Equivalence of field theories: Crane-Yetter and the shadow

Abstract

This work solves a 28-year conjecture by showing that two major invariants of smooth 4-manifolds, the shadow model (motivated by statistical mechanics [Tur91]) and the simplicial Crane-Yetter model (motivated by topological quantum field theory [CY93]), 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 \`a la Turaev, and suggest once again that the semisimple models have reached their limits.