

  • 4d TQFT

    • 4d Chern-Simons theory

References

• Kevin Walker, On Witten’s 3-manifold invariants (1991) pdf

• Scott Carter, Louis Kauffman, Masahico Saito, Diagrams, Singularities, and Their Algebraic Interpretations, in “10th Brazilian Topology Meeting, S. Carlos, July 22 26, 1996 (pdf (dead))

• Kursat Sozer, Alexis Virelizier, 3d TQFTs and 3-manifold invariants, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2401.10587]

Discussion via the cobordism hypothesis with fusion categories as fully dualizable objects is in

• Chris Douglas, Chris Schommer-Pries, Noah Snyder, The Structure of Fusion Categories via 3D TQFTs (talk pdf)

• Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories (arXiv:1312.7188)

• Bruce Bartlett, Christopher Douglas, Christopher Schommer-Pries, Jamie Vicary, Extended 3-dimensional bordism as the theory of modular objects (arXiv:1411.0945)

Constructions in the generality of non-semisimple tensor categories:

• Christian Blanchet, Marco De Renzi, Modular Categories and TQFTs Beyond Semisimplicity (arXiv:2011.12932)