nLab D=4 TQFT

Idea

TQFT of dimension 4.

Constructions

In analogy to how D=3 TQFTs are induced from quantum groups/Hopf algebras/ and generally bialgebras (hence 3-modules, the higher space of quantum states assigned to the point which by the cobordism theorem defines the theory) one may build D=4 TQFTs from higher analogs of these, namely models of 4-modules given by algebraic structures such as trialgebras and Hopf categories.

Examples

• Yetter model

• Kapustin-Witten TQFT

• D=4 Chern-Simons theory

• Walker-Wang model

• 3d toric code

Related concepts

• loop braid group

• TQFT

  • D=0 TQFT

  • D=2 TQFT

    • D=2 Chern-Simons theory

  • D=3 TQFT

    • D=3 Chern-Simons theory

  • D=4 TQFT

    • D=4 Chern-Simons theory

References

Original 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)

• Louis Crane, Igor Frenkel, Four dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994) 5136-5154, (arXiv:hep-th/9405183)

See also

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

Construction via factorization homology from braided tensor categories (with relation to double affine Hecke algebras) is discussed in

• David Ben-Zvi, Adrien Brochier, David Jordan, Integrating quantum groups over surfaces: quantum character varieties and topological field theory (arXiv:1501.04652)

From fully dualizable braided tensor categories, via the cobordism hypothesis:

• Adrien Brochier, David Jordan, Pavel Safronov, Noah Snyder, Invertible braided tensor categories (arXiv:2003.13812)

As descriptions of topological insulators/topological phases of matter:

• Kevin Walker, Zhenghan Wang, (3+1)-TQFTs and Topological Insulators, Frontiers of Physics volume 7, pages 150–159 (2012) (arXiv:1104.2632, doi:10.1007/s11467-011-0194-z)

  (see Walker-Wang model)

• Clement Delcamp, Excitation basis for (3+1)d topological phases, Journal of High Energy Physics 2017 (2017) 128 (arXiv:1709.04924 doi:10.1007/JHEP12(2017)128)

Many more references should eventually go here…