The Many Faces of Non-invertible Symmetries

TL;DR

The paper develops a unified framework linking fusion-category (categorical) non-invertible symmetries to algebraic weak Hopf algebra actions via Tannaka–Krein duality, emphasizing the non-uniqueness of the dual and its physical implications. It constructs an explicit symmetrization (conditional expectation) map with an index giving universal bounds on entropic order parameters, connecting symmetry breaking to quantum-information measures. By passing from fusion algebras to strip algebras and employing Rieffel induction, the authors provide a practical algorithm to realize WHA symmetries on extended systems and to quantify breaking across 2D QFTs, TQFTs, and BCFTs. The paper then illustrates the framework through concrete examples (Fibonacci, 2D TQFT, and BCFT/Ising^2 with H8), deriving explicit entropic bounds and showing how non-uniqueness of the categorical-to-algebraic reconstruction manifests in different symmetry-breaking patterns. These results offer a robust toolkit for analyzing generalized symmetries in quantum field theory and quantum gravity, with potential applications to holography, anomalies, and non-invertible gauging.

Abstract

We investigate the interplay between algebraic and categorical notions of non-invertible symmetries. In particular, a fusion categorical symmetry $\mathcal{C}$ is shown to induce an algebraic symmetry encoded in a weak Hopf algebra $H$ which is Tannaka-Krein dual to $\mathcal{C}$ in the sense that $\mathcal{C} = \text{Rep}(H^*)$. The latter duality is not unique, and consequently the algebraic symmetry acts on an extended system relative to the categorical one. We present an approach to analyzing the symmetry breaking patterns of weak Hopf algebraic non-invertible symmetries. The central ingredient is a certain conditional expectation, which serves as the analog of a group averaging map for a non-invertible symmetry. The index of this conditional expectation emerges as a quantum information theoretic quantity that determines the extent to which the underlying symmetry can be broken. Ambiguities which ensue from the non-uniqueness of the categorical reconstruction lead to distinct properties of symmetry breaking compared to the invertible case. Finally, we exemplify our approach through topological and conformal quantum field theories in which non-invertible symmetries are naturally interpreted as defect operators and boundary conditions.

Figures (2)

• Figure 1: The regulated setup for defining the density operator $\rho$ for a state on half-line with physical boundary $|{\mathcal{B}}\rangle$ and regulating boundary $|{\mathcal{B}}_{\rm reg}\rangle$.
• Figure 2: The replicated annulus partition $Z_{{\mathcal{B}}'|{\mathcal{B}}_{\rm reg}}(q^n)[\phi']$ for $n=3$. Here the gray curly lines represent branch cuts connecting the three sheets. The colored lines represent algebra elements in the WHA symmetry that act on each sheet via the symmetrization map \ref{['rhosym']}.The black dots represent possible (non-topological) boundary changing operators on the symmetry breaking boundary $|{\mathcal{B}}'\rangle$ and the white dots represent the topological operator insertions.