nLab Yetter model

Idea

The 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 Yetter model is not the same as the Crane-Yetter model.

Definition

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

  $$[\alpha] \in H_{Grpd}^4(G, U(1)) \,.$$

The Yetter-model is the ∞-Dijkgraaf-Witten theory induced by this data.

Related concepts

• ∞-Chern-Simons theory

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

  • 2d Chern-Simons theory

  • 3d Chern-Simons theory

  • 4d Chern-Simons theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

  • infinite-dimensional Chern-Simons theory

  • AKSZ sigma-model

References

The model without a background gauge field/cocycle was considered in

• David Yetter, TQFTs from homotopy 2-types , Journal of Knot Theory and its Ramifications 2 (1993), 113-123.

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)

  .

The reinterpretation of the “state sum” equation used in the above publications as giving homomorphisms of simplicial sets/topological spaces is given in

• Tim Porter, Interpretations of Yetter’s notion of $G$-coloring : simplicial fibre bundles and non-abelian cohomology, Journal of Knot Theory and its Ramifications 5 (1996) 687-720,

and then extended to colorings in homotopy n-types in

• Tim Porter, Topological Quantum Field Theories from Homotopy n-types, Journal of the London Math. Soc. (2) 58 (1998) 723-732.

See also

• João Faria Martins and Tim Porter, On Yetter’s invariants and an extension of the Dijkgraaf-Witten invariant to categorical groups, Theory and Applications of Categories, Vol. 18, 2007, No. 4, pp 118-150.

  (TAC)

which has some remarks about higher (2-)group cocycles towards the end.