We develop a classification of

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1$ under $\SU(2)_R$. In the latter case, the extra operators $\oi'$ have conformal dimension $\Delta(\oi')>1$. ]]>

1$, such that $\gf$ remains the isometry of the Coulomb branch. In order to address this question, let us employ the following claim, which results from the work on monopole operators on the $3d~\mathcal N=4$ Coulomb branch~\cite{Borokhov-ml-2002cg,Borokhov-ml-2002ib,GW09,CHZ13,CFHM14}: a gauge node of a \3d quiver determines the presence of operators $\mathcal O_i$ with $\Delta(\mathcal{O}_i)=1$ in the Coulomb branch in the following way: \begin{itemize} \item If the node has \emph{excess} $e>0$, it contributes with a single Casimir operator $\phi_i$, such that $\Delta (\phi_i)=1$. \item A set of nodes with excess $e=0$, in the form of a Dynkin diagram of a Lie group $G$, contributes with a number of operators $\mathcal{O}_i$ (with $\Delta(\mathcal O_i)=1$) equal to the dimension of $G$. There will be one Casimir operator $\phi_i$ per gauge node. The remaining operators are \emph{bare monopole operators} $V_i$ that correspond to the different roots of the algebra~$\Lie(G)$. \item If the quiver has no flavor nodes, one Casimir operator with $\Delta(\mathcal \phi_i) = 1$ needs to be removed from the counting, corresponding to the adjoint representation of the decoupled $\U(1)$ \emph{center of mass}. \end{itemize} Two different cases of \3d quivers with isometry $\gf$ on the Coulomb branch $\mathcal C$ can be readily identified employing this claim: \begin{enumerate} \item {\bfseries Nilpotent orbit's closure:} $\mathcal C = \bar{\mathcal{O}}_\lambda\subset \Lie(\gf)$. All the generators of $\mathcal C$ have dimension $\Delta(\mathcal O_i) = 1$. The gauge nodes of the quiver form the Dynkin diagram of $\Lie(\gf)$, for any classical or exceptional Lie group $\gf$. All gauge nodes of the quiver are balanced. Flavor nodes are added to ensure such balance condition. Moreover, the rank of the flavor nodes always follows the pattern of the \emph{weighted Dynkin diagram}~\cite{CM93} of the corresponding nilpotent orbit $\mathcal O_\lambda$. This can realise nilpotent orbit's closures of height $\OPht(\mathcal O_\lambda) = 2$.\footnote{The \emph{height} of a nilpotent orbit is defined as in~\cite[section~2]{Panyushev-ml-1999on}. Note that for $\gf$ of $A$-type, this construction can be extended to nilpotent orbits of all heights $\OPht(\mathcal O_\lambda)$, where the flavor nodes are determined by the partition $\lambda$ of the nilpotent orbit. See~\cite{GW09,HK16,CH16} for examples.} \item {\bfseries Minimally unbalanced quiver:} the gauge nodes of the quiver form a minimal extension of the Dynkin diagram of $\Lie(\gf)$. By minimal extension we mean that there is a single extra gauge node, connected to the other gauge nodes that form the Dynkin diagram. There are no flavor nodes. All gauge nodes in the Dynkin diagram are balanced (with $e=0$). The the extra node is unbalanced, i.e.\ it has excess $e>0$.\footnote{In the following sections the cases with $e=-1$ are discussed separately from generic cases with $e<0$. If all nodes have $e=0$, the quiver forms an affine or twisted affine Dynkin diagram of the global symmetry $\gf$ and these cases are also discussed separately. We refrain from discussing the pathological case of the $A^{(2)}_2$ twisted affine Dynkin diagram.} \end{enumerate} Examples of the first case are the theories in equations~\eqref{eq:nil3} and~\eqref{eq:nil10}. Equation~\eqref{eq:muq10} is an example of the second case. Both cases have a number of generators $\mathcal O_i$, with $\Delta(\mathcal O_i)=1$, equal to the dimension of $\gf$. In all three examples the Coulomb branch has the same isometry $\gf$. The difference is that the first case has no extra generators of $\mathcal C$, while the second case has extra generators $\mathcal O_i'$ with $\Delta(\mathcal O'_i)>1$. As mentioned before, $3d~\mathcal{N}=4$ quiver gauge theories whose Coulomb branches are closures of nilpotent orbits have already been extensively studied (note the recent progress for exceptional $\gf$ in~\cite{Hanany-ml-2017ooe}). In this note we present a classification of all minimally unbalanced quivers, for any classical Lie group $\gf$. We emphasize that all quivers presented in this paper are in the basic form such that the ranks are the lowest possible. Other theories can be obtained by multiplying the basic forms of the quivers by an integer number (this will not modify the isometry of the Coulomb branch). \paragraph{Minimally unbalanced quiver.} We are in the position to present the general solution for finding all minimally unbalanced quivers with a Coulomb branch isometry $\gf$, where $\gf$ contains a single factor. The remaining sections of the paper contain the specific results for all the different types of Lie groups. As the first step, consider a \3d quiver $\q$ with the shape of a particular Dynkin diagram and with an extra node attached to it in the simplest fashion.\footnote{Simplest fashion means that the extra node is attached by a simply laced edge to only one of the nodes of the balanced Dynkin diagram.} All nodes are $\U(N_{i})$ gauge nodes, where $N_i$ is the number of colors of the $i$-th node. The nodes in the Dynkin diagram need to be balanced (with excess $e=0$). In order to impose the balancing condition one can remember the vectors $\vec{v}$ and $\vec{w}$ on Nakajima's quiver varieties~\cite{N94}. In this case they are used slightly differently. Let $\vec v$ be the vector with the ranks of all the nodes of the part of the quiver that forms the (balanced) Dynkin diagram. Let $C$ be the corresponding Cartan matrix. Then, the vector $\vec{w}$ is defined as: \begin{equation} \vec{w} := C\cdot \vec{v}\,. \end{equation} Note that $\vec w$ measures the excess in each of the nodes in $\vec v$ in the presence of no other nodes in the quiver. Now, one sets to zero the all components of $\vec w$ except of one. The non-zero component can be set to $k$. This corresponds to attaching an extra node of rank $k$ (the node that will become \emph{minimally unbalanced}) at the position of the non-zero element of $\vec w$ and simultaneously balancing all nodes in $\vec v$. After fixing the rank of the imbalanced node (node with $e\neq 0$) to $k$, the ranks of the balanced nodes are uniquely determined:\footnote{Note that the existence of the inverse of the Cartan matrix is guaranteed since we are dealing with finite-dimensional Lie algebras.} \begin{equation}\label{eq:Cartan} \vec{v} = C^{-1}\cdot \vec{w}\,. \end{equation} Finally, the value of $k$ can be chosen to be the smallest value such that all the other ranks are integer numbers. In the following sections we perform this computation for all different choices of the position of the non-zero component of $\vec w$. In this way, we obtain all possible minimally unbalanced quivers with a balanced subset of nodes corresponding to a certain Dynkin diagram. ]]>

0$& Good & $\Delta \geq 1$ \\ \hline $e=0$ & Good & $\Delta= 1$ \\ \hline $e=-1$ & Ugly &$\Delta > 0$ \\ \hline $e<-1$ & Bad & not applicable \\ \hline \end{tabular} ]]>

0$ the global symmetry of the Coulomb branch is $\SU(n)$, where $n=a+b$, and one says that the quiver is \emph{minimally unbalanced} with positive excess. Therefore, the quiver in the first row in table~\ref{tab:A8} with $e(5,4)=2$ corresponds to a good theory with positive excess. The quiver in the second row with $e(2,1)=0$ represents a good theory that is \emph{fully balanced} since all nodes have excess zero. The two bad theories with $e(7,2)=-4$ and $e(0,1)=-10$ are contained in the third and the fourth row of table~\ref{tab:A8}, respectively. Minimally unbalanced quivers with the unbalanced node with excess $e=-1$ have either the entire or a part of the Coulomb branch freely generated.\footnote{See observation~3.1 in~\cite{FG08}.} Equation~\eqref{eq:e} defines a function: \begin{equation} \begin{aligned} e: \mathbb{N}\times \mathbb{N}&\rightarrow \mathbb Z\\ (a,b)&\mapsto e(a,b) \end{aligned} \end{equation} that maps the two parameters of the family $a$ and $b$ to the excess of the top node of the corresponding quiver. This function can be visualized by defining a matrix $M$, with elements: \begin{equation} M_{ab} = e(a,b)\,. \end{equation} Let $a$ and $b$ run from 1 to 16, then $M$ is $16\times 16$: \begin{equation} M=\left( \begin{array}{cccccccccccccccc} -3 & -4 & -5 & -6 & -7 & -8 & -9 & \bm{-10} & -11 & -12 & -13 & -14 & -15 & -16 & -17 & -18 \\ -4 & -2 & -4 & -2 & -4 & -2 & \bm{-4} & -2 & -4 & -2 & -4 & -2 & -4 & -2 & -4 & -2 \\ -5 & -4 & -1 & -2 & -1 & \bm{0} & 1 & 2 & 1 & 4 & 5 & 2 & 7 & 8 & 3 & 10 \\ -6 & -2 & -2 & 0 & \bm{2} & 2 & 6 & 2 & 10 & 6 & 14 & 4 & 18 & 10 & 22 & 6 \\ -7 & -4 & -1 & \bm{2} & 1 & 8 & 11 & 14 & 17 & 4 & 23 & 26 & 29 & 32 & 7 & 38 \\ -8 & -2 & \bm{0} & 2 & 8 & 2 & 16 & 10 & 8 & 14 & 32 & 6 & 40 & 22 & 16 & 26 \\ -9 & \bm{-4} & 1 & 6 & 11 & 16 & 3 & 26 & 31 & 36 & 41 & 46 & 51 & 8 & 61 & 66 \\ \bm{-10} & -2 & 2 & 2 & 14 & 10 & 26 & 4 & 38 & 22 & 50 & 14 & 62 & 34 & 74 & 10 \\ -11 & -4 & 1 & 10 & 17 & 8 & 31 & 38 & 5 & 52 & 59 & 22 & 73 & 80 & 29 & 94 \\ -12 & -2 & 4 & 6 & 4 & 14 & 36 & 22 & 52 & 6 & 68 & 38 & 84 & 46 & 20 & 54 \\ -13 & -4 & 5 & 14 & 23 & 32 & 41 & 50 & 59 & 68 & 7 & 86 & 95 & 104 & 113 & 122 \\ -14 & -2 & 2 & 4 & 26 & 6 & 46 & 14 & 22 & 38 & 86 & 8 & 106 & 58 & 42 & 34 \\ -15 & -4 & 7 & 18 & 29 & 40 & 51 & 62 & 73 & 84 & 95 & 106 & 9 & 128 & 139 & 150 \\ -16 & -2 & 8 & 10 & 32 & 22 & 8 & 34 & 80 & 46 & 104 & 58 & 128 & 10 & 152 & 82 \\ -17 & -4 & 3 & 22 & 7 & 16 & 61 & 74 & 29 & 20 & 113 & 42 & 139 & 152 & 11 & 178 \\ -18 & -2 & 10 & 6 & 38 & 26 & 66 & 10 & 94 & 54 & 122 & 34 & 150 & 82 & 178 & 12 \\ \end{array} \right) . \end{equation} The elements in bold are those that correspond to the quivers of length $a+b-1=8$, i.e.\ those in table~\ref{tab:A8}. One can see that for a generic quiver the excess is positive. A theory with $a+b-1>8$ is \emph{bad} (negative excess) only if one of the two parameters is either 1 or 2. Furthermore, there are only three cases where the extra node is also balanced, i.e.\ excess $e(a,b)=0$. These are: $(a,b)=(3,6)$, $(a,b)=(4,4)$ and $(a,b)=(6,3)$. The first and last cases correspond to an enhancement of the global symmetry of the Coulomb branch from $\SU(9)$ to $E_8$. The case $(a,b)=(4,4)$ sees a similar enhancement, this time from $\SU(8)$ to $E_7$. The three cases with $e=-1$ are obtained for $(a,b)\in\{(3,3), (3,5), (5,3)\}$. For $(a,b)=(3,3)$ the greatest common divisor is $\gcd(3,3)=3$, therefore, the quiver takes the~form: \begin{equation} \label{eq:quiver12321w2on3} \q_{(3,3)} = ~\node{}1 -\node{}{2} -\node{\tnoder 2}{3}-\node{}{2}-\node{}{1} \end{equation} and the Coulomb branch of this quiver is a freely generated variety (see~3.10 in~\cite{Hanany-ml-2018vph}):\footnote{Since the balanced sub-quiver corresponds to $A_5$ global symmetry, but $\mathbb{H}^{10}$ has isometry $\Sp(10)$, we find an embedding: $\SU(6) \hookleftarrow \Sp(10)$. In particular, the pseudo-real fundamental rep of $\Sp(10)$ projects to the pseudo-real $3^{\rm rd}$ rank antisymmetric rep of $\SU(6)$: $[1,0,0,0,0]_{\Sp(10)} \hookrightarrow [0,0,1,0,0]_{\SU(6)}$.} \begin{equation} \mathcal{C}=\mathbb{H}^{10}. \end{equation} For $(a,b)=(3,5)$ (or equivalently $(a,b)=(5,3)$) the quiver takes the form: \begin{equation} \label{eq:quiver3691215105w8on15} \q_{(3,5)} = ~\node{}3 -\node{}{6} -\node{}{9} -\node{}{12}-\node{\tnoder 8}{15}-\node{}{10}-\node{}{5}\,. \end{equation} The quaternionic dimension of the Coulomb branch in~\ref{eq:quiver3691215105w8on15} is $67$. The unbalanced node connects to the Dynkin node that corresponds to the $\SU(8)$ representation with Dynkin labels $[0,0,0,0,1,0,0]$ and with dimension $56$. Drawing intuition from the quiver in~\ref{eq:quiver12321w2on3} one would expect $112$ new operators transforming in the $[0,0,0,0,1,0,0]$ and its complex conjugate rep $[0,0,1,0,0,0,0]$. Although the excess is $e=-1$ (i.e.\ same as in freely generated~\ref{eq:quiver12321w2on3}) the Coulomb branch of~\ref{eq:quiver3691215105w8on15} seems to be more complicated (i.e.\ has both a freely generated as well as a non-trivial part) and we leave its explicit computation for future study. A formula for the HWG for minimally unbalanced $A$-type quivers with $a=b, s=1$ (i.e.\ with outer $\mathbb{Z}_2$ automorphism symmetry) is given by equation~(23) in~\cite{HPsic17}. Quivers of this type also show up in the study of Higgs branches of $5d$ $\mathcal{N}=1$ theories with $8$ supercharges~\cite{FHMZ17}. ]]>

1$. In such scenario, the global symmetry takes the form: \begin{equation} G_{\rm global} = \prod_{i} G_i \times \U(1)^{N-1}, \end{equation} where $G_i$ are the groups corresponding to the Dynkin sub-diagrams formed by the subset of balanced nodes. The number of the $\U(1)$ Abelian factors in the global symmetry is one less than the number of unbalanced nodes. Quivers with more than one unbalanced node appear in various contexts in the study of $5d$ and $6d$ Higgs branches~\cite{FHMZ17,Hanany-ml-2018vph,DGHZ18}. In the classification of this paper, we find a raft of quiver theories that are not studied in any existing literature. This opens a large and possibly fructiferous domain for extensive future investigations. \emph{Quaerite et invenietis ordinem.} ]]>

0$ for all $m\in \Gamma _{\hat G}$ which guarantees that the monopole formula can be applied to calculate the Coulomb branch of the moduli space.\footnote{In fact, there are special cases of balanced quivers with moduli spaces that are not hyperK\"ahler varieties, hence the monopole cannot be applied. Thus, it seems that balance is necessary but not sufficient condition for a quiver to be well behaved and treatable by the currently known methods.} The global symmetry of the Coulomb branch is determined by the operators with $\Delta=1$. From the quiver one can quickly write a set of operators with $\Delta=1$ such that they correspond to the roots of the Dynkin diagram formed by nodes that are balanced. Extra operators with $\Delta=1$ might exist, which would enhance the global symmetry. In the previous pages we are restricted to quivers where only one node is unbalanced, and the remaining nodes form the Dynkin diagram of either a classical or an exceptional Lie algebra. The Higgs branch of $5$d theories at infinite coupling is given by the Coulomb branch of a $3$d quiver. Physically, we can motivate this with a use of 3d mirror symmetry~\cite{IS96} and the presence of $8$ supercharges in both theories. Considering a reduction of the $5$d SCFT on a torus leads to a $3$d Higgs branch that is unchanged thanks to the amount of supersymmetry. In addition, many $3$d theories have mirror duals for which the Coulomb and Higgs branch are exchanged. In general a dual theory can lack a Lagrangian description, however, it was argued in~\cite{BTX10} that specific class of $5$d theories described by intersecting D5, NS5 and $(1,1)$-branes reduces to $A$-type class $S$ theories compactified on a circle. It was further argued that reducing class $S$ theories on circle to $3$d leads to $3$d SCFTs with Lagrangian mirrors whose shape is a three-legged unitary quiver. For a $\SU(n)$ theory with fundamental matter the bound for the number of flavors is: $N_f>2n$. $5$d SCFTs with enough matter belong to this class. This is a strong motivation for the approach of this paper. ]]>