From Regular to Irregular: A Unified Origin for Argyres-Douglas Theories

TL;DR

This work establishes that Argyres-Douglas theories of type $D_p(\mathrm{SU}(N))$ and $(A_{p-1},A_{N-1})$ share a unified origin from a regular-puncture class $\mathcal{S}$ ancestor, via a sequence of mass deformations whose 4d effects are captured by FI deformations on the 3d mirror. The authors develop a constructive framework based on the Euclidean algorithm to define a chain of deformations and then reverse it to build a star-shaped 3d mirror that serves as the parent theory; this parent ultimately descends from torus compactification of 6d $\mathcal{N}=(1,0)$ orbi-instanton theories. They substantiate the program with explicit examples, including deformations that connect $D_p(\mathrm{SU}(N))$ to $D_N(\mathrm{SU}(p))$ and further to $(A_{N-1},A_{p-1})$, thus unifying the UV origins of these AD theories under a single geometric and quiver-theoretic mechanism. The approach provides a practical method for exploring the 4d SCFT landscape of Type A class $\mathcal{S}$ theories via 3d mirrors and FI moves, and suggests a generalizable route to other AD families and puncture configurations. Overall, the paper offers a concrete, algorithmic bridge between regular-puncture ancestors and irregular-puncture AD theories, with broad implications for geometric engineering and the organization of 4d $\mathcal{N}=2$ theories.

Abstract

We propose that Argyres-Douglas theories of type $D_p(\mathrm{SU}(N))$ and $(A_{p-1}, A_{N-1})$ - both realizable as Type A class $\mathcal{S}$ theories with irregular punctures - can be obtained via a sequence of mass deformations from a common ancestor: a class $\mathcal{S}$ theory with only regular punctures. Building on our previous work, this result establishes that these theories ultimately originate from 6d $\mathcal{N}=(1,0)$ orbi-instanton theories compactified on a torus. The requisite 4d mass deformations are realized as tractable Fayet-Iliopoulos deformations on the 3d mirror quiver. The core of our method is a constructive procedure that utilizes the Euclidean algorithm to define a chain of deformations connecting different $D_p(\mathrm{SU}(N))$ theories. By reversing this chain, we recursively build a "parent" star-shaped quiver for any given $(N,p)$. This quiver is the 3d mirror theory of the required class $\mathcal{S}$ ancestor. We substantiate our general claims with several detailed examples that explicitly illustrate the deformation procedure.