Non-invertible symmetries and selection rules for RG flows of coset models

Abstract

We analyze superselection sectors, non-invertible symmetries and selection rules for RG flows triggered via perturbations of a UV two-dimensional conformal field theory (CFT$_2$). To this end we describe a method whose input is the local data, and whose output is the set of submodels of the modular invariant completions. We explain how this output set provides a classification of superselection sectors (DHR categories and Q-systems) and of topological defect lines, leading to a unified and potentially complete approach to selection rules for RG flows. This method is applied to scenarios in which the UV is a coset or a parafermion model. For these CFT$_2$ we explicitly find all submodels of the diagonal modular invariants. Our results gives selection rules that unify several known facts about such RG flows, while also allowing us to find new aspects.

Figures (5)

• Figure 1: Classification of submodels of the coset model D($k,l$) \ref{['coset-gko']} for $k,l$ odd, with each denoted by its DHR category.
• Figure 2: Classification of submodels of the Parafermion diagonal modular invariant \ref{['coset-parafermions']} for $k$ prime. Each is denoted by their DHR category.
• Figure 3: Classification of submodels of the coset model D($k,l$) \ref{['coset-gko']} diagonal modular invariant for $k$ odd and $l$ even (left) and for $k$ even and $l$ odd (right), with each denoted by its DHR category. This includes the corresponding relations between $\mathfrak{su}(2)_k$, $\mathfrak{su}(2)_l$ and even projections.
• Figure 4: Conservation of extensions and DHR categories in the RG flow from D$(k,l)$ UV (left) and D$(k-l,l)$ IR (right) for $k$ and $l$ odd.
• Figure 5: Classification of submodels of the Parafermion diagonal modular invariant \ref{['coset-parafermions']} for $k=9$ (left) and $k=10$ (right). Each is denoted by their DHR category.