Unit inclusion in a non-semisimple braided tensor category and non-compact relative TQFTs

Abstract

The inclusion of the unit in a braided tensor category $\mathcal{V}$ induces a 1-morphism in the Morita 4-category of braided tensor categories $BrTens$. We give criteria for the dualizability of this morphism. When $\mathcal{V}$ is a semisimple (resp. non-semisimple) modular category, we show that the unit inclusion induces under the Cobordism Hypothesis a (resp. non-compact) relative 3-dimensional topological quantum field theory. Following Jordan-Safronov, we conjecture that these relative field theories together with their bulk theories recover Witten-Reshetikhin-Turaev (resp. De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel) theories, in a fully extended setting. In particular, we argue that these theories can be obtained by the Cobordism Hypothesis.

Significant changes from the first version including conjectures for SO(4) and SO(5) homotopy fixed point structures and a more appropriate statement of the cobordism hypothesis for twisted field theories. 35 pages. To appear in Geometry&Topology

References in corpus (2)

• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• Invertible braided tensor categories