Extension of the fully extended oriented 3D TQFT from spherical fusion categories to the Turaev–Viro state sum TQFT

Determine whether, for any spherical fusion category C of nonzero dimension, the fully extended oriented 3-dimensional topological quantum field theory associated to C extends the Turaev–Viro state sum TQFT |·|_C defined from C, meaning that the restriction of the fully extended theory to the 3-dimensional bordism category reproduces the assignments of |·|_C.

Background

Extended TQFTs aim to refine standard 3-dimensional TQFTs by assigning algebraic data not only to closed surfaces and 3-cobordisms but also to lower-dimensional strata, fitting into the framework of the cobordism hypothesis. In this survey, the authors discuss a candidate target 3-category (TC) and recall results identifying spherical fusion categories as homotopy SO(3)-fixed points and as appropriate fully dualizable objects, which yield fully extended oriented 3-dimensional TQFTs.

Within this framework, a natural expectation is that the fully extended oriented 3D TQFT associated to a spherical fusion category C should agree with the Turaev–Viro state sum TQFT |·|_C when restricted to the usual 3-dimensional bordism category. The authors explicitly indicate that this agreement is conjectural, highlighting an unresolved question concerning the precise relationship between the fully extended and state sum constructions.