Decorated TQFTs and their Hilbert spaces

Abstract

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.