Topological lattice gauge theory enriched by non-invertible symmetry

We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry.

For invertible symmetry enriched topological order, there is an established axiomatisation in terms of a G-crossed braided fusion category.

We lay the foundations for a generalisation of this notion.

By condensing an arbitrary algebra of charges in a quantum double model, we demonstrate that the category of localised excitations in the resulting theory forms a hypergroup-graded extension of the category of deconfined excitations.

For every element in the hypergroup, the associated domain wall acts in a typically non-invertible way on these localised excitations.

Both this action and the monoidal structure are compatible with the hypergroup grading.

The actual categorical action is encoded in a Hopf monad on the category of localised excitations, and gauging the non-invertible symmetry amounts to computing the category of modules over this Hopf monad.

Finally, we outline how this framework naturally extends to theories obtained by condensing algebras in a generic string-net model.