Equivalence of field theories: Crane–Yetter and the shadow

This work solves a 28-year conjecture by showing that two major invariants of smooth 4-manifolds, the shadow model (motivated by statistical mechanics [V. G. Turaev, Topology of shadows (1991)]) and the simplicial Crane–Yetter model (motivated by topological quantum field theory [L. Crane and D. Yetter, A categorical construction of 4D topological quantum field theories, in Quantum Topology (World Scientific, 1993), pp. 120–130, https://doi.org/10.1142/9789812796387_0005 ]), are in fact equal. These invariants, both of which degenerate to the 3D Witten–Reshetikhin–Turaev model in a special case, had been open for years to clarify their relationship. Despite the seeming difference in their origins and formal constructions, we prove their equivalence. Along the way, we sketch a dictionary between the two models, provide a brief survey of the shadow construction à la Turaev, and suggest once again that the semisimple models have reached their limits.