For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
physics, mathematical physics, philosophy of physics
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Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
The Yetter model or Crane-Yetter model is a 4d TQFT sigma-model quantum field theory whose target space is a discrete 2-groupoid and whose background gauge field is a circle 4-bundle.
Together with the Dijkgraaf-Witten model these form the first two steps in filtering of target spaces by homotopy type truncation of ∞-Chern-Simons theory with discrete target spaces. It is hence also an example of a 4d Chern-Simons theory.
The 3d TQFT Turaev-Viro model is a boundary field theory of the Yetter model (Barrett&Garci-Islas&Martins 04, theorem 2). Related discussion is in Freed “4-3-2 8-7-6”.
Fix
$G$ a discrete 2-group; write $\mathbf{B}G$ for its delooping 2-groupoid;
$\alpha : \mathbf{B}G \to \mathbf{B}^4 U(1)$ a characteristic class with coefficients in the circle 4-group. This is equivalently a cocycle in degree $4$ group cohomology
The Yetter-model is the ∞-Dijkgraaf-Witten theory induced by this data.
The model without a background gauge field/cocycle was considered in
The effect of having a nontrivial group 4-cocycle was considered (but now only on a 1-group) in
D. Birmingham, M. Rakowski, On Dijkgraaf-Witten Type Invariants, Lett. Math. Phys. 37 (1996), 363.
Marco Mackaay, Spherical 2-categories and 4-manifold invariants, Adv. Math. 153 (2000), no. 2, 353–390. (arXiv:math/9805030) .
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)
The reinterpretation of the “state sum” equation used in the above publications as giving homomorphisms of simplicial sets/topological spaces is given in
and then extended to colorings in homotopy n-types in
See also
which has some remarks about higher (2-)group cocycles towards the end.