Equivalence of field theories: Crane-Yetter and the shadow

TL;DR

This paper proves the long-standing conjecture that the shadow model and the Crane-Yetter invariant, two major invariants of smooth 4-manifolds, are in fact equivalent, unifying different approaches in topological quantum field theory.

Contribution

It establishes the equivalence of the shadow and Crane-Yetter invariants, clarifying their relationship and providing a dictionary between the models.

Findings

Proved the equivalence of the shadow and Crane-Yetter invariants.

Provided a survey of the shadow construction and a dictionary between models.

Highlighted the limitations of semisimple models in topological quantum field theory.

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.