Evaluating TQFT invariants from $G$-crossed braided spherical fusion categories via Kirby diagrams with 3-handles

Abstract

A family of TQFTs parametrised by $G$-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of a $G$-crossed braided spherical fusion category.

This reformulation is very efficient for explicit calculations, and the manifold invariant is calculated for several examples. It is also shown that in most cases, the invariant is multiplicative under connected sum, which implies that it does not detect exotic smooth structures.