Categorical Time-Reversal Symmetries

Abstract

The classification of phases using categorical symmetries has greatly expanded the landscape of gapped and gapless phases. So far, however, these developments have largely been restricted to phases with unitary (higher-)categorical symmetries over $\mathbb{C}$. In this work, we incorporate anti-unitary symmetries, such as time-reversal symmetry $\mathbb{Z}_2^T$, and show that the relevant physical structures are naturally described by fusion categories over $\mathbb{R}$. A class of real fusion categories, which we call Galois-real fusion categories, provides the correct categorical model for anti-unitary symmetries. A simple example is the time-reversal symmetry $\mathbb{Z}_2^T$ itself. We discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories $(G^T)^ω$ and $\mathsf{Rep}(G^T)$ associated to anti-linear groups $G^T$, as well as non-invertible time-reversal symmetries described by a real analogue of Tambara--Yamagami fusion categories. We then classify gapped phases enriched with anti-linear symmetries in terms of module categories over Galois-real fusion categories. We furthermore apply the categorical formulation to prove dualities (i.e. gauge or Morita equivalences) of anti-linear symmetries generated by gauging subgroups. Complementing this, we also develop a Symmetry Topological Field Theory (SymTFT) framework, in which Galois-real fusion categories arise as boundary conditions of a $\mathbb{Z}_2^T$-enriched SymTFT. Morita equivalent anti-linear symmetries are shown to arise as different boundaries of the same $\mathbb{Z}_2^T$-enriched SymTFT.

Figures (7)

• Figure 1: Full structure of $(1+1)$d gapped phases with ${\mathbb Z}_4^T$ symmetry. Each vertex is a gapped phase. Each arrow is a category of symmetric domain walls. The blue categories are Morita equivalent symmetry (i.e. Galois-real) categories. The category $\mathsf{Rep}({\mathbb Z}_4^T)$ in contrast is not an admissible symmetry category even though it is Morita equivalent to $\mathsf{Vec}_{{\mathbb Z}_4^T}$, as it is an $\mathbb{R}$-real fusion category.
• Figure 2: A simple line $a$ in a Galois-real fusion category can act on scalars as either identity or complex conjugation, depending on the Galois indicator $s(a)$.
• Figure 3: We can move a scalar $z$ from the LHS to the RHS in two ways, yielding relation $s(a)s(b)=s(c)$, assuming the junction operator is nonzero.
• Figure 4: A defect network for a generalized anti-linear symmetry. The network is a diagram valued in the symmetry category $\mathcal{C}_1\oplus \mathcal{C}_T$. The lines in blue/red are liner/anti-linear defects. The theories on different sides of an anti-linear defect are complex conjugate of each other. If one traces along the anti-linear defects, one gets a cycle (red line) Poincaré dual to $w_1(M)$.
• Figure 5: T-enriched SymTFT quiche for the Galois-real category $\mathcal{C}_1 \oplus \mathcal{C}_T$: The fusion category $\mathcal{C}_1$ ($\mathcal{C}_1^*$) is the canonical boundary of the Levin-Wen topological order $\mathfrak{Z}(\mathcal{C}_1)$ ($\mathfrak{Z}(\mathcal{C}_1^*)$). The ${\mathbb Z}_2^T$ (anti-unitary) automorphism defect in the bulk of the SymTFT ends in $\mathcal{C}_T$, which is an invertible $\mathcal{C}_1$-$\mathcal{C}_1^*$ bimodule. The boundary of the $D_T$ surface is a domain wall between two vacua on the boundary (denoted $|\uparrow\rangle$ and $|\downarrow\rangle$).

Theorems & Definitions (36)

• Theorem : \ref{['thm_gauge_subgroup']}
• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Theorem 3.2
• Definition 3.3
• Theorem 3.4
• Theorem 4.1
• Theorem 4.2
• Example 4.3