More Three Dimensional Mirror Pairs

TL;DR

This work extends three-dimensional ${\cal N}=4$ mirror symmetry to a broad class of generalized quiver theories arising from 6d ${\cal N}=(2,0)$ compactifications, by adding fundamentals, gauging ${U}(1)$ flavor symmetries, introducing general quiver tails, and incorporating irregular singularities. By formulating a graph-based representation of 3d theories and applying S-duality, the authors construct new mirrors and systematically extract the weakly coupled four-dimensional ${\cal N}=2$ dual frames and matter content across duality frames. The results cover ${A}_{N-1}$ and ${D}_{N}$ theories, provide a practical rank-manipulation procedure for bad mirrors, and connect irregular Hitchin systems to Argyres-Douglas points through explicit quiver constructions. The paper lays out a versatile toolkit for mapping 3d mirrors to 4d dual data and outlines multiple avenues for future exploration, including outer automorphisms and irregular singularities. Overall, it significantly broadens the landscape of known 3d mirrors and their correspondence with 4d S-duality structures.

Abstract

We found a lot of new three dimensional N = 4 mirror pairs generalizing previous considerations on three dimensional generalized quiver gauge theories. We recovered almost all previous discovered mirror pairs with these constructions. One side of these mirror pairs are always the conventional quiver gauge theories. One of our result can also be used to determine the matter content and weakly coupled gauge groups of four dimensional N = 2 generalized quiver gauge theories derived from six dimensional A_N and D_N theory, therefore we explicitly constructed four dimensional S-duality pairs.

Figures (18)

• Figure 1: (a)One S-duality frame for four dimensional ${\cal N}=2$ SU(2) with four fundamentals, each puncture carries a SU(2) flavor symmetry. (b) Three dimensional version of (a), which is derived by compactifying (a) on a circle. we represent it as $N=4$ SYM on the graph. (c)The graph mirror of (b), which is simply derived by gluing the SU(2) flavor symmetry of four quiver tails.
• Figure 2: (a) The addition of a "D5" brane to the internal leg of the graph. (b)Its mirror.
• Figure 3: (a) The naive mirror for adding one fundamental to one of weakly coupled gauge group; We assume that the left central node is bad. (b)We replace the rank of the left central node with $N_c^{'}=N_f-N=\sum_{k=1}^jn_k$.
• Figure 4: (a) The rank and excess number of a quiver tail associated with a central node, we assume that $e+e_1<0$ for this quiver tail. (b)The rank and excess number of the quiver tail after the manipulation is finished. It stops at the $j$th node, from the condition, we have $e_j>0$, this shows that no balanced node is lost, and they balanced chain is not altered. The new excess number is $e_{j-1}^{'}=-(e+e_1+e_2+...e_{j-1})>0$, $e_{j}^{'}=e+e_1+e_2+...e_j\geq 0$, so it is only possible for one more balanced node to appear. If a new balanced node appears, it shows that there are already fundamentals exist; The new rank is $n_{i}^{'}=n_i+e+e_1+...e_i$.
• Figure 5: (a)The weakly coupled duality frame with SU(3) gauge group on the left, there are two type of punctures: the cross represents the simple puncture with Young tableaux $[2,1]$, the circle cross represents the full puncture with tableaux $[1,1,1]$. The graph representation for three dimensional theory is shown on the right. (b) The weakly coupled duality frame with SU(2) gauge group on the left, graph representation for three dimensional theory on the right. (c) The mirror for theory (a) and (b), they are identical.