Unoriented 3d TFTs

Abstract

This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We show how Turaev-Viro construction can be generalized to unoriented 3d TFTs. On the other hand, it is known that the "fermionic" versions of oriented TFTs, known as Spin-TFTs, can be constructed in terms of "shadow" TFTs which are ordinary oriented TFTs with an anomalous $\mathbb{Z}_2$ 1-form symmetry. We generalize this correspondence to Pin$^+$-TFTs by showing that they can be constructed in terms of ordinary unoriented TFTs with anomalous $\mathbb{Z}_2$ 1-form symmetry having a mixed anomaly with time-reversal symmetry. The corresponding Pin$^+$-TFT does not have any anomaly for time-reversal symmetry however and hence it can be unambiguously defined on a non-orientable manifold. In case a Pin$^+$-TFT admits a topological boundary condition, one can combine the above two statements to obtain a Turaev-Viro-like construction of Pin$^+$-TFTs. As an application of these ideas, we construct a large class of Pin$^+$-SPT phases.