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.

41 pages, 31 figures, v2: additional references, v3: minor revision

References in corpus (3)

• Generalized Global Symmetries
• The "Parity" Anomaly On An Unorientable Manifold
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

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