3d Modularity Revisited

The three-manifold topological invariants $\hat Z$ capture the half-index of the three-dimensional theory with ${\mathcal{N}}=2$ supersymmetry obtained by compactifying the M5 brane theory on the closed three-manifold. In 2019, surprising general relations between the $\hat Z$-invariants, quantum modular forms, and vertex algebras have been proposed. In the meanwhile, an extensive array of examples have been studied, but several important general structural questions remain. First, for many three-manifolds we have seen hints of concrete $\widetilde {\rm SL}_2(\mathbb{Z})$ representations underlying the different $\hat Z$-invariants of the given manifolds. At the same time, these invariants appear to only span a subspace of the representation, and the role of the latter remains mysterious. We elucidate the meaning of the modular group representation, realized as vector-valued quantum modular forms, by first proposing the analogue $\hat Z$-invariants with supersymmetric defects, and subsequently showing that the full vector-valued quantum modular form for $\widetilde {\rm SL}_2(\mathbb{Z})$ is precisely the object capturing all the $\hat Z$-invariants of a given three-manifold, when the newly defined defects $\hat Z$-invariants are included. Second, it was expected that matching radial limits is a key feature of $\hat Z$-invariants when changing the orientation of the plumbed three-manifold, suggesting the relevance of mock modularity. We substantiate the conjecture by providing explicit proposals for such $\hat Z$-invariants for three three-manifolds and verify their mock modularity and limits. Third, we initiate the study of the vertex algebra structure of the mock type invariants by showcasing a systematic way to construct cone vertex operator algebras associated to these mock invariants, which can be viewed as the partner of logarithmic vertex operator algebras in this context.