Categorical Symmetries via Operator Algebras

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 ${\text{Rep}}^k(C_0(G))$. 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 $C^{\ast }$-algebra of the Fell line bundle ${\Sigma }_k$ over the conjugation action groupoid $G/{/}_{\mathrm{Ad}}G$, denoted by $\text{Rep}(C^{\ast }(G/{/}_{\mathrm{Ad}}G,{\Sigma }_k))$, where the twist is characterized by the transgression $\tau (k)\in H^2(G/{/}_{\mathrm{Ad}}G,U(1))$. 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.