Decorated TQFTs and their Hilbert Spaces

We discuss topological quantum field theories that compute topological invariants which depend on additional structures (or decorations) on three-manifolds. The $q$-series invariant $\widehat{Z}(q)$ proposed by Gukov, Pei, Putrov and Vafa is an example of such an invariant. We describe how to obtain these decorated invariants by cutting and gluing, and make a proposal for Hilbert spaces that are assigned to two-dimensional surfaces in the $\widehat{Z}$-TQFT.