Decorated TQFTs and their Hilbert Spaces

TL;DR

This paper explores decorated topological quantum field theories that compute refined invariants of three-manifolds, proposing a framework for their Hilbert spaces and construction methods.

Contribution

It introduces a method to derive decorated invariants via cutting and gluing, and proposes a new approach to assign Hilbert spaces to surfaces within the T framework.

Findings

Description of how to obtain decorated invariants through cutting and gluing

Proposal for Hilbert spaces associated with surfaces in T

Connection to the  invariant 

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 $\hat{Z}(q)$ 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 $\hat{Z}$-TQFT.