Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

TL;DR

This work extends the notion of symmetry to fusion category actions in 1+1D, framing RG flows and phase structure through anomaly in-flow from a 2+1D Turaev-Viro/Levin-Wen theory. It establishes a precise anomaly-vanishing criterion: a 1+1D fusion-category symmetry is non-anomalous and supports a symmetric gapped phase if and only if the fusion category admits a fiber functor, linking symmetric gapped phases to $\,\mathcal{A}$-module categories and SPT-like fiber-functor data. The authors develop gauge-theoretic techniques to classify gapped phases for Tambara-Yamagami categories, including self-duality, duality-twisted sectors, and edge modes between fiber functors, with concrete examples from ${\rm Rep}(D_8)$, ${\rm Rep}(H_8)$, and Ising$^2$ CFT. They apply these ideas to finite gauge theories, mapping gapped phases to Higgs/deconfined patterns and deriving detailed phase structures for $D_8$ and $Q_8$, before analyzing six Ising$^2$-related self-dualities and triality phenomena on the $c=1$ moduli space. The results illuminate how non-invertible symmetries constrain RG flows, yield edge-mode protected phases, and motivate exploration of higher fusion-category symmetries and lattice realizations via MPOs.

Abstract

We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.

Figures (7)

• Figure 1: Pentagon identity for the F-symbols (the label $c$ is summed over the simple TDLs).
• Figure 2: A generic 6-fold junction of surfaces. The six $\mathcal{A}$ labels live along the green and brown half planes and along the four yellow quadrants. The Turaev-Viro state-sum weight is a product over all such junctions, each contributing their $F$-symbol. Fusion labels are included along the four three-fold junctions (legs of the cross on the yellow plane) if there are multiplicities. In the triangulation version of the state-sum, this point-like singularity occurs inside a Poincaré dual tetrahedron, drawn in black. The four three-fold junctions are dual to the triangles and the six planar regions are dual to the edges (and intersect their corresponding edge in the figure).
• Figure 3: When the boundary (grey plane) absorbs a six-fold junction (see Fig. \ref{['fig6junction']}), the boundary lines (black) perform an $F$-move (compare Eq. \ref{['eqnFmove']}). This is also a picture of the pentagon equation for module categories (which classify general gapped boundary conditions, see Section \ref{['subsecanomvanishing']}) where the left picture contributes two $\mu$'s, the center picture contributes an $F$, and the right picture contributes two more $\mu$'s.
• Figure 4: An $\mathcal{A}$ defect $a$ (red curve) passes from the fixed boundary condition on the left to a symmetry-preserving boundary condition on the right, where it becomes invisible. The crossing point gives a map from $a \otimes J \to J$, where $J$ is any boundary-changing junction from the fixed boundary condition to the symmetry-preserving one (blue curve). This proves that the associated module category ${\rm hom}(\mathbb{A},\mathbb{B})$ of these boundary-changing junctions has one simple object.
• Figure 5: A simple string-net state for the annulus with inner boundary in the fixed boundary condition and outer boundary (red) labelled by simples $m,m' \in \mathcal{M}$. At the right junction we have a morphism $m \to a \otimes m'$. We clean up this state by fusing the $a$ line into the outer boundary. Then, we lift the $a'$ line off the inner boundary and fuse it to the other boundary, leaving the inner boundary with label $1$ and the outer boundary in a superposition of $\mathcal{M}$ labels. Such states thus form a basis of the ground states in this geometry.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3