Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization

TL;DR

The paper provides a comprehensive framework to analyze symmetries in 2+1D topological orders using string-net models, unifying invertible (group) and noninvertible (categorical) symmetries as gauge invariances or global symmetries. It develops a two-step construction: first classifying Morita-equivalent string-net inputs with duality maps, then, when inputs are isomorphic, composing with isomorphisms to realize symmetry within the same model. A key result is the identification of the first noninvertible categorical gauge invariance—the Fibonacci 2-category gauge symmetry—inside the doubled Fibonacci order, realized through a noninvertible projection in an enlarged Hilbert space. The framework also provides a concrete criterion to distinguish gauge invariances from global symmetries and outlines extensions to local symmetry actions and SET phases, highlighting potential applications in bosonic anyon condensation and gauging procedures.

Abstract

We develop a systematic framework for understanding symmetries in topological phases in 2+1 dimensions using the string-net model, encompassing both gauge symmetries that preserve anyon species and global symmetries permuting anyon species, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge symmetry of topological orders in 2+1 dimensions: the Fibonacci gauge symmetry of the doubled Fibonacci topological order, described by the Fibonacci fusion 2-category. Our approach involves two steps: first, establishing duality between different string-net models with Morita equivalent input fusion categories that describe the same topological order; and second, constructing symmetry transformations within the same string-net model when the dual models have isomorphic input fusion categories, achieved by composing duality maps with isomorphisms of degrees of freedom between the dual models.

Figures (5)

• Figure 1: Sketch of the construction: [1] Duality map $\mathcal{D}$ between distinct but equivalent models describing the same topological order; [2] Isomorphism $\varphi$ between the input UFCs of the equivalent models; Combining [1] and [2] yields a symmetry transformation of the string-net model, as well as the topologgical order.
• Figure 2: Part of the string-net model lattice. A tail (wavy line) is attached to an arbitrary edge of every plaquette.
• Figure 3: (a) The physical basic degrees of freedom $1$ and $\tau$ embedded in the $\{1, \tau_1, \tau_2\}$ enlarged Hilbert space. The black lines refer to the physical degrees of freedom $1$ and $\tau$. (2) The action of the unitary symmetry transformations $\mathcal{G}$ (the blue vectors) and the symmetry transformation $\tilde{\mathcal{G}}$ (the orange vectors) for anyon species $\tau\bar{1}, 1\bar{\tau}$. (c) The action of the unitary symmetry transformations $\mathcal{G}$ (the blue vectors) and the symmetry transformation $\tilde{\mathcal{G}}$ (the orange vectors) for anyon species $\tau\bar{\tau}$.
• Figure 4: (a) In the local dual model, each plaquette $P$ is equipped with a Frobenius algebra $\mathcal{A}_P$. The tail in $P$ carries simple $\mathcal{A}_P$ bimodule objects and the edge between adjacent plaquettes $P, Q$ carries simple $\mathcal{A}_P\text{-}\mathcal{A}_Q$ bimodules. (b) Under a local global symmetry transformation, different global symmetry sectors (red and blue regions) are separated by symmetric gapped domain walls---simple $\mathcal{A}_{\text{red}}-\mathcal{A}_{\text{blue}}$ bimodules on the edges, where $\mathcal{A}_{\text{red}}-\mathcal{A}_{\text{blue}}$ bimodule category is not isomorphic to ${\mathsf{Bimod}}_\mathscr{F}(\mathcal{A}_{\text{red}})\cong{\mathsf{Bimod}}_\mathscr{F}(\mathcal{A}_{\text{blue}})$.
• Figure 5: Contract a plaquette with a tail labeled by the trivial bimodule $\mathcal{A}$ in the original model.