Fully-Dualizable and Invertible ${\mathcal{E}}_n$-Algebras

We prove a conjecture of Brochier, Jordan, Safronov, and Snyder [BJSS21], first formulated by Lurie [Lur09b], characterizing fully-dualizable and invertible ${\mathcal{E}}_n$-algebras viewed as objects in the higher Morita categories ${\mathbf{M}\mathbf{o}\mathbf{r}}_n(\mathcal{V})$ [Lur09b, Sch14, Hau17b, Hau23]. In other words, we characterize those ${\mathcal{E}}_n$-algebras which give rise to $(n+1)$-dimensional topological quantum field theories (TQFT), and those which give rise to invertible theories.