Anomalies of $(1+1)D$ categorical symmetries

TL;DR

The work develops a bulk, Drinfeld-center–based criterion for anomalies of fusion category symmetries by introducing magnetic Lagrangian algebras in $\mathcal{Z}[\mathcal{A}]$ that can realize symmetric gapped $(1+1)D$ phases; anomaly-freedom is equivalent to the existence of a magnetic Lagrangian algebra $\mathcal{L}_m$ intersecting the electric one trivially. It recasts anomaly detection beyond fiber functors by exploiting obstructions such as insufficient bosons in the center, and applies the framework to Tambara–Yamagami categories, reproducing known fiber-functor results and providing computable criteria via boson counting. The paper also analyzes the gauging of duality symmetries to obtain centers $\mathcal{Z}[\mathrm{TY}_G^{\chi,\epsilon}]$, deriving explicit anomaly patterns for $G=\mathbb{Z}_N\times\mathbb{Z}_N$ with diagonal and off-diagonal bicharacters and various $\epsilon$, thereby connecting algebraic data to physical 1+1D SPT classifications. Overall, the approach offers a scalable, bulk-oriented route to classifying and understanding anomalies of non-invertible and abelian fusion category symmetries with potential extensions to higher dimensions and lattice realizations.

Abstract

We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category $\mathcal{A}$ describes a $(2+1)D$ topological order whose gapped boundaries enumerate all $(1+1)D$ gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the \emph{electric} Lagrangian algebra, that describes a phase with $\mathcal{A}$ fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in $\mathcal{A}$. We observe that if there exists a different gapped boundary, given by a \emph{magnetic} Lagrangian algebra, then there exists a gapped phase where $\mathcal{A}$ is not spontaneously broken at all, which means that $\mathcal{A}$ is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to $\mathcal{A}$ being anomaly-free. As an application, we consider the Drinfeld centers of $\mathbb{Z}_N\times\mathbb{Z}_N$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.

Figures (3)

• Figure 1: The pentagon equation is a consistency equation ensuring the two paths to the same configuration given by the top part of the pentagon and the bottom part of the pentagon match. This puts strict constraints on the allowed $F$ symbols. For fusion categories describing invertible symmetries, the pentagon equation is equivalent to the cocycle condition on the $F$ symbol. Note that we have omitted the arrows on the anyon lines for clarity of the figure.
• Figure 2: We study the fusion category symmetry $\mathcal{A}$ using the Drinfeld center $\mathcal{Z}[\mathcal{A}]$ which describes a $(2+1)D$ topological order. Gapped phases of a $(1+1)D$ system with symmetry $\mathcal{A}$, which may be spontaneously broken, are in one-to-one correspondence with Lagrangian algebras $\mathcal{L}$ of $\mathcal{Z}[\mathcal{A}]$. $\mathcal{L}$ describes which anyons can be annihilated at the boundary. In this figure, $a$ is one such anyon.
• Figure 3: The boundary pentagon equation is a consistency equation ensuring the two paths to the same configuration given match. We have again omitted the arrows on the anyon lines for clarity of the figure.