Role of non-invertible one-form symmetries in non-Abelian topological transitions

Determine how the non-invertible one-form symmetry structure governs transitions out of non-Abelian topological order in microscopic quantum models.

Background

The paper contrasts Abelian and non-Abelian anyon condensation. In Abelian topological orders, anyon worldlines generate invertible one-form symmetries whose spontaneous breaking and restoration organize deconfined and confined phases. For non-Abelian anyons, the corresponding line operators obey fusion algebras rather than group multiplication and therefore generate non-invertible one-form symmetries.

The authors develop a sign-free path-integral and Monte Carlo framework for quantum doubles $\mathcal D(G)$, together with generalized Fredenhagen–Marcu order parameters, and apply it to $\mathcal D(S_3)$. These results provide examples of how non-invertible symmetry data can diagnose electric-anyon condensation and confinement, but the broader relationship between non-invertible symmetry structure and microscopic transitions out of non-Abelian topological order is explicitly identified as unresolved.