Recovering Mochida’s invariants from graded-fusion 2-categories

Determine whether, in appropriate cases, the invariants of Mochida constructed from finite-type involutory quasitriangular Hopf G-algebras can be recovered as state-sum invariants arising from a spherical (1 → 1 → G)-graded fusion 2-category.

Background

The paper constructs quantum homotopy invariants of closed oriented 4-manifolds equipped with flat 3-bundles, using spherical fusion 2-categories graded by a 2-crossed module. Section 8.6 compares this construction with Mochida’s invariants of flat G-bundles, which are obtained from finite-type involutory quasitriangular Hopf G-algebras through a four-dimensional Hennings-type construction.

The authors suggest that, in suitable cases, the delooping of the representation category of such a Hopf G-algebra should produce a spherical graded fusion 2-category. They explicitly formulate as a conjecture the unresolved question of whether Mochida’s invariants arise from the paper’s state-sum construction in this way.