Breaking bad theories of class $\mathcal{S}$

We study weakly-coupled descriptions/channel decompositions of the 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $\mathfrak{s}\mathfrak{u}(N)$, from the perspective of the 3d $\mathcal{N}=4$ mirror duals of their circle compactifications. This is a delicate problem when the channel decomposition produces pathological, or bad, 4d configurations that correspond to spheres with non-maximal punctures. The star-shaped quivers, describing the 3d mirrors associated with such bad 4d configurations, are bad 3d $\mathcal{N}=4$ theories. Leveraging recent results regarding 3d bad theories, we identify a new and interesting family of bad theories, which we coin $\mathit{\text{broken}}$ theories, that naturally arise in this context. Using these broken theories, we develop a systematic and analytic method that determines the generically non-Lagrangian matter sectors and the weakly-coupled gauge groups in such channel decompositions. We understand these weakly-coupled descriptions as emerging dynamically via Higgs mechanisms triggered by operators acquiring vacuum expectation values.