Unoriented 3d TFTs

TL;DR

The paper develops a framework to define unoriented 3d TFTs by using orientation-reversing defects and a twisted, G-graded fusion-category, enabling TV-like state-sum constructions on non-orientable manifolds. It then extends the shadow formalism from Spin to Pin^+-theories, showing how Pin^+-TFTs arise from unoriented shadows with a mixed time-reversal anomaly and how to take products and build Pin^+-SPT phases. It provides explicit constructions and classifications for Pin^+-SPTs, including Gu-Wen-type phases with symmetry G and Ising-like examples without symmetry, and it introduces the notion of twisted Drinfeld centers and shadow products to handle nontrivial orientation-reversal data. Overall, the work furnishes a practical, algebraic toolkit for constructing and classifying Pin^+-invariant topological phases in 3d and establishes a path toward a systematic treatment of unoriented and fermionic topological field theories.

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.

Figures (31)

• Figure 1: (a) A boundary line $L$. (b) The dual line $L^*$ is defined by reversing the orientation of $L$.
• Figure 2: A morphism $m$ between outgoing lines $A_1$, $A_2$ and $A_3$ corresponds to a state $m$ in the Hilbert space on a disk with boundary punctures $A_1$, $A_2$ and $A_3$. Consider on a hemisphere geometry with a boundary on the spherical part and the disk shown in (b) being the cross-section. The state shown in (b) is produced on the cross-section if the boundary has the graph shown on (a) inserted on it such that $A_i$ end on their respective punctures.
• Figure 3:
• Figure 4: Canonical maps: (a) Associator $a(A,B,C)$, (b) Evaluation $e_L$, and (c) Co-evaluation $i_L$.
• Figure 5: (a) Graphical representation of the chosen basis. (b) Graphical representation of the dual basis.