Is Crane--Yetter fully extended?

TL;DR

This paper investigates whether the Crane-Yetter TQFT can be fully extended using tools from stable homotopy theory, revealing a classification involving roots of unity and infinite classes of extensions.

Contribution

It provides a classification of fully extended invertible 4D TQFTs related to Crane-Yetter, using homotopy-theoretic methods and identifying an infinite family parametrized by a  extension of the Witt group.

Findings

Classifies extensions of invertible 4D TQFTs with specific homotopy properties.

Shows infinitely many equivalence classes of fully extended TQFTs reproducing Crane-Yetter.

Clarifies previous claims and raises questions about natural choices of fixed point data.

Abstract

We revisit the question of whether the Crane-Yetter topological quantum field theory (TQFT) associated to a modular tensor category admits a fully extended refinement. More specifically, we use tools from stable homotopy theory to classify extensions of invertible four-dimensional TQFTs to theories valued in symmetric monoidal 4-categories whose Picard spectrum has nontrivial homotopy only in degrees 0 and 4. We show that such extensions are classified by two pieces of data: an equivalence class of an invertible object in the target and a sixth root of unity. Applying this result to the 4-category $\mathbf{BrFus}$ of braided fusion categories, we find that there are infinitely many equivalence classes of fully extended invertible TQFTs reproducing the Crane-Yetter partition function on top-dimensional manifolds, parametrized by a $\mathbb{Z}/6$-extension of the Witt group of nondegenerate braided fusion categories. This analysis clarifies common claims in the literature and raises the question of how to naturally pick out the $SO(4)$-fixed point data on the framed TQFT which assigns the input braided fusion category to the point so that it selects the Crane-Yetter state-sum.