Decorated TQFTs and their Hilbert spaces

bstract We discuss topological quantum field theories that compute topological invariants which depend on additional structures (or decorations) on three-manifolds. The q -series invariant Ẑ ( 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 Ẑ -TQFT.