# nLab Yetter model

### Context

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

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.

## Properties

### Relation to Turaev-Viro model on the boundary

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 Freed4-3-2 8-7-6”.

## 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.

## 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

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.

• 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.

Revised on October 6, 2013 22:31:00 by Urs Schreiber (195.37.209.182)