\begin{array}{cc}
r & 1-r \\
1-r & r \\
\end{array}
\right]   \,.
\esp
\end{align}

Ignoring the decoupled matter node,  the dual pairs in (<ref>), from RHS to LHS, are $\A, \E_{r,s}$ and $ \B $ respectively.  Note that the first and the third line in (<ref>) are the same, which implies that in this case
$\B= \A^{-1}$.  On the other hand,  the second line is the same as (<ref>), which implies $\E_r=\A^{-1}, \E_s=\B, \E_r =ST\circ \E_s$. We emphasize that (<ref>) includes all possible dual pairs in this partial gauging.

§.§.§ Gauging $U(1)_y$

Analogously we can consider partial gauging for $U(1)_y$, keeping $U(1)_z$ as a flavor symmetry, so that  the parameter $z$ is viewed as a mass parameter, $\langle z\rangle =\tilde{m}$.  This operation is represented by the integral

\begin{align}
& \int \mathbf{gauge}(y	):=\int dy \, 	e^{ -\pi i  a\, y^2  + 2\pi i\, \xi_y y}
\end{align}

on both sides of (<ref>), so that the equivalence of sphere partition functions takes form

\begin{align}
\bsp
\boxed{ \int \textbf{gauge}(y)
\int dx\, e^{ - 2 \pi i\, \tilde{m} x  }  		s_b \le(  -\frac{i Q}{2}  + 2y\r)
% s_b\le( \frac{iQ}{2} -x -y \r)
s_b\le( \frac{iQ}{2} \pm x -y \r)
=  \int \textbf{gauge}(y)	~ s_b(y\pm \tilde{m})   } \\
\esp
\end{align}

The corresponding plumbing graphs can be obtained by deleting the gauge node $U(1)_z$ and all lines connecting to this node:

\begin{align}\label{higgsinggraph2}
\bsp
\includegraphics[width=3in]{Figures/higgsing2}
\esp
\end{align}

Interestingly,  in this partial gauging the adjoint matter does not decouple, and hence it is still a 2-3 move. The two-node graph is the same as the two-node graph in (<ref>), so we copy effective levels for the orbit of $ST$-moves

\begin{align}\label{partical2rks}
\bsp
&\text{RHS:} ~~	\left[
\begin{array}{cc}
a+1 & 1 \\
1 & 1 \\
\end{array}
\right] \,,~
\left[
\begin{array}{cc}
a & -1 \\
-1 & 0 \\
\end{array}
\right] \,, ~
\left[
\begin{array}{cc}
\frac{a}{a+1} & -\frac{1}{a+1} \\
-\frac{1}{a+1} & \frac{a}{a+1} \\
\end{array}
\right] \,,  ~
\left[
\begin{array}{cc}
\frac{a-1}{a} & \frac{1}{a} \\
\frac{1}{a} & -\frac{1}{a} \\
\end{array}
\right]\,, ~
\left[
\begin{array}{cc}
\frac{1}{1-a} & \frac{1}{1-a} \\
\frac{1}{1-a} & \frac{1}{1-a} \\
\end{array}
\right] \,.
\esp
\end{align}

The left graph (three-node graph) in (<ref>)  leads to an orbit of theories with the following effective levels

\begin{align}\label{partialtriangle}
\bsp
(1) : ~\left[
\begin{array}{ccc}
a+1 & a+2 & 2 \\
a+2 & a+4 & 2 \\
2 & 2 & 1 \\
\end{array}
\right],\left[
\begin{array}{ccc}
a-3 & a-2 & -2 \\
a-2 & a & -2 \\
-2 & -2 & 0 \\
\end{array}
\right],
\left[
\begin{array}{ccc}
\frac{1}{1-a} & \frac{a-2}{a-1} & -\frac{2}{a-1} \\
\frac{a-2}{a-1} & \frac{1}{1-a} & -\frac{2}{a-1} \\
-\frac{2}{a-1} & -\frac{2}{a-1} & -\frac{4}{a-1} \\
\end{array}
\right],
\left[
\begin{array}{ccc}
\frac{a-4}{a} & \frac{2}{a}-1 & -\frac{4}{a} \\
\frac{2}{a}-1 & \frac{a-1}{a} & \frac{2}{a} \\
-\frac{4}{a} & \frac{2}{a} & -\frac{4}{a} \\
\end{array}
\right], \\
(2) : ~ \left[
\begin{array}{ccc}
\frac{a}{a+1} & -\frac{a+2}{a+1} & -\frac{2}{a+1} \\
-\frac{a+2}{a+1} & \frac{a}{a+1} & -\frac{2}{a+1} \\
-\frac{2}{a+1} & -\frac{2}{a+1} & \frac{a-3}{a+1} \\
\end{array}
\right],
\left[
\begin{array}{ccc}
-\frac{4}{a} & \frac{a+2}{a} & -\frac{4}{a} \\
\frac{a+2}{a} & -\frac{1}{a} & \frac{2}{a} \\
-\frac{4}{a} & \frac{2}{a} & \frac{a-4}{a} \\
\end{array}
\right],   \\
(3) : ~\left[
\begin{array}{ccc}
-\frac{4}{a-4} & \frac{a-2}{a-4} & \frac{4}{a-4} \\
\frac{a-2}{a-4} & \frac{1}{4-a} & -\frac{2}{a-4} \\
\frac{4}{a-4} & -\frac{2}{a-4} & -\frac{4}{a-4} \\
\end{array}
\right],
\left[
\begin{array}{ccc}
-\frac{1}{a+3} & \frac{a+2}{a+3} & \frac{2}{a+3} \\
\frac{a+2}{a+3} & -\frac{1}{a+3} & \frac{2}{a+3} \\
\frac{2}{a+3} & \frac{2}{a+3} & \frac{a-1}{a+3} \\
\end{array}
\right],
\left[
\begin{array}{ccc}
\frac{a}{a+4} & -\frac{a+2}{a+4} & \frac{4}{a+4} \\
-\frac{a+2}{a+4} & \frac{a+3}{a+4} & -\frac{2}{a+4} \\
\frac{4}{a+4} & -\frac{2}{a+4} & \frac{a}{a+4} \\
\end{array}
\right],
\left[
\begin{array}{ccc}
\frac{a-4}{a-3} & \frac{2-a}{a-3} & \frac{2}{a-3} \\
\frac{2-a}{a-3} & \frac{a-4}{a-3} & \frac{2}{a-3} \\
\frac{2}{a-3} & \frac{2}{a-3} & -\frac{4}{a-3} \\
\end{array}
\right], \\
(4) : ~ \left[
\begin{array}{ccc}
\frac{a+4}{4} & \frac{a+2}{4} & -1 \\
\frac{a+2}{4} & \frac{a+4}{4} & -1 \\
-1 & -1 & 1 \\
\end{array}
\right],
\left[
\begin{array}{ccc}
\frac{a}{4} & \frac{a-2}{4} & 1 \\
\frac{a-2}{4} & \frac{a}{4} & 1 \\
1 & 1 & 0 \\
\end{array}
\right]\,.
\esp
\end{align}

We can now pick up integral matrices from RHS and LHS above and form dual pairs; they are in fact identified as linking ($\mathbf{B}$) and unlinking ($\mathbf{A}$) respectively:

\begin{align}\label{partial2pair2}
\bsp
\text{RHS } ~~&\longleftrightarrow ~~\text{LHS}: \\
\left[
\begin{array}{cc}
a+1 & 1 \\
1 & 1 \\
\end{array}
\right] \quad
\quad
\left[
\begin{array}{ccc}
a+1 & a+2 & 2 \\
a+2 & a+4 & 2 \\
2 & 2 & 1 \\
\end{array}
\right] \,,   \\
\left[
\begin{array}{cc}
a & -1 \\
-1 & 0 \\
\end{array}
\right] \quad
\quad
\left[
\begin{array}{ccc}
a-3 & a-2 & -2 \\
a-2 & a & -2 \\
-2 & -2 & 0 \\
\end{array}
\right]
\esp
\end{align}

In addition to these dual pairs, one can also introduce the following parameters to the last three matrices of the two-node graph (RHS) in (<ref>) to make them formally integer

\begin{align}\label{relation2}
a = -1 + \frac{1}{n} \,,\quad a=\frac{1}{n} \,,\quad a= 1+\frac{1}{n} \,,\quad \forall \, n \in \mathbf{Z}\,.
\end{align}

For $n=\pm 1$ we get $a= -2, 0, \pm1$, which are special since these values are allowed for the two-node graph in (<ref>), which describes the theory $U(1)_a+1\F+1\AF$.
We can also use another parameter $r=- n$ for these three matrices,  which yields

\begin{align} \label{rarelation2}
r = \frac{a}{1+a} \,,~~ -\frac{1}{a}\,,~~\frac{1}{1-a}  ~~\in ~\mathbf{Z}  .
\end{align}

We rewrite the above $2\times 2$ matrices in terms of $r$ and find they are dual to $3 \times 3$ matrices in the first $(1)$ and second $(2)$ line of LHS in (<ref>):

\begin{align}\label{partial2pair}
\bsp
\text{RHS } ~~&\longleftrightarrow ~~\text{LHS}: \\
\left[
\begin{array}{cc}
r & r-1 \\
r-1 & r \\
\end{array}
\right]  ~~&\longleftrightarrow~~
\left[
\begin{array}{ccc}
r & r-2 & 2 (r-1) \\
r-2 & r & 2 (r-1) \\
2 (r-1) & 2 (r-1) & 4 r-3 \\
\end{array}
\right] \,,\\
\left[
\begin{array}{cc}
r+1 & -r \\
-r & r \\
\end{array}
\right]
\left[
\begin{array}{ccc}
r+1 & -2 r-1 & -2 r \\
-2 r-1 & 4 r+1 & 4 r \\
-2 r & 4 r & 4 r \\
\end{array}
\right]\,,~\left[
\begin{array}{ccc}
r & 1-2 r & -2 r \\
1-2 r & 4 r & 4 r \\
-2 r & 4 r & 4 r+1 \\
\end{array}
\right]
\,,\\
\left[
\begin{array}{cc}
r & r \\
r & r \\
\end{array}
\right]
\left[
\begin{array}{ccc}
r & r+1 & 2 r \\
r+1 & r & 2 r \\
2 r & 2 r & 4 r \\
\end{array}
\right]\,,
\esp
\end{align}

These dual pairs are identified respectively as $\A, \E,\B$ operations.

Furthermore,  preferred parameters for the third line $(3)$ of LHS are respectively:

\begin{align} \label{relation1}
a= 4 +\frac{1}{n}
\,, 	\quad
a= -3 +\frac{1}{n}
\,, 	\quad
a= -4 +\frac{1}{n}
\,, 	\quad
a= 3 +\frac{1}{n} \,, \quad \forall \, n \in \mathbf{Z} \,.
\end{align}

Rewriting the matrices in the third line in terms of these parameters does not give rise to literally integer matrices,  however they are in the same $ST$-orbit of $\A,\B,\E$ triangles (we do not show these matrices here).

For $n=\pm 1$ in (<ref>), we get special values $a= \pm2, \pm3, \pm4,\pm5$, which are allowed for the theory $U(1)_a+1\F+1\AF$ in (<ref>).
There is only one overlapping value $a=-2$ for both (<ref>) and (<ref>), so $a=-2$ should be a special value that leads to the maximal number of dual graphs with matrices:

\begin{align}
\left[
\begin{array}{cc}
-1 & 1 \\
1 & 1 \\
\end{array}
\right] \,,
\left[
\begin{array}{cc}
-2 & -1 \\
-1 & 0 \\
\end{array}
\right] \,,
\left[
\begin{array}{cc}
2 & 1 \\
1 & 2 \\
\end{array}
\right]
~\longleftrightarrow	~	\left[
\begin{array}{ccc}
-1 & 0 & 1 \\
0 & -1 & 2 \\
1 & 2 & -3 \\
\end{array}
\right] \,,
\left[
\begin{array}{ccc}
2 & 0 & 2 \\
0 & 2 & 2 \\
2 & 2 & 5 \\
\end{array}
\right] \,,
\left[
\begin{array}{ccc}
-1 & 0 & 2 \\
0 & 2 & 2 \\
2 & 2 & 1 \\
\end{array}
\right],\left[
\begin{array}{ccc}
-5 & -4 & -2 \\
-4 & -2 & -2 \\
-2 & -2 & 0 \\
\end{array}
\right] \,. \nn
\end{align}

Note that matrices in the fourth line $(4)$ of LHS of (<ref>)  cannot be integral for any $a$, so they are meaningless in this context.

§.§.§ External nodes

The 2-2 move and the 2-3 move that we found by partial gauging can be also coupled to external nodes.  Let us show that in the presence of external nodes, the 2-2 move arising from partial gauging is a special case of the results found by full gauging discussed in section <ref>.  For the 2-2 move, adding an external node $U(1)_{c_i}$ to (<ref>) leads to

\begin{align}\label{higgsinggeneric}
\bsp
\includegraphics[width=3.5in]{Figures/higgsinggeneric}
\esp
\end{align}

Similarly to the full gauging, we do not need to apply a Kirby move to the external node $U(1)_{c_i}$. After scanning the orbit of theories given by $ST$-moves of the right graph in (<ref>), we find many matrices with submatrices of the form (<ref>).   One subset of this orbit of theories is given by the following matrices

\begin{align}\label{RHShiggs}
&\text{RHS:}\quad  \left[
\begin{array}{c|cc}
{c_i} & \pm {b_i} & 0 \\ \hline
\pm	{b_i} & b+1 & -1 \\
0 & -1 & 1 \\
\end{array}
\right] \,,~~\left[
\begin{array}{c|cc}
{c_i} & \pm{b_i} & 0 \\  \hline
\pm {b_i} & b & 1 \\
0 & 1 & 0 \\
\end{array}
\right]
\end{align}

which are linear graphs given by $ST$-moving the $\AF$ using the triality (<ref>). Moreover, since these four matrices are in the same orbit of $ST$-moves, it seems that $ST$-moves change the sign of $b_i$. We will discuss in the next section that this implies the possibility of Higgsing a pair of $1\F+1\AF$.

The graph on LHS of (<ref>) leads to four plumbing graphs in the orbit of $ST$-moves:

\begin{align}\label{LHShiggs}
\left[
\begin{array}{c|cc}
c_i & \pm b_i & \pm b_i \\ \hline
\pm b_i & b+1 & b \\
\pm b_i & b & b \\
\end{array}
\right]
\end{align}

which also are triangle graphs. The RHS relates to LHS through unlinking and linking as well, up to an element coming from the decoupled adjoint chiral multiplet on the LHS. If this decoupled matter node is added, the LHS is encoded in $3\times 3$ matrices representing triangle graphs:

\begin{align}\label{generalhiggsingpari}
\bsp
\left[
\begin{array}{c|cc}
{c_i} & \pm {b_i} & 0 \\ \hline
\pm	{b_i} & b+1 & -1 \\
0 & -1 & 1 \\
\end{array}
\right]  &~~ \xlongrightarrow{~\text{linking}~}~~
\left[
\begin{array}{c|c|cc}
{c_i} &~0~ & \pm {b_i} &  \pm {b_i} \\ \hline
0&1&0&0\\ \hline
\pm	{b_i}&0 & b+1 & b \\
\pm {b_i}  &0& b & b \\
\end{array}
\right]  \,,   \\
\left[
\begin{array}{c|ccc}
{c_i} & \pm {b_i} & ~0~ \\ \hline
\pm	{b_i} & b & 1 \\
0 & 1 & 0 \\
\end{array}
\right]   & ~~\xlongrightarrow{\text{unlinking}}~~
\left[
\begin{array}{c|c|cc}
{c_i} &~0~ & \pm {b_i} &  \pm {b_i}  \\ \hline
0&0&0&0\\ \hline
\pm	{b_i}&0 & b+1 & b \\
\pm {b_i}  &0& b & b \\
\end{array}
\right]   \,.
\esp
\end{align}

These are therefore dual pairs between linear graphs and triangle graphs, generalizing the dual pairs (<ref>) by coupling to an external node. One can also go from the right to the left of (<ref>), which would be identified with exotic operations; however, the element $1$ for the  decoupled chiral multiplet may be replaced equivalently by $0$ to satisfy the relations  for exotic triangles in (<ref>).

Let us discuss the generalization of the dual pair (<ref>) to include external nodes. In this case,  both graphs in  (<ref>) can be transformed into linear or triangle graphs through $ST$- and Kirby moves:

\begin{align}\label{lineartriangle}
\bsp
\includegraphics[width=3in]{Figures/higgsingRHS}
\esp
\end{align}

The corresponding dual pairs are represented by matrices, which in appropriate parametrization take form

\begin{align}
\bsp
\text{RHS} ~~&\longleftrightarrow~~ \text{LHS} \,,\\
\left[
\begin{array}{c|cc}
{C_i} & B_i & -{B_i} \\ \hline
{B_i} & r & 1-r \\
-{B_i} & 1-r & r \\
\end{array}
\right]
\left[
\begin{array}{c|cc}
C_i & B_i & -B_i \\  \hline
B_i & r & -r \\
-B_i & -r & r \\
\end{array}
\right]  \,,\\
\left[
\begin{array}{c|cc}
C_i & \pm B_i & \pm B_i \\  \hline
\pm B_i & r+1 & r \\
\pm B_i & r & r \\
\end{array}
\right]
\left[
\begin{array}{c|cc}
C_i & \pm B_i & 0 \\    \hline
\pm B_i & r+1 & -1 \\
0 & -1 & 1 \\
\end{array}
\right] \,,~~\left[
\begin{array}{c|cc}
C_i & \pm B_i & 0 \\    \hline
\pm B_i & r & 1 \\
0 & 1 & 0 \\
\end{array}
\right] \,,\\
\left[
\begin{array}{c|cc}
C_i & B_i & -B_i \\  \hline
B_i & r & -r \\
-B_i & -r & r \\
\end{array}
\right]
\left[
\begin{array}{c|cc}
C_i & B_i & -B_i \\  \hline
B_i & r & 1-r \\
-B_i & 1-r & r \\
\end{array}
\right] \,.
\esp
\end{align}

Note that the first and the third line are the same,  and represent a dual pair of triangle graphs. The second line represents a duality between a triangle and linear graphs.

Analogously one can discuss 2-3 move given by partial gauging the other flavor symmetry $U(1)_y$. In this case the results are a generalization of (<ref>) and (<ref>), as an interested reader can check.

§.§ Higgsing

Finally, we discuss Higgsing of matter fields, which happens when mass parameters are tuned to some specific values,  so that the multiplet $\F$ cancels with $\AF$

\[
\text{Higgsing:} \quad 1\F + 1\AF \rightarrow 1
\]

In this case we can delete from the graph these two matter nodes and any lines connecting to them

\begin{align}
\bsp
\begin{tikzpicture}
\draw[thick] (0,0)--(0.3,-1)node[midway,left	]{\small$+1$}--(0.6,0) node[midway,right	]{\small$-1$}  ;
\node[rectangle,fill=gray] at (0,0) {};
\node[rectangle,fill=gray] at (0.6,0) {};
\node at (2,-0.5){$=~~\mathbf{1}$};
\end{tikzpicture} \,.
\esp
\end{align}

The lesson one can learn from brane webs and vortex partition functions is that the Higgsing for 3d theories can be engineered by adjusting the mass parameters of these two matter fields: $m_{\F} =m_{\AF} \pm \epsilon_{-}$, where $\epsilon_- = \epsilon_1 -\epsilon_2$ and $\pm$ depends on the assignment of Omega deformation parameters $\epsilon_{1,2}$ on the spacetime of 3d theories $\mathbf{R}_{\epsilon_i}^2 \subset \mathbf{R}_{\epsilon_i}^2 \times S^1$, and then the identity $s_b( x)s_b(-x)=1$ leads to  the cancellation of these two matter fields. From the perspective of topological strings, this Higgsing is a manifestation of the geometric transition [23].

We now illustrate that Higgsing is consistent with gauged dualities, by applying it to dual graphs and showing that they lead to the same reduced dual theories. We  consider partially gauged graphs in (<ref>), which we copy here for convenience

\begin{align}\label{higgsinggenericcopy}
\bsp
\includegraphics[width=3in]{Figures/higgsinggenericcopy}
\esp
\end{align}

Note that $U(1)_{c_i}+1\F$ is the external node that we can add to the dual pair. The sphere partition functions for this dual pair can be easily written down by simply adding the contribution of the external node to both sides of the equivalence (<ref>).  For $m=\frac{iQ}{2}$, Higgsing happens on the right graph (RHS), which surprisingly reduces to the external node $U(1)_{c_i}+1\F$, while the left graph (LHS) does not change. One can read off this conclusion from the sphere partition functions

\begin{align}\label{RHShiggsing}
Z^{LHS} &=\int d z_i e^{-\pi i\, c_i z_i^2 - 2\pi i\, b_i z z_i }s_b\le(\frac{i Q}{2}-z_i \r)
\int dz  \, 	e^{ -\pi\, i   b\, z^2 + 2 \pi i\, \xi_z z   }  ~s_b \le( \frac{i\,Q}{2} \pm z \r) \,,\\
Z^{RHS} &=	s_b \le( \frac{i\, Q}{2}\r) \int d z_i e^{-\pi i\, c_i z_i^2 - 2\pi i\, b_i z z_i }s_b\le(\frac{i Q}{2}-z_i \r)	\int dz  \, 	e^{ -\pi\, i   b\, z^2 +2 \pi i \, \xi_z z   } 	\int dx \, e^{ - 2 \pi i \, z x} s_b(\pm x)
\nn \\
& =	s_b \le( \frac{i\, Q}{2}\r) \int d z_i e^{-\pi i\, c_i z_i^2  } s_b\le(\frac{i Q}{2}-z_i \r) \,,
\end{align}

where integrating out $x$ gives a delta function $\delta(z)$ which imposes a constraint $z=0$. After this Higgsing, the two matter nodes in RHS graph are deleted and then we get two pure gauge nodes $U(1)_0 \times U(1)_b$. Using the second rule of Kirby moves for gauge nodes in (<ref>), one can reduce the RHS graph to $U(1)_{c_i} +1\F$. Note that this reduction only happens when we Higgs these two matter nodes. If one does not turn on this special value $m=i Q/2$ by hand, then the two matter nodes cannot be deleted and the RHS graph does not reduce.  Since $Z^{LHS} =Z^{RHS}$, we are left with

\begin{align}
\int dz  \, 	e^{ -\pi\, i   b\, z^2 + 2 \pi i \xi_z z   }  ~s_b \le( \frac{i\,Q}{2} \pm z \r)  \times \text{(external node)}~ \sim ~s_b \le( \frac{i\, Q}{2}\r) \times \text{(external node)}	  \,.
\end{align}

The reduced dual pair arising from Higgsing matter nodes on the RHS graph is represented by the following graphs

\begin{align}\label{higgs22move}
\bsp
\includegraphics[width=2in]{Figures/higgs22move}
\esp
\end{align}

and it looks like a 3-1 move. Note that after Higgsing, the mass parameters for the $\F$ and $\AF$ on the LHS graph vanish, see (<ref>), but $\F$ does not cancel with $\AF$, so the LHS graph does not reduce. However, although the $\F$ and $\AF$ on the RHS graph have a small mass parameter $ iQ/2$, they cancel with each other, thereby reducing the graph. Because of the reduced dual pair (<ref>), although the sub-graph $U(1)_b+1\F+1\AF$ on LHS survives, one can just delete this part. From this perspective, when $m= i Q/2$, the LHS graph also gets reduced. In this process, the gauge node $U(1)_b$ is removed with these two matter fields $1\F+1\AF$. This phenomenon is similar to what happens when  Higging a bifundamental hypermultiplet in 5d brane webs, in which the Higgsing is interpreted as lifting a D5-brane  to infinity and hence the rank of the gauge group also reduces by one, see e.g. [23, 39].

Moreover,  we can also set $m=0$ to directly Higgs the $\F$ and $\AF$ on the left graph (LHS) of the dual pair (<ref>). However,  in this case we find an undecorated gauge node $U(1)_z$ left, which needs to be integrated out to produce the same reduced dual pair (<ref>). The left and the right graph reduce in the following way:

\begin{align}\label{higgsingmzero}
\bsp
\includegraphics[width=4in]{Figures/higgsingmzero}
\esp
\end{align}

In the last step of the above operations, one ends up with the same dual pair in (<ref>), up to the shift of levels.

We emphasize that mixed levels can also indicate the Higgsing.  A hint is encoded in (<ref>) and (<ref>), where the signs of $\pm b_i$ are flipped by $ST$-moves, which implies that lines with $\pm b_i$ could be cut and then the external node $U(1)_{c_i}$ decouples from plumbing graphs. In other words, the part $U(1)_b+1\F+1\AF$  does not really couple to the external node.   In addition, since generic Kirby moves and Higgsing can be applied commutatively on graphs in (<ref>) to get some linear or triangle graphs (<ref>) as we have discussed in section <ref>, lines $\pm B_i$ in these graphs can also be cut.

Then the dual pair would be between a triangle and a one-node graph, which is also a 3-1 move, instead of (<ref>).

Apart from Higgsing, one can also consider decoupling of matter nodes from dual pairs. This should simply lead to the triality (<ref>) that we discussed before. We leave this discussion as an exercise for interested readers.

§ EXCHANGE SYMMETRY

In refined topological string theory and refined theory [91, 39] there is an exchange symmetry ${\q} \leftrightarrow \t^{-1}$, which in the unrefined limit reduces to $\q\leftrightarrow \q^{-1}$. The 3d theory is realized on a surface defect of 5d spacetime $\mathbf{R}^2 \times S^1 \subset \mathbf{R}_\q^2 \times \mathbf{R}^2_{\t^{-1}} \times S^1$. This exchange symmetry is equivalent to exchanging the surface defect from $\mathbf{R}^2_\q$ to $\mathbf{R}^2_{\t^{-1}}$. In other words, it exchanges $\bar{\q}$-brane and $\t$-brane. In the knot theory context,  this exchange symmetry is called mirror symmetry and it transposes Young diagrams that label colored knot polynomials.  Here we note that this exchange symmetry suggests a duality for abelian 3d theories,  which amounts to flipping the signs of bare levels. We confirm this by considering vortex partition functions, or more completely holomorphic blocks of 3d theories [26, 31, 92, 38].

For instance, the theory $U(1)_k+N_f \F +N_a \AF$ has the vortex partition function on a vacuum in Higgs branch:

\begin{align}
Z^{\text{vortex}}( z, {\a},{\b}	) = \frac{  \prod_{j=1}^{N_f} ( \b_j;\q)_\inf }{ \prod_{i=1}^{N_a}  (\a_i ;\q)_\inf } \cdot \sum_{n=0}^{\inf} \le(  -\sqrt{\q}\r)^{k^{\eff} n^2 } z^n \cdot \frac{ \prod_{i=1}^{N_a}  (\a_i ;\q)_n }{  \prod_{j=1}^{N_f} ( \b_j;\q)_n }	 \,,
\end{align}

where contributions of $\AF$ and $\F$ are $(\a_i; \q)_n/(\a_i; \q)_\inf$ and $ (\b_j;\q)_\inf/(\b_j;\q)_n$ respectively, and $\a$ and $\b$ are associated with mass parameters $, ≃̱e^m$.
Using the identity
( \a ;\q)_n = ( \a^{-1};\q^{-1})_n \le(- \sqrt{\q^{-1}}\r)^{-n^2} \le( \sqrt{\q^{-1}} \a \r)^n \,,$$
one can derive the vortex partition function of a dual theory given by the exchange symmetry:
\begin{align}\label{exchangeU1}
Z^{\text{vortex}}(z, \a,\b) ~\xrightarrow{ \q \,\rightarrow\, \q^{-1}}~
Z^{\text{vortex}}( \tilde{z}, \a^{-1},\b^{-1})  \,,
\end{align}
where the vortex partition function and the effective \CS level for the dual theory are\footnote{Note that we have a normalization factor $\eta$, which can be ignored using  \eqref{normal}. For the definition of $q$-Pochharmmer symbols and theta functions, see the appendix \ref{pochappendix}.}
\begin{align}\label{thetaappear}
\bsp
&Z^{\text{vortex}}(\tilde{z}, \a^{-1},\b^{-1})= \eta \cdot 	\frac{  \prod_{j=1}^{N_f} ( \b_j^{-1};\q)_\inf }{ \prod_{i=1}^{N_a}  (\a_i^{-1} ;\q)_\inf } \cdot \sum_{n=0}^{\inf} \le(  -\sqrt{\q}\r)^{ \widetilde{k}^{\eff} n^2 } \ {\tilde{z}}^n \cdot \frac{ \prod_{i=1}^{N_a}  (\a_i^{-1} ;\q)_n }{  \prod_{j=1}^{N_f} ( \b_j^{-1};\q)_n }	 \,, \\
&	 \widetilde{k}^{\eff} = -k^{\eff}-N_a +N_f =-\le(k +\frac{N_f-N_a}{2} \r) - N_a+N_f= -k +\frac{N_f-N_a}{2} \,,\\
& \widetilde{z} =	z \cdot \frac{ \prod\limits_{i=1}^{N_a}\sqrt{\q} \a_i  }{ \prod\limits_{j=1}^{N_f} \sqrt{\q} \b_j  }  \,,\quad  \eta =\frac{ \prod\limits_{i=1}^{N_a} \theta( - \q^{1/2} \a_i )  }{ \prod\limits_{i=1}^{N_f} \theta( - \q^{1/2} \b_i )  } \,.
\esp
\end{align}
The term $η$ contains theta functions, which arise from exchanging boundary conditions for all chiral multiplets [92, 31, 93, 38]. Note that 3d theories should couple to 2d $(0,2)$ theories on their boundaries to cancel anomalies. These theta functions are contributions of 2d multiplets. Adding some theta functions implies changing the matter content of the 2d theory on the boundary. Here we see that the exchange symmetry is sensitive to the boundary conditions, so considering holomorphic blocks would be more appropriate, however the physical argument would be the same.
%This is unusual and different to $ST$-transformations, which are not sensitive to boundaries conditions.
In short, the equivalence in \eqref{exchangeU1} suggests a dual pair
\begin{align}\label{dualpariu1k}
\bsp
U(1)_k + N_f \F+N_a\AF &~~\longleftrightarrow~~	U(1)_{-k} + N_f \F+N_a\AF \\
( k\,, m_i) &~~\longleftrightarrow~~	( -k\,, -m_i)
\esp
\end{align}
In particular,  for self-dual theories \CS level should vanish, $k=0$.

For balanced plumbing theories $(U(1)+1)_k_ij^N_f$, namely the theories that contain only decorated gauge nodes, the exchange symmetry also  suggests a dual pair.  This can be deduced from the form of the vortex partition function on a Higgs vacuum:
\begin{align}
Z^{\text{vortex}}( {z,\b}) =\prod_{j=1}^{N_f} ( \b_j;\q)_\inf \cdot \sum_{n_1\,,\cdots\,, n_{N_f} =0 }^{\inf} ~ \le( -\sqrt{\q}\r)^{\sum\limits_{i,j=1}^{N_f} k^{\eff}_{ij} n_i n_j } \cdot \prod_{i=1}^{N_f} \frac{ z_i^{n_i} }{ (\b_i;\q)_{n_i} } \,,
\end{align}
where $z_i$ is the FI parameter for each gauge node. In [69], it is shown that theories of this type become massless if one absorbs mass parameters into FI parameters, and then their contributions reduce from $(_̱j;)^-1_n$ to $(;)^-1_n$. The exchange symmetry also leads to a dual pair by applying $↔^-1$ on this vortex partition function:
\begin{align}\label{dualpairSTthry}
\bsp
(U(1)+1\F)_{k_{ij}}^{N_f}  ~~&\longleftrightarrow~~	(U(1)+1\F)_{-k_{ij}}^{N_f}  \\
( \,k_{ij} \,, ~z_i)~~\,&\longleftrightarrow~~ (-k_{ij}  \,,~ \tilde{z}_i )
\esp
\end{align}
The effective \CS levels are $k^_ij= k_ij + 1/2 δ_ij$.
The effective \CS levels for the dual pair are related by
\begin{align}\label{mixedCSexchange}
~k_{ij}^{\eff}~~&\longleftrightarrow~~ -k_{ij}^{\eff} +\delta_{ij} \,.
\end{align}
This duality can be confirmed by performing generic Kirby moves on plumbing theories, as we consider in examples in section \ref{secexample}.
%Moreover, we comment that if two theories are dual to each other by the exchange symmetry; then after performing generic Kirby moves on both theories, the mixed CS levels also related to each other by \eqref{mixedCSexchange}.
%We will show some examples in section \ref{secexample}.
In particular, when $k_ij=0$, the theory is self-dual and trivially coupled, as it consists of only some free theories without interactions $(U(1)_0+1)^ ⊗N_f$.

This exchange symmetry has non-trivial implications. For example, integrating out the undecorated gauge node $U(1)_k±1/2$ of the gauged triality \eqref{blowupdownmatter} leads to $U(1)_ ∓3+2k /2±4 k  +$, which should be equivalent to $U(1)_-k+1$, as suggested by the exchange symmetry. Then the equivalence $ ∓3+2k /2±4 k  =-k$ leads to solutions:
\begin{align}\label{specialk1F}
k = \pm \frac{1}{2} \,, ~\pm \frac{3}{2} .
\end{align}
In this way, these values are selected by exchange symmetry. Perhaps, they are special because the theories with these values can be related by $𝒮$-duality of type IIB string theory from the perspective of brane webs, which we discuss in  section \ref{Sduality}. Because of the factor of $η$ that involves theta functions, the exchange symmetry and $𝒮$-duality do not leave vortex partition functions invariant. This again confirms that vortex partition functions are not complete and it is more appropriate to consider holomorphic blocks, which include contributions from the 2d boundary theory.

\section{Effective superpotential}   \label{seceffective}

In this section we briefly discuss the structure of Coulomb branches of moduli spaces of plumbing theories, which are given by effective superpotentials.

The effective superpotential is the leading term of the vortex partition function $e^/ħ+…$.  It can be also obtained by taking the classical limit $= e^ħ →1$ of a holomorphic block.	 For balanced theories $( U(1)+1)_k_ij^N_f$, the effective superpotential
takes form\footnote{If we replace $Y_i= \log\, y_i$, then twisted effective effective superpotential is written as
$	\widetilde{\W}^{\eff}(\mathbf{y}, \mathbf{\xi})= \sum_{i=1}^{N_f} \le(  \Li_2(y_i) + \xi^{\eff}_i \log \,y_i  \r) + \sum_{i,j=1}^{N_f} \frac{ k_{ij}^{\eff}}{2} \,\log \,y_i \,\log \,y_j   $.	}
\begin{align}
\widetilde{\W}^{\eff}(\mathbf{Y}, \mathbf{\xi})= \sum_{i=1}^{N_f} \le(  \Li_2(e^{Y_i +m_i}) + \xi^{\eff}_i Y_i  \r) + \sum_{i,j=1}^{N_f} \frac{ k_{ij}^{\eff}}{2} \, Y_i Y_j \,,
\end{align}
where $Y_i$ is the variable for the $i$-th gauge node $U(1)_k_i$,  the term $1/2 k_ij  Y_iY_j$ comes from the \CS coupling between $i$-th and $j$-th gauge node, and $m_i$ is the real mass parameter for the $i$-th chiral multiplet.  Each chiral multiplet contributes a dilogarithm term $_2≤(e^Y_i+m_i)̊$. In [94, 95] the vacuum equations are found to take form
\begin{align}
\exp\left( \frac{d\,\Weff}{d\, Y_i} \right) =1\,,\quad \text{for}~\forall ~i = 1 \,, \cdots\,,N_f ,
\end{align}
or more explicitly:
\begin{align}\label{vacuaeqn}
\sum_{j=1}^{N_f} k_{ij}^{\eff} \,Y_j + \xi_i^{\eff} =\log \left( 1- e^{Y_i +m_i}  \right)
\,,\quad \text{for}~\forall ~i = 1 \,, \cdots\,,N_f \,.
\end{align}
The terms on the right hand side come from  chiral multiplets as $d   _2(e^Y_i+m_i)/d Y_i  =-log( 1- e^Y_i+m_i)$, so we can ignore these terms when $m_i$ are very large, and then the moduli space of vacua becomes the Coulomb branch discussed in [66], in which the vacua are given by the scalar potential.

In [38], the  $ST$-moves (transformations) of holomorphic blocks have been discussed. $ST$-moves could be represented as some operations on effective superpotentials. By looking at the gauged mirror triality \eqref{ststF}, one can see how
$ST$-moves act on effective superpotentials.  $ST$-moves only transform terms representing matter fields:
\begin{align}\label{STformatter}
(ST)^{\pm}:~~\Li_2\left(e^{Y}\right)  ~\rightarrow~ \Li_2\left(e^{Z}\right) \pm \frac{Z^2}{2} \pm Y Z  \,,
\end{align}
where the sign $±$ corresponds respectively to $ST$-move and $(ST)^-1$-move (or equivalently $(ST)^2$-move).  The same conclusion follows from analysis of sphere partition functions,  upon taking advantage of identities for double-sine functions from section \ref{gaugetri}.

If the FI parameter $ξ$ for $Y$ does not vanish, then one should shift $Z→Z ∓ξ$ to get
\begin{align}
(ST)^{\pm}:~~	\Li_2\left(e^{Y+m}\right) +\xi Y ~ \rightarrow ~\Li_2\left(e^{Z \mp \xi}\right) \pm \frac{Z^2}{2} \pm Y Z  +\left(  -\xi \pm m\right) Z
\pm \frac{\xi^2}{2}  \,.
\end{align}
Hence the FI parameter $ξ$ becomes the mass parameter and also contributes to the FI parameter for $Z$. The last term can be ignored as it is not dynamical. Note that the sign of FI parameter $ξ$ is  flipped by  $ST$-moves, which is consistent with decoupling of matter nodes that we discussed in section \ref{decouplematter}. The flip is interesting as it can be illustrated using 3d brane webs [38].
Moreover, the property
\begin{align}
\Li_2\left(e^{X} \right)+ \Li_2\left(e^{-X} \right)= -\frac{X^2}{2} -\frac{\pi^2}{6}
\end{align}
exchanges $$ and $$, and hence one obtains the ST-moves \eqref{ststAF} for anti-fundamental matter. Using the relation $	_2≤(e^X )̊  =
±X^2/2$ for $X  →±inf$, one can also see how to decouple heavy chiral multiplets in effective superpotentials.
% which is consistent with the discussion in section \ref{decouplematter}.

We emphasize that dual theories should have the same propotential that encodes BPS invariants. 3d propotential can be obtained by integrating over all gauge nodes:
\begin{align}\label{OVformula}
\exp\le(	\frac{\W^{3d} ( \mathbf{\xi})}{\hbar}  \r)=\int  \prod_i d Y_i \, \exp{\left(\frac{\Weff} {\hbar } \right) } \,,
\end{align}
which is also equivalent to the prepotential of open topological string partition functions. This prepotential encodes Ooguri-Vafa invariants $N^OV_𝒞$ of open topological string theories that engineer 3d theories. Since the parameters for gauge groups are $Y_i$, all undecorated gauge nodes can be integrated out,  and only decorated gauge nodes are left, as it is hard to integrate out the term $_2(e^Y)$ from matter fields. It is obvious from \eqref{OVformula} that theories related by Kirby moves,  which only differ by some undecorated gauge nodes, have the same prepotential. Hence theories related by Kirby moves have the same contracted Ooguri-Vafa invariants, which are given by summing over all spin indices $N^OV_𝒞 = ∑_j,r ∈𝐙/2 (-1)^2j +2 r N_𝒞^(j,r)$.  For more details see [96, 69]. Moreover,  in terms of the Ooguri-Vafa prepotential $^3d$, the $ST$-moves on matter fields \eqref{STformatter} can be viewed as introducing Lagrangian multipliers.

\section{Examples: $U(1)_k+N_f\F$ theories}  \label{secexample}

In this section we illustrate our considerations in more involved examples of $U(1)_k+N_f$ theories. The theory with $N_f=1$ has been discussed in section \ref{gaugetri}, so in this section we analyze in more detail theories with $N_f=2,3$.

\subsection{$U(1)_k + 2\F$ }

A plumbing graph of this theory consists of two matter nodes connected to a gauge node. Performing $ST$-moves on matter nodes leads to a chain of theories:
\begin{align}\label{U1_2Fplumb}
\begin{split}
\includegraphics[]{Figures/U1_2Fplumb}
\end{split}
\end{align}
Although this chain contains infinitely many graphs, integrating out undecorated gauge nodes leads to several two-node graphs with different mixed \CS levels:
\begin{align}\label{2Fmatrix}
k_{ij}^{\eff} ~ =~
\begin{bmatrix}
k+1 &~ 1 \\
~1~& ~1
\end{bmatrix} \,,\quad
\begin{bmatrix}
k & -1 \\
-1& 0
\end{bmatrix} \,,\quad
\begin{bmatrix}
\frac{k}{1+k} &	-\frac{1}{1+k} \\
-\frac{1}{1+k} & 	\frac{k}{1+k}
\end{bmatrix} \,,
\quad
\begin{bmatrix}
\frac{k-1}{k} &~	\frac{1}{k} \\
~	\frac{1}{k} & 	-\frac{1}{k}
\end{bmatrix} \,, \quad
\begin{bmatrix}
\frac{1}{1-k} &	\frac{1}{1-k} \\
\frac{1}{1-k} & 	\frac{1}{1-k}
\end{bmatrix} \,.
\end{align}
These theories are related by $ST$-moves and hence dual to each other.  To make the last three matrices integral, $k$ needs to satisfy the following conditions respectively:
\begin{align}
\bsp
k = \frac{n+1}{n} \,, ~\frac{1}{n}\,, ~\frac{n-1}{n} \,, ~~\text{where}~n\in \mathbf{Z} \setminus  \{ 0\}  \,.
\esp
\end{align}
For the original theory $U(1)_k+2$, the condition is simply $k ∈𝐙$. Imposing these two conditions, the common solutions are $k= 0, ±1, ±2$. More explicitly, for $k ∈𝐙$ the first two plumbing graphs in \eqref{2Fmatrix} have obviously integral \CS levels, however other matrices may not be integral. However, for $k= ±1 , ±2$ there is a maximal numbers of equivalent graphs labeled by integral matrices:
\begin{align}
\bsp
& k=1 \,,\quad ~~k^{\eff}=
\begin{bmatrix}
~0 &~ ~	1 \\
~1& -1 ~
\end{bmatrix} \,,~~
\begin{bmatrix}
~2 &~ 1~ \\
~1~& ~1 ~
\end{bmatrix}
\,,~~
\begin{bmatrix}
~1 &-1~ \\
-1~& ~0	 ~
\end{bmatrix} \,,\\
& k=-1 \,,\quad k^{\eff}=
\begin{bmatrix}
2 & -1 \\
-1& 1
\end{bmatrix}
\,,~~
\begin{bmatrix}
~0~ &~ 1~ \\
~1~& ~1 ~
\end{bmatrix} \,,~~
\begin{bmatrix}
-1&-1~ \\
-1~& ~0~
\end{bmatrix} \,,
\\
& k=2 \,,\quad~~ k^{\eff}=
\begin{bmatrix}
-1 & -1 \\
-1& -1
\end{bmatrix} \,,~~
\begin{bmatrix}
~3 &~ 1~ \\
~1~& ~1 ~
\end{bmatrix}
\,,~~
\begin{bmatrix}
~2 &- 1~ \\
-1~& ~0 ~
\end{bmatrix} \,,
\\
&k=-2 \,,\quad 		k^{\eff}=
\begin{bmatrix}
~2 ~& ~1~ \\
~1~& ~2~
\end{bmatrix}
\,,~~
\begin{bmatrix}
~1~ &~ 1~ \\
~1~& -1
\end{bmatrix}
\,,~~
\begin{bmatrix}
-2 &- 1~ \\
-1~& ~0 ~
\end{bmatrix} \,.
\esp
\end{align}
For other values of $k$ only the first two matrices are integral and thus meaningful:
\begin{align}
& k=0 \,, \pm3\,,\pm4\,,\cdots\,, 	\quad
\begin{bmatrix}
k+1 &~ 1~ \\
~1~& ~1~
\end{bmatrix} \,,
\quad
\begin{bmatrix}
k & -1 \\
-1& 0
\end{bmatrix}
\end{align}
and corresponding plumbing graphs are
\begin{align}\label{U1_2Fpm}
\begin{split}
\includegraphics[]{Figures/U1_2Fpm}
\end{split}
\end{align}
For $k = -2 , 0  , 2$, plumbing graphs are symmetric:
\begin{align}
\begin{bmatrix}
2 &~ 1~ \\
~1~& ~2~
\end{bmatrix} \,,
\quad
\begin{bmatrix}
1~ &1~ \\
1~& 1~
\end{bmatrix}\,  ~\text{and}
\begin{bmatrix}
0 &-1 \\
-1& 0
\end{bmatrix} \,,
\quad
\begin{bmatrix}
-1 &-1 \\
-1& -1
\end{bmatrix}
\,.
\end{align}
We will reconsider these special values $k=0,±1, ±2$ from the perspective of brane webs in section \ref{secbranewebs}.

It is easy to check that matrices of \CS levels in this example satisfy relations \eqref{dualpariu1k} and \eqref{dualpairSTthry} arising from the exchange symmetry $↔^-1$. Moreover, since graphs for these theories have two nodes, we can couple them to one additional decorated gauge node to get many dual triangle graphs satisfying unlinking, linking and exotic relations \eqref{trianglesgraph}. It would be nice to generate in this way infinitely many equivalent graphs.

\subsection{$U(1)_k + 3\F$}

Theories with $N_f=3$ have different properties than for $N_f=2$.  We now provide their classification based on \CS level $k$ and some properties of plumbing graphs. The $ST$-moves of this theory lead to the orbit:
\begin{align}\label{3fgraph}
\begin{split}
\includegraphics[]{Figures/U1_3Fplumb}
\end{split}
\end{align}
Note that the third diagram in this chain was found and discussed in [69] using brane webs. This graph has \CS levels
\begin{align}\label{branewen3F}
\begin{bmatrix}
k+\frac{3}{2}    & 1  &1\\
1 & 1 &0\\
1& 0 &1
\end{bmatrix}  \,.
\end{align}
The fourth and all following graphs contain undecorated gauge nodes that are not attached with matter.  Integrating them out results in balanced triangle graphs, which we now classify as symmetric, partially symmetric, and bifundamental.
%Let us discuss these special values of $k$. Just like the last example, we note that all  triangle graphs for the $N_f=3$ theory are meaningful, if we define appropriate parameters. We classify these triangles into very symmetric, partial symmetric, and bifundamental cases, based on a naive observation.

\subsubsection*{Symmetric cases}

Scanning all balanced triangle graphs in \eqref{3fgraph} we find that there are two  symmetric matrices of \CS levels:
\begin{align}
\begin{bmatrix}
\frac{2}{3-2k}   & 	\frac{2}{3-2k}  &	\frac{2}{3-2k}   \\
\frac{2}{3-2k}   & 	\frac{2}{3-2k}  &	\frac{2}{3-2k}   \\
\frac{2}{3-2k}   & 	\frac{2}{3-2k}  &	\frac{2}{3-2k}
\end{bmatrix} \,,	\quad
\begin{bmatrix}
\frac{2k+1}{2k+3}   & -\frac{2}{2k+3}  &-\frac{2}{2k+3}  \\
-\frac{2}{2k+3}  & ~\frac{2k+1}{2k+3}    &-\frac{2}{2k+3}  \\
-\frac{2}{2k+3}  & -\frac{2}{2k+3}   &
\end{bmatrix}
\end{align}
Effective \CS levels for the first matrix are integer if
\begin{align}
k=\frac{3}{2}+ \frac{1}{n} \,,\quad n \in \mathbf{Z} \setminus  \{ 0\}   \,.\label{cond1}
\end{align}
Let us pick up some special values:
\[k=\frac{1}{2}, 1, 2,\frac{5}{2}, \cdots \]
Their corresponding matrices are
\begin{align}
\begin{bmatrix}
~1   & ~1  & ~1~\\
~1   & ~1  & ~1~\\
~1   & ~1  & ~1~
\end{bmatrix}  \,,\quad
\begin{bmatrix}
~2   & ~2  & ~2~\\
~2   & ~2  & ~2~\\
~2   & ~2  & ~2~
\end{bmatrix} \,,\quad
\begin{bmatrix}
-2   & -2  & -2~\\
-2   & -2  & -2~\\
-2   & -2  & -2~
\end{bmatrix} \,,\quad
\begin{bmatrix}
-1   & -1  & -1\\
-1   & -1  & -1\\
-1   & -1  & -1
\end{bmatrix}  \,,\quad \cdots \,.
\end{align}
The vaules $k=1/2, 5/2$ are quite special, since they also ensure that the original theory $U(1)_k+3$ is free of parity anomaly. For these values there is also a brane web description.

For the second matrix, the integrality condition takes form
\begin{align}
k= -\frac{3}{2}  + \frac{1}{n'}  \,\quad n' \in \mathbf{Z} \setminus  \{ 0\}   \,.	\label{cond2}
\end{align}
We choose some special values:
\[k=-\frac{5}{2},-2,-1, \frac{1}{2}, \cdots\]
for which the corresponding matrices are
\begin{align}
\begin{bmatrix}
~	2~   & 1~  & 1~\\
~	1~ & 2~ & 1~\\
~	1~ & 1~ & 2~
\end{bmatrix}  \,,\quad
\begin{bmatrix}
~	3~   & 2~  & 2~\\
~	2~ & 3~ & 2~\\
~	2~ & 2~ & 3~
\end{bmatrix}  \,,\quad
\begin{bmatrix}
-1~   & -2~  & -2~\\
-	2~ & -1~ & -2~\\
-	2~ & -2~ & -1~
\end{bmatrix}  \,,\quad
\begin{bmatrix}
~0~   & -1~  & -1~\\
-	1~ & ~0~ & -1~\\
-	1~ & -1~ & ~0~
\end{bmatrix}  \,,\quad \cdots \,.
\end{align}
For $k= -5/2 ,-1/2$, we have $k^= -1,1$, so the original theory $U(1)_k+3$ is also free of anomaly and there is a brane web description. Note that there is no common value satisfying both conditions \eqref{cond1} and \eqref{cond2}. Interestingly, these special values $k=- 5/2, - 1/2$ lead also to symmetric plumbing graphs. More explicitly, the fourth graph in \eqref{3fgraph} can be turned into a balanced triangle  after integrating out the undecorated gauge node.  For $k= -5/2$ we get a triangle
\begin{align}\label{triangle3F52}
\bsp
\includegraphics[width=2.5in]{Figures/triangle3F52.pdf}
\esp
\end{align}
with effective \CS levels
\begin{align}
\begin{bmatrix}
~	2~   & 1~  & 1~\\
~	1~ & 2~ & 1~\\
~	1~ & 1~ & 2~
\end{bmatrix}  \,.
\end{align}
For $k =-1/2$ the corresponding triangle is
\begin{align}
\bsp
\includegraphics[width=2.5in]{Figures/triangle3F12.pdf}
\esp
\end{align}
which has effective \CS levels
\begin{align}
\begin{bmatrix}
0   & -1  &-1\\
-1 & 0 &-1\\
-1& -1 & 0
\end{bmatrix}  \,.
\end{align}

\subsubsection*{Partially symmetric}

Apart from symmetric triangle graphs, we also find partially symmetric ones
\begin{align}
\begin{bmatrix}\begin{array}{ccc}
\frac{2k-1}{2k+1}   & 	-\frac{2}{2k+1}  &	\frac{2}{2k+1}   \\
-\frac{2}{2k+1}   & 	\frac{2k-1}{2k+1}    &	\frac{2}{2k+1}  \\
\frac{2}{2k+1}   & 	\frac{2}{2k+1}  &	-\frac{2}{2k+1}
\end{array}
\end{bmatrix} \,,	\quad
\begin{bmatrix}\begin{array}{ccc}
\frac{2k-3}{2k-1}   & 	\frac{2}{2k-1}  &	\frac{2}{2k-1}   \\
\frac{2}{2k-1}   & 	\frac{2}{1-2k}    &	\frac{2}{1-2k}  \\
\frac{2}{2k-1}   & 	\frac{2}{1-2k}  &	\frac{2}{1-2k}
\end{array}
\end{bmatrix}  \,.
\end{align}
Integrality condition for the first matrix takes form
\begin{align}
k=-\frac{1}{2}  + \frac{1}{n} \,,\quad n \in \mathbf{Z} \setminus  \{ 0\} \,.\label{condbi1}
\end{align}
For special values:
\[k=-\frac{3}{2}, 1, 0,\frac{1}{2}, \ldots \]
the corresponding matrices are
\begin{align}\label{U13F32}
\begin{bmatrix}
\begin{array}{ccc}
2 & 1 & -1 \\
1 & 2 & -1 \\
-1 & -1 & 1 \\
\end{array}
\end{bmatrix}  \,,\quad
\begin{bmatrix}
\begin{array}{ccc}
3 & 2 & -2 \\
2 & 3 & -2 \\
-2 & -2 & 2 \\
\end{array}
\end{bmatrix} \,,\quad
\begin{bmatrix}
\begin{array}{ccc}
-1 & -2 & 2 \\
-2 & -1 & 2 \\
2 & 2 & -2 \\
\end{array}
\end{bmatrix} \,,\quad
\begin{bmatrix}
\begin{array}{ccc}
0 & -1 & 1 \\
-1 & 0 & 1 \\
1 & 1 & -1 \\
\end{array}
\end{bmatrix}  \,, \quad \cdots
\end{align}

For the second matrix, the integrality condition takes form
\begin{align}
k= \frac{1}{2}  + \frac{1}{n}  \,\quad n \in \mathbf{Z} \setminus  \{ 0\}   \,.	\label{condbi2}\end{align}
For some special values:
\[k=-\frac{1}{2},0,1, \frac{3}{2},\ldots \]
the corresponding matrices are
\begin{align}
\begin{bmatrix}
\begin{array}{ccc}
2 & -1 & -1 \\
-1 & 1 & 1 \\
-1 & 1 & 1 \\
\end{array}
\end{bmatrix}  \,,\quad
\begin{bmatrix}
\begin{array}{ccc}
3 & -2 & -2 \\
-2 & 2 & 2 \\
-2 & 2 & 2 \\
\end{array}
\end{bmatrix}  \,,\quad
\begin{bmatrix}
\begin{array}{ccc}
-1 & 2 & 2 \\
2 & -2 & -2 \\
2 & -2 & -2 \\
\end{array}
\end{bmatrix}  \,,\quad
\begin{bmatrix}
\begin{array}{ccc}
0 & 1 & 1 \\
1 & -1 & -1 \\
1 & -1 & -1 \\
\end{array}
\end{bmatrix}  \,,\quad \cdots
\end{align}
Note that there are some overlaps $k=±1/2, ±1$, which are allowed for both symmetric and partially symmetric matrices. This implies the equivalence of theories labeled by corresponding triangle graphs.

Let us analyze partially symmetric triangle graphs for special values $k=±3/2, ±1/2$, as these also allow brane web descriptions of the original theory.  For $k= ±3/2$, plumbing graphs degenerate to theories with bifundamental chiral multiplets.
In particular,  for $k=-3/2$ we obtain a mirror pair found in [66], with the following matrix of \CS levels
\begin{align}
\begin{bmatrix}
2   & 1  &-1\\
1 & 2 &-1\\
-1& -1 & 1
\end{bmatrix}  \,.\end{align}
By comparing  its triangle graph with $ST$-moved graphs for a bifundamental matter in \eqref{bifundgraph},  we see that this triangle encodes a bifundamental multiplet. Then the plumbing graph can be reduced to
\begin{align}
\bsp
\includegraphics[width=1.8in]{Figures/bifund3Fplus}
\esp
\end{align}
with effective \CS levels
\begin{align}
k^{\eff} =	\begin{bmatrix}
2   & ~1  \\
1 &~ 2
\end{bmatrix} \,.
\end{align}
For $k=3/2$ one can also get a theory labeled by a triangle graph, with mixed \CS levels
\begin{align}	k^{\eff}=
\begin{bmatrix}
-1 & -1 &~1\\
-1& -1 & ~1 \\
1   & 1  &~0
\end{bmatrix}   \,.  \end{align}
By comparing with \eqref{bifundgraph}, we see that it also encodes a bifundamental chiral multiplet, and the corresponding plumbing graph is
\begin{align}
\bsp
\includegraphics[width=1.8in]{Figures/bifund3Fminus}
\esp
\end{align}
with effective \CS levels
\begin{align}
k^{\eff} =	\begin{bmatrix}
-1   & -1  \\
-1 & -1
\end{bmatrix} \,.
\end{align}

\subsubsection*{Classification}

%From the above discussion, one can see that for different values of $k$, the graphs show different patterns. One can somehow use these pattern to classify theories, although the full classification of generic plumbing graphs could be complicated, as there are many choices for $k$ that lead to integral $k^{\eff}_{ij}$.

Based on the above results, in the following table we summarize some patterns of plumbing graphs for theories $U(1)_k+3$, for several values of $k$:
\begin{align}
\begin{array}{c| c| c| c| c |c| c| c |c |c| c| c }\hline
\text{level}~ k=	&~ -\frac{5}{2}~&\,{\color{red}-2~}&\,-\frac{3}{2}~& \,{\color{red}-1}~& ~~-\frac{1}{2}~&~~ ~{\color{red} 0}~~&~~ ~\frac{1}{2}~~&~ ~{\color{red}1} ~~& ~~\frac{3}{2}~~&~~{ \color{red} 2}~~& ~~\frac{5}{2}~~~   \\ \hline
\text{symmetric}&\checkmark&\checkmark &&\checkmark&\checkmark&&\checkmark&\checkmark&&\checkmark&\checkmark \\ \hline
\text{partially sym.}&&&\checkmark&\checkmark&\checkmark&\checkmark\checkmark&\checkmark&\checkmark&\checkmark&& \\ \hline
\text{ bifund.}&&&\checkmark&&&&&&\checkmark&& \\ \hline
\text{brane web}&\checkmark&&\checkmark&&\checkmark&&\checkmark&&\checkmark&&\checkmark \\ \hline
\end{array}
\end{align}
For $k=0$ there are two matrices, so we marks this value twice.  The values of $k$ marked in red have parity anomaly for the original theory $U(1)_k+3$, but we can keep them if we only consider the corresponding triangle graphs.  Note that patterns for theories with $k$ and $-k$ are the same, which confirms the existence of a duality between $U(1)_k+3$ and $U(1)_-k+3$, which is a manifestation of the exchange symmetry $↔^-1$, see \eqref{dualpariu1k} and \eqref{dualpairSTthry}.

\section{Brane webs}\label{secbranewebs}

In this section we analyze properties of branes webs that can be used to engineer some theories labeled by plumbing graphs, in particular the theory $U(1)_k+N_f +N_a$. In brane web constructions we use a blue node to denote D3-brane along a perpendicular direction, which we do not draw explicitly. Hence brane webs contain two layers of 5-branes, and a D3-brane is a segment connecting these layers, see [2, 38, 97, 68] for more details on 3d brane webs.
%In this section, we use both plumbing graphs and brane webs to analyze some examples, and shows the correspondence between them.
In particular,  in what follows we argue that the symmetry $↔^-1$ that exchanges the bare \CS levels $k ↔-k$ can be interpreted as the reflection of a brane web along the horizontal line:
\begin{align}
\q\leftrightarrow\q^{-1} ~~ =	~~ 	\text{reflection} \,,
\end{align}
which transforms $(p,q)$-brane into $(p,-q)$-brane.  On the other hand, the rotation of a brane web by an angle $π/2$ transforms $(p,q)$-brane into $(q,p)$-brane. Furthermore,  $𝒮$-duality in type IIB string theory is the operation that exchanges $(p,q)$-brane and $(-q,p)$-brane, so it is basically a combination of reflection and rotation of brane webs:
\begin{align}
~~ 	\text{rotation} + \text{reflection} ~~ =	~~ \text{$\mathcal{S}$-duality}.
\end{align}
In practice, one can simply view a rotation as the $𝒮$-duality operation, as it is always easy to reflect brane webs.

In some special cases, the reflection and $𝒮$-duality give rise to the same brane web.  %If two brane webs are related by the rotation, we can simply say they are S-dual to each other, since the reflection can always be applied.
Some brane webs are even self-dual, i.e.  invariant under $𝒮$-duality.  For theories $U(1)_k+N_f+N_a$,   brane webs are self-dual only for some special values of $k$. This conclusion also follows from analysis of plumbing graphs. In the manipulation of brane webs, a crucial role is played by the movement of 5-branes plays, which makes some brane webs symmetric along the diagonal direction and suggests $𝒮$-duality between theories.

\subsection{$U(1)_k + 1\F$}\label{Sduality}

To start with, we discuss special values of $k$ given in \eqref{specialk1F}. These values are related by the exchange symmetry $↔^-1$. Other values of $k$ are also meaningful, but their brane webs are not symmetric.
%We find these dual pairs can be interpreted by $S$-duality in type IIB string theory.
For $k=±1/2$ one gets two dual theories of the mirror triality \eqref{triviliaty} that we discussed before:
\begin{align}
U(1)_{\frac{1}{2}} + 1 \F  ~~\longleftrightarrow~~ 	U(1)_{-\frac{1}{2}} + 1\F  \,.
%	 ~~\longleftrightarrow~~  1 \F  \,.
\end{align}
The corresponding plumbing graphs are
\begin{align}\label{freenode}
\bsp
\btik
\begin{scope}
\filldraw (0,0) node[below]{$1$}circle(2pt);
\draw[thick] (0,0)--(0,0.3) ;
\node at (0,0.3) {$\graybox$};
\end{scope}
\node[xshift=1cm] at (0,0) {$\leftrightarrow$};
\begin{scope}[xshift=2cm]
\filldraw (0,0) node[below]{$0$}circle(2pt);
\draw[thick] (0,0)--(0,0.3) ;
\node at (0,0.3) {$\graybox$};
\end{scope}
%\node[xshift=3cm] at (0,0) {$\leftrightarrow$};
%	\node at (0,0.3) {$\graybox$};
\etik
\esp
\end{align}
where the numbers denote effective \CS levels.
%Because of the triality, the $U(1)_{\pm \frac{1}{2}}+1\F$ can be viewed as a free chiral multiplet. One can also view this mirror duality as the gauging the flavor symmetry of free chiral multiplet. One cannot gauge the flavor symmetry in the mirror triality, as there is no free mass parameter. The matters in the mirror triality are massless.
Brane webs for these theories take form
\begin{align}\label{U11Fdual}
\bsp
\begin{tikzpicture}[scale=0.8]
\draw[thick] (0,0)--(2,0)--(3,-1); \draw[line width=2 pt, gray] (0.5,1.5)-- (0.5,-1); \draw[thick](2,0)--(2,1.5);
\draw[blue,fill] (0.5,0) circle(3 pt){}
\draw[dashed, orange] (-0.5,0)--(3,0);
\node at (1.5,-2) { $ U(1)_{1/2} + 1 \F$  with $k^{\eff} =1$} ;
\node at (5	,0.2 ) {$\xlongleftrightarrow{\text{reflection}}$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.8]
\draw[thick] (0,0)--(2,0)--(2,-1.5); \draw[line width=2 pt, gray] (0.5,1)-- (0.5,-1.5); \draw[thick](2,0)--(3,1);
\draw[blue,fill] (0.5,0) circle(3 pt){}
\draw[dashed, orange] (-0.5,0)--(3,0);
\node at (1.5,-2.5) { $ U(1)_{-1/2} + 1 \F$  with $k^{\eff} =0$} ;
\end{tikzpicture}
\esp
\end{align}
This mirror pair is related by reflecting the 3d brane web along the horizontal line (the dashed orange line) where the D3-brane is located. Notice that the D3-brane is invariant under such a reflection.
%This reflection exchanges $(p,q)$-branes to $(p,-q)$-branes.

One can perform the $𝒮$-duality by rotating clockwise the right graph in \eqref{U11Fdual} by $π/2$. Here we view the D3-brane as accompanying a local conifold through a geometric transition, and from a perspective of 5d $=1$ brane webs, this D3-brane is also a 5-brane. Then this  $𝒮$-duality is the same as for 5d brane webs:
%The right brane web in \eqref{U11Fdual} is obtained by $S$-duality and a clockwise rotation of ${\pi}/{2}$.
% rotate the $\mathbf{C}^3$ or exchanging (0,1)-brane and (1,0)-brane.
\begin{align}\label{U11Fdual2}
\bsp
\begin{tikzpicture}[scale=0.8]
\begin{scope}
\draw[thick] (0,0)--(2,0)--(2,-1.5); \draw[line width=2 pt, gray] (0.5,1)-- (0.5,-1.5); \draw[thick](2,0)--(3,1);
%	\draw[dashed, orange] (-0.5,0)--(3,0);
\draw[blue,fill] (0.5,0) circle(3 pt){}
\node at (5	,0 ) {$\xlongrightarrow{\text{rotation}}$};
%	\node at (1.5,-2.5) { $ U(1)_{-1/2} + 1 \F$  with $k^{\eff} =0$} ;
\end{scope}
\begin{scope}[rotate around={-90:(5,-3.7)}]
\draw[thick] (0,0)--(2,0)--(2,-1.5); \draw[line width=2 pt, gray] (0.5,1)-- (0.5,-1.5); \draw[thick](2,0)--(3,1);
\draw[dashed] (2,0) --(1,-1) ;
%	\draw[dashed, orange] (-0.5,0)--(3,0);
\draw[blue,fill] (0.5,0) circle(3 pt){}
%	\node at (1.5,-2.5) { $ U(1)_{-1/2} + 1 \F$  with $k^{\eff} =0$} ;
\end{scope}
\end{tikzpicture}
\esp
\end{align}
It follows that the left graph in \eqref{U11Fdual} is $𝒮$-dual to the right graph in \eqref{U11Fdual2}. We emphasize that although these two brane webs are manifestly $𝒮$-dual to each other, the theory labeled by the right graph in \eqref{U11Fdual2} is hard to identify. An interesting point is that the $𝒮$-duality only changes the position of the thick gray line and the D3-brane.

Other special values for \CS levels are $k=±3/2$, which correspond to a dual pair in \eqref{equivalence032}
\begin{align}
\bsp
U(1)_{\frac{3}{2}} + 1 \F  ~~\longleftrightarrow~~ 	U(1)_{-\frac{3}{2}} + 1\F \,,
\esp
\end{align}
whose associated plumbing graphs are
\begin{align}
\bsp
\btik
\begin{scope}
\filldraw (0,0) node[below]{$2$}circle(2pt);
\draw[thick] (0,0)--(0,0.3) ;
\node at (0,0.3) {$\graybox$};
\end{scope}
\node[xshift=1cm] at (0,0) {$\longleftrightarrow$};
\begin{scope}[xshift=2cm]
\filldraw (0,0) node[below]{$-1	$}circle(2pt);
\draw[thick] (0,0)--(0,0.3) ;
\node at (0,0.3) {$\graybox$};
\end{scope}
%	\node[xshift=3cm] at (0,0) {$\leftrightarrow$};
%	\filldraw (0,0) node[below]{$\frac{1}{2}	$}circle(2pt);
%	\draw[thick] (0,0)--(0,0.3) ;	\node at (0,0.3) {$\graybox$};\end{scope}
\etik
\esp
\end{align}
%where the third graph is for the theory $U(1)_0 + 1 \F$ which breaks parity anomaly as its effective CS level is $k^{\eff}=1/2$.
Their brane webs are related by the reflection along the dashed orange line:
\begin{align}\label{U11Fdual32}
\bsp
\begin{tikzpicture}[scale=0.8]
\draw[dashed,orange] (-0.8,0 )--(3, 0)  ;
\draw[thick] (0,0)--(2,0)--(2,2); \draw[line width=2 pt, gray] (0,-0.5)-- (2.5,2); \draw[thick](2,0)--(2.5,-0.5);
\draw[blue,fill] (0.5,0) circle(3 pt){}
\node at (1.2,-1.5) { $ U(1)_{3/2} + 1 \F$  with $k^{\eff} =2$} ;
\node at (5,0.5) {$\xlongleftrightarrow{\text{reflect}}$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.8]
\draw[dashed,orange] (-0.8,2 )--(3, 2)  ;
\draw[thick] (0,2)--(2,2)--(2,0); \draw[line width=2 pt, gray] (0,2+0.5)-- (2.5,0); \draw[thick](2,2)--(2.5,2.5);
\draw[blue,fill] (0.5,2) circle(3 pt){}
\node at (1.5,-1) { $ U(1)_{-3/2} + 1 \F$  with $k^{\eff} =-1$} ;
\end{tikzpicture}
\esp
\end{align}
Analogously, one can also rotate the brane web for $U(1)_-3/2+1$ to identify the $𝒮$-dual brane web. In comparison to the last example, the $𝒮$-dual operation only changes the position of the D3-brane from one intersection point to another one. %this dual brane web pair can also be related by $S$-duality after a clockwise rotation of ${\pi}/{2}$. In this time, S-duality only obviously changes the position of the D3-brane.

The two brane webs in \eqref{U11Fdual} can be obtained by decoupling a D5-brane of \eqref{U102F} or \eqref{SQEDweb} respectively, i.e. sending its position to infinity. In what follows we show that brane webs in  \eqref{U11Fdual} and  \eqref{U11Fdual32} are special, as they are sub-webs of many dual theories. By adding more 5-branes to these sub-webs, one can produce many other dual theories.

%We expect that if the coincidence
%\begin{align}	\text{reflection } ~~ =	~~ \text{S-duality}
%	\end{align}
%is satisfied by certain brane webs, then their sub-webs containing the D3-brane should always be \eqref{U11Fdual} or \eqref{U11Fdual32}.

\subsection{$U(1)_k + 2\F$}

In turn,  we consider the theory $U(1)_0 + 2 $, which can be engineered by two equivalent brane webs
\begin{align}\label{U102F}
\bsp
\btik
\begin{scope}
\draw[thick] (-0.2,0)--( 1,0)--(1,1)--(-0.2,1)
(1,0)--( 1.5, -0.5)
(1,1)--(1.5,1.5) ;
\draw [line width=2 pt, gray](0.2,-0.5)--(0.2,1.5) ;
\draw[blue,fill] (0.2,0) circle (3pt);
\node at (3,0.5) {$\xlongrightarrow{\text{move} ~\F}$};
\end{scope}
\begin{scope}[xshift=4.5cm]
\draw[thick] (-0.2,0)--( 1.2,0)--(1.2,1.5)
(1.2,0)--( 1.7, -0.5)
\draw [line width=2 pt, gray](0.2,-0.5)--(0.2,1)
\draw[blue,fill] (0.2,0) circle (3pt);
\end{scope}
\etik
\esp
\end{align}
for which the effective \CS level $k^ =0 +2 ×1/2 =1$.
The right brane web has an interesting double layer structure. It is obtained by moving a fundamental D5-brane vertically along the direction of D3-brane from the thin sub-web to the thick gray line, which we do not draw explicitly. After this move, projecting these two sub-webs along the direction of the D3-brane leads to the right brane web in \eqref{U102F}. See  [38] for more details of this move. For this theory, one can see that the $𝒮$-duality of type IIB string theory changes the vacuum of the D3-brane on brane webs in \eqref{U102F}, which is the exchange of the blue node from one intersection point to another. The plumbing graphs in the orbit of $ST$-moves are
\begin{align}\label{2Fkeql0}
\bsp
\includegraphics[]{Figures/2Fkeql0}
\esp
\end{align}
We obtain these graphs by moving the two matter fields respectively, using \eqref{ststF}.

It is also easy to draw brane webs for $k= ±1, ±2$.  Such brane webs are related by reflections of branes along the horizontal line. For these \CS values, one can get corresponding brane webs by simply adding one more D5-brane to brane webs in  \eqref{U11Fdual} and  \eqref{U11Fdual32}.

\subsection{$U(1)_k + 1\F+1\AF$}

The next example we consider is the theory $U(1)_k + 1 + 1 $. The effective \CS level $k^ =k$ if both matter fields have positive masses. The corresponding plumbing graphs are
\begin{align}\label{1F1AFk}
\bsp
\includegraphics[]{Figures/1F1AFk}
\esp
\end{align}
where we use the dual triality \eqref{ststAF} to move the matter. In particular,  for $k=0$  we get the theory  $U(1)_0 +1+1$, which is the SQED in the dual pair SQED-XYZ. We show its brane webs below
\begin{align}\label{SQEDweb}
\bsp
\btik
\begin{scope}
\draw[thick,scale=0.8] (-0.2,0)--( 1,0)--(2,1)--(2.8,1)
(1,0)--( 1, -0.8)
(2,1)--(2,1.8) ;
\draw [line width=2 pt, gray,scale=0.8](0.2,-0.8)--(0.2,1.8) ;
\draw[blue,fill] (0.16,0) circle (3pt);
\node at (3.5,0.2) {$\xlongrightarrow{\text{move}~ \AF }  $};
\end{scope}
\begin{scope}[xshift=5.5cm]
\draw[thick,scale=0.8] (-0.5,0)--( 1.2,0)--(2,0.8)
(1.2,0)--( 1.2, -0.8)
\draw [line width=2 pt, gray,scale=0.8](0.2,-0.8)--(0.2,1) --(-0.5,1)
\draw[blue,fill] (0.16,0) circle (3pt);
\end{scope}
\etik
\esp
\end{align}
which corresponds to $k^=k +1/2-1/2=tanθ=0 $, where $θ$ is the relative angle between NS5-brane and NS5'-brane.  See e.g.[38] for more details on the relative angles.  These two brane webs are also related by moving a D5-brane perpendicularly; equivalently, from a perspective of 5d theories, it corresponds to moving horizontally a 7-brane attached at the infinity of this D5-brane.

The plumbing graphs for the mirror pair SQED-XYZ  are
\begin{align}\label{SQEDXYZdualpari}
\bsp
\includegraphics[]{Figures/SQEDXYZplumb}
\esp
\end{align}
where the last graph denotes the XYZ model. The graphs in the middle are the two-node graphs, and the last graph is the three-node graph for the 2-3 move discussed in section \ref{secgauging}. One can use the superpotential triangles \eqref{trianglesgraph} to check that the SQED-XYZ dual pair only satisfies the unlinking relation. This example shows that superpotential triangles could have no mixed \CS levels between gauge nodes.
%Note that the linking and exotic dual of the two-node graphs are triangles with lines connecting gauge nodes.

Moreover, the right brane webs in \eqref{U102F} and \eqref{SQEDweb} are obviously $𝒮$-dual to themselves. One can decorate these $𝒮$-dual brane webs by adding 5-branes to get other self-dual brane webs. For example, the theory $U(1)_0 +2+2$ is also $𝒮$-dual to itself.  Its two equivalent brane webs take form
\begin{align}
\bsp
\btik
\begin{scope}[xshift=-4cm]
\draw[thick] (-0.2,0)--( 1.2,0)--(1.2,1.5)
(1.2,0)--( 1.7, -0.5)
\draw [line width=2 pt, gray](0.2,-0.5)--(0.2,1)
(-0.8,1.5) -- (-0.2,1.5 )--(-0.2,2 )
\draw[blue,fill] (0.2,0) circle (3pt);
\end{scope}
\begin{scope}
\draw[thick,scale=0.8] (-0.5,0)--( 1.2,0)--(2,0.8)
(1.2,0)--( 1.2, -0.8)
\draw [line width=2 pt, gray,scale=0.8](0.2,-0.8)--(0.2,1) --(-0.5,1)
( 1.8,1.8)--(1,1.8)--(1,2.8)
\draw[blue,fill] (0.16,0) circle (3pt);
\end{scope}
\etik
\esp
\end{align}
Note that this theory can be viewed as a self-dual 3d $=4$ theory with gauge group $U(1)_0$ and two fundamental hypermultiplets [1]. Adding more segments would lead to non-toric intersections,  however we postulate that $𝒮$-dual theories should be $U(1)_0 + N_a + N_f$ with the condition that $N_a+N_f$ is an even number.

\subsection{Mirror pairs}\label{secmirrorpair}

Mirror symmetry exchanges Coulomb and Higgs branches of the moduli spaces of vacua of dual theories. In this section we show that some abelian mirror dual theories in [66] have the same plumbing graphs, and some mirror dual pairs can be simply interpreted as a manifestation of $𝒮$-duality of type IIB string theory. We have discussed a mirror pair with one matter nodes before, so we start from theories with two matter nodes.

A mirror pair found in [66] is
\begin{align}\label{U12Fdual1}
U(1)_{+1} + 1 \F +1 \AF  ~~\xlongleftrightarrow{~S~}~~ 	U(1)_{-1} + 2\F.
\end{align}
A brane web for $U(1)_+1 + 1 +1 $ theory takes form
\begin{align}\label{1F1AF2F}
\bsp
\includegraphics[width=2.5in]{Figures/Uplus11F1AF.pdf}
\esp
\end{align}
where we first attach a 7-brane at infinity of the anti-fundamental D5-brane and then Hanany-Witten move it to the left infinity to produce an equivalent brane web.
%Note that the right brane web of \eqref{1F1AF2F} is the same for the theory $U(1)_{+1}+2 \F$ which is dual to $U(1)_{-1}+2\F$ through reflection.
%This can also interpreted as moving the D5-brane from one layer of 5-brane web to another layer of 5-brane web.
This is equivalent to moving the 5-brane for $$ perpendicularly.
A brane web for $U(1)_-1 + 2$ theory is
\begin{align}\label{U12Fdual22}
\bsp
\includegraphics[width=2.5in]{Figures/Uminus1_2F.pdf}
\esp
\end{align}
where we also move the fundamental D5-brane to the sub-web along a gray layer.  It is obvious that brane webs on the right in \eqref{U12Fdual1} and \eqref{U12Fdual22} are related by $𝒮$-duality. Since these two theories are related by mirror symmetry,  in this case it can be interpreted as the $𝒮$-duality of type IIB string theory.

%It can be noticed that the brane webs for these two theories are related by exchanging $(p,q)$-brane and $(q,p)$-brane, so it is S-duality in Type-IIB string theory. Hence this mirror pair  is interpreted as the S-duality.

The above example can be also presented in terms of plumbing graphs. Although it is not clear whether plumbing graphs may be used to distinguish and exchange different branches of moduli spaces, we find that for this mirror pair they are the same:
\begin{align}\label{U12Fkeql1}
\bsp
\includegraphics[]{Figures/U12Fkel1}
\esp
\end{align}
where we use \eqref{ststAF} to move a matter node. Similarly, the exchange symmetry of plumbing graphs suggests another mirror dual pair
\begin{align}\label{U12Fdual2}
U(1)_{-1} + 1 \F + 1 \AF ~~\xlongleftrightarrow{~~}~~ 	U(1)_{+1} + 2\F \,,
\end{align}
so that corresponding plumbing graphs are also equivalent:
\begin{align}\label{U12Fkelminus1}
\bsp
\includegraphics[]{Figures/U12Fkelminus1}
\esp
\end{align}
In addition,  the exchange symmetry $↔^-1$ (reflection) relates the dual pairs \eqref{U12Fdual1} and \eqref{U12Fdual2}. This extends the original mirror pair to a big dual pair containing four theories ${U(1)_±1+2 ,  U(1)_±1+1+1 }$, which are dual to each other through $𝒮$-duality and reflection. Brane webs for this dual pair take form:
\begin{align}\label{mirrorwebsum}
\bsp
\includegraphics[width=4.5in]{Figures/fourmirrorpair.pdf}  \,.
\esp
\end{align}
Note that for the theory $U(1)_±1 +2 $, the two Higgs branches of vacua correspond to two intersections on brane webs.

By adding a half-horizontal 5-brane on the brane webs in  \eqref{mirrorwebsum}, one can produce a new dual pair:
\begin{align}\label{dual2F1AF}
U(1)_{- \frac{1}{2}} + 2 \F+1 \AF ~~\xlongleftrightarrow{~S~}~~	U(1)_{ \frac{1}{2}} + 1\F+2 \AF \,,
\end{align}
for which the effective \CS levels for both theories also vanish $k^ =0$. One can also derive this dual pair by adding a matter node $$ to both sides of the dual graphs in \eqref{higgsinggraph} (or equivalently \eqref{higgsinggeneric}) and set $b=-1/2$ (or $b=-1$). The mirror triality in \eqref{ststAF} could turn this $$ into $$. Then one gets corresponding graphs  for the dual pair in \eqref{dual2F1AF}. Note that here the operation in \eqref{higgsinggraph} should be $𝒮$-duality in the language of 3d $=4$ brane webs, so in total the operation we encounter for \eqref{dual2F1AF} is $𝒮 ∘(ST)^2$. The dual theories in \eqref{dual2F1AF} cannot be directly related through $ST$-moves only. We note that this dual pair is given by the $𝒮$-duality of 3d brane webs:
\begin{align}\label{U12F1AFmirror}
\bsp
\includegraphics[width=4in]{Figures/anothermirror.pdf}
\esp
\end{align}
Applying reflection leads to another dual pair
\begin{align}
U(1)_{ \frac{1}{2}} + 2 \F+1 \AF ~~\xlongleftrightarrow{~S~}~~	U(1)_{- \frac{1}{2}} + 1\F+2 \AF ,
\end{align}
so that corresponding brane webs are the reflection of brane webs in \eqref{U12F1AFmirror}.

It appears to us that it is not possible to produce other dual pairs for theories with a single gauge group, i.e. $U(1)_k+N_f+N_a$. In principle it is possible to apply $𝒮$-duality to any brane web,  but it may not be easy to identify the corresponding theories from dual brane webs, or dual brane webs may even not have physical interpretation.
%This dual pair can also be obtained by decoupling the
%If looking at the plumbing graphs, one can get a more generic dual pair
%	U(1)_{\pm \le( k- \frac{1}{2}  \r) } + 2 \F+1 \AF ~~\xlongleftrightarrow{~~}~~	U(1)_{\pm \le( k+ \frac{1}{2}  \r) } + 1\F+2 \AF \,.

\subsubsection*{Abelian mirror pairs}

Generic mirror dual theories contain bifundamental matter.  It is possible to show equivalence of mirror abelian theories found in [66].  The first theory, referred to as theory A, is defined as
\begin{align}
&\text{theory A}:~~ U(1)_{-\frac{N_f}{2}} + N_f \F .
\end{align}
The  mirror dual theory B is a linear quiver theory with bifundamental chiral multiplets between gauge nodes and two fundamental matter fields on two tails.  Its conventional quiver diagram takes form
\begin{align}\label{quivertheoryB}
&	\text{theory B}:~~
% U(1)^{N_f-1}_{k_{ij}  } + N_f\Phi
\end{align}
for which the $a$-th chiral multiplet has charge $(δ_a, i, δ_a, i+1)$ under the $i$-th and $(i+1)$-th gauge nodes. This theory also has mixed \CS levels $ k_ij=δ_ij-1/2δ_i,j-1 -1/2δ_i-1,j$ with $i=1 ,2 ,⋯ , N_f-1$ and $j=1 ,⋯ , N_f$, which are not shown in the quiver in  \eqref{quivertheoryB}.

Let us show equivalence of these two theories. First, we perform $ST$-move for each chiral multiplet of theory A, which produces a star graph with undecorated gauge node $U(1)_0$ in the center. Then, we separate this $U(1)_0$ gauge node by introducing many new gauge nodes $U(1)_0$ using Kirby moves \eqref{deltasplit}. After integrating out some particular $U(1)_0$,  we get a plumbing graphs for theory B. We illustrate this in the example of $U(1)_-2+4$ theory:
\begin{align}
\bsp
\includegraphics[width=5.2in]{Figures/4Fmirror1}
\esp
\end{align}
We have also made a similar check for some other mirror dual theories with generic charges, and hope that all mirror dual abelian theories have equivalent plumbing graphs.
% \footnote{We would like to thank Lakshya Bhardwaj for suggesting us to check dual theories with generic charges}.

%\section{Monopole operators and monopole potentials}

%As one can learn from [3, 4], the basic dual pair SQED-XYZ in \eqref{SQEDXYZ} could have rich moduli  structures from monopole operators. Let us review the discussion in the following. To begin with, the SQED as a $U(1)_0$ theory with two fundamental chiral multiplets $Q, \tilde{Q}$ of charge $\pm1$ respectively, whose chiral ring has monople operators $V^{\pm}$, and a meson $M=Q\tilde{Q}$. This theory is dual to the XYZ-model, and the map is $( V^+,V^-,M ) \leftrightarrow (X, Y, Z)$. Therefore, the magnetic theory XYZ is often denoted by monopole operators which are chiral fields coupled by $\W=MV_+V_-$. Note that one should not view $V_\pm$ as monopole of the magnetic theory. Generically, monopole operators in electric theories and magnetic theories are different. The basic duality in  \eqref{triviliaty}  are decoupling limits of SQED-XYZ. For instance, one basic dual pair $U(1)_{-1/2} +1\F \leftrightarrow 1\F$ can be denoted in details as  $U(1)_{-1/2} +1Q \leftrightarrow V_+$, where monopole operator $V_+$ becomes the dual free chiral multiplet $\F$.

% In the above discussion, monopole operators have not really entered 3d dualities. However, monopole operators indeed could appear in superpotentials of 3d theories, and play crucial roles. However, this would involve more complicated superpotential and dualities, and is not addressed yet in our context, as by now we only considered the most simple cubic superpotential $\W=XYZ$. By appropriately adding monopole operator terms to the superpotential, even for the cubic type $\W = XYZ + \mathcal{M}^* + \cdots$, more dualities can be generated. Some examples are discussed in [22]. Their basic ideas is the following: because of mirror symmetry,  monopole operators in the electric theory would map to a product of chiral multiplets in the magnetic theory, namely $ \mathcal{M}^*~\leftrightarrow~ \prod_i \Phi_i $, and vice versa, so adding a meaningful superpotential term (which can be a cubic term $\Phi_1\Phi_2\Phi_3$) by doing some operations on the magnetic theory could correspondingly lead an additional monopole term $\mathcal{M}^*$ to the superpotential of electric theory. We prefer to left the analysis of monopole superpotentials in the context this note as a future work.

\acknowledgments

We thank Lakshya Bhardwaj, Jin Chen,  Sung-Soo Kim, Ioannis Lavdas, Satoshi Nawata for helpful discussions.  This work has been supported by the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00- 00-5C55/17-00). The work of S.C. has been also supported by NSFC Grant No.11850410428.

\bigskip

\newpage
\appendix

\section{$q$-Pochhammer symbol}\label{pochappendix}

In various computations we use various identities satisfied by $q$-Pochhammer symbol $≤(x; )̊_n$:
\begin{align}
&	\le(x; \q  \r)_n := \prod_{k=0}^{n-1} ( 1-x \q^k) \,, \quad 	(x;\q)_n \sim e^{ \frac{1}{\hbar} \le(   \Li_2(x) -\Li_2\le( x \q^n \r) \r) }  \,,~~ \q := e^{\hbar}.\\
& \le(x;\q^{-1}  \r)_\inf = \frac{1}{  \le( x \q;\q \r)_\inf} \,,\qquad  \le( x;\q \r)_n = \frac{ 1}{ \le( x \q^n;\q  \r)_{-n} }
\,,\qquad  (x;\q)_n = \frac{ (x;\q)_\inf}{ (x \q^n ;\q)_\inf } \,,
\\
&	\Li_2(x)+ \Li_2(x^{-1}) = -\frac{1}{2} \le( \log(-x) \r)^2 -\frac{\pi^2}{6} \,,\quad 	\Li_2(1) =\frac{\pi^2}{6}	\,, \\
&	\theta( x;\q )  := \frac{  \sum_{n \in \mathbf{Z}} \q^{\frac{n^2}{2}} x^n  }{(\q;\q)_\inf  }  \,,	 \quad 	\frac{\theta(\q^n z;\q) }{\theta(z;\q) } =\q^{-\frac{n^2}{2}} z^{-n} \,, \quad
\theta(x;\q) \sim e^{-\frac{\pi^2}{6 \hbar} +\frac{\hbar}{24}}   \, e^{ \frac{1}{ \hbar} \le( -\frac{1}{2} ( \log\, x)^2 \r) }
\\
&		\theta ( -\q^{ \frac{1}{2}} x ) = (\q x;\q)_\inf (x^{-1};\q)_\inf \,, \quad			\theta ( -q^{ -\frac{1}{2}} x ) = (x;\q)_\inf (\q x^{-1};\q)_\inf \,, \\
& \theta(x;\q) :=\theta(x^{-1};\q)= \le(-\q^{1/2} x;\q\r)_\inf \le(-q^{1/2} x^{-1};\q\r)_\inf = \frac{ \le(-\q^{1/2} x;\q\r)_\inf}{ \le(-\q^{-1/2} x^{-1};\q^{-1}\r)_\inf } \,.
\end{align}
$q$-Pochhammer symbol is also related to MacMahon function:
\begin{align}
&  M(Q,\q) =\prod_{i,j =1}^{\inf} ( 1- Q \q^{i+j-1})  \,,\\
& (\q z;\q)_\inf = \frac{ M(z^{-1};\q) }{M( \q z^{-1};\q) } =\frac{ M(z;\q) }{M( \q z;\q) } \,,  \quad M(z;\q):= \PE\bigg[ - \frac{\q z}{ (1-\q)^2} \bigg]  \,,   \label{normal} \\
&  \frac{ M(z;\q)}{M( z^{-1};\q) } =  \frac{ M(\q z;\q)}{M( \q z^{-1};\q) } =\text{constant} \,.
\end{align}
The doube-sine function is defined as
\begin{align}
&	s_b(x) :=   \prod_{n_1, n_2 \geqslant 0 } \frac{ n_1 b + n_2 b^{-1} + Q/2 -i\, x  }{  n_1 b + n_2 b^{-1} + Q/2 + i\, x        } \,, ~
&	s_b\left(  \frac{i Q}{2}  \pm x \right)	= \frac{ 1}{ 2\, \cosh\left( \pi Q x \right)   } \,.
\end{align}

\section{$SL(2, \mathbf{Z})$ 	}\label{SL2Zappendex}

For non-abelian plumbing theories with gauge group $SU(2)$, the edge denotes the duality wall 	$T[SU(2)]$ theory [41, 87], which for some manifolds such as lens spaces are interpreted as the $S$-transformation in $SL(2,𝐙)$.  $SL(2, 𝐙)$ is a global symmetry of 3d gauge theories. The polarization of three manifolds $M_3$ that engineer 3d theories can be interpreted in terms of $T$ and $S$ generators in this group [71, 67].

In other notation we choose
\begin{align}
S=	\begin{bmatrix}
~0~  & -1  \\
~1~ & ~0 \\
\end{bmatrix} \,,\quad
T=	\begin{bmatrix}
~1  & ~0~  \\
~1 & ~1 ~\\
\end{bmatrix} \,,
\end{align}
so that standard relations hold: $(ST)^3 =1$, $S^2=-1$, and $S^4=1$.  Notice that there is an equivalence
\begin{align} \label{equivalence}
(ST)^2 = ST ST =(ST)^{-1} = T^{-1} S^{-1} = - T^{-1} S \,,
\end{align} where
\begin{align}
ST=	\begin{bmatrix}
-1  & -1  \\
~1~ & ~0 \\
\end{bmatrix} \,,\quad
TS=	\begin{bmatrix}
~0 & ~-1~  \\
~1 & ~-1 ~\\
\end{bmatrix} \,,
\quad
(ST)^2=	\begin{bmatrix}
~0 & ~1~  \\
-1 & -1 ~\\
\end{bmatrix} \,.
\end{align}
For linear plumbing graphs, some Kirby moves can be interpreted using the equivalence \eqref{equivalence}. For example, the Kirby move
\eqref{example1integ} can be represented as
\begin{align}
T^{k_1} (S T^{\mp 1}S )	T^{k_2}  ~~\rightleftarrows ~~ T^{k_1 \pm 1} S^{\pm 1} T^{ k_2 \pm 1}  \,,
\end{align}
which is obviously correct because of the equivalence
\begin{align}
ST^{-1} S=TST\,,\quad STS=T^{-1}S^{-1} T^{-1}  \,. \end{align}
%	We can rewrite $T^{k_1 \pm 1} S^{\pm 1} T^{ k_2 \pm 1} = T^{k_1} ( T^{ \pm 1} S^{\pm 1} T^{ \pm 1} ) T^{ k_2 }$.
%	Other types of Kirby moves involve more delicate operations on  plumbing manifolds.

%The blow-up of matters in \eqref{blowupdownmatter} is the transformation that
%	T^{k}  T^{\frac{1}{2}} \rightarrow 	T^{k+\frac{1}{2}} S 	T^{\frac{1}{2}} 	T^{\frac{1}{2}} =
%where $k^{\eff} =k+1/2$.

\newpage

\bibliographystyle{JHEP}
\providecommand{\href}[2]{#2}\begingroup\raggedright\begin{thebibliography}{10}

[1]
K.~A. Intriligator and N.~Seiberg, {\it {Mirror symmetry in three-dimensional
gauge theories}},  {\em Phys. Lett.} {\bf B387} (1996) 513--519,
[\href{http://arxiv.org/abs/hep-th/9607207}{{\tt hep-th/9607207}}].

[2]
J.~de~Boer, K.~Hori, Y.~Oz, and Z.~Yin, {\it Branes and mirror symmetry in n=2
supersymmetric gauge theories in three dimensions},  {\em Nucl.Phys.B} {\bf
502} (1997) 107--124, [\href{http://arxiv.org/abs/hep-th/9702154}{{\tt

[3]
O.~Aharony, A.~Hanany, K.~Intriligator, N.~Seiberg, and M.~J. Strassler, {\it
Aspects of n=2 supersymmetric gauge theories in three dimensions},  {\em
Nucl.Phys.B} {\bf 499} (1997) 67--99,
[\href{http://arxiv.org/abs/hep-th/9703110}{{\tt hep-th/9703110}}].

[4]
K.~Intriligator and N.~Seiberg, {\it {Aspects of 3d N=2 Chern-Simons-Matter
Theories}},  {\em JHEP} {\bf 07} (2013) 079,
[\href{http://arxiv.org/abs/1305.1633}{{\tt arXiv:1305.1633}}].

[5]
O.~Aharony, {\it {IR duality in d = 3 N=2 supersymmetric USp(2N(c)) and U(N(c))
gauge theories}},  {\em Phys. Lett. B} {\bf 404} (1997) 71--76,
[\href{http://arxiv.org/abs/hep-th/9703215}{{\tt hep-th/9703215}}].

[6]
H.~Ooguri and C.~Vafa, {\it {Knot invariants and topological strings}},  {\em
Nucl. Phys.} {\bf B577} (2000) 419--438,
[\href{http://arxiv.org/abs/hep-th/9912123}{{\tt hep-th/9912123}}].

[7]
A.~Giveon and D.~Kutasov, {\it {Seiberg Duality in Chern-Simons Theory}},  {\em
Nucl. Phys. B} {\bf 812} (2009) 1--11,
[\href{http://arxiv.org/abs/0808.0360}{{\tt arXiv:0808.0360}}].

[8]
A.~Kapustin and M.~J. Strassler, {\it {On mirror symmetry in three-dimensional
Abelian gauge theories}},  {\em JHEP} {\bf 04} (1999) 021,
[\href{http://arxiv.org/abs/hep-th/9902033}{{\tt hep-th/9902033}}].

[9]
D.~Tong, {\it {Dynamics of N=2 supersymmetric Chern-Simons theories}},  {\em
JHEP} {\bf 07} (2000) 019, [\href{http://arxiv.org/abs/hep-th/0005186}{{\tt

[10]
O.~Aharony, S.~S. Razamat, N.~Seiberg, and B.~Willett, {\it {3d dualities from
4d dualities}},  {\em JHEP} {\bf 07} (2013) 149,
[\href{http://arxiv.org/abs/1305.3924}{{\tt arXiv:1305.3924}}].

[11]
F.~Benini, C.~Closset, and S.~Cremonesi, {\it Comments on 3d seiberg-like
dualities},  {\em JHEP} {\bf 1110} (2011) 075,
[\href{http://arxiv.org/abs/1108.5373}{{\tt arXiv:1108.5373}}].

[12]
F.~Benini, S.~Benvenuti, and S.~Pasquetti, {\it {SUSY monopole potentials in
2+1 dimensions}},  {\em JHEP} {\bf 08} (2017) 086,
[\href{http://arxiv.org/abs/1703.08460}{{\tt arXiv:1703.08460}}].

[13]
K.~Nii, {\it {Generalized Giveon-Kutasov duality}},  {\em JHEP} {\bf 08} (2021)
130, [\href{http://arxiv.org/abs/2005.04858}{{\tt arXiv:2005.04858}}].

[14]
T.~Okazaki and D.~J. Smith, {\it {Web of Seiberg-like dualities for 3D N=2
quivers}},  {\em Phys. Rev. D} {\bf 105} (2022), no.~8 086023,
[\href{http://arxiv.org/abs/2112.07347}{{\tt arXiv:2112.07347}}].

[15]
T.~Okazaki and D.~J. Smith, {\it {Seiberg-like dualities for orthogonal and
symplectic 3d $ \mathcal{N} $ = 2 gauge theories with boundaries}},  {\em
JHEP} {\bf 07} (2021) 231, [\href{http://arxiv.org/abs/2105.07450}{{\tt

[16]
K.~Nii, {\it {3d duality with adjoint matter from 4d duality}},  {\em JHEP}
{\bf 02} (2015) 024, [\href{http://arxiv.org/abs/1409.3230}{{\tt

[17]
N.~Kubo and K.~Nii, {\it {3d $ \mathcal{N} $ = 3 generalized Giveon-Kutasov
duality}},  {\em JHEP} {\bf 04} (2022) 158,
[\href{http://arxiv.org/abs/2111.13366}{{\tt arXiv:2111.13366}}].

[18]
A.~Amariti and M.~Fazzi, {\it {Dualities for three-dimensional $\mathcal{N} =
2$ $SU(N_c)$ chiral adjoint SQCD}},  {\em JHEP} {\bf 11} (2020) 030,
[\href{http://arxiv.org/abs/2007.01323}{{\tt arXiv:2007.01323}}].

[19]
J.-E. Bourgine, {\it {Engineering 3D N=2 theories using the quantum affine
sl(2) algebra}},  {\em Nucl. Phys. B} {\bf 985} (2022) 115979,
[\href{http://arxiv.org/abs/2107.10063}{{\tt arXiv:2107.10063}}].

[20]
C.~Hwang, S.~Pasquetti, and M.~Sacchi, {\it {Rethinking mirror symmetry as a
local duality on fields}},  {\em Phys. Rev. D} {\bf 106} (2022), no.~10
105014, [\href{http://arxiv.org/abs/2110.11362}{{\tt arXiv:2110.11362}}].

[21]
F.~Nieri, Y.~Pan, and M.~Zabzine, {\it 3d mirror symmetry from s-duality},
{\em Phys. Rev. D} {\bf 98} (2018) 126002,
[\href{http://arxiv.org/abs/1809.00736}{{\tt arXiv:1809.00736}}].

[22]
S.~Benvenuti and S.~Pasquetti, {\it {3d $ \mathcal{N} $ = 2 mirror symmetry,
pq-webs and monopole superpotentials}},  {\em JHEP} {\bf 08} (2016) 136,
[\href{http://arxiv.org/abs/1605.02675}{{\tt arXiv:1605.02675}}].

[23]
T.~Dimofte, S.~Gukov, and L.~Hollands, {\it {Vortex Counting and Lagrangian
3-manifolds}},  {\em Lett. Math. Phys.} {\bf 98} (2011) 225--287,
[\href{http://arxiv.org/abs/1006.0977}{{\tt arXiv:1006.0977}}].

[24]
S.~Benvenuti and S.~Pasquetti, {\it {3D-partition functions on the sphere:
exact evaluation and mirror symmetry}},  {\em JHEP} {\bf 05} (2012) 099,
[\href{http://arxiv.org/abs/1105.2551}{{\tt arXiv:1105.2551}}].

[25]
C.~Hwang, H.-C. Kim, and J.~Park, {\it {Factorization of the 3d superconformal
index}},  {\em JHEP} {\bf 08} (2014) 018,
[\href{http://arxiv.org/abs/1211.6023}{{\tt arXiv:1211.6023}}].

[26]
C.~Beem, T.~Dimofte, and S.~Pasquetti, {\it {Holomorphic Blocks in Three
Dimensions}},  {\em JHEP} {\bf 12} (2014) 177,
[\href{http://arxiv.org/abs/1211.1986}{{\tt arXiv:1211.1986}}].

[27]
F.~Benini and W.~Peelaers, {\it Higgs branch localization in three dimensions},
{\em JHEP} {\bf 1405} (2014) 030,
[\href{http://arxiv.org/abs/1312.6078}{{\tt arXiv:1312.6078}}].

[28]
S.~Pasquetti, {\it {Factorisation of N = 2 Theories on the Squashed 3-Sphere}},
{\em JHEP} {\bf 04} (2012) 120, [\href{http://arxiv.org/abs/1111.6905}{{\tt

[29]
A.~Nedelin, S.~Pasquetti, and Y.~Zenkevich, {\it {T[SU(N)] duality webs: mirror
symmetry, spectral duality and gauge/CFT correspondences}},  {\em JHEP} {\bf
02} (2019) 176, [\href{http://arxiv.org/abs/1712.08140}{{\tt

[30]
F.~Aprile, S.~Pasquetti, and Y.~Zenkevich, {\it {Flipping the head of
$T[SU(N)]$: mirror symmetry, spectral duality and monopoles}},  {\em JHEP}
{\bf 04} (2019) 138, [\href{http://arxiv.org/abs/1812.08142}{{\tt

[31]
Y.~Yoshida and K.~Sugiyama, {\it {Localization of three-dimensional
$\mathcal{N}=2$ supersymmetric theories on $S^1 \times D^2$}},  {\em PTEP}
{\bf 2020} (2020), no.~11 113B02, [\href{http://arxiv.org/abs/1409.6713}{{\tt

[32]
Y.~Pan and W.~Peelaers, {\it {Intersecting Surface Defects and Instanton
Partition Functions}},  {\em JHEP} {\bf 07} (2017) 073,
[\href{http://arxiv.org/abs/1612.04839}{{\tt arXiv:1612.04839}}].

[33]
C.~Closset, H.~Kim, and B.~Willett, {\it {Supersymmetric partition functions
and the three-dimensional A-twist}},  {\em JHEP} {\bf 03} (2017) 074,
[\href{http://arxiv.org/abs/1701.03171}{{\tt arXiv:1701.03171}}].

[34]
D.~Gaiotto and H.-C. Kim, {\it {Duality walls and defects in 5d $ \mathcal{N}=1
$ theories}},  {\em JHEP} {\bf 01} (2017) 019,
[\href{http://arxiv.org/abs/1506.03871}{{\tt arXiv:1506.03871}}].

[35]
Y.~Fan, {\it {3D\textendash{}3D correspondence from Seifert fibering
operators}},  {\em J. Phys. A} {\bf 54} (2021), no.~22 225401,
[\href{http://arxiv.org/abs/2008.13202}{{\tt arXiv:2008.13202}}].

[36]
M.~Manabe, S.~Terashima, and Y.~Terashima, {\it {The colored Jones polynomials
as vortex partition functions}},  {\em JHEP} {\bf 12} (2021) 197,
[\href{http://arxiv.org/abs/2110.05662}{{\tt arXiv:2110.05662}}].

[37]
T.~Kimura and F.~Nieri, {\it {Intersecting defects and supergroup gauge
theory}},  {\em J. Phys. A} {\bf 54} (2021), no.~43 435401,
[\href{http://arxiv.org/abs/2105.02776}{{\tt arXiv:2105.02776}}].

[38]
S.~Cheng, {\it {3d $ \mathcal{N} $ = 2 brane webs and quiver matrices}},  {\em
JHEP} {\bf 07} (2022) 107, [\href{http://arxiv.org/abs/2108.03696}{{\tt

[39]
S.~Cheng and P.~Su\l{}kowski, {\it {Refined open topological strings
revisited}},  {\em Phys. Rev. D} {\bf 104} (2021), no.~10 106012,
[\href{http://arxiv.org/abs/2104.00713}{{\tt arXiv:2104.00713}}].

[40]
Y.~Terashima and M.~Yamazaki, {\it 3d {N}=2 theories from cluster algebras},
{\em Prog.Theor.Exp.Phys.} {\bf 023} (2014) B01,
[\href{http://arxiv.org/abs/1301.5902}{{\tt arXiv:1301.5902}}].

[41]
Y.~Terashima and M.~Yamazaki, {\it {SL(2,R) Chern-Simons, Liouville, and Gauge
Theory on Duality Walls}},  {\em JHEP} {\bf 08} (2011) 135,
[\href{http://arxiv.org/abs/1103.5748}{{\tt arXiv:1103.5748}}].

[42]
S.~Cecotti, C.~Cordova, and C.~Vafa, {\it {Braids, Walls, and Mirrors}},
\href{http://arxiv.org/abs/1110.2115}{{\tt arXiv:1110.2115}}.

[43]
D.~Gang, N.~Kim, M.~Romo, and M.~Yamazaki, {\it {Aspects of Defects in 3d-3d
Correspondence}},  {\em JHEP} {\bf 10} (2016) 062,
[\href{http://arxiv.org/abs/1510.05011}{{\tt arXiv:1510.05011}}].

[44]
A.~Gadde, S.~Gukov, and P.~Putrov, {\it {Fivebranes and 4-manifolds}},  {\em
Prog. Math.} {\bf 319} (2016) 155--245,
[\href{http://arxiv.org/abs/1306.4320}{{\tt arXiv:1306.4320}}].

[45]
M.~Yamazaki, {\it Quivers, {YBE} and 3-manifolds},  {\em JHEP} {\bf 1205}
(2012) 147, [\href{http://arxiv.org/abs/1203.5784}{{\tt arXiv:1203.5784}}].

[46]
F.~Benini, C.~Closset, and S.~Cremonesi, {\it {Quantum moduli space of
Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3}},  {\em JHEP} {\bf 09}
(2011) 005, [\href{http://arxiv.org/abs/1105.2299}{{\tt arXiv:1105.2299}}].

[47]
D.~Gang and K.~Yonekura, {\it {Symmetry enhancement and closing of knots in
3d/3d correspondence}},  {\em JHEP} {\bf 07} (2018) 145,
[\href{http://arxiv.org/abs/1803.04009}{{\tt arXiv:1803.04009}}].

[48]
S.~Choi, D.~Gang, and H.-C. Kim, {\it {Infrared phases of 3D class R
theories}},  {\em JHEP} {\bf 11} (2022) 151,
[\href{http://arxiv.org/abs/2206.11982}{{\tt arXiv:2206.11982}}].

[49]
D.~Gaiotto, A.~Kapustin, N.~Seiberg, and B.~Willett, {\it {Generalized Global
Symmetries}},  {\em JHEP} {\bf 02} (2015) 172,
[\href{http://arxiv.org/abs/1412.5148}{{\tt arXiv:1412.5148}}].

[50]
J.~Eckhard, H.~Kim, S.~Schafer-Nameki, and B.~Willett, {\it {Higher-Form
Symmetries, Bethe Vacua, and the 3d-3d Correspondence}},  {\em JHEP} {\bf 01}
(2020) 101, [\href{http://arxiv.org/abs/1910.14086}{{\tt arXiv:1910.14086}}].

[51]
O.~Bergman, Y.~Tachikawa, and G.~Zafrir, {\it {Generalized symmetries and
holography in ABJM-type theories}},  {\em JHEP} {\bf 07} (2020) 077,
[\href{http://arxiv.org/abs/2004.05350}{{\tt arXiv:2004.05350}}].

[52]
M.~van Beest, D.~S.~W. Gould, S.~Schafer-Nameki, and Y.-N. Wang, {\it {Symmetry
TFTs for 3d QFTs from M-theory}},  {\em JHEP} {\bf 02} (2023) 226,
[\href{http://arxiv.org/abs/2210.03703}{{\tt arXiv:2210.03703}}].

[53]
N.~Mekareeya and M.~Sacchi, {\it {Mixed anomalies, two-groups, non-invertible
symmetries, and 3d superconformal indices}},  {\em JHEP} {\bf 01} (2023) 115,
[\href{http://arxiv.org/abs/2210.02466}{{\tt arXiv:2210.02466}}].

[54]
J.~A. Damia, R.~Argurio, and L.~Tizzano, {\it {Continuous Generalized
Symmetries in Three Dimensions}},
\href{http://arxiv.org/abs/2206.14093}{{\tt arXiv:2206.14093}}.

[55]
J.~Kaidi, K.~Ohmori, and Y.~Zheng, {\it {Symmetry TFTs for Non-Invertible
Defects}},  \href{http://arxiv.org/abs/2209.11062}{{\tt arXiv:2209.11062}}.

[56]
S.~Gukov, P.~Koroteev, S.~Nawata, D.~Pei, and I.~Saberi, {\it {Branes and DAHA
Representations}},  \href{http://arxiv.org/abs/2206.03565}{{\tt

[57]
A.~Antinucci, F.~Benini, C.~Copetti, G.~Galati, and G.~Rizi, {\it {The
holography of non-invertible self-duality symmetries}},
\href{http://arxiv.org/abs/2210.09146}{{\tt arXiv:2210.09146}}.

[58]
H.-J. Chung, T.~Dimofte, S.~Gukov, and P.~Su{\l}kowski, {\it 3d-3d
correspondence revisited},  {\em JHEP} {\bf 1604} (2016) 140,
[\href{http://arxiv.org/abs/1405.3663}{{\tt arXiv:1405.3663}}].

[59]
S.~Gukov, P.~Putrov, and C.~Vafa, {\it {Fivebranes and 3-manifold homology}},
{\em JHEP} {\bf 07} (2017) 071, [\href{http://arxiv.org/abs/1602.05302}{{\tt

[60]
S.~Gukov, D.~Pei, P.~Putrov, and C.~Vafa, {\it {BPS spectra and 3-manifold
invariants}},  {\em J. Knot Theor. Ramifications} {\bf 29} (2020), no.~02
2040003, [\href{http://arxiv.org/abs/1701.06567}{{\tt arXiv:1701.06567}}].

[61]
S.~Gukov and D.~Pei, {\it {Equivariant Verlinde formula from fivebranes and
vortices}},  {\em Commun. Math. Phys.} {\bf 355} (2017), no.~1 1--50,
[\href{http://arxiv.org/abs/1501.01310}{{\tt arXiv:1501.01310}}].

[62]
S.~Gukov and C.~Manolescu, {\it {A two-variable series for knot complements}},
{\em Quantum Topol.} {\bf 12} (2021), no.~1 1--109,
[\href{http://arxiv.org/abs/1904.06057}{{\tt arXiv:1904.06057}}].

[63]
H.-J. Chung, {\it {Index for a Model of 3d-3d Correspondence for Plumbed
3-Manifolds}},  {\em Nucl. Phys. B} {\bf 965} (2021) 115361,
[\href{http://arxiv.org/abs/1912.13486}{{\tt arXiv:1912.13486}}].

[64]
H.-J. Chung, {\it {BPS Invariants for Seifert Manifolds}},  {\em JHEP} {\bf 03}
(2020) 113, [\href{http://arxiv.org/abs/1811.08863}{{\tt arXiv:1811.08863}}].

[65]
M.~C.~N. Cheng, S.~Chun, F.~Ferrari, S.~Gukov, and S.~M. Harrison, {\it {3d
Modularity}},  {\em JHEP} {\bf 10} (2019) 010,
[\href{http://arxiv.org/abs/1809.10148}{{\tt arXiv:1809.10148}}].

[66]
N.~Dorey and D.~Tong, {\it {Mirror symmetry and toric geometry in
three-dimensional gauge theories}},  {\em JHEP} {\bf 05} (2000) 018,
[\href{http://arxiv.org/abs/hep-th/9911094}{{\tt hep-th/9911094}}].

[67]
E.~Witten, {\it {SL(2,Z) action on three-dimensional conformal field theories
with Abelian symmetry}},  \href{http://arxiv.org/abs/hep-th/0307041}{{\tt

[68]
T.~Kitao, K.~Ohta, and N.~Ohta, {\it Three-dimensional gauge dynamics from
brane configurations with (p,q)-fivebrane},  {\em Nucl.Phys.} {\bf B539}
(1999) 79--106, [\href{http://arxiv.org/abs/hep-th/9808111}{{\tt

[69]
S.~Cheng, {\it {Mirror symmetry and mixed Chern-Simons levels for Abelian 3D
N=2 theories}},  {\em Phys. Rev. D} {\bf 104} (2021), no.~4 046011,
[\href{http://arxiv.org/abs/2010.15074}{{\tt arXiv:2010.15074}}].

[70]
C.~Closset and O.~Khlaif, {\it {Twisted indices, Bethe ideals and 3d
$\mathcal{N}=2$ infrared dualities}},
\href{http://arxiv.org/abs/2301.10753}{{\tt arXiv:2301.10753}}.

[71]
T.~Dimofte, D.~Gaiotto, and S.~Gukov, {\it {Gauge Theories Labelled by
Three-Manifolds}},  {\em Commun. Math. Phys.} {\bf 325} (2014) 367--419,
[\href{http://arxiv.org/abs/1108.4389}{{\tt arXiv:1108.4389}}].

[72]
Y.~Choi, C.~Cordova, P.-S. Hsin, H.~T. Lam, and S.-H. Shao, {\it
{Non-invertible Condensation, Duality, and Triality Defects in 3+1
Dimensions}},  \href{http://arxiv.org/abs/2204.09025}{{\tt

[73]
L.~Bhardwaj, Y.~Lee, and Y.~Tachikawa, {\it {$SL(2,\mathbb{Z})$ action on QFTs
with $\mathbb{Z}_2$ symmetry and the Brown-Kervaire invariants}},  {\em JHEP}
{\bf 11} (2020) 141, [\href{http://arxiv.org/abs/2009.10099}{{\tt

[74]
J.~Kaidi, K.~Ohmori, and Y.~Zheng, {\it {Kramers-Wannier-like Duality Defects
in (3+1)D Gauge Theories}},  {\em Phys. Rev. Lett.} {\bf 128} (2022), no.~11
111601, [\href{http://arxiv.org/abs/2111.01141}{{\tt arXiv:2111.01141}}].

[75]
P.~Kucharski, M.~Reineke, M.~Stosic, and P.~Su\l{}kowski, {\it {Knots-quivers
correspondence}},  {\em Adv. Theor. Math. Phys.} {\bf 23} (2019) 1849--1902,
[\href{http://arxiv.org/abs/1707.04017}{{\tt arXiv:1707.04017}}].

[76]
P.~Kucharski, M.~Reineke, M.~Stosic, and P.~Su{\l}kowski, {\it {BPS states,
knots and quivers}},  {\em Phys. Rev.} {\bf D96} (2017), no.~12 121902,
[\href{http://arxiv.org/abs/1707.02991}{{\tt arXiv:1707.02991}}].

[77]
M.~Panfil and P.~Su{\l}kowski, {\it {Topological strings, strips and quivers}},
{\em JHEP} {\bf 01} (2019) 124, [\href{http://arxiv.org/abs/1811.03556}{{\tt

[78]
T.~Ekholm, P.~Kucharski, and P.~Longhi, {\it {Physics and geometry of
knots-quivers correspondence}},  {\em Commun. Math. Phys.} {\bf 379} (2020),
no.~2 361--415, [\href{http://arxiv.org/abs/1811.03110}{{\tt

[79]
T.~Ekholm, P.~Kucharski, and P.~Longhi, {\it {Multi-cover skeins, quivers, and
3d $\mathcal{N}=2$ dualities}},  {\em JHEP} {\bf 02} (2020) 018,
[\href{http://arxiv.org/abs/1910.06193}{{\tt arXiv:1910.06193}}].

[80]
T.~Ekholm, A.~Gruen, S.~Gukov, P.~Kucharski, S.~Park, and P.~Su\l{}kowski, {\it
{${\widehat{Z}}$ at Large N: From Curve Counts to Quantum Modularity}},  {\em
Commun. Math. Phys.} {\bf 396} (2022), no.~1 143--186,
[\href{http://arxiv.org/abs/2005.13349}{{\tt arXiv:2005.13349}}].

[81]
J.~Jankowski, P.~Kucharski, H.~Larragu\'\i{}vel, D.~Noshchenko, and
P.~Su\l{}kowski, {\it {Permutohedra for knots and quivers}},  {\em Phys. Rev.
D} {\bf 104} (2021), no.~8 086017,
[\href{http://arxiv.org/abs/2105.11806}{{\tt arXiv:2105.11806}}].

[82]
M.~C.~N. Cheng, S.~Chun, B.~Feigin, F.~Ferrari, S.~Gukov, S.~M. Harrison, and
D.~Passaro, {\it {3-Manifolds and VOA Characters}},
\href{http://arxiv.org/abs/2201.04640}{{\tt arXiv:2201.04640}}.

[83]
T.~Ekholm, P.~Kucharski, and P.~Longhi, {\it {Knot homologies and generalized
quiver partition functions}},  \href{http://arxiv.org/abs/2108.12645}{{\tt

[84]
T.~Ekholm, A.~Gruen, S.~Gukov, P.~Kucharski, S.~Park, M.~Stosi\'c, and
P.~Su\l{}kowski, {\it {Branches, quivers, and ideals for knot complements}},
{\em J. Geom. Phys.} {\bf 177} (2022) 104520,
[\href{http://arxiv.org/abs/2110.13768}{{\tt arXiv:2110.13768}}].

[85]
J.~Jankowski, P.~Kucharski, H.~Larragu\'\i{}vel, D.~Noshchenko, and
P.~Su\l{}kowski, {\it {Quiver diagonalization and open BPS states}},
\href{http://arxiv.org/abs/2212.04379}{{\tt arXiv:2212.04379}}.

[86]
L.~F. Alday, P.~Benetti~Genolini, M.~Bullimore, and M.~van Loon, {\it {Refined
3d-3d Correspondence}},  {\em JHEP} {\bf 04} (2017) 170,
[\href{http://arxiv.org/abs/1702.05045}{{\tt arXiv:1702.05045}}].

[87]
D.~Gaiotto and E.~Witten, {\it {S-Duality of Boundary Conditions In N=4 Super
Yang-Mills Theory}},  {\em Adv. Theor. Math. Phys.} {\bf 13} (2009), no.~3
721--896, [\href{http://arxiv.org/abs/0807.3720}{{\tt arXiv:0807.3720}}].

[88]
N.~Hama, K.~Hosomichi, and S.~Lee, {\it {SUSY Gauge Theories on Squashed
Three-Spheres}},  {\em JHEP} {\bf 05} (2011) 014,
[\href{http://arxiv.org/abs/1102.4716}{{\tt arXiv:1102.4716}}].

[89]
A.~Kapustin, B.~Willett, and I.~Yaakov, {\it {Exact Results for Wilson Loops in
Superconformal Chern-Simons Theories with Matter}},  {\em JHEP} {\bf 03}
(2010) 089, [\href{http://arxiv.org/abs/0909.4559}{{\tt arXiv:0909.4559}}].

[90]
V.~Jejjala, Y.~Lei, S.~van Leuven, and W.~Li, {\it {Modular factorization of
superconformal indices}},  \href{http://arxiv.org/abs/2210.17551}{{\tt

[91]
M.~Aganagic and S.~Shakirov, {\it {Refined Chern-Simons Theory and Topological
String}},  \href{http://arxiv.org/abs/1210.2733}{{\tt arXiv:1210.2733}}.

[92]
T.~Dimofte, D.~Gaiotto, and N.~M. Paquette, {\it {Dual boundary conditions in
3d SCFT\textquoteright{}s}},  {\em JHEP} {\bf 05} (2018) 060,
[\href{http://arxiv.org/abs/1712.07654}{{\tt arXiv:1712.07654}}].

[93]
A.~Gadde, S.~Gukov, and P.~Putrov, {\it {Walls, Lines, and Spectral Dualities
in 3d Gauge Theories}},  {\em JHEP} {\bf 05} (2014) 047,
[\href{http://arxiv.org/abs/1302.0015}{{\tt arXiv:1302.0015}}].

[94]
N.~A. Nekrasov, {\it {Seiberg-Witten prepotential from instanton counting}},
{\em Adv.Theor.Math.Phys.} {\bf 7} (2004) 831--864,
[\href{http://arxiv.org/abs/hep-th/0206161}{{\tt hep-th/0206161}}].

[95]
S.~Shadchin, {\it On {F}-term contribution to effective action},  {\em Journal
of High Energy Physics} {\bf 2007} (Aug, 2007) 052--052,
[\href{http://arxiv.org/abs/hep-th/0611278}{{\tt hep-th/0611278}}].

[96]
M.~Aganagic and C.~Vafa, {\it {Mirror symmetry, D-branes and counting
holomorphic discs}},  \href{http://arxiv.org/abs/hep-th/0012041}{{\tt

[97]
S.~Cremonesi, {\it {Type IIB construction of flavoured ABJ(M) and fractional M2
branes}},  {\em JHEP} {\bf 01} (2011) 076,
[\href{http://arxiv.org/abs/1007.4562}{{\tt arXiv:1007.4562}}].

\end{thebibliography}\endgroup

%\author[a]{Shi Cheng,}
% \author[a,b]{Piotr Su\l kowski}

%\affiliation[a]{Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland}
% \affiliation[b]{Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA}

%<EMAIL_ADDRESS>
\end{document}