nLab Crane-Yetter model

Idea

The Crane-Yetter model is a 4d TQFT (not to be confused with the Yetter model).

It was defined originally as a state sum in State-Sum Invariants of 4-Manifolds. One labels the elements of a triangulation with objects and morphisms of a ribbon fusion category, usually the representation category of a quasitriangular Hopf algebra, or quantum group.

(The Yetter model, on the contrary, is defined for a finite 2-group.)

Relation to Turaev-Viro model on the boundary

The 3d TQFT Turaev-Viro model is a boundary field theory of the Crane-Yetter model (Barrett, Garcia-Islas, & Martins 2004, theorem 2; Tham 2021 §9.3).

As a fully extended TQFT

It can be understood as an extended topological quantum field theory, see 4-3-2 8-7-6.

Relation to BF-theory and Dijkgraaf-Witten theory

It is expected that the classical limit of the Crane-Yetter TQFT is BF theory (Baez 1995, cf. this blog post).

A state sum model like the Crane-Yetter model, but with a classical Lie group, would be the Ooguri model?. We could understand the Crane-Yetter as a Dijkgraaf-Witten theory with a quantum group instead of a classical group, whereas the Yetter model is a Dijkgraaf-Witten theory with a 2-group.

References

• Louis Crane, Louis Kauffman, David Yetter, State-Sum invariants of 4-manifolds I, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997) (arXiv:hep-th/9409167, pdf, doi:10.1142/S0218216597000145)

Relation to the Turaev-Viro model:

• John Barrett, J. Garcia-Islas, João Faria Martins, Observables in the Turaev-Viro and Crane-Yetter models, J. Math. Phys. 48:093508, 2007 (arXiv:math/0411281)

• Ying Hong Tham: On the Category of Boundary Values in the Extended Crane-Yetter TQFT, PhD thesis, Stony Brook (2021) [arXiv:2108.13467]

See also:

• John Baez: 4-Dimensional BF Theory as a Topological Quantum Field Theory [arXiv:q-alg/9507006]