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 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.

Automated summary

This article discusses topological quantum field theories that compute topological invariants dependent on additional structures or decorations on three-manifolds. An example of such an invariant is the q-series invariant Ẑ(q) proposed by Gukov, Pei, Putrov, and Vafa. The article describes the process of obtaining these decorated invariants through cutting and gluing, and proposes Hilbert spaces assigned to two-dimensional surfaces in the Ẑ-TQFT.