Higgs Branches of Argyres-Douglas theories as Quiver Varieties

TL;DR

This work provides a systematic route to realize the Higgs branches of circle-reduced Argyres-Douglas theories as quiver varieties by constructing 3d N=4 Lagrangians for the IR SCFTs. Central to the approach is leveraging class S mirrors for AD theories and applying sequences of Abelian S-type operations to produce Lagrangian duals, yielding explicit unitary-quiver (often non-ADE) Lagrangians for the reductions of the (A_{p-N-1},A_{N-1}) and D_p(SU(N)) families. The authors establish an infinite class of IR dualities by matching sphere partition functions and map Coulomb/Higgs branch symmetries across dual pairs, with concrete checks in examples like D_4(SU(6)) and D_9(SU(3)). The results unify AD circle reductions with Lagrangian 3d mirrors, clarify how Coulomb symmetries emerge in IR, and relate to existing constructions (e.g., Closset et al.) via IR dualities, broadening the toolkit for exact descriptions of AD Higgs branches. The methodology opens pathways to extend to additional AD families and to classify AD theories whose 3d mirrors yield Lagrangian IR descriptions.

Abstract

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.

Figures (22)

• Figure 1: Generating new dual pairs using an elementary $S$-type operation.
• Figure 2: A quiver gauge theory with $G=U(N_1) \times U(N_2) \times SU(N_3) \times SU(N_4)$ and hypermultiplets in bifundamental representations, with the various conventions listed on the RHS.
• Figure 3: The gauging, flavoring and identification quiver operations on a generic quiver $X$ of class ${\cal U}$. Each operation involves two steps. In the first step, one splits the flavor node $U(M_\alpha)$ into two, corresponding to $U(r_\alpha) \times U(M_\alpha - r_\alpha)$ flavor nodes. In the second step, one gauges, adds hypermultiplets to, or identifies the $U(r_\alpha)$ flavor node respectively. The identification operation may involve two or more $U(r_\alpha)$ flavor nodes.
• Figure 4: Generating new dual pairs using an elementary $S$-type operation.
• Figure 5: The quiver gauge theory corresponding to the case with $m=3$ and generic $m_G$. The quiver on the right is obtained by decoupling a $U(1)$ vector multiplet associated with one of the vertices of the complete graph on the left.

Theorems & Definitions (3)

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