Fully local Reshetikhin-Turaev theories

We define a symmetric tensor enhancement $\mathrm{E}\mathbb{F}$ with full duals of the 3-category $\mathbb{F}$ of fusion categories in which every Reshetikhin--Turaev theory has a fully local realization. Our $\mathrm{E}\mathbb{F}$ is a direct sum of invertible $\mathbb{F}$-modules, indexed by a $\mu_6$-extension of the Witt group $W$ of non-degenerate braided fusion categories. Similarly, we enhance the 3-category $S\mathbb{F}$ of fusion super-categories to a symmetric tensor 3-category $\mathrm{E} S\mathbb{F}$ with full duals, which is a sum of invertible $S\mathbb{F}$-modules, indexed by an extension of the super-Witt group $SW$ with kernel the Pontrjagin dual of the stable stem $\pi_3^s$. The unit spectrum of $\mathrm{E}S\mathbb{F}$ is the connective cover of the Pontrjagin dual of $\mathbb{S}^{-3}$. We discuss tangential structures and central charges of the resulting TQFTs. We establish Spin-invariance of fusion supercategories and relate SO-invariance structures to modular and spherical structures. This confirms some conjectures from arXiv:1312.7188.