Categorical Symmetries via Operator Algebras

TL;DR

This paper develops a categorical framework for symmetries in 2D quantum field theories with anomalies, linking them to operator algebras and groupoid representations, and explores implications for anyon braiding and gauging symmetries.

Contribution

It introduces a novel operator algebraic description of symmetry categories with anomalies and computes their Drinfeld centers and braiding structures.

Findings

Categorical symmetry associated to 2D QFTs with anomalies is modeled by twisted measurable fields of Hilbert spaces.

The Drinfeld center of the symmetry category relates to unitary representations of a groupoid C*-algebra with a twist.

Explicit calculations of anyon braiding and physical examples for abelian and non-abelian groups are provided.

Abstract

We propose that the symmetry category associated to a 2D quantum field theory with 0-form $G$-symmetry with 't Hooft anomaly $k\in H^4(BG,\mathbb{Z})$ for a large class of Lie groups $G$ is the category of twisted measurable fields of Hilbert spaces over $G$ denoted by $\mathrm{Hilb}^k(G)$, which is equivalent to the category of unitary representations of $C_0(G)$ with convolution product twisted by a multiplicative bundle gerbe labeled by $k$ denoted by $\textbf{Rep}^k(C_0(G))$. We find that the Drinfeld center of the symmetry category $\mathcal{Z}(\mathrm{Hilb}^{k}(G))$ equivalent to the category of unitary representations of the groupoid $C^*$-algebra of the Fell line bundle $\Sigma_k$ over the conjugation action groupoid $G//_{\rm Ad} G$, denoted by $\textbf{Rep}(C^*(G//_{\rm Ad}G,\Sigma_k))$, where the twist is characterized by the transgression $\tau(k)\in H^2(G//_{\rm Ad}G,U(1))$. To the full generality, our framework applies to a Lie group $G$ that is a direct product of a compact connected Lie group and a number of $\mathbb{R}$ or $GL(1,\mathbb{C})$ factors. We compute the braiding of anyon lines in the bulk 3D SymTFT from this formalism. Finally we provide physical examples for abelian and non-abelian $G$, and discuss the physical consequences of flat gauging continuous global symmetries.