Fully local Reshetikhin-Turaev theories

Abstract

Completing an arc of research initiated by Reshetikhin--Turaev and Witten, we construct fully local 3-dimensional topological field theories from non-degenerate braided fusion categories. We do so by defining 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 point generator. This $\mathrm{E}\mathbb{F}$ is a direct sum of invertible $\mathbb{F}$-modules, indexed by a ${\mu }_6$-extension of the Witt group 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 a ${\mu }_{24}$-extension of the super-Witt group. 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 super-categories, and relate SO-invariance structures to modular and spherical structures, confirming some conjectures from arXiv:1312.7188.

arXiv comment33 pages, 4 figures. Version 2 improves the exposition, replaces an overly vague argument in Section 7, and removes Section 8 (on projective symmetry, now relegated to a future paper). Version 3 has only minor edits