Non-abelian quantum double models from iterated gauging
David Blanik, José Garre-Rubio
TL;DR
This work extends the gauging framework to non-abelian finite groups by embedding global $\mathsf{Rep}\,G$ symmetries into a categorical gauging procedure based on matrix product operators (MPOs). By classifying Frobenius algebras internal to $\mathsf{Rep}\,G$ and constructing the explicit emergent dual symmetry, the authors show how all $(2+1)$D quantum double models with boundary arise from iterated gauging of a one-dimensional input state. They additionally introduce a gauging prescription for 1-form $\mathsf{Rep}\,G$ symmetries on two-dimensional lattices, enabling extension to $(3+1)$D quantum doubles via iteration. The results unify gauge-duality concepts with tensor-network constructions and offer a practical route to realizing non-abelian topological orders, while outlining open questions on higher Hopf-algebraic symmetries and anomalous boundaries.
Abstract
We reconstruct all (2+1)D quantum double models of finite groups from their boundary symmetries through the repeated application of a gauging procedure, extending the existing construction for abelian groups. We employ the recently proposed categorical gauging framework, based on matrix product operators (MPOs), to derive the appropriate gauging procedure for the $\mathsf{Rep}\, G$ symmetries appearing in our construction and give an explicit description of the dual emergent $G$ symmetry, which is our main technical contribution. Furthermore, we relate the possible gapped boundaries of the quantum double models to the quantum phases of the one-dimensional input state to the iterated gauging procedure. Finally, we propose a gauging procedure for 1-form $\mathsf{Rep}\, G$ symmetries on a two-dimensional lattice and use it to extend our results to the construction of (3+1)D quantum doubles models through the iterative gauging of (2+1)-dimensional symmetries.