Higgs Branches of Argyres-Douglas theories as Quiver Varieties

We present a general prescription for constructing 3d $\mathcal{N}=4$ Lagrangians for the IR SCFTs that arise from the circle reduction of a large class of Argyres-Douglas theories. The resultant Lagrangian gives a realization of the Higgs branch of the 4d SCFT as a quiver variety, up to a set of decoupled interacting SCFTs with empty Higgs branches. As representative examples, we focus on the families $(A_{p-N-1}, A_{N-1})$ and $D_p(SU(N))$. The Lagrangian in question is generically a non-ADE-type quiver gauge theory involving only unitary gauge nodes with fundamental and bifundamental hypermultiplets, as well as hypermultiplets which are only charged under the $U(1)$ subgroups of certain gauge nodes. Our starting point is the Lagrangian 3d mirror of the circle-reduced Argyres-Douglas theory, which can be read off from the class $\mathcal{S}$ construction. Using the toolkit of the $S$-type operations, developed in \cite{Dey:2020hfe}, we show that the mirror of the 3d mirror for any Argyres-Douglas theory in the aforementioned families is guaranteed to be a Lagrangian theory of the above type, up to some decoupled free sectors. We comment on the extension of this procedure to other families of Argyres-Douglas theories. In addition, for the case of $D_p(SU(N))$ theories, we compare these 3d Lagrangians to the ones found in \cite{Closset:2020afy} and propose that the two are related by an IR duality. We check the proposed IR duality at the level of the three-sphere partition function for specific examples. In contrast to the 3d Lagrangians in \cite{Closset:2020afy}, which are linear chains involving unitary-special unitary nodes, we observe that the Coulomb branch global symmetries are manifest in the 3d Lagrangians that we find.