Equivalence of field theories: Crane-Yetter and the shadow

Abstract: It has been open for years to clarify the relationship between two smooth 4-manifolds invariants, the shadow model (motivated by statistical mechanics [Tur91]) and the simplicial Crane-Yetter model (motivated by topological quantum field theory [CY93]), both of which degenerate to the 3D Witten-Reshetikhin-Turaev model in a special case. Despite the seeming difference in their origins and formal constructions, we show that they are in fact equal. Along the way, we sketch a dictionary between the shadow model and the Crane-Yetter model, provide a brief survey to the shadow construction a la Turaev, and suggest once again that the semisimple models have reached their limits.