[Paper Review] Anomalies of $(1+1)D$ categorical symmetries

Carolyn Zhang, Clay Córdova|arXiv (Cornell University)|Apr 3, 2023

Homotopy and Cohomology in Algebraic TopologyMathematics30 citations

TL;DR

The paper introduces a bulk (2+1)D approach using Drinfeld centers and magnetic Lagrangian algebras to detect anomalies of fusion category symmetries in (1+1)D, with obstructions and examples from Tambara-Yamagami categories.

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 $\mathbb{Z}_N imes\mathbb{Z}_N$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.

Motivation & Objective

• motivate and classify ’t Hooft anomalies for fusion category symmetries in (1+1)D
• develop a bulk (2+1)D criterion using magnetic Lagrangian algebras in Z[A] to detect anomalies
• connect anomaly existence to obstructions related to boson content in the center and fiber functor data
• apply the framework to Tambara-Yamagami fusion categories, especially ZN×ZN cases, and recover known results from fiber functors
• illustrate an edge- vs bulk-based perspective on non-anomalous vs anomalous fusion category symmetries

Proposed method

• use Drinfeld center Z[A] of a fusion category A to describe a (2+1)D topological order whose gapped boundaries enumerate (1+1)D phases with symmetry A
• identify electric Lagrangian algebra L_e describing fully symmetry-broken boundary and magnetic Lagrangian algebra L_m describing a symmetric, gapped boundary
• require the existence of a magnetic Lagrangian algebra L_m in Z[A] that intersects trivially with L_e to certify anomaly-freedom; obstructions to L_m imply anomalies
• derive computable obstructions to anomaly-freedom from properties of Lagrangian algebras and from boson content in the TQFT (e.g., number of bosons)
• analyze Tambara-Yamagami categories TY_G^χ,ε (especially G ≅ Z_N × Z_N for N=2 and N>2, N odd) and compare with known fiber functor results
• discuss abelian vs non-abelian centers and related boundary condensations and Lagrangian subgroups

Experimental results

Research questions

• RQ1 When does a fusion category symmetry A admit a magnetic Lagrangian algebra in Z[A] that yields a symmetric, gapped (1+1)D boundary?
• RQ2 Can obstructions to the existence of L_m in Z[A] be efficiently computed to detect anomaly of A without constructing all Lagrangian algebras?
• RQ3 How do Tambara-Yamagami categories, especially TY_{Z_N×Z_N}^{χ,ε}, exhibit anomalies or anomaly-free behavior under the magnetic Lagrangian algebra criterion?
• RQ4 What is the relationship between fiber functors, Lagrangian algebras, and the existence of (1+1)D SPTs realizing A?
• RQ5 How do invertible vs non-invertible symmetries manifest anomalies in the Drinfeld center framework?

Key findings

• A fusion category symmetry A is anomaly-free if Z[A] admits a magnetic Lagrangian algebra L_m that intersects trivially with the electric Lagrangian algebra L_e.
• Obstructions to the existence of L_m can yield highly computable criteria for anomaly, even without full L_m construction.
• For abelian centers Z[Vec_G^ω], certain type-I and type-II cocycles lead to anomalies, aligning with known cocycle classifications.
• In Tambara-Yamagami categories with G = Z_N × Z_N, specific choices of N and F symbols produce anomalies consistent with prior results.
• An explicit (illustrative) example shows Vec_{Z_2}^ω is anomaly-free for the trivial ω but anomalous for the nontrivial ω, via the presence/absence of a magnetic Lagrangian subgroup in the corresponding Z center.
• The framework connects (1+1)D SPT classifications and anomaly structure to bulk (2+1)D topological orders and their gapped boundaries.