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))

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)