§ INTRODUCTION

Three dimensional (3d) $\mathcal{N}=4$ SCFT has very interesting mirror symmetry properties  [1], and there are nontrivial maps between physical quantities of two mirror theories $A$ and $B$;
For  example, the Coulomb branch of theory $A$ is identified with the Higgs branch of theory $B$, and vice versa [The moduli space of 3d $\mathcal{N}=4$ SCFT could have two different branches depending on
the transformation properties under the $SU(2)_1\times SU(2)_2$ R symmetry. The name “Coulomb" branch comes from the fact that the theory could be given as the IR limit  of a quiver gauge theory and this branch
is identified with the limit of the  Coulomb branch of the quiver gauge theory; Similar meaning is applied to  the name “Higgs" branch.].
This mirror symmetry is quite useful as the physical properties are easier to compute  in one theory than  the other one.

The mirror pairs studied in [1] have purely 3d description: theories $A$ and $B$ could arise from the IR limit of purely 3d theories. Another way of getting 3d $\mathcal{N}=4$ SCFT is to start with a higher dimensional theory with eight supercharges and
study its compactification down to 3d. For  example, if a four dimensional (4d) $\mathcal{N}=2$ superconformal field theory (SCFT) is compactified on a circle, one can get three dimensional (3d) $\mathcal{N}=4$ SCFT [2] in the IR limit, which we call it theory  $A$.
It would then be interesting to find the 3d mirror theory $B$. If theory $B$ has a simpler description, i.e. it is given by the IR limit of a 3d quiver gauge theory, then $B$ is very useful to learn
interesting properties of theory $A$ (and original 4d theory) which is hard to obtain through other methods. For example, one can find the Higgs branch of the original 4d theory by studying the Coulomb branch of theory $B$.

In general, there is no systematic way to find 3d mirror pairs. Type IIB Brane construction is powerful in finding mirror for 3d linear (or cyclic) quivers [3, 4, 5, 6]. The mirror theory for 3d abelian gauge theories was studied in [7]. One can also  find mirror pairs
by studying circle compactification of  4d $\mathcal{N}=2$  class $\mathcal{S}$ theories, where theory $B$ is given by the IR limit of star-shaped quiver [8].
The mirror for circle compactification of some 4d Argyes-Douglas (AD) theories are found in [9, 10] (see [11] for related mathematical study).
These 3d mirrors are important tools  to study properties of 4d theory, i.e. the Higgs branch of 4d theory [12] and the $S$ duality property [13].

The 3d mirrors of most AD theories found in [10, 14, 15] are not found [The author proposed 3d mirror for $(A_1, A_2)$ theory in [9], some other examples are found in [12], and see [16] for discussion of the 3d mirror of $(A_1, A_{2N})$ and $(A_1, D_{2N+1})$ theories. See also [17, 18, 19, 20] for 3d mirror of some AD theories. ].
The purpose of this paper is to fill this gap and find the 3d mirror for all the AD theories constructed from 6d $(2,0)$ theories.
The punctured Riemann surface construction [21] for class ${\cal S}$ theory is very useful to find its 3d mirror [8]: one associates a quiver tail for each puncture and the full mirror quiver is derived by gluing these quiver tails. The most important discovery of this paper is that similar strategy works for all the other AD theories, and the crucial ingredient is the punctured
sphere representation used in studying S duality [22, 23]. Given the punctured sphere representation of AD theories, one associates a quiver tail for each puncture and the mirror quiver $B$ is constructed by gluing these quiver tails!
We summarize the detailed strategy of finding 3d mirror for AD theories [When we talk about 3d mirror for AD theories, we always mean the corresponding 3d $\mathcal{N}=4$ SCFT derived by compactifying 4d  AD theory on a circle.]:

* Find a punctured Riemann sphere representation of AD theory [22, 23]. There are typically three kinds of punctures: black, blue, and red. Each puncture has a label, i.e. a Young Tableaux for $A$ type, see section 2 for more details. See figure. <ref> for  an example [Notice that this is not the punctured sphere where one uses to engineer AD theory from 6d $(2,0)$ SCFT.].

* Attach a quiver tail for each puncture. The quiver tail of red and black puncture is quite similar to that of [8], which is often a linear quiver. The quiver tail for the black puncture is more subtle: the adjoint matters are needed. See figure. <ref>.

* Glue above quiver tails together. This is the most difficult part of the construction, and the rule is found by using  various predictions from 3d mirror: the match of Higgs (Coulomb) branch of theory $A$  and Coulomb (Higgs) branch of theory $B$, etc. See the rule listed in figure. <ref>.

Several examples are listed in the figure. <ref>. The above strategy is similar to what is used in finding 3d mirror for class ${\cal S}$ theories [8], and the stories presented here is much more general and the case [8] can  be regarded as a special case.  If the AD theories are constructed using 6d $(2,0)$ theory of $A$ and  $D$ type, the 3d mirror has a quiver gauge theory description, and more generally the 3d mirror is constructed by gauging strongly coupled 3d SCFTs. It is quite satisfactory that
there is a simple and uniform way of finding  3d mirror for all 4d $\mathcal{N}=2$ theories constructed using 6d $(2,0)$ theories.

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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 $(a,n_1,n,k)$.

The 3d mirror of AD theory is quite useful in studying the properties of 4d AD theory: a): it confirms the S duality conjectures of these theories [22, 23]: the S duality could be found by decomposing the 3d mirror into various pieces corresponding to the mirror of AD matter, see [13] for how to use 3d mirror to find S duality of AD theory; b): it predicts the Higgs branch  of 4d theories:  the Coulomb branch of the mirror quiver can be computed and they agree with
the Higgs branch result of AD theories found in [24]. It also has interesting applications for the study of 3d $\mathcal{N}=4$ SCFTs: we find a large class of new interesting theories and some of them can be useful to construct new Chern-Simons matter theory.

This paper is organized as follows: section 2 reviews the 4d AD theory constructed in [10, 14, 15]; section 3 discusses the 3d mirror for general AD theories; section 4 discusses some applications of 3d mirror of AD theories; finally, a conclusion is given in section 5.

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(481,140) .. controls (481,134.48) and (485.48,130) .. (491,130) .. controls (496.52,130) and (501,134.48) .. (501,140) .. controls (501,145.52) and (496.52,150) .. (491,150) .. controls (485.48,150) and (481,145.52) .. (481,140) – cycle ;

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[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (470,139.5) .. controls (470,138.12) and (471.12,137) .. (472.5,137) .. controls (473.88,137) and (475,138.12) .. (475,139.5) .. controls (475,140.88) and (473.88,142) .. (472.5,142) .. controls (471.12,142) and (470,140.88) .. (470,139.5) – cycle ;

(411,140) – (421,140) ;

(550,274) .. controls (550,268.48) and (554.48,264) .. (560,264) .. controls (565.52,264) and (570,268.48) .. (570,274) .. controls (570,279.52) and (565.52,284) .. (560,284) .. controls (554.48,284) and (550,279.52) .. (550,274) – cycle ;

(550,234) .. controls (550,228.48) and (554.48,224) .. (560,224) .. controls (565.52,224) and (570,228.48) .. (570,234) .. controls (570,239.52) and (565.52,244) .. (560,244) .. controls (554.48,244) and (550,239.52) .. (550,234) – cycle ;

(550,194) .. controls (550,188.48) and (554.48,184) .. (560,184) .. controls (565.52,184) and (570,188.48) .. (570,194) .. controls (570,199.52) and (565.52,204) .. (560,204) .. controls (554.48,204) and (550,199.52) .. (550,194) – cycle ;

(560,264) – (560,244) ;

(499,146.41) – (560,184) ;

(592,64) .. controls (592,58.48) and (596.48,54) .. (602,54) .. controls (607.52,54) and (612,58.48) .. (612,64) .. controls (612,69.52) and (607.52,74) .. (602,74) .. controls (596.48,74) and (592,69.52) .. (592,64) – cycle ;

(592,64) .. controls (595,42) and (605,19.7) .. (612,64) ;

(619,122) .. controls (619,116.48) and (623.48,112) .. (629,112) .. controls (634.52,112) and (639,116.48) .. (639,122) .. controls (639,127.52) and (634.52,132) .. (629,132) .. controls (623.48,132) and (619,127.52) .. (619,122) – cycle ;

(619,122) .. controls (622,100) and (632,77.7) .. (639,122) ;

[color=rgb, 255:red, 248; green, 231; blue, 28   ,draw opacity=1 ]   (551,87.55) – (592,67.55) ;

(540,97.41) – (560,184) ;

(598,72.41) – (560,184) ;

(623,130.41) – (560,183) ;

[color=rgb, 255:red, 248; green, 231; blue, 28   ,draw opacity=1 ]   (606,74.55) – (619,122.55) ;

[color=rgb, 255:red, 248; green, 231; blue, 28   ,draw opacity=1 ]   (551,93.55) – (620,122.55) ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (558,209.03) .. controls (558,207.65) and (559.12,206.53) .. (560.5,206.53) .. controls (561.88,206.53) and (563,207.65) .. (563,209.03) .. controls (563,210.41) and (561.88,211.53) .. (560.5,211.53) .. controls (559.12,211.53) and (558,210.41) .. (558,209.03) – cycle ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (558,219.03) .. controls (558,217.65) and (559.12,216.53) .. (560.5,216.53) .. controls (561.88,216.53) and (563,217.65) .. (563,219.03) .. controls (563,220.41) and (561.88,221.53) .. (560.5,221.53) .. controls (559.12,221.53) and (558,220.41) .. (558,219.03) – cycle ;

(500,135.27) – (534,94.41) ;

(501,139.41) – (595,72.41) ;

(500,142.41) – (621,127.41) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (361,22.41) – (360,341.41) ;

(215,75.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(276,52.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(303,110.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(98,130.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{1}$;

(11,131.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(233,174.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(179,176.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{1}$;

(263,174.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(303,174.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(152,89.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(188,108.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(231,127.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(225,195.34) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(242,193.34) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(258,195.34) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(242,54.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $nk$;

(291,69.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $nk$;

(266,91.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $nk$;

(277,12.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(314,88.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(215,40.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(232,295.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(556,267.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(538,46.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(637,94.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(600,18.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(590,104.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $nk$;

(609,81.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $nk$;

(565,66.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $nk$;

(573,138.34) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(544,139.34) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(595,148.34) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(527,130.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(528,110.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(509,105.62) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(398,133.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(485,132.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{1}$;

(626,116.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(599,58.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(538,81.21) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

Left: the gluing rule for the quiver tails listed in figure. <ref>:  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 ${(n-1)(k-1)\over2}$.

every picture/.style=line width=0.75pt

(73,184.43) .. controls (73,160.92) and (106.8,141.86) .. (148.5,141.86) .. controls (190.2,141.86) and (224,160.92) .. (224,184.43) .. controls (224,207.94) and (190.2,227) .. (148.5,227) .. controls (106.8,227) and (73,207.94) .. (73,184.43) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (143,157.86) .. controls (143,156.2) and (144.34,154.86) .. (146,154.86) .. controls (147.66,154.86) and (149,156.2) .. (149,157.86) .. controls (149,159.51) and (147.66,160.86) .. (146,160.86) .. controls (144.34,160.86) and (143,159.51) .. (143,157.86) – cycle ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 208; green, 2; blue, 27   ,fill opacity=1 ] (192,197.36) .. controls (192,195.42) and (193.57,193.86) .. (195.5,193.86) .. controls (197.43,193.86) and (199,195.42) .. (199,197.36) .. controls (199,199.29) and (197.43,200.86) .. (195.5,200.86) .. controls (193.57,200.86) and (192,199.29) .. (192,197.36) – cycle ;

[fill=rgb, 255:red, 80; green, 227; blue, 194   ,fill opacity=1 ] (101,197.36) .. controls (101,195.42) and (102.57,193.86) .. (104.5,193.86) .. controls (106.43,193.86) and (108,195.42) .. (108,197.36) .. controls (108,199.29) and (106.43,200.86) .. (104.5,200.86) .. controls (102.57,200.86) and (101,199.29) .. (101,197.36) – cycle ;

(389,170.75) .. controls (389,159.84) and (397.84,151) .. (408.75,151) .. controls (419.66,151) and (428.5,159.84) .. (428.5,170.75) .. controls (428.5,181.66) and (419.66,190.5) .. (408.75,190.5) .. controls (397.84,190.5) and (389,181.66) .. (389,170.75) – cycle ;

(389,170.75) .. controls (389.5,112) and (424.5,93) .. (428.5,170.75) ;

(83,388.43) .. controls (83,364.92) and (116.8,345.86) .. (158.5,345.86) .. controls (200.2,345.86) and (234,364.92) .. (234,388.43) .. controls (234,411.94) and (200.2,431) .. (158.5,431) .. controls (116.8,431) and (83,411.94) .. (83,388.43) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (153,361.86) .. controls (153,360.2) and (154.34,358.86) .. (156,358.86) .. controls (157.66,358.86) and (159,360.2) .. (159,361.86) .. controls (159,363.51) and (157.66,364.86) .. (156,364.86) .. controls (154.34,364.86) and (153,363.51) .. (153,361.86) – cycle ;

[color=rgb, 255:red, 0; green, 0; blue, 0   ,draw opacity=1 ][fill=rgb, 255:red, 208; green, 2; blue, 27   ,fill opacity=1 ] (202,401.36) .. controls (202,399.42) and (203.57,397.86) .. (205.5,397.86) .. controls (207.43,397.86) and (209,399.42) .. (209,401.36) .. controls (209,403.29) and (207.43,404.86) .. (205.5,404.86) .. controls (203.57,404.86) and (202,403.29) .. (202,401.36) – cycle ;

[fill=rgb, 255:red, 80; green, 227; blue, 194   ,fill opacity=1 ] (111,401.36) .. controls (111,399.42) and (112.57,397.86) .. (114.5,397.86) .. controls (116.43,397.86) and (118,399.42) .. (118,401.36) .. controls (118,403.29) and (116.43,404.86) .. (114.5,404.86) .. controls (112.57,404.86) and (111,403.29) .. (111,401.36) – cycle ;

(460.5,399.75) .. controls (460.5,388.84) and (469.34,380) .. (480.25,380) .. controls (491.16,380) and (500,388.84) .. (500,399.75) .. controls (500,410.66) and (491.16,419.5) .. (480.25,419.5) .. controls (469.34,419.5) and (460.5,410.66) .. (460.5,399.75) – cycle ;

(460.5,399.75) .. controls (461,341) and (496,322) .. (500,399.75) ;

(290,404.5) .. controls (290,396.49) and (296.49,390) .. (304.5,390) .. controls (312.51,390) and (319,396.49) .. (319,404.5) .. controls (319,412.51) and (312.51,419) .. (304.5,419) .. controls (296.49,419) and (290,412.51) .. (290,404.5) – cycle ;

(340,404.5) .. controls (340,396.49) and (346.49,390) .. (354.5,390) .. controls (362.51,390) and (369,396.49) .. (369,404.5) .. controls (369,412.51) and (362.51,419) .. (354.5,419) .. controls (346.49,419) and (340,412.51) .. (340,404.5) – cycle ;

(398,403) .. controls (398,392.51) and (406.51,384) .. (417,384) .. controls (427.49,384) and (436,392.51) .. (436,403) .. controls (436,413.49) and (427.49,422) .. (417,422) .. controls (406.51,422) and (398,413.49) .. (398,403) – cycle ;

(319,404.5) – (340,404.5) ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (378,403.86) .. controls (378,402.2) and (379.34,400.86) .. (381,400.86) .. controls (382.66,400.86) and (384,402.2) .. (384,403.86) .. controls (384,405.51) and (382.66,406.86) .. (381,406.86) .. controls (379.34,406.86) and (378,405.51) .. (378,403.86) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (389,403.86) .. controls (389,402.2) and (390.34,400.86) .. (392,400.86) .. controls (393.66,400.86) and (395,402.2) .. (395,403.86) .. controls (395,405.51) and (393.66,406.86) .. (392,406.86) .. controls (390.34,406.86) and (389,405.51) .. (389,403.86) – cycle ;

(436,403) – (461.5,403) ;

(198,183.57) node  [font=] [align=left] trivial;

(148.5,184.43) node    [font=] [align=left] $(k,n)$;

(99,185.57) node  [font=] [align=left] trivial;

(117,96) node [anchor=north west][inner sep=0.75pt]    $( A_{n-1} ,A_{k-1})$ theory;

(403,162) node [anchor=north west][inner sep=0.75pt]    $1$;

(364,96) node [anchor=north west][inner sep=0.75pt]  [font=]  $l=( n-1)( k-1) /2$;

(209,387.57) node  [font=] [align=left] full;

(161.5,393.43) node    [font=] [align=left] $(k,n)$;

(109,389.57) node  [font=] [align=left] trivial;

(129,308) node [anchor=north west][inner sep=0.75pt]  [font=]  $D_{n+k}( SU( n))$ theory;

(476,390) node [anchor=north west][inner sep=0.75pt]    $1$;

(480,322) node [anchor=north west][inner sep=0.75pt]  [font=]  $l=( n-1)( k-1) /2$;

(299,397) node [anchor=north west][inner sep=0.75pt]  [font=]  $1$;

(351,399) node [anchor=north west][inner sep=0.75pt]  [font=]  $2$;

(404,396) node [anchor=north west][inner sep=0.75pt]  [font=]  $n-1$;

(444,389) node [anchor=north west][inner sep=0.75pt]  [font=]  $n$;

(18,183) node [anchor=north west][inner sep=0.75pt]    $1)$;

(19,376) node [anchor=north west][inner sep=0.75pt]    $2)$;

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 $[1]$.

§ AD THEORIES FROM 6D $(2,0)$ SCFT

§.§ Basic construction

One can engineer a large class of 4d $\mathcal{N}=2$ SCFTs by putting a 6d $(2,0)$ theory of type $\mathfrak{j}=ADE$ on a sphere with an irregular singularity and a regular singularity [21, 25, 10, 14, 15], see figure. <ref>.
The Coulomb branch is captured by a Hitchin system with singular boundary conditions near the singularity. The Higgs field of the Hitchin system near the irregular singularity takes the following form,

\begin{equation}
\Phi={T\over z^{2+{k\over b}}}+\ldots.
\label{ire}
\end{equation}

Here $T$ is determined by a positive principle grading of Lie algebra $\mathfrak{j}$ [26], and is a regular semi-simple element of $\mathfrak{j}$. $k>-b$ and  is an integer. Subsequent terms of the Higgs field are chosen
such that they are compatible with the leading order term (essentially the grading determines the choice of these terms). We call them $J^{(b)}[k]$ type irregular puncture. Theories constructed using only above irregular singularity can also be engineered using a three dimensional singularity in type IIB string theory as summarized in table <ref>[27].
One can add another regular singularity which is labeled by a nilpotent orbit $f$ of $\mathfrak{j}$ (We use Nahm labels such that the trivial orbit corresponding to regular puncture with maximal flavor symmetry). A detailed discussion about these defects can be found in  [28].
So a theory is specified by four labels $<\mathfrak{j}, b, k, f>$, here $\mathfrak{j}$ denotes type of 6d $(2,0)$ SCFT, $b,k$ denotes irregular singularity, and $f$ denotes regular singularity.

every picture/.style=line width=0.75pt

(125.52,135.74) .. controls (125.52,95.57) and (158.08,63) .. (198.26,63) .. controls (238.43,63) and (271,95.57) .. (271,135.74) .. controls (271,175.92) and (238.43,208.48) .. (198.26,208.48) .. controls (158.08,208.48) and (125.52,175.92) .. (125.52,135.74) – cycle ;

at (195,40) $\mathfrak{j}$;
(192,89.48) – (203.52,89.48) – (203.52,101) – (192,101) – cycle ;
at (178,172) $f$;
(192.53,165.18) – (203.58,175.9)(203.49,164.94) – (192.62,176.13) ;

at (178,97) $\Phi$;

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>, and
the regular singularity is labeled by $f$.

$ \mathfrak{j}$    $b$      Singularity
$A_{N-1}$   $N$    $x_1^2+x_2^2+x_3^N+z^k=0$
$~$      $N-1$     $x_1^2+x_2^2+x_3^N+x_3 z^k=0$
$D_N$     $2N-2$     $x_1^2+x_2^{N-1}+x_2x_3^2+z^k=0$
$~$      $N$    $x_1^2+x_2^{N-1}+x_2x_3^2+z^k x_3=0$
$E_6$   12      $x_1^2+x_2^3+x_3^4+z^k=0$
$~$    9     $x_1^2+x_2^3+x_3^4+z^k x_3=0$
$~$     8     $x_1^2+x_2^3+x_3^4+z^k x_2=0$
$E_7$    18      $x_1^2+x_2^3+x_2x_3^3+z^k=0$
$~$   14       $x_1^2+x_2^3+x_2x_3^3+z^kx_3=0$
$E_8$    30       $x_1^2+x_2^3+x_3^5+z^k=0$
$~$     24      $x_1^2+x_2^3+x_3^5+z^k x_3=0$
$~$      20      $x_1^2+x_2^3+x_3^5+z^k x_2=0$

Three-fold isolated quasi-homogenous singularities of cDV type corresponding to the $J^{(b)}[k]$  irregular punctures of the regular-semisimple type in [14]. These 3d singularity is very useful in extracting the Coulomb branch spectrum, see [27].

$  j $     $A_{2N}$    $A_{2N-1}$     $D_{N+1}$     $E_6$   $D_4$
Outer-automorphism $o$     $Z_2$    $Z_2$    $Z_2$      $Z_2$   $Z_3$
Invariant subalgebra  $\fg^\vee$    $B_N$   $C_N$    $B_{N}$      $F_4$   $G_2$
Flavor symmetry $\fg$    $C_N^{(1)}$   $B_N$    $C_{N}^{(2)}$      $F_4$   $G_2$

Outer-automorphisms of simple Lie algebras $ j$, its invariant subalgebra $ g^\vee$ and flavor symmetry $ g$ from the Langlands dual of $g^\vee$.

$j$ with twist     $b_t$     SW geometry at SCFT point     $\Delta[z]$
$A_{2N}/Z_2$     $2N+1$    $x_1^2+x_2^2+x^{2N+1}+z^{k+{1\over2}}=0$     ${4N+2\over 4N+2k+3}$
$2N$     $x_1^2+x_2^2+x^{2N+1}+xz^k=0$     ${2N\over k+2N}$
$A_{2N-1}/Z_2$    $2N-1$     $x_1^2+x_2^2+x^{2N}+xz^{k+{1\over2}}=0$     ${4N-2\over 4N+2k-1}$
$2N$    $x_1^2+x_2^2+x^{2N}+z^{k}=0$     ${2N\over 2N+k}$
$D_{N+1}/Z_2$    $N+1$     $x_1^2+x_2^{N}+x_2x_3^2+x_3z^{k+{1\over2}}=0$     ${2N+2\over 2k +2N+3}$
$2N$    $x_1^2+x_2^{N}+x_2x_3^2+z^{k}=0$     ${2N\over k+2N}$
$D_4/Z_3$     $4$    $x_1^2+x_2^{3}+x_2x_3^2+x_3z^{k\pm {1\over3}}=0$     ${12\over 12+3k\pm1}$
$6$    $x_1^2+x_2^{3}+x_2x_3^2+z^{k}=0$     ${6\over 6+k}$
$E_6/Z_2$    $9$    $x_1^2+x_2^{3}+x_3^4+x_3z^{k+{1\over2}}=0$     ${18\over 18+2k+1}$
$12$    $x_1^2+x_2^{3}+x_3^4+z^{k}=0$     ${12\over 12+k}$
$8$    $x_1^2+x_2^{3}+x_3^4+x_2z^{k}=0$     ${8\over 12+k}$

SW geometry of twisted theories at the SCFT point. Here we also list the scaling dimension of coordinate $z$. All $k$'s in this table are integer valued and the power of $z$ coordinate in singularity is equal to  $k_t$ used in equation <ref>.

To get non-simply laced flavor groups, we need to consider the outer-automorphism twist of ADE Lie algebra and its Langlands dual. A systematic study of these AD theories
was performed in [15]. Denoting the twisted Lie algebra of $\mathfrak{j}$ as $\fg^{\vee}$ and its Langlands dual as $\fg$, outer-automorphisms and twisted algebras of $\mathfrak{j}$ are summarized in table <ref>. The irregular singularity of regular semi-simple type is also classified as in table <ref> with the following form,

\begin{equation}
\Phi={T^t\over z^{2+{k_t\over b_t}}}+\ldots
\label{eq:irregularPunctureTwisted}
\end{equation}

Here $T^t$ is an element of Lie algebra $\fg^{\vee}$ or other parts of the decomposition of $\mathfrak{j}$ under outer automorphism. $k_t>-b_t$, and
the novel thing  is that $k_t$  take half-integer value or
in  thirds ($\fg=G_2$). We could again add
a twisted regular puncture labeled  by a nilpotent orbit $f$ of $\fg$.  A theory is then labeled by following data $<\mathfrak{j}, o, b_t, k_t, f>$. Here $j$ is the type of 6d $(2,0)$ SCFT, $o$ is the outer automorphism twist we use, $b_t,k_t$ denotes the irregular singularity, and $f$ denotes
regular singularity.

§.§ Two  generalizations and geometric representation

It was found in [22] that one can find more SCFTs by using other kinds of irregular singularities which are different from those listed in last subsection. This class of theories are necessary for studying S dualities of these theories.
The generalizations are the following:

* The irregular singularity takes the following block diagonal form

\begin{equation}
\Phi=\left[\begin{array}{cc}
\Phi_1&0 \\
\end{array}\right]=\left[\begin{array}{cc}
{T_1\over z}&0 \\
0&{T_2\over z^{2+{k\over n}}}\\
\end{array}\right]+\ldots
\label{general}
\end{equation}

Here $\Phi_1$ has regular singularity, and $\Phi_2$ is an irregular singularity listed in last subsection. This type of singularity can be regarded as the combination of regular and irregular singularity.

* For the irregular singularity listed in the last subsection, if $k$ and $b$ are not co-prime, it is possible to consider
the degenerating case, i.e. some of the coefficients of the higher order pole is set to be the same. For example, let's consider $\mathfrak{j}=A_3, k=6, b=4$. The generic coefficients of Higgs field considered in last section take the from

\begin{equation}
\Phi=\left( \begin{array}{cc}
0& a_2\left(\begin{array}{cc}
1&0\\0&-1\end{array}\right) \\
\end{array}\right){1\over  z^{1+{6\over 4}}}+\ldots
\end{equation}

The leading order term has two different coefficients $a_1$  and $a_2$.
For the degenerating case, the leading order coefficients take the following form instead:

\begin{equation}
\Phi=\left( \begin{array}{cc}
0& a\left(\begin{array}{cc}
1&0\\0&-1\end{array}\right) \\
\end{array}\right){1\over  z^{1+{6\over 4}}}+\ldots
\end{equation}

and here the leading order term has only one coefficient.  If we'd like to get a conformal field theory, it was shown in [22] that the irregular part should have the same pattern of degeneracy.  The first order part, on the other hand, could take arbitrary degenerating form.

Given the generalizations of  irregular singularities, it is possible to represent our theory by an auxiliary punctured Riemann sphere (notice that this is an extra sphere), see figure. <ref>.
Here we take $\mathfrak{j}=A_{N-1}$ type as example, other cases are similar.
The original sphere in the $(2,0)$ construction involves an irregular singularity and a regular singularity. The basic idea of finding an auxiliary sphere is to represent a single irregular singularity by several punctures of an extra sphere.
We use a red puncture to denote the regular singularity, a blue puncture for the regular part inside the irregular singularity <ref>, and
several black punctures for the  irregular blocks in irregular singularity.  Each puncture has a Young Tableaux with different size.  The rule for assigning a Young Tableaux to a red and blue puncture is the same as
that found in [21], as they are both just regular singularities. For the irregular singularity, they take the following general form

\begin{equation}
\Phi={1\over z^{2+{np\over nq}}}diag(\underbrace{a_1j_q,\ldots, a_1j_q}_{n_1},\ldots, \ldots, \underbrace{a_sj_q,\ldots, a_sj_q}_{n_s})+\ldots
\end{equation}

Here $j_q$ is a standard diagonal matrix depending on integer $q$, and the detailed form is not important here (see [10]).
We represent this irregular singularity by $s$ black punctures, and each puncture has a size $n_1$.  The detailed form of the Young Tableaux at each puncture is determined by
the form of first order coefficients!

The above representation is similar to the Class ${\cal S}$ theory where there are only one type of puncture (red puncture) [21]. Here we need three kinds of punctures, and
there are some further important differences:

* There is only one red and one blue puncture (both of them could be trivial), and arbitrary number of black puncture (there should be at  least one nontrivial black puncture).

* One need an extra pair of co-prime integers $(p,q)$ to indicate the theory type, this pair gives the slope of the irregular part of the irregular singularity (${k\over b}={p\over q}$), see <ref>.

* When $(p,q)=(k,1),~k>0$, the blue puncture and black puncture are of the same type. When $(p,q)=(0,1)$, all three punctures are of the same type,  this actually represents class $S$ theory engineered using only regular punctures on sphere  [21].

* The size of Young tableaux is related as follows

\begin{equation}
Y_{red}=Y_{blue}+q \sum_{black} Y_i
\label{sizecons}
\end{equation}

The basic theory (called AD matter in [22]) is represented by a sphere with three punctures: one red, one blue and one black puncture.

The weakly coupled gauge theory description of general AD theory is found by degenerating above punctured Riemann surface into three punctured spheres [22, 23]: the rule is that the blue puncture of one three sphere is connected with
the red puncture of the other sphere, and each degenerating three sphere should be the allowed type: in the general case, it should have one blue, one red and one black puncture.

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(370,205.4) node [anchor=north west][inner sep=0.75pt]    $( p,q)$;

(399,78.4) node [anchor=north west][inner sep=0.75pt]    $j$;

A configuration shown in figure. <ref> 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.

§ 3D MIRROR FOR AD THEORIES

Let's now put 4d $\mathcal{N}=2$ theory discussed in last section on a circle with finite radius. The resulting low energy theory on the Coulomb branch can be described by the moduli space of the Hitchin system ${\cal M}_{Hit}$ [29].
The Hitchin moduli space has a Hitchin map $\pi: {\cal M}_{Hit} \rightarrow B$ [30], where
the generic fibre of this map is an open set inside an abelian variety (the abelian variety has half dimension of the Hitchin moduli space) [30]. The Coulomb branch of both 4d and 3d theory can be described by Hitchin fiberation.
For four dimensional theory, only the base $B$ describes the Coulomb branch moduli, and
the low energy photon coupling is given by the complex structure of the abelian variety. For three dimensional theory, the fibre is also regarded as
the vacua moduli (the reason is that in three dimension, one can perform duality on abelian gauge fields to get scalar fields which parameterize the fibre).  The Higgs branch of 3d theory does not receive quantum corrections and
is the same as the Higgs branch of the parent 4d theory.

We are interested in the most singular point of $M_{Hit}$ which should be a three dimensional $\mathcal{N}=4$ SCFT, and we call it theory $A$.  One might get the properties of this 3d SCFT by taking the radius of the compactified circle  to be zero, and flow to the deep IR.
Conjecturally, the Coulomb branch of this 3d SCFT $A$ is given by the dense open set ${\cal M}_{Hit}^*$ of ${\cal M}_{Hit}$.
We would like to find a 3d SCFT $B$ which is a mirror theory of theory $A$.  In some cases, the mirror SCFT can be described as the IR SCFT of a quiver gauge theory (We want to emphasize that this is not always possible);
If this is the case, the Higgs branch of theory $B$ has a classical description as the hyperkahler quotient, which is often called Nakajima quiver variety [31].  In this case we have following useful map:

\begin{equation}
{\cal M}_{Hit}^*=quiver~variety
\end{equation}

Mathematically, some examples of above equivalence were shown in [32]. Physically, the results in [32] is interpreted as 3d mirror symmetry [8, 10].

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(160,125.4) node [anchor=north west][inner sep=0.75pt]    $M^{A}_{Higgs}$;

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(146,208.4) node [anchor=north west][inner sep=0.75pt]    $A$;

(522,207.4) node [anchor=north west][inner sep=0.75pt]    $B$;

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(537,293.4) node [anchor=north west][inner sep=0.75pt]    $M^{B}_{Higgs}$;

The basic feature of 3d mirror symmetry: the Higgs branch of theory $A$ is identified with the Coulomb branch of theory $B$, and vice versa.

Let's recall some basic maps of 3d mirror symmetry [1]:

* The Coulomb branch of theory $A$ is identified with Higgs branch of theory $B$, and vice verse.  The simple checks are: the dimension should match, and the flavor symmetries should also match.

* The mass deformation which would deform the Coulomb branch (usually lift the Higgs branch), mapped to the FI parameters on the Higgs branch of the mirror theory.

The map is shown schematically in figure. <ref>.  Notice that it is possible that the theory $A$ has just one type of moduli space, and it is still possible to find its mirror theory. An example is the IR SCFT of $U(1)$  gauge theory coupled with a single hypermultiplet,  which has only Coulomb branch but no Higgs branch; its mirror theory is just a free hypermultiplet, which has only a Higgs branch but no Coulomb branch.

The 3d mirror for class ${\cal S}$ theory is successfully found in [8] (This corresponds to $(0,1)$ class of theories discussed in last section).  The idea is  the following (here we  take $\mathfrak{j}=A_{N-1}$ as examples). Recall that the geometric representation for this type of theories has only regular singularities and so there are only red punctures (see <ref>); and the 3d mirror is found as follows:

* For each puncture with label $Y$, one associates a linear quiver tail whose structure is determined by $Y$. There is a $U(N)$ flavor node at the end.  For example, if the red puncture has the label $[h_1, h_2, \ldots, h_s]$, then the quiver tail would be

\begin{equation}
\boxed{U(N)}-U(r_1)-U(r_2)-\ldots-U(r_{s-1})
\label{atail}
\end{equation}

with $r_i=\sum_{j=i+1}^{j=s} h_j$.

* The mirror is formed by gauging the diagonal $U(N)$ of the flavor quiver nodes of these quiver tails, and we get a star-shaped quiver.

See figure. <ref> for an example.

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(713,242) – (723,230) ;

(810,288) – (820,298) ;

(768,240) – (756,228) ;

(499,181) – (638,181) ;
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(714,210) .. controls (714,196.19) and (725.19,185) .. (739,185) .. controls (752.81,185) and (764,196.19) .. (764,210) .. controls (764,223.81) and (752.81,235) .. (739,235) .. controls (725.19,235) and (714,223.81) .. (714,210) – cycle ;

(124,189.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $( 0,1)$;

(550,153.4) node [anchor=north west][inner sep=0.75pt]    $glue$;

Left: a 4d $\mathcal{N}=2$ theory is engineered by putting $A_{N-1}$ $(2,0)$ theory on a three punctured sphere. Middle: We associate a linear quiver tail for each red puncture, whose form is given in <ref>, and the flavor node of the quiver tail is $U(N)$. Right: the quiver tails are glued together by gauging
diagonally the flavor nodes of the quiver tails, and this gives the 3d mirror for the left theory.

The 3d mirror for $(k,1)$ class of theories is found in [10]. The 3d mirror is found as follows: first attaching a complicated quiver for
the irregular singularity, and a quiver tail for the regular singularity; second, the 3d mirror is found by gluing the above two quivers together. The 6d construction shown in figure. <ref> is very useful in extracting the 3d mirror.
One might try to generalize the above construction to other  AD theories: assigning a quiver for the general irregular singularity and regular singularity, and then glue the quiver together.
However, we find it quite difficult to do so as there is no simple rule for assigning a quiver for an arbitrary irregular singularity.
To overcome the difficulty of assigning a quiver for general irregular singularity, we found that re-interpreting the result in [10] by using the geometric representation of
figure. <ref> is very illumilating. This class of theory (type $(k,1)$) is represented by a punctured sphere with two kinds of punctures: red one and black one.
We reinterpret the rule of finding its 3d mirror as follows:

* One has a quiver tail for a red puncture, and the rule is the same as <ref>.
One has a quiver tail for black puncture according to its Young Tableaux. The rule is also the same as  that listed in <ref>, and the  difference is that the flavor quiver node in <ref> is also gauged: for instance, if
a black puncture has label $[1]$, the associated quiver is a gauged $U(1)$ node.

* The gluing rule is following: there are $k$ edges between the end nodes of quiver tails of black puncture. For the red puncture, we choose following subgroup of its flavor quiver node $\prod_i U(n_i)$ where $n_i$ is
the size of Young Tableaux of $i$th black puncture; Here we have $\sum n_i=N$.
and the gluing is simply identifying these flavor nodes with the end quiver nodes of black punctures.

* The gluing rule is consistent due to the constraint shown in equation <ref> [Since there is no blue puncture and $q=1$ in formula <ref>, we have $N=\sum n_i$, here $N$ is the size of the Young Tableaux of red puncture, and
$n_i$ is the size of the Young Tableaux of the $i$th black puncture. ].

An example is shown in figure. <ref>.

every picture/.style=line width=0.75pt

every picture/.style=line width=0.75pt

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(389.5,83) – (390,100) ;

(352,190) – (376,190) – (376,214) – (352,214) – cycle ;

(390.5,169) – (376,190) ;

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(442.5,274.75) .. controls (442.5,268.26) and (447.76,263) .. (454.25,263) .. controls (460.74,263) and (466,268.26) .. (466,274.75) .. controls (466,281.24) and (460.74,286.5) .. (454.25,286.5) .. controls (447.76,286.5) and (442.5,281.24) .. (442.5,274.75) – cycle ;

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(293,319) – (304,307) ;

(489,315) – (499,325) ;

(72,197) .. controls (72,171.04) and (102.67,150) .. (140.5,150) .. controls (178.33,150) and (209,171.04) .. (209,197) .. controls (209,222.96) and (178.33,244) .. (140.5,244) .. controls (102.67,244) and (72,222.96) .. (72,197) – cycle ;

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(727,102.5) .. controls (727,96.15) and (732.15,91) .. (738.5,91) .. controls (744.85,91) and (750,96.15) .. (750,102.5) .. controls (750,108.85) and (744.85,114) .. (738.5,114) .. controls (732.15,114) and (727,108.85) .. (727,102.5) – cycle ;

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(738.5,114) – (739,131) ;

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(664.5,275.5) .. controls (664.5,269.15) and (669.65,264) .. (676,264) .. controls (682.35,264) and (687.5,269.15) .. (687.5,275.5) .. controls (687.5,281.85) and (682.35,287) .. (676,287) .. controls (669.65,287) and (664.5,281.85) .. (664.5,275.5) – cycle ;

(692.5,249.5) .. controls (692.5,243.15) and (697.65,238) .. (704,238) .. controls (710.35,238) and (715.5,243.15) .. (715.5,249.5) .. controls (715.5,255.85) and (710.35,261) .. (704,261) .. controls (697.65,261) and (692.5,255.85) .. (692.5,249.5) – cycle ;

(817.5,306.5) .. controls (817.5,300.15) and (822.65,295) .. (829,295) .. controls (835.35,295) and (840.5,300.15) .. (840.5,306.5) .. controls (840.5,312.85) and (835.35,318) .. (829,318) .. controls (822.65,318) and (817.5,312.85) .. (817.5,306.5) – cycle ;

(789.5,280.5) .. controls (789.5,274.15) and (794.65,269) .. (801,269) .. controls (807.35,269) and (812.5,274.15) .. (812.5,280.5) .. controls (812.5,286.85) and (807.35,292) .. (801,292) .. controls (794.65,292) and (789.5,286.85) .. (789.5,280.5) – cycle ;

(763.5,247.75) .. controls (763.5,241.26) and (768.76,236) .. (775.25,236) .. controls (781.74,236) and (787,241.26) .. (787,247.75) .. controls (787,254.24) and (781.74,259.5) .. (775.25,259.5) .. controls (768.76,259.5) and (763.5,254.24) .. (763.5,247.75) – cycle ;

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(657,297) – (668,285) ;

(710,239) – (739.5,200) ;

(810,288) – (820,298) ;

(768,240) – (739.5,200) ;

(499,181) – (638,181) ;
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(404,191) – (428,191) – (428,215) – (404,215) – cycle ;

(390.5,169) – (404,191) ;

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(124,189.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $( k,1)$;

(550,153.4) node [anchor=north west][inner sep=0.75pt]    $glue$;

(331,262.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{1}$;

(446,265.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{2}$;

(354,193.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{1}$;

(406,194.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{2}$;

(697,242.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{1}$;

(767,242.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $n_{2}$;

(732,248.4) node [anchor=north west][inner sep=0.75pt]    $k$;

Left: a three punctured sphere of $(k,1)$ type theories. Middle: Attach a quiver tail for each puncture. There is a flavor $U(N)$ node for  the quiver tail of red puncture and we decompose it into $U(n_1)\times U(n_2)$; notice that the flavor node $U(n_i)$ for the quiver tail of the black puncture is gauged. Right: the gluing rule is following: a): there are $k$ edges between
the end nodes of the black punctures; b): the $U(N)$ flavor node of the red puncture is sprayed as $ U(n_1) \times U(n_2)$, and we identify the flavor nodes with the end nodes of black punctures.

The construction of 3d mirror of above two class of theories are quite suggestive, since it shows the following pattern:

* Find the quiver tails for the red, blue, and black punctures.

* The 3d mirror is formed by finding appropriate gluing rules, which would glue together the quiver tails of each puncture.

Once we realize the above patterns, it is quite natural to try to implement above strategy for other type of AD theories.
Very amusingly, the above ideas work perfectly for all the AD theories constructed using 6d $(2,0)$ theories.

§.§ A type theories

Let's now try to find the 3d mirror for general $A$ type AD theory. These theories are represented by a punctured sphere with three  kinds of punctures: red, black and blue. Each puncture has a Young Tableaux and there is also
an extra label $(k,n)$ for this class of theories.

To find
its 3d mirror, we need to find the quiver tail for each puncture, and then find a gluing rule. The quiver tail for the red puncture is the one given in formula <ref>. The quiver tail for the blue puncture is the same as
the tail for the red puncture with only one difference: the end node is gauged. The extra gauging can be understood as follows: the flavor symmetry for the blue puncture is actually $U(m)$ while that of red puncture is
$SU(N)$. Since the number of $U(1)$ factors (this gives the number of FI parameter) of the mirror should be the same the number of mass parameters of the original theory, the quiver tail for the blue puncture should have
an extra  gauged $U(1)$, which explains the fact that the end node of blue puncture should be gauged. To match with Coulomb branch contribution of the blue puncture, we also need to add extra hypermultiplets on the $U(m)$ node,
and the number is given by

\begin{equation}
k\sum n_i
\end{equation}

Here $n_i$ is the size of the Young Tableaux of the $i$th black puncture.

So what is the quiver tail for the black puncture? To answer this question, we consider the simplest AD theory which is represented by a sphere with just one black puncture whose Young Tableaux is $[1]$ (the blue and red puncture are trivial), and the type of
the theory is labeled by $(k,n)$.
This theory is engineered by a six dimensional $A_{n-1}$ $(2,0)$ theory on a sphere with only an irregular singularity of the type:

\begin{equation}
\Phi={T\over z^{2+{k\over n}}}+\ldots
\end{equation}

Here $T$ is regular semi-simple, and here we assume $n\geq 0$, and $k,n$ are co-prime. This class of theory is also called $(A_{n-1}, A_{k-1})$ theory [33]. This theory has no Higgs branch, and the Coulomb branch dimension is equal to ${(k-1)(n-1)\over 2}$ [10].  If there is indeed a mirror
quiver for this theory, it must have just a single $U(1)$ quiver node [If there is no fundamental matter, the overall $U(1)$ of the quiver gauge theory is decoupled, so if the quiver has only one quiver node, its Coulomb branch is empty.] .
Now to match the Coulomb branch dimension of the original theory, we must add extra $l={(k-1)(n-1)\over 2}$ adjoint on the $U(1)$ quiver node, see figure. <ref>.  We now propose that the quiver shown in figure. <ref>
is indeed the 3d mirror for the black puncture with Young Tableaux $[1]$. A simple check is that it gives the correct answer for $n=1$. More checks would be given later.

every picture/.style=line width=0.75pt

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(520,188.4) node [anchor=north west][inner sep=0.75pt]    $1$;

(525,131.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $l=\frac{( n-1)( k-1)}{2}$;

The quiver tail for a black puncture with label $[1]$. The punctured sphere has the label $(k,n)$.

For the general black puncture, we can form a quiver tail (the end node is gauged) according to its Young Tableaux, and the only difference is to add an extra $l=\frac{( n-1)( k-1)}{2}$ adjoints on the end node. See figure. <ref>.  With this proposal, we
now have the quiver tails for all kinds of punctures, see figure. <ref>.

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(99,181.4) node [anchor=north west][inner sep=0.75pt]    $(k,n)$;

(558,35.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $l=\frac{( n-1)( k-1)}{2}$;

Quiver tails for three kinds of punctures of type $(k,n)$ theories. The flavor node of the red tail is $N$, and the flavor tail of the blue tail is $k \sum n_i$ with $n_i$ the size of Young Tableaux of the $i$th black puncture. We also have the relation $N=m+n\sum n_i$.

The next question is the gluing rule.  Let's denote the rank of the end node of the red tail as $N$, that of the blue tail as $m$, and the ranks of the black tails as $n_1,\ldots, n_s$. There is a constraint on these numbers (see formula <ref>):

\begin{equation}
N=m+n\sum_{i=1}^s n_s
\end{equation}

Now let's state the gluing rule:

* The Higgs branch dimension of the mirror should be equal to the Coulomb branch of the original theory, which is computed in [10]. By computing some examples, we find following rules: 1) there are $nk$ edges between end nodes of black tails; 2) there are $k$ edges
between end nodes of blue and black nodes.  See figure. <ref>.

* We decompose the flavor node of the red puncture as $U(N)=\textcolor{blue}{U(m)}\times U(n_1)\times \ldots\times U(n_s)$, and the number of edges for black flavor symmetries are $n$. The gluing rule is to
simply gauge these nodes with the end nodes of black and blue tails.

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(488,97.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(394,45.4) node [anchor=north west][inner sep=0.75pt]    $nk$;

(393,121.4) node [anchor=north west][inner sep=0.75pt]    $k$;

(279,353.28) node [anchor=north west][inner sep=0.75pt]    $n$;

(272,385.28) node [anchor=north west][inner sep=0.75pt]    $n$;

(60,63.4) node [anchor=north west][inner sep=0.75pt]    $A$;

(62,143.4) node [anchor=north west][inner sep=0.75pt]    $B$;

(61,352.4) node [anchor=north west][inner sep=0.75pt]    $C$;

The gluing rule for $(k,n)$ type theory: a): There are $nk$ edges between black quiver tails; b): There are $k$ edges between black and blue tails. c): The flavor node of the red tail is decomposed as $U(m)\times U(n_1)\times \ldots \times
U(n_s)$, and the multiplicity of the black flavor node is $n$.

Using above rules ,we can find 3d mirror for several interesting class of theories, see figure. <ref>.

every picture/.style=line width=0.75pt

(154,159) .. controls (154,142.43) and (167.43,129) .. (184,129) .. controls (200.57,129) and (214,142.43) .. (214,159) .. controls (214,175.57) and (200.57,189) .. (184,189) .. controls (167.43,189) and (154,175.57) .. (154,159) – cycle ;

(273,160) .. controls (273,143.43) and (286.43,130) .. (303,130) .. controls (319.57,130) and (333,143.43) .. (333,160) .. controls (333,176.57) and (319.57,190) .. (303,190) .. controls (286.43,190) and (273,176.57) .. (273,160) – cycle ;

(214,159) – (273,160) ;

(171,131) .. controls (184,47) and (195,93) .. (199,132) ;

(292,132) .. controls (305,48) and (316,95) .. (320,134) ;

(217,252) .. controls (217,235.43) and (230.43,222) .. (247,222) .. controls (263.57,222) and (277,235.43) .. (277,252) .. controls (277,268.57) and (263.57,282) .. (247,282) .. controls (230.43,282) and (217,268.57) .. (217,252) – cycle ;

(190,188) – (222.5,236) ;

(292.5,189) – (266.5,229) ;

(234,278) .. controls (244.5,377) and (253,309) .. (260.5,278) ;

(613,154) .. controls (613,137.43) and (626.43,124) .. (643,124) .. controls (659.57,124) and (673,137.43) .. (673,154) .. controls (673,170.57) and (659.57,184) .. (643,184) .. controls (626.43,184) and (613,170.57) .. (613,154) – cycle ;

(732,155) .. controls (732,138.43) and (745.43,125) .. (762,125) .. controls (778.57,125) and (792,138.43) .. (792,155) .. controls (792,171.57) and (778.57,185) .. (762,185) .. controls (745.43,185) and (732,171.57) .. (732,155) – cycle ;

(673,154) – (732,155) ;

(630,126) .. controls (643,42) and (654,88) .. (658,127) ;

(751,127) .. controls (764,43) and (775,90) .. (779,129) ;

(676,247) .. controls (676,230.43) and (689.43,217) .. (706,217) .. controls (722.57,217) and (736,230.43) .. (736,247) .. controls (736,263.57) and (722.57,277) .. (706,277) .. controls (689.43,277) and (676,263.57) .. (676,247) – cycle ;

(649,183) – (681.5,231) ;

(751.5,184) – (725.5,224) ;

(693,273) .. controls (703.5,372) and (712,304) .. (719.5,273) ;

(874,88.25) .. controls (874,79.83) and (880.83,73) .. (889.25,73) .. controls (897.67,73) and (904.5,79.83) .. (904.5,88.25) .. controls (904.5,96.67) and (897.67,103.5) .. (889.25,103.5) .. controls (880.83,103.5) and (874,96.67) .. (874,88.25) – cycle ;

(833,116.25) .. controls (833,107.83) and (839.83,101) .. (848.25,101) .. controls (856.67,101) and (863.5,107.83) .. (863.5,116.25) .. controls (863.5,124.67) and (856.67,131.5) .. (848.25,131.5) .. controls (839.83,131.5) and (833,124.67) .. (833,116.25) – cycle ;

(514,91.25) .. controls (514,82.83) and (520.83,76) .. (529.25,76) .. controls (537.67,76) and (544.5,82.83) .. (544.5,91.25) .. controls (544.5,99.67) and (537.67,106.5) .. (529.25,106.5) .. controls (520.83,106.5) and (514,99.67) .. (514,91.25) – cycle ;

(549,119.25) .. controls (549,110.83) and (555.83,104) .. (564.25,104) .. controls (572.67,104) and (579.5,110.83) .. (579.5,119.25) .. controls (579.5,127.67) and (572.67,134.5) .. (564.25,134.5) .. controls (555.83,134.5) and (549,127.67) .. (549,119.25) – cycle ;

(756,302.25) .. controls (756,293.83) and (762.83,287) .. (771.25,287) .. controls (779.67,287) and (786.5,293.83) .. (786.5,302.25) .. controls (786.5,310.67) and (779.67,317.5) .. (771.25,317.5) .. controls (762.83,317.5) and (756,310.67) .. (756,302.25) – cycle ;

(796,338.25) .. controls (796,329.83) and (802.83,323) .. (811.25,323) .. controls (819.67,323) and (826.5,329.83) .. (826.5,338.25) .. controls (826.5,346.67) and (819.67,353.5) .. (811.25,353.5) .. controls (802.83,353.5) and (796,346.67) .. (796,338.25) – cycle ;

(540,102) – (552.5,112) ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (586,133.25) .. controls (586,131.46) and (587.46,130) .. (589.25,130) .. controls (591.04,130) and (592.5,131.46) .. (592.5,133.25) .. controls (592.5,135.04) and (591.04,136.5) .. (589.25,136.5) .. controls (587.46,136.5) and (586,135.04) .. (586,133.25) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (599,143.25) .. controls (599,141.46) and (600.46,140) .. (602.25,140) .. controls (604.04,140) and (605.5,141.46) .. (605.5,143.25) .. controls (605.5,145.04) and (604.04,146.5) .. (602.25,146.5) .. controls (600.46,146.5) and (599,145.04) .. (599,143.25) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (819,132.25) .. controls (819,130.46) and (820.46,129) .. (822.25,129) .. controls (824.04,129) and (825.5,130.46) .. (825.5,132.25) .. controls (825.5,134.04) and (824.04,135.5) .. (822.25,135.5) .. controls (820.46,135.5) and (819,134.04) .. (819,132.25) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (802,144.25) .. controls (802,142.46) and (803.46,141) .. (805.25,141) .. controls (807.04,141) and (808.5,142.46) .. (808.5,144.25) .. controls (808.5,146.04) and (807.04,147.5) .. (805.25,147.5) .. controls (803.46,147.5) and (802,146.04) .. (802,144.25) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (736,269.25) .. controls (736,267.46) and (737.46,266) .. (739.25,266) .. controls (741.04,266) and (742.5,267.46) .. (742.5,269.25) .. controls (742.5,271.04) and (741.04,272.5) .. (739.25,272.5) .. controls (737.46,272.5) and (736,271.04) .. (736,269.25) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (746.5,283) .. controls (746.5,281.07) and (748.07,279.5) .. (750,279.5) .. controls (751.93,279.5) and (753.5,281.07) .. (753.5,283) .. controls (753.5,284.93) and (751.93,286.5) .. (750,286.5) .. controls (748.07,286.5) and (746.5,284.93) .. (746.5,283) – cycle ;

(783,313) – (800.5,329) ;

(877,96) – (861.5,107) ;

(184,159) node  [font=] [align=left] 1;

(303,160) node  [font=] [align=left] 1;

(242,148) node  [font=] [align=left] $nk$;

(197,214) node  [font=] [align=left] $nk$;

(290,218) node  [font=] [align=left] $nk$;

(185,66) node  [font=] [align=left] $l$;

(307,62) node  [font=] [align=left] $l$;

(248,342) node  [font=] [align=left] $l$;

(247,252) node  [font=] [align=left] 1;

(762,155) node  [font=] [align=left] ;

(701,143) node  [font=] [align=left] $nk$;

(656,209) node  [font=] [align=left] $nk$;

(749,213) node  [font=] [align=left] $nk$;

(644,61) node  [font=] [align=left] $l$;

(766,57) node  [font=] [align=left] $l$;

(707,337) node  [font=] [align=left] $l$;

(762,155) node  [font=]  $n_{1}$;

(706,247) node  [font=]  $n_{2}$;

(643,154) node  [font=]  $n_{3}$;

Here $l={(n-1)(k-1)\over 2}$. Left: The 4d theory is represented by a sphere with three black punctures with type $[1]$, and one trivial red puncture, and one trivial blue puncture. This theory
is actually $(A_{3n-1}, A_{3k-1})$ theory, and the 3d mirror was found here. Right: The original theory is represented by a sphere with three generic black punctures of type $(k,n)$, and here the 3d mirror is given.

every picture/.style=line width=0.75pt

(287,233) .. controls (287,212.01) and (314.98,195) .. (349.5,195) .. controls (384.02,195) and (412,212.01) .. (412,233) .. controls (412,253.99) and (384.02,271) .. (349.5,271) .. controls (314.98,271) and (287,253.99) .. (287,233) – cycle ;

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[fill=rgb, 255:red, 208; green, 2; blue, 27   ,fill opacity=1 ] (387,245.5) .. controls (387,243.57) and (385.43,242) .. (383.5,242) .. controls (381.57,242) and (380,243.57) .. (380,245.5) .. controls (380,247.43) and (381.57,249) .. (383.5,249) .. controls (385.43,249) and (387,247.43) .. (387,245.5) – cycle ;

[fill=rgb, 255:red, 80; green, 227; blue, 194   ,fill opacity=1 ] (311,246.5) .. controls (311,244.57) and (312.57,243) .. (314.5,243) .. controls (316.43,243) and (318,244.57) .. (318,246.5) .. controls (318,248.43) and (316.43,250) .. (314.5,250) .. controls (312.57,250) and (311,248.43) .. (311,246.5) – cycle ;

[color=rgb, 255:red, 80; green, 227; blue, 194   ,draw opacity=1 ] (549.46,259.45) .. controls (549.46,247.87) and (558.95,238.49) .. (570.65,238.49) .. controls (582.35,238.49) and (591.83,247.87) .. (591.83,259.45) .. controls (591.83,271.03) and (582.35,280.42) .. (570.65,280.42) .. controls (558.95,280.42) and (549.46,271.03) .. (549.46,259.45) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][fill=rgb, 255:red, 255; green, 255; blue, 255   ,fill opacity=1 ] (680.17,258.74) .. controls (680.17,247.16) and (689.65,237.78) .. (701.35,237.78) .. controls (713.05,237.78) and (722.54,247.16) .. (722.54,258.74) .. controls (722.54,270.32) and (713.05,279.71) .. (701.35,279.71) .. controls (689.65,279.71) and (680.17,270.32) .. (680.17,258.74) – cycle ;

(608.35,188.38) .. controls (608.35,176.8) and (617.84,167.41) .. (629.54,167.41) .. controls (641.24,167.41) and (650.72,176.8) .. (650.72,188.38) .. controls (650.72,199.96) and (641.24,209.35) .. (629.54,209.35) .. controls (617.84,209.35) and (608.35,199.96) .. (608.35,188.38) – cycle ;

(614.81,204.37) – (582.5,241.33) ;

(644.26,204.37) – (685.19,244.17) ;

(591.83,259.45) – (680.17,258.74) ;

(480.44,323.85) .. controls (480.44,316.39) and (486.55,310.35) .. (494.09,310.35) .. controls (501.63,310.35) and (507.73,316.39) .. (507.73,323.85) .. controls (507.73,331.31) and (501.63,337.36) .. (494.09,337.36) .. controls (486.55,337.36) and (480.44,331.31) .. (480.44,323.85) – cycle ;

(451,351.57) .. controls (451,344.11) and (457.11,338.07) .. (464.65,338.07) .. controls (472.18,338.07) and (478.29,344.11) .. (478.29,351.57) .. controls (478.29,359.03) and (472.18,365.07) .. (464.65,365.07) .. controls (457.11,365.07) and (451,359.03) .. (451,351.57) – cycle ;

(764.19,351.5) .. controls (764.19,344.04) and (770.3,337.99) .. (777.84,337.99) .. controls (785.37,337.99) and (791.48,344.04) .. (791.48,351.5) .. controls (791.48,358.95) and (785.37,365) .. (777.84,365) .. controls (770.3,365) and (764.19,358.95) .. (764.19,351.5) – cycle ;

(736.18,322.36) .. controls (736.18,314.9) and (742.29,308.85) .. (749.83,308.85) .. controls (757.36,308.85) and (763.47,314.9) .. (763.47,322.36) .. controls (763.47,329.81) and (757.36,335.86) .. (749.83,335.86) .. controls (742.29,335.86) and (736.18,329.81) .. (736.18,322.36) – cycle ;

(758.45,331.6) – (768.5,342.97) ;

(473.98,342.33) – (484.04,333.8) ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (510.76,309.5) .. controls (510.76,308.33) and (511.73,307.37) .. (512.92,307.37) .. controls (514.11,307.37) and (515.07,308.33) .. (515.07,309.5) .. controls (515.07,310.68) and (514.11,311.64) .. (512.92,311.64) .. controls (511.73,311.64) and (510.76,310.68) .. (510.76,309.5) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (528.1,293.69) .. controls (528.1,292.51) and (529.06,291.55) .. (530.25,291.55) .. controls (531.44,291.55) and (532.41,292.51) .. (532.41,293.69) .. controls (532.41,294.86) and (531.44,295.82) .. (530.25,295.82) .. controls (529.06,295.82) and (528.1,294.86) .. (528.1,293.69) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (731.87,304.59) .. controls (731.87,303.41) and (732.84,302.45) .. (734.03,302.45) .. controls (735.22,302.45) and (736.18,303.41) .. (736.18,304.59) .. controls (736.18,305.76) and (735.22,306.72) .. (734.03,306.72) .. controls (732.84,306.72) and (731.87,305.76) .. (731.87,304.59) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (726.13,296.06) .. controls (726.13,294.88) and (727.09,293.93) .. (728.28,293.93) .. controls (729.47,293.93) and (730.44,294.88) .. (730.44,296.06) .. controls (730.44,297.24) and (729.47,298.19) .. (728.28,298.19) .. controls (727.09,298.19) and (726.13,297.24) .. (726.13,296.06) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (718.95,285.4) .. controls (718.95,284.22) and (719.91,283.26) .. (721.1,283.26) .. controls (722.29,283.26) and (723.26,284.22) .. (723.26,285.4) .. controls (723.26,286.57) and (722.29,287.53) .. (721.1,287.53) .. controls (719.91,287.53) and (718.95,286.57) .. (718.95,285.4) – cycle ;

(613.38,173.81) .. controls (616.25,146.09) and (637.8,102.02) .. (647.13,175.23) ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (543.1,281.69) .. controls (543.1,280.51) and (544.06,279.55) .. (545.25,279.55) .. controls (546.44,279.55) and (547.41,280.51) .. (547.41,281.69) .. controls (547.41,282.86) and (546.44,283.82) .. (545.25,283.82) .. controls (544.06,283.82) and (543.1,282.86) .. (543.1,281.69) – cycle ;

(302,239) node    $A$;

(400,240) node    $A$;

(360,207) node    $A$;

(349.5,233) node    $(k,n)$;

(701.35,258.74) node  [font=]  $N-1$;

(570.65,259.45) node  [font=]  $n_{1}$;

(629.54,188.38) node    $1$;

(622,114.11) node    $l={(n-1)(k-1)\over2}$;

(574.34,213.35) node    $k$;

(681.32,213.19) node    $n$;

(777.84,351.5) node  [font=]  $1$;

(464.65,351.57) node  [font=]  $1$;

This theory has flavor symmetry $SU(N)\times U(n_1)$, and is represented by a sphere with one maximal blue puncture ($Y=[1^{n_1}]$), one maximal red puncture ($Y=[1^{N}$]), and one simple black puncture ($Y=[1]$). The 3d mirror is shown on the right. Here $N=n_1+n$.

As an important example, we found the 3d mirror for AD matter found in [22].  This class of theory is engineered using the irregular singularity  (here $\mathfrak{g}=A_{N-1}$):

\begin{equation}
\Phi=\left[\begin{array}{cc}
\Phi_1&0 \\
\end{array}\right]=\left[\begin{array}{cc}
{T_1\over z}&0 \\
0&{T_2\over z^{2+{k\over n}}}\\
\end{array}\right]+\ldots
\label{general}
\end{equation}

Here $T_1$ has size $n_1$, and there is a maximal regular singularity $f$, see figure. <ref>. This theory has flavor symmetry $U(n_1)\times SU(N)$.  It is represented by a sphere with one maximal blue puncture, one  maximal red puncture, and
one black puncture of type $[1] $. This is a type $(k,n)$ theory. According to our proposal, its 3d mirror is found in figure. <ref>.
Let's now make several checks for the mirror quiver shown in figure. <ref>:

* The Coulomb branch dimension of original 4d theory is (see page 28 of [34]):

\begin{equation}
n_C={(n+k-1)(2n_1+n+1)\over 4}
\label{admattercoulomb}
\end{equation}

This equals to the Higgs branch dimension of the quiver shown in figure. <ref>. We assume the mirror quiver has a pure Higgs branch, and its dimension is given by the the difference of hypermultiplets and vectormultiplets. There is no flavor symmetry on
the Coulomb branch of original theory, and so there is no flavor symmetry on the Higgs branch of the mirror theory. This implies that the mirror quiver should have no flavor quiver node, which is the case for the quiver shown in figure. <ref>.

* The Higgs branch of 4d AD matter theory is in general a mixed branch, i.e. the generic point of the Higgs branch consists of free hypermultiplets and an interacting AD theory which has no Higgs branch [24]. The Higgs branch part are described by following variety:

\begin{equation}
\overline{{\cal O}_q}\cap S_f;
\end{equation}

Here $ {\cal O}_q$ is the nilpotent orbit with partition $[\underbrace{n+k, \ldots, n+k}_{n_1}, n]$, and $f=[(n+k-1)^{n_1},1^{n+n_1}]$.  The dimension of Higgs branch is simply the difference of the dimension of  $ \overline{{\cal O}_q}$ and $S_f$, and direct computation gives us that

\begin{equation}
n_H=n_1^2+n n_1+{n^2\over2}-{n\over2}.
\label{amatterhiggs}
\end{equation}

The interacting theory part is given by $(A_{n-1}, A_{k-1})$ theory, and this theory has only a Coulomb branch with dimension ${(n-1)(k-1)\over 2}$.

We now show that the mirror theory of the quiver shown in figure. <ref> recovers the mixed branch structure found in [24]. Firstly,
one can check that the Coulomb branch dimension of the mirror shown in figure. <ref> is the same as the number in <ref>.
The mirror theory does not have a pure Coulomb branch, as the adjoint matter on $U(1)$ quiver node decouples; One can give the vevs to these matter at the generic point of
Coulomb branch, and the dimension of the Higgs factor is just ${(n-1)(k-1)\over2}$, which is the same as the Coulomb factor of the mixed branch of original 4d theory.

* In fact, the  above proposal can be checked using the known information of Higgs branch of original 4d AD theory when $k=1$. The idea is following:  for the 4d theory, the Higgs branch is pure and is given by the following variety

\begin{equation}
\overline{{\cal O}_q}\cap S_f
\end{equation}

One can find a 3d quiver whose Coulomb branch is the same as above variety using the brane construction [3, 35] (See [34] for the detailed computation of this example), and
the result agrees with the mirror shown in figure. <ref>.

Case $k<0$: The above proposal for the mirror quiver is only good for $k\geq 0$.
To find the mirror for theories with $-n<k<0$, we look at the AD matter again.
The Higgs factor of the mixed branch of original 4d theory is given by $\overline{{\cal O}}_q\cap S_f$ [36, 34]. Here

\begin{equation}
{\cal O}_q=[\underbrace{n+k,\ldots,n+k}_{n_1},\underbrace{n+k,\ldots,n+k}_{a},b],~f=[(n+k-1)^{n_1},1^{n+n_1}];
\label{brane}
\end{equation}

and we have $a(n+k)+b=n$ with
$b<(n+k)$. Notice that $a,b$ is uniquely determined by the pair of numbers $(n,k)$.

To find the 3d mirror, we use following strategy: firstly, we use brane construction of [3, 35] to construct a quiver so that its Coulomb branch is given by ${\cal O}_q\cap S_f$, see  formula <ref>, and the quiver is given in figure. <ref> (without the adjoint matter).
Then we look at the Higgs branch of the mirror quiver, whose dimension should match the Coulomb branch dimension, see formula. <ref>. Similarly to our previous case, we conjecture that this requires the
addition of extra adjoint matter on $U(1)$ node, and the number can be easily computed.
The mirror  then takes the form shown in figure. <ref> [For $n+k=1$, we have $z=0$, but there are still $a$ quiver nodes with rank $n_1$ on the blue tail.].

Example: Let's take $n+k=2$, and the corresponding 4d theory has enhanced flavor symmetry $SU(n+2n_1)$, and is actually equivalent to $D_2(SU(n+2n_1)$ theory (This can be verified by comparing  the Coulomb branch spectrum).
The mirror of this theory was found in [37].
Now look at our quiver in figure. <ref>, we have $b=1, a={{n-1}\over 2}$ [$n,k$ coprime, and $n+k=2$ implies that $n$,$k$ are both odd integers.], and $z=1,x={n-1\over2}+n_1$. Using these numbers, the quiver in figure. <ref>
is the same as that given in [37].

every picture/.style=line width=0.75pt

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(348,255) – (356.5,246) ;

(458,127) node  [font=]  $1$;

(548,233) node  [font=]  $x$;

(376.5,232) node  [font=]  $n_1+az$;

(643.75,373.75) node  [font=]  $1$;

(533,162) node    $b$;

(472,46) node    ${(b-1)(n+k-b-1)\over 2}$;

(612.75,328.75) node  [font=]  $2$;

(252.75,358.75) node  [font=]  $n_{1}$;

(398,162) node   [font=]   $n+k-b$;

(171.75,450.75) node  [font=]  $1$;

(208.75,407.75) node  [font=]  $2$;

(289.75,315.75) node  [font=]  $n_1+z$;

(336.75,265.75) node  [font=]  $n_1+(a-1)z$;

The 3d mirror for AD matter of type $(k,n)$ with $k<0$. The flavor symmetry of AD matter is $U(n_1)\times SU(n+n_1)$. We have $a(n+k)+b=n$, with $b<(n+k)$.
Here $x=n+n_1-(a+1),~z=n+k-1$.

For general case, it is now easy to get the rules for finding the 3d mirror (assuming there is $m$ simplest black punctures, a maximal blue and red puncture): a) The quiver tail for red and blue puncture are the same as the bottom part of quiver. <ref> (removing the $U(1)$ quiver node with adjoint matter), with the red and blue quiver nodes
as the maximal ones, the parameters are changed as follows: $x=n_1+\textbf{m}n-(a+1),~~z=\textbf{m}(n+k)-1$;  b): For each black puncture, there is a quiver tail, and the number of adjoints are modified as ${(b-1)(n+k-b-1)\over 2}$. Finally, there are $b$ edges between black and red nodes, and
$n+k-b$ edges between blue and black nodes, and $b(n+k-b)$ edges between black nodes.

Degenerating case: Up to this point, we have only considered the simplest black puncture. We now verify our proposal for the degenerating case. The example is represented by a $(3,2)$ type
sphere with one trivial red puncture, and one trivial blue, and a black puncture of type $[1,1]$. This theory is studied in [22] (see section 5 of that paper and it is claimed that this is the rank two $H_0$ theory), and its Coulomb branch has dimension 2. The 3d mirror for the theory is
shown in figure. <ref> and its Higgs branch indeed has dimension 2.

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(327.5,147.27) .. controls (339.5,111.27) and (346.5,101.27) .. (360.5,148.27) ;

(143,157) node [anchor=north west][inner sep=0.75pt]   [align=left] [font=] (3,2);

(146,107.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $[ 1,1]$;

(395,153.4) node [anchor=north west][inner sep=0.75pt]    $1$;

(337,152.4) node [anchor=north west][inner sep=0.75pt]    $2$;

Left: a type $(3,2)$ theory with one trivial red, one trivial blue and one black puncture with Young Tableaux $[1,1]$; Right: the 3d mirror for the theory on the left.

§.§ $D_{N}$ type theory