3d mirror for Argyres-Douglas theories

TL;DR

This work provides a uniform construction of 3d mirrors for all Argyres-Douglas theories engineered from 6d (2,0) theories by representing each AD theory with a punctured sphere, assigning a quiver tail to every puncture, and gluing the tails to obtain the mirror. The approach extends the class S strategy to A-, D-, twisted A-, and exceptional-type AD theories, and yields practical tools for accessing the Higgs branch of 4d theories, as well as for exploring S-duality and generating new 3d N=4 SCFTs. Key contributions include explicit mirror quivers for broad families of AD theories, detailed gluing rules (Coulomb-branch gluing on the mirror side matching Higgs-branch data of the original theory), and applications to Higgs-branch decompositions, mixed branches, and new 3d matter sectors. The results deepen the connection between 4d AD physics and 3d mirror symmetry, providing a versatile framework for both physical and mathematical investigations of moduli spaces and dualities.

Abstract

3d mirrors for all 4d $\mathcal{N}=2$ Argyres-Douglas (AD) theories engineered using 6d $(2,0)$ theory are found. The basic steps are: 1): Find a punctured sphere representation for the AD theories (this is achieved in our previous studies of S duality); 2): Attach a 3d theory for each puncture; 3): Glue together the 3d theory for each puncture. We found the 3d mirror quiver gauge theory for the AD theories engineered using 6d $A$ and $D$ type theories. These 3d mirrors are useful for studying the properties of original 4d theory such as Higgs branch, S-duality, etc; We also construct many new 3d $\mathcal{N}=4$ SCFTs.

Figures (34)

• Figure 1: 1): A 4d Argyres-Douglas theory engineered using 6d $A_{N-1}$$(2,0)$ theories can be represented by a punctured sphere, and there are three types of punctures: blue, black, and red; Here the black puncture is taken to be the simplest one (with Young Tableaux $[1]$), and blue and red puncture are taken to be the maximal ones: the blue one has flavor symmetry $U(n_1)$ and the red one has $SU(N)$ flavor symmetry. There is also an extra label $(k,n)$ on the punctured sphere. 2): The quiver tail for each type of puncture; and there is adjoint matter for black puncture. The numeric numbers are related as $N=n a+n_1$, with $a$ the number of simple black punctures, so the theory is specified by four numbers $(a,n_1,n,k)$.
• Figure 2: Left: the gluing rule for the quiver tails listed in figure. \ref{['intro']}: a) there are $nk$ edges between black tails; b): there are $k$ edges between a blue tail and a black tail; c): We spray the flavor quiver node for the red tail so the flavor symmetry is $U(n_1)\times U(1)^a$, and there are $n$ edges for every $U(1)$ flavor node; the $U(n_1)$ flavor node is glued with the blue tail, and $U(1)$s are glued with black tail. Right: the final mirror quiver. $l$ is given by ${(n-1)(k-1)\over2}$.
• Figure 3: 1): The 3d mirror for $(A_{n-1}, A_{k-1})$ theory, here $n,k$ is coprime (($n,k)=1$). 2): The 3d mirror for $D_{n+k}(SU(n))$ theory with $(n,k)=1$ and $k>0$, the 3d mirror for the case $k<0$ is given in section 3. In both examples, the black puncture is the simplest one with label $[1]$.
• Figure 4: A 4d Argyres-Douglas theory is constructed by putting a 6d $(2,0)$ theory of type $\mathfrak{j}$ on a sphere with one irregular singularity and one regular singularity. The irregular singularity is labeled by $\Phi$, see \ref{['ire']}, and the regular singularity is labeled by $f$.
• Figure 5: A configuration shown in figure. \ref{['6dconstruct']} is now represented by a different punctured sphere: here the red puncture represents the regular singularity, the blue puncture represents the regular part in irregular singularity, and the black punctures represent the irregular blocks inside irregular singularity.