\end{tikzpicture}}$;
(01)–(0120)–(20) (20)–(2012)–(12) (12)–(1231)–(31) (31)–(3150)–(50) (12)–(1204)–(04) (31)–(3123)–(23) (50)–(5042)–(42) (04)–(0423)–(23) (23)–(2342)–(42) (42)–(4261)–(61);
(23)–(1*-0.3*0.5*,2.66*+0.3*1*) (42)–(2*-0.3*0.5*,2.66*+0.3*1*) (61)–(3*-0.3*0.5*,2.66*+0.3*1*) (61)–(3*+0.4*,2.66*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{PSL}(3,\mathbb{C})}{}{[0,1]}$.

§.§ The B-type affine Grassmannian

For this setup, we require a ${\color{orange}\widetilde{\textnormal{ON}}^-}$ plane. In the main text we always draw the full covering space and refer to half branes simply as branes.

Component $0\in\mathbb{Z}_2$. The affine Grassmannian of $\mathrm{Spin}(2n+1,\mathbb{C})$ is the $0$-th component of the affine Grassmannian of $\mathrm{SO}(2n+1,\mathbb{C})$. The brane system for $\w{\mathrm{SO}(2n+1,\mathbb{C})}{}{[0,0\dots,0]}$ is proposed to be

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[orange] (2,0)--(2,4);
\node at (2,-0.5) {$\widetilde{\textnormal{ON}}^-$};
\end{tikzpicture}
}} \, ,
\end{equation}

which is depicted at the origin, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{\begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=below:{$0$}] (g4) at (4,0) {};
\node[flavour,label=above:{$0$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=above:{$0$}] (f4) at (4,1) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4);
\draw (g1)--(g2)--(g)--(g3);
\draw[transform canvas={yshift=-2pt}] (g3)--(g4);
\draw[transform canvas={yshift=2pt}] (g3)--(g4);
\draw (3.45,0.2)--(3.55,0)--(3.45,-0.2);
\end{tikzpicture}}
\end{equation}

Component $1\in\mathbb{Z}_2$. The brane system for the $1$-st component of the affine Grassmannian of $\mathrm{SO}(2n+1,\mathbb{C})$, is proposed to be
\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[orange] (2,0)--(2,4);
\node at (2,-0.5) {$\widetilde{\textnormal{ON}}^-$};
\node[D5] at (-2.5,2) {};
\node[D5] at (6.5,2) {};
\end{tikzpicture}
}} \, ,
\end{equation}

which is depicted at the origin, with all D3 branes and D5 branes suppressed. The quiver representing the origin is

\begin{equation}
\raisebox{-.5\height}{\begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=below:{$0$}] (g4) at (4,0) {};
\node[flavour,label=above:{$1$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=above:{$0$}] (f4) at (4,1) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4);
\draw (g1)--(g2)--(g)--(g3);
\draw[transform canvas={yshift=-2pt}] (g3)--(g4);
\draw[transform canvas={yshift=2pt}] (g3)--(g4);
\draw (3.45,0.2)--(3.55,0)--(3.45,-0.2);
\end{tikzpicture}} \, .
\end{equation}

This is indeed the quiver representing the origin of $\w{\mathrm{SO}(2n+1,\mathbb{C})}{}{[1,0,\dots,0]}$.

Minimal transitions. When activating D3 branes between D5 branes, one can follow the same procedure as for the A-type affine Grassmannian, when considering branes away from the ON (keeping in mind that in the covering space one has to move branes together with their mirror images). If however one wants to activate a D3 brane between a D5 brane and its image, in order to move the D3 brane along an NS5 brane and its image, one has to keep in mind the boundary conditions reviewed in Appendix <ref>. The D3 brane ending on a D5 brane and its image needs to be accompanied by its own mirror image. The associated transition is $b_n$ if the D5 branes are separated by $n$ NS5 branes (and their mirror images).

\begin{equation}
\vcenter{\hbox{
% \scalebox{0.70}{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-1,0)--(-1,4) (0,0)--(0,4) (4,0)--(4,4) (5,0)--(5,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1.5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (0.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5.5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(3.9,-0.25);
\draw[orange] (2,0)--(2,4);
\node[D5] (1) at (-4,2) {};
\node[D5] (2) at (8,2) {};
\draw[transform canvas={yshift=-1.5pt}] (1)--(-3,2) (7,2)--(2);
\draw[transform canvas={yshift=1.5pt}] (1)--(-1,2) (5,2)--(2);
\draw (-3,1.5)--(-1,1.5) (5,1.5)--(7,1.5);
\draw[double] (0,1)--(4,1);
\node at (8,3.5) {$b_n$, $n>2$};
\end{tikzpicture}
% }
% \qquad.
\end{equation}

In the case of $n=1$ one has to be slightly more careful. If we attempt to move the D3 brane and its mirror along the NS5 brane we would obtain a set-up breaking the S-rule:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\node[D5] (1) at (-1,2) {};
\node[D5] (2) at (5,2) {};
\draw[double] (1)--(0,2) (4,2)--(2) (0,1)--(4,1);
\node at (7,2) {Breaks S-rule!};
\end{tikzpicture}}}
\end{equation}

We therefore need to include another D5 brane and its mirror image and can now perform a $b_1=A_1$ transition:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\node[D5] (1) at (-1.5,2.1) {};
\node[D5] (2) at (-1,1.9) {};
\node[D5] (3) at (5,1.9) {};
\node[D5] (4) at (5.5,2.1) {};
\draw (1)--(0,2.1) (4,2.1)--(4) (2)--(0,1.9) (4,1.9)--(3);
\draw[double] (0,1)--(4,1);
\node at (8,2) {$b_1=A_1$};
\end{tikzpicture}}}\qquad.
\label{eq:b1trans}
\end{equation}

Drawing the D5 branes at different vertical positions is purely for convenience. In (<ref>) more D5 branes could be present between the NS5 and the $\widetilde{\textnormal{ON}}^-$ plane leading to $A_k$ transitions.

When dealing with the $1$-st component for $\mathrm{SO}(5,\mathbb{C})$, i.e. $n=2$, one also has to be careful with the first transition. Activating a minimal number of moduli without breaking supersymmetry gives the configuration

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (2,0)--(2,4) (6,0)--(6,4) (8,0)--(8,4);
\draw[orange] (4,0)--(4,4);
\node[D5] (1) at (-1,2) {};
\node[D5] (2) at (9,2) {};
\node[D5] at (1,2.5) {};
\node[D5] at (7,2.5) {};
\draw[double] (1)--(0,2) (8,2)--(2) (2,1)--(6,1);
\draw (0,2)--(2,2) (6,2)--(8,2);
\draw (0,1.5)--(2,1.5) (6,1.5)--(8,1.5);
\node at (10.5,2) {$ac_2$};
\end{tikzpicture}}}\quad.
\label{eq:ab2trans}
\end{equation}

This transition is called $ac_2$ as it appears also in the type $C$ affine Grassmannians, and we use the isomorphism $C_2 = B_2$. For $n>2$ no new slices arise.

§.§.§ The B1 affine Grassmannian

Component $0 \in \mathbb{Z}_2$. The affine Grassmannian of $\mathrm{Spin}(3,\mathbb{C})$, i.e. component 0 of the affine Grassmannian of $\mathrm{SO}(3,\mathbb{C})$, is given by the brane system:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\end{tikzpicture}}}\qquad,
\end{equation}

depicted at the origin of $\w{\mathrm{SO}(3,\mathbb{C})}{}{[0]}$. We can now move onto the lowest non-trivial leaf through a $b_1=A_1$ transition as in (<ref>). After a Hanany-Witten transition we can read off the quiver:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\node[D5] (1) at (0.7,2) {};
\node[D5] (2) at (1.3,2) {};
\node[D5] (3) at (2.7,2) {};
\node[D5] (4) at (3.3,2) {};
\draw[double] (0,1)--(4,1);
\end{tikzpicture}}}
\qquad
\vcenter{\hbox{
\begin{tikzpicture}
\node[gauge,label=below:{$1$}] (1) at (0,0) {};
\node[flavour,label=above:{$2$}] (2) at (0,1) {};
\draw (1)--(2);
\end{tikzpicture}
}} \, .
\end{equation}

The only way to keep moving onto higher leaves in $\w{\mathrm{SO}(3,\mathbb{C})}{}{[0]}$ is by repeating the same operation, however the D5 branes already placed between the NS5 branes make the transverse slices $A_{2k-1}$ for the $k$-th transition of this type. A general leaf and the corresponding quiver are

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\node[D5] (1) at (0.5,2) {};
\node at (1,2.3) {$2k$};
\node at (1,2) {$\cdots$};
\node[D5] (2) at (1.5,2) {};
\node[D5] (3) at (2.5,2) {};
\node at (3,2.3) {$2k$};
\node at (3,2) {$\cdots$};
\node[D5] (4) at (3.5,2) {};
\draw[double] (0,1)--(4,1);
\node at (1,0.75) {$\vdots$ $k$};
\node at (3,0.75) {$\vdots$ $k$};
\draw[double] (0,0.3)--(4,0.3);
\end{tikzpicture}}}
\qquad
\vcenter{\hbox{
\begin{tikzpicture}
\node[gauge,label=below:{$k$}] (1) at (0,0) {};
\node[flavour,label=above:{$2k$}] (2) at (0,1) {};
\draw (1)--(2);
\end{tikzpicture}
}} \, ,
\label{eq:B1_1st}
\end{equation}

which completely agrees with the construction in Section <ref>.

Component $1\in\mathbb{Z}_2$. For the second connected component of $\mathrm{SO}(3,\mathbb{C})$ we start with:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\node[D5] at (1,2) {};
\node[D5] at (3,2) {};
\end{tikzpicture}}}\qquad,
\end{equation}

depicting the origin of $\w{\mathrm{SO}(3,\mathbb{C})}{}{[1]}$
The transverse slice for the $k$-th transition is $A_{2k}$ and the brane system and quiver for the leaf are:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[orange] (2,0)--(2,4);
\node[D5] (1) at (0.5,2) {};
\node at (1,2.3) {$2k+1$};
\node at (1,2) {$\cdots$};
\node[D5] (2) at (1.5,2) {};
\node[D5] (3) at (2.5,2) {};
\node at (3,2.3) {$2k+1$};
\node at (3,2) {$\cdots$};
\node[D5] (4) at (3.5,2) {};
\draw[double] (0,1)--(4,1);
\node at (1,0.75) {$\vdots$ $k$};
\node at (3,0.75) {$\vdots$ $k$};
\draw[double] (0,0.3)--(4,0.3);
\end{tikzpicture}}}
\qquad
\vcenter{\hbox{
\begin{tikzpicture}
\node[gauge,label=below:{$k$}] (1) at (0,0) {};
\node[flavour,label=above:{$2k+1$}] (2) at (0,1) {};
\draw (1)--(2);
\end{tikzpicture}
}} \, ,
\label{eq:B1_2nd}
\end{equation}

which is again in agreement with Section <ref>.

§.§.§ The B2 affine Grassmannian

Component $0\in\mathbb{Z}_2$. This is depicted in Figure <ref>.

Component $1\in\mathbb{Z}_2$. This is depicted in Figure <ref>.

(00) at (0*,0*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\end{tikzpicture}}$;
(01) at (0*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (3,1.5) {};
\draw (0,0.5)--(2,0.5);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\end{tikzpicture}}$;
(20) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1.5,1.5) {};
\draw (0,0.5)--(2,0.5) (0,0.8)--(2,0.8);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\end{tikzpicture}}$;
(02) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (2.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\draw (0,0.5)--(2,0.5) (0,0.8)--(2,0.8);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\draw (2,0.6) .. controls (4.7,0.5) .. (2,0.4);
\end{tikzpicture}}$;
(21) at (1*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1.5,1.5) {};
\node[D5] at (3,1.5) {};
\draw (0,0.5)--(2,0.5) (0,0.8)--(2,0.8) (0,1.1)--(2,1.1);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\draw (2,0.6) .. controls (4.7,0.5) .. (2,0.4);
\end{tikzpicture}}$;

[draw,label=left:$b_2$] (0001) at ($(00)!0.5!(01)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5o] (2) at (8.5,1.5) {};
\draw[olive,double,thick] (1)--(0,1.5) (2,0.3)--(6,0.3) (8,1.5)--(2);
\draw[olive,thick] (0,1.5)--(2,1.5) (0,0.5)--(2,0.5) (6,0.5)--(8,0.5) (8,1.5)--(6,1.5);
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (0120) at ($(01)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5] (2) at (3,1.5) {};
\node[D5] (3) at (5,1.5) {};
\node[D5o] (4) at (8.5,1.5) {};
\draw[olive,thick] (1)--(0,1.5) (0,0.8)--(2,0.8) (2,1.5)--(2) (3)--(6,1.5) (6,0.8)--(8,0.8) (8,1.5)--(4);
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[double] (2,0.3)--(6,0.3);
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (2002) at ($(20)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5] (1) at (0.5,1.6) {};
\node[D5] (2) at (1.5,1.4) {};
\node[D5] (3) at (6.5,1.4) {};
\node[D5] (4) at (7.5,1.6) {};
\draw[olive,thick] (1)--(2,1.6) (2)--(2,1.4) (6,1.4)--(3) (6,1.6)--(4);
\draw[olive,thick,double] (2,0.6)--(6,0.6);
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5) (0,0.8)--(2,0.8) (6,0.8)--(8,0.8);
\draw[double] (2,0.3)--(6,0.3);
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (0221) at ($(02)!0.5!(21)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5] (2) at (2.5,1.5) {};
\node[D5] (3) at (5.5,1.5) {};
\node[D5o] (4) at (8.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\node[D5] at (4.5,1.5) {};
\draw[olive,thick] (1)--(0,1.5) (2)--(2,1.5) (6,1.5)--(3) (8,1.5)--(4) (0,1.1)--(2,1.1) (6,1.1)--(8,1.1);
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5) (0,0.8)--(2,0.8) (6,0.8)--(8,0.8);
\draw[double] (2,0.3)--(6,0.3) (2,0.6)--(6,0.6);
\end{tikzpicture}}$;

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{SO}(5,\mathbb{C})}{}{[0,0]}$.

(00) at (0*,0*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\end{tikzpicture}}$;
(01) at (0*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (3,1.5) {};
\node[D5] at (1,1.5) {};
\draw (0,0.5)--(2,0.5);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\end{tikzpicture}}$;
(20) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (1.5,1.5) {};
\draw (0,0.5)--(2,0.5) (0,0.8)--(2,0.8);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\end{tikzpicture}}$;
(02) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\node[D5] at (2.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\draw (0,0.5)--(2,0.5) (0,0.8)--(2,0.8);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\draw (2,0.6) .. controls (4.7,0.5) .. (2,0.4);
\end{tikzpicture}}$;
(21) at (1*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (1.5,1.5) {};
\node[D5] at (3,1.5) {};
\draw (0,0.5)--(2,0.5) (0,0.8)--(2,0.8) (0,1.1)--(2,1.1);
\draw (2,0.3) .. controls (4.7,0.2) .. (2,0.1);
\draw (2,0.6) .. controls (4.7,0.5) .. (2,0.4);
\end{tikzpicture}}$;

[draw,label=left:$ac_2$] (0001) at ($(00)!0.5!(01)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (1,1.8) {};
\node[D5] at (7,1.8) {};
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5o] (2) at (8.5,1.5) {};
\draw[olive,double,thick] (1)--(0,1.5) (2,0.3)--(6,0.3) (8,1.5)--(2);
\draw[olive,thick] (0,1.5)--(2,1.5) (0,0.5)--(2,0.5) (6,0.5)--(8,0.5) (8,1.5)--(6,1.5);
\end{tikzpicture}}$;
[draw,label=below:$A_2$] (0120) at ($(01)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5] at (1,1.8) {};
\node[D5] at (7,1.8) {};
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5] (2) at (3,1.5) {};
\node[D5] (3) at (5,1.5) {};
\node[D5o] (4) at (8.5,1.5) {};
\draw[olive,thick] (1)--(0,1.5) (0,0.8)--(2,0.8) (2,1.5)--(2) (3)--(6,1.5) (6,0.8)--(8,0.8) (8,1.5)--(4);
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[double] (2,0.3)--(6,0.3);
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (2002) at ($(20)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5] (1) at (1,1.6) {};
\node[D5] (2) at (1.5,1.4) {};
\node[D5] at (0.5,1.5) {};
\node[D5] at (7.5,1.5) {};
\node[D5] (3) at (6.5,1.4) {};
\node[D5] (4) at (7,1.6) {};
\draw[olive,thick] (1)--(2,1.6) (2)--(2,1.4) (6,1.4)--(3) (6,1.6)--(4);
\draw[olive,thick,double] (2,0.6)--(6,0.6);
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5) (0,0.8)--(2,0.8) (6,0.8)--(8,0.8);
\draw[double] (2,0.3)--(6,0.3);
\end{tikzpicture}}$;
[draw,label=below:$A_2$] (0221) at ($(02)!0.5!(21)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[orange] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5] (2) at (2.5,1.5) {};
\node[D5] (3) at (5.5,1.5) {};
\node[D5o] (4) at (8.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (7,1.5) {};
\node[D5] at (3.5,1.5) {};
\node[D5] at (4.5,1.5) {};
\draw[olive,thick] (1)--(0,1.5) (2)--(2,1.5) (6,1.5)--(3) (8,1.5)--(4) (0,1.1)--(2,1.1) (6,1.1)--(8,1.1);
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5) (0,0.8)--(2,0.8) (6,0.8)--(8,0.8);
\draw[double] (2,0.3)--(6,0.3) (2,0.6)--(6,0.6);
\end{tikzpicture}}$;

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{SO}(5,\mathbb{C})}{}{[1,0]}$.

§.§ The C-type affine Grassmannian

For this setup, we require a ${\color{green}\textnormal{ON}^+}$ plane. In the main text we draw the full covering space and refer to half branes simply as branes.

Component $0\in\mathbb{Z}_2$. The brane system for the affine Grassmannian of $\mathrm{Sp}(2n,\mathbb{C})$, i.e. the $0$-th component of the affine Grassmannian of $ \mathrm{PSp}(2n,\mathbb{C})$, is proposed to be

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[green] (2,0)--(2,4);
\node at (2,-0.5) {$\textnormal{ON}^+$};
\end{tikzpicture}
}} \, .
\end{equation}

Depicted is the origin of $\w{\mathrm{PSp}(2n,\mathbb{C})}{}{[0,\dots,0,0]}$, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{\begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=below:{$0$}] (g4) at (4,0) {};
\node[flavour,label=above:{$0$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=above:{$0$}] (f4) at (4,1) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4);
\draw (g1)--(g2)--(g)--(g3);
\draw[transform canvas={yshift=-2pt}] (g3)--(g4);
\draw[transform canvas={yshift=2pt}] (g3)--(g4);
\draw (3.55,0.2)--(3.45,0)--(3.55,-0.2);
\end{tikzpicture}} \, .
\end{equation}

Component $1\in\mathbb{Z}_2$. The brane system for the $1$-st component of the affine Grassmannian of $ \mathrm{PSp}(2n,\mathbb{C})$, is proposed to be
\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[green] (2,0)--(2,4);
\node at (2,-0.5) {$\textnormal{ON}^+$};
\node[D5] at (2,2) {};
\end{tikzpicture}
}}  \, .
\end{equation}

Depicted is the origin of $\w{\mathrm{PSp}(2n,\mathbb{C})}{}{[0,\dots,0,1]}$, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{ \begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=below:{$0$}] (g4) at (4,0) {};
\node[flavour,label=above:{$0$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=above:{$1$}] (f4) at (4,1) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4);
\draw (g1)--(g2)--(g)--(g3);
\draw[transform canvas={yshift=-2pt}] (g3)--(g4);
\draw[transform canvas={yshift=2pt}] (g3)--(g4);
\draw (3.55,0.2)--(3.45,0)--(3.55,-0.2);
\end{tikzpicture}}
\end{equation}

Minimal transitions. In the $C$-type brane systems, we may perform the same transitions as in the $A$-type case. Additionally there are two more minimal transitions:

\begin{equation}
\vcenter{\hbox{
% \scalebox{0.70}{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-1,0)--(-1,4) (0,0)--(0,4) (4,0)--(4,4) (5,0)--(5,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (4.5,2) {$\cdots$};
\draw[green] (2,0)--(2,4);
\node[D5] (1) at (-4,2) {};
\node[D5] (2) at (8,2) {};
\draw (1)--(-3,2) (-3,1.5)--(-1,1.5) (0,1)--(4,1) (5,1.5)--(7,1.5) (7,2)--(2);
\draw [decorate,decoration={brace,amplitude=5pt}] (0.1,-0.25)--(-3.1,-0.25);
\node at (-1.5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(3.9,-0.25);
\node at (5.5,-0.6) {$n$};
\node at (8,3.5) {$c_n$};
\end{tikzpicture}
% }
% \qquad.
\end{equation}

\begin{equation}
\vcenter{\hbox{
% \scalebox{0.70}{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-1,0)--(-1,4) (0,0)--(0,4) (4,0)--(4,4) (5,0)--(5,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (4.5,2) {$\cdots$};
\draw[green] (2,0)--(2,4);
\node[D5] (1) at (-4,2) {};
\node[D5] at (2,2) {};
\node[D5] (2) at (8,2) {};
\draw (1)--(-3,2) (-3,1.5)--(-1,1.5) (0,1)--(4,1) (5,1.5)--(7,1.5) (7,2)--(2);
\draw [decorate,decoration={brace,amplitude=5pt}] (0.1,-0.25)--(-3.1,-0.25);
\node at (-1.5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(3.9,-0.25);
\node at (5.5,-0.6) {$n$};
\node at (8,3.5) {$ac_n$};
\end{tikzpicture}
% }
% \qquad.
\end{equation}

For $n=1$ there are transitions corresponding to Kleinian singularities studied in detail in the next section.

§.§.§ The C1 affine Grassmannian

Component $0 \in \mathbb{Z}_2$. The affine Grassmannian of $ \mathrm{PSp}(2,\mathbb{C})$ is given by the brane system:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[green] (2,0)--(2,4);
\end{tikzpicture}}}\qquad,
\end{equation}

depicted at the origin of $\w{\mathrm{PSp}(2,\mathbb{C})}{}{[0]}$. We can now move onto the lowest non-trivial leaf by activating a D3 brane between two D5 branes, which are mirror images, and moving the D3 brane along the NS5 branes, corresponding to a $A_1$ transition:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[green] (2,0)--(2,4);
\node[D5] (1) at (-1,2) {};
\node[D5] (2) at (5,2) {};
\draw (1)--(0,2) (4,2)--(2) (0,1)--(4,1);
\end{tikzpicture}}}\qquad.
\end{equation}

After a Hanany-Witten transition we can read off the quiver:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[green] (2,0)--(2,4);
\node[D5] (1) at (1,2) {};
\node[D5] (2) at (3,2) {};
\draw (0,1)--(4,1);
\end{tikzpicture}}}
\qquad
\vcenter{\hbox{
\begin{tikzpicture}
\node[gauge,label=below:{$1$}] (1) at (0,0) {};
\node[flavour,label=above:{$2$}] (2) at (0,1) {};
\draw (1)--(2);
\end{tikzpicture}
\end{equation}

The only way to keep moving onto higher leaves in the affine Grassmannian is by repeating the same operation, however the D5 branes already placed between the NS5 branes make the transverse slices $A_{2k-1}$ for the $k$-th transition of this type. A general leaf and the corresponding quiver are:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[green] (2,0)--(2,4);
\node[D5] (1) at (0.5,2) {};
\node at (1,2.3) {$k$};
\node at (1,2) {$\cdots$};
\node[D5] (2) at (1.5,2) {};
\node[D5] (3) at (2.5,2) {};
\node at (3,2.3) {$k$};
\node at (3,2) {$\cdots$};
\node[D5] (4) at (3.5,2) {};
\draw (0,1)--(4,1);
\node at (1,0.75) {$\vdots$ $k$};
\node at (3,0.75) {$\vdots$ $k$};
\draw (0,0.3)--(4,0.3);
\end{tikzpicture}}}
\qquad
\vcenter{\hbox{
\begin{tikzpicture}
\node[gauge,label=below:{$k$}] (1) at (0,0) {};
\node[flavour,label=above:{$2k$}] (2) at (0,1) {};
\draw (1)--(2);
\end{tikzpicture}
\label{eq:C1_1st}
\end{equation}

This is in agreement with the construction in both Section <ref> and <ref>.

Component $1 \in \mathbb{Z}_2$. For the second connected component for $\mathrm{PSp}(2,\mathbb{C})$ we start with:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[green] (2,0)--(2,4);
\node[D5] at (2,2) {};
\end{tikzpicture}}}\qquad.
\end{equation}

Depicting the origin of $\w{\mathrm{PSp}(2,\mathbb{C})}{}{[1]}$. The transverse slice for the $k$-th transition is $A_{2k}$ and the brane system and quiver for the leaf are:

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (0,0)--(0,4) (4,0)--(4,4);
\draw[green] (2,0)--(2,4);
\node[D5] at (2,2) {};
\node[D5] (1) at (0.5,2) {};
\node at (1,2.3) {$k$};
\node at (1,2) {$\cdots$};
\node[D5] (2) at (1.5,2) {};
\node[D5] (3) at (2.5,2) {};
\node at (3,2.3) {$k$};
\node at (3,2) {$\cdots$};
\node[D5] (4) at (3.5,2) {};
\draw (0,1)--(4,1);
\node at (1,0.75) {$\vdots$ $k$};
\node at (3,0.75) {$\vdots$ $k$};
\draw (0,0.3)--(4,0.3);
\end{tikzpicture}}}
\qquad
\vcenter{\hbox{
\begin{tikzpicture}
\node[gauge,label=below:{$k$}] (1) at (0,0) {};
\node[flavour,label=above:{$2k+1$}] (2) at (0,1) {};
\draw (1)--(2);
\end{tikzpicture}
\label{eq:C1_2nd}
\end{equation}

This is in agreement with the construction in both Section <ref> and <ref>.

§.§.§ The C2 affine Grassmannian

The two components are represented in Figures <ref> and <ref>.

(00) at (0*,0*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\end{tikzpicture}}$;
(10) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(4,0.3);
\end{tikzpicture}}$;
(02) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (3,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6);
\end{tikzpicture}}$;
(20) at (2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1.5,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6);
\end{tikzpicture}}$;
(12) at (1*,3*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\node[D5] at (3,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9);
\end{tikzpicture}}$;
(30) at (3*,3*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (1.5,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (0,1)--(2,1) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9);
\end{tikzpicture}}$;
(04) at (0*,4*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (2.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9) (2,1.2)--(4,1.2);
\end{tikzpicture}}$;
(22) at (2*,4*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1.5,1.5) {};
\node[D5] at (3,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (0,1)--(2,1) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9) (2,1.2)--(4,1.2);
\end{tikzpicture}}$;

[draw,label=left:$c_2$] (0010) at ($(00)!0.5!(10)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5o] (2) at (8.5,1.5) {};
\draw[olive,thick] (1)--(0,1.5) (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (8,1.5)--(2);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (1002) at ($(10)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] (1) at (1,1.5) {};
\node[D5] (2) at (7,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4);
\draw[olive,thick] (1)--(2,1.5) (2,0.7)--(6,0.7) (6,1.5)--(2);
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (0220) at ($(02)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,1.5) {};
\node[D5o] (2r) at (8.5,1.5) {};
\node[D5] (1) at (3,1.5) {};
\node[D5] (2) at (5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6);
\draw[olive,thick] (1l)--(0,1.5) (0,0.7)--(2,0.7) (2,1.5)--(1) (2)--(6,1.5) (6,0.7)--(8,0.7) (8,1.5)--(2r);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (2012) at ($(20)!0.5!(12)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] at (0.5,1.5) {};
\node[D5] at (7.5,1.5) {};
\node[D5] (1) at (1.5,1.5) {};
\node[D5] (2) at (6.5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7);
\draw[olive,thick] (1)--(2,1.5) (2,0.9)--(6,0.9) (6,1.5)--(2);
\end{tikzpicture}}$;
[draw,label=below:$A_2$] (1230) at ($(12)!0.5!(30)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,1.5) {};
\node[D5o] (2r) at (8.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (7,1.5) {};
\node[D5] (1) at (3,1.5) {};
\node[D5] (2) at (5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9);
\draw[olive,thick] (1l)--(0,1.5) (0,1)--(2,1) (2,1.5)--(1) (2)--(6,1.5) (6,1)--(8,1) (8,1.5)--(2r);
\end{tikzpicture}}$;
[draw,label=left:$A_3$] (1204) at ($(12)!0.5!(04)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] (1) at (1,1.5) {};
\node[D5] (2) at (7,1.5) {};
\node[D5] at (3,1.5) {};
\node[D5] at (5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9);
\draw[olive,thick] (1)--(2,1.5) (2,1.2)--(6,1.2) (6,1.5)--(2);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (3022) at ($(30)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] at (0.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] (1) at (1.5,1.5) {};
\node[D5] (2) at (6.5,1.5) {};
\node[D5] at (7,1.5) {};
\node[D5] at (7.5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9) (0,1)--(2,1) (6,1)--(8,1);
\draw[olive,thick] (1)--(2,1.5) (2,1.2)--(6,1.2) (6,1.5)--(2);
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (0422) at ($(04)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,1.5) {};
\node[D5o] (2r) at (8.5,1.5) {};
\node[D5] (1) at (2.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\node[D5] at (5.5,1.5) {};
\node[D5] (2) at (4.5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9) (2,1.2)--(6,1.2);
\draw[olive,thick] (1l)--(0,1.5) (0,1)--(2,1) (2,1.5)--(1) (2)--(6,1.5) (6,1)--(8,1) (8,1.5)--(2r);
\end{tikzpicture}}$;
(22)–(2*-0.3*,4*+0.3*) (22)–(2*+0.7*,4*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{ \mathrm{PSp}(4,\mathbb{C})}{}{[0,0]}$.

(00) at (0*,0*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(10) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(4,0.3);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(02) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (3,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(20) at (2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1.5,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(12) at (1*,3*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\node[D5] at (3,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(30) at (3*,3*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (1.5,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (0,1)--(2,1) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(04) at (0*,4*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (2.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9) (2,1.2)--(4,1.2);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(22) at (2*,4*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[green] (4,0)--(4,3);
\node[D5] at (0.5,1.5) {};
\node[D5] at (1.5,1.5) {};
\node[D5] at (3,1.5) {};
\draw (0,0.4)--(2,0.4) (0,0.7)--(2,0.7) (0,1)--(2,1) (2,0.3)--(4,0.3) (2,0.6)--(4,0.6) (2,0.9)--(4,0.9) (2,1.2)--(4,1.2);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;

[draw,label=left:$ac_2$] (0010) at ($(00)!0.5!(10)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,1.5) {};
\node[D5o] (2) at (8.5,1.5) {};
\draw[olive,thick] (1)--(0,1.5) (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (8,1.5)--(2);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (1002) at ($(10)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] (1) at (1,1.5) {};
\node[D5] (2) at (7,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4);
\draw[olive,thick] (1)--(2,1.5) (2,0.7)--(6,0.7) (6,1.5)--(2);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (0220) at ($(02)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,1.5) {};
\node[D5o] (2r) at (8.5,1.5) {};
\node[D5] (1) at (3,1.5) {};
\node[D5] (2) at (5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6);
\draw[olive,thick] (1l)--(0,1.5) (0,0.7)--(2,0.7) (2,1.5)--(1) (2)--(6,1.5) (6,0.7)--(8,0.7) (8,1.5)--(2r);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (2012) at ($(20)!0.5!(12)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] at (0.5,1.5) {};
\node[D5] at (7.5,1.5) {};
\node[D5] (1) at (1.5,1.5) {};
\node[D5] (2) at (6.5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7);
\draw[olive,thick] (1)--(2,1.5) (2,0.9)--(6,0.9) (6,1.5)--(2);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=below:$A_2$] (1230) at ($(12)!0.5!(30)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,1.5) {};
\node[D5o] (2r) at (8.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] at (7,1.5) {};
\node[D5] (1) at (3,1.5) {};
\node[D5] (2) at (5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9);
\draw[olive,thick] (1l)--(0,1.5) (0,1)--(2,1) (2,1.5)--(1) (2)--(6,1.5) (6,1)--(8,1) (8,1.5)--(2r);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_4$] (1204) at ($(12)!0.5!(04)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] (1) at (1,1.5) {};
\node[D5] (2) at (7,1.5) {};
\node[D5] at (3,1.5) {};
\node[D5] at (5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9);
\draw[olive,thick] (1)--(2,1.5) (2,1.2)--(6,1.2) (6,1.5)--(2);
\node[D5] at (4,1.5) {};
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (3022) at ($(30)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node at (-0.5,1.5) {};
\node at (8.5,1.5) {};
\node[D5] at (0.5,1.5) {};
\node[D5] at (1,1.5) {};
\node[D5] (1) at (1.5,1.5) {};
\node[D5] (2) at (6.5,1.5) {};
\node[D5] at (7,1.5) {};
\node[D5] at (7.5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9) (0,1)--(2,1) (6,1)--(8,1);
\draw[olive,thick] (1)--(2,1.5) (2,1.2)--(6,1.2) (6,1.5)--(2);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
[draw,label=below:$A_1$] (0422) at ($(04)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[green] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,1.5) {};
\node[D5o] (2r) at (8.5,1.5) {};
\node[D5] (1) at (2.5,1.5) {};
\node[D5] at (3.5,1.5) {};
\node[D5] at (5.5,1.5) {};
\node[D5] (2) at (4.5,1.5) {};
\draw (0,0.4)--(2,0.4) (2,0.3)--(6,0.3) (6,0.4)--(8,0.4) (2,0.6)--(6,0.6) (0,0.7)--(2,0.7) (6,0.7)--(8,0.7) (2,0.9)--(6,0.9) (2,1.2)--(6,1.2);
\draw[olive,thick] (1l)--(0,1.5) (0,1)--(2,1) (2,1.5)--(1) (2)--(6,1.5) (6,1)--(8,1) (8,1.5)--(2r);
\node[D5] at (4,1.5) {};
\end{tikzpicture}}$;
(22)–(2*-0.3*,4*+0.3*) (22)–(2*+0.7*,4*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{ \mathrm{PSp}(4,\mathbb{C})}{}{[0,1]}$.

§.§ The D-type affine Grassmannian

For this setup, we require a ${\color{cyan}\textnormal{ON}^-}$ plane. In the main text we draw the full covering space and refer to half branes simply as branes.

Component $(0,0)\in\mathbb{Z}_2\times\mathbb{Z}_2$ or $0\in\mathbb{Z}_4$. The brane system for the affine Grassmannian of $\mathrm{Spin}(2n,\mathbb{C})$, i.e. the $(0,0)$-th (for $n$ even) or $0$-th component (for $n$ odd) of the affine Grassmannian of $ \mathrm{PSO}(2n,\mathbb{C})$, is proposed to be

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[cyan] (2,0)--(2,4);
\node at (2,-0.5) {$\textnormal{ON}^-$};
\end{tikzpicture}
\end{equation}

Depicted is at the origin of $\w{\mathrm{Spin}(2n,\mathbb{C})}{}{[0,0,\dots,0,0,0]}$, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{ \begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=above:{$0$}] (g4) at (4,0.5) {};
\node[gauge,label=below:{$0$}] (g5) at (4,-0.5) {};
\node[flavour,label=above:{$0$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=right:{$0$}] (f4) at (5,0.5) {};
\node[flavour,label=right:{$0$}] (f5) at (5,-0.5) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4) (f5)--(g5);
\draw (g1)--(g2)--(g)--(g3)--(g4) (g3)--(g5);
\end{tikzpicture}} \, .
\end{equation}

There are three more components.

Component $(1,1)\in\mathbb{Z}_2\times\mathbb{Z}_2$ or $2\in\mathbb{Z}_4$.

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[cyan] (2,0)--(2,4);
\node at (2,-0.5) {$\textnormal{ON}^-$};
\node[D5] at (-2.5,2) {};
\node[D5] at (6.5,2) {};
\end{tikzpicture}
\end{equation}

Depicted at the origin of $\w{\mathrm{Spin}(2n,\mathbb{C})}{}{[1,0,\dots,0,0,0]}$, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{ \begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=above:{$0$}] (g4) at (4,0.5) {};
\node[gauge,label=below:{$0$}] (g5) at (4,-0.5) {};
\node[flavour,label=above:{$1$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=right:{$0$}] (f4) at (5,0.5) {};
\node[flavour,label=right:{$0$}] (f5) at (5,-0.5) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4) (f5)--(g5);
\draw (g1)--(g2)--(g)--(g3)--(g4) (g3)--(g5);
\end{tikzpicture}} \, .
\end{equation}

Component $(1,0)\in\mathbb{Z}_2\times\mathbb{Z}_2$ or $1\in\mathbb{Z}_4$.

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[cyan] (2,0)--(2,4);
\node at (2,-0.5) {$\textnormal{ON}^-$};
\node[D5] (d) at (2,2) {};
\draw (1,2)--(d)--(3,2);
\end{tikzpicture}
\end{equation}

Depicted at the origin of $\w{\mathrm{Spin}(2n,\mathbb{C})}{}{[0,0,\dots,0,1,0]}$, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{ \begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=above:{$0$}] (g4) at (4,0.5) {};
\node[gauge,label=below:{$0$}] (g5) at (4,-0.5) {};
\node[flavour,label=above:{$0$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=right:{$1$}] (f4) at (5,0.5) {};
\node[flavour,label=right:{$0$}] (f5) at (5,-0.5) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4) (f5)--(g5);
\draw (g1)--(g2)--(g)--(g3)--(g4) (g3)--(g5);
\end{tikzpicture}} \, .
\end{equation}

Component $(0,1)\in\mathbb{Z}_2\times\mathbb{Z}_2$ or $3\in\mathbb{Z}_4$.

\begin{equation}
\vcenter{\hbox{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-2,0)--(-2,4) (1,0)--(1,4) (3,0)--(3,4) (6,0)--(6,4) (7,0)--(7,4);
\node at (-0.5,2) {$\cdots$};
\node at (-1,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (1.1,-0.25)--(-3.1,-0.25);
\node at (4.5,2) {$\cdots$};
\node at (5,-0.6) {$n$};
\draw [decorate,decoration={brace,amplitude=5pt}] (7.1,-0.25)--(2.9,-0.25);
\draw[cyan] (2,0)--(2,4);
\node at (2,-0.5) {$\textnormal{ON}^-$};
\node[D5] at (2,2) {};
\end{tikzpicture}
\end{equation}

Depicted at the origin of $\w{\mathrm{Spin}(2n,\mathbb{C})}{}{[0,0,\dots,0,0,1]}$, with all D3 branes and D5 branes suppressed. The quiver representing the origin is:

\begin{equation}
\raisebox{-.5\height}{ \begin{tikzpicture}
\node[gauge,label=below:{$0$}] (g1) at (0,0) {};
\node[gauge,label=below:{$0$}] (g2) at (1,0) {};
\node (g) at (2,0) {$\cdots$};
\node[gauge,label=below:{$0$}] (g3) at (3,0) {};
\node[gauge,label=above:{$0$}] (g4) at (4,0.5) {};
\node[gauge,label=below:{$0$}] (g5) at (4,-0.5) {};
\node[flavour,label=above:{$0$}] (f1) at (0,1) {};
\node[flavour,label=above:{$0$}] (f2) at (1,1) {};
\node[flavour,label=above:{$0$}] (f3) at (3,1) {};
\node[flavour,label=right:{$0$}] (f4) at (5,0.5) {};
\node[flavour,label=right:{$1$}] (f5) at (5,-0.5) {};
\draw (f1)--(g1) (f2)--(g2) (f3)--(g3) (f4)--(g4) (f5)--(g5);
\draw (g1)--(g2)--(g)--(g3)--(g4) (g3)--(g5);
\end{tikzpicture}} \, .
\end{equation}

Minimal transitions. In the $D$-type brane systems, we may perform the same transitions as in the $A$-type case. Additionally there is one more minimal transition:

\begin{equation}
\vcenter{\hbox{
% \scalebox{0.70}{
\begin{tikzpicture}
\draw[red] (-3,0)--(-3,4) (-1,0)--(-1,4) (0,0)--(0,4) (2,0)--(2,4) (6,0)--(6,4) (8,0)--(8,4) (9,0)--(9,4) (11,0)--(11,4);
\node at (-0.5,2) {$\cdots$};
\node at (-0.5,2.3) {$n$};
\node at (8.5,2) {$\cdots$};
\node at (8.5,2.3) {$n$};
\draw[cyan] (4,0)--(4,4);
\node[D5] (1) at (-4,2) {};
\node[D5] (2) at (12,2) {};
\draw[transform canvas={yshift=-1pt}] (1)--(-3,2) (9,2)--(2);
\draw[transform canvas={yshift=1pt}] (1)--(-1,2) (11,2)--(2);
\draw[transform canvas={yshift=-1pt}] (0,1.5)--(6,1.5);
\draw[transform canvas={yshift=1pt}] (2,1.5)--(8,1.5);
\draw (-3,1)--(-1,1) (9,1)--(11,1) (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\node at (12,3.5) {$d_n$};
\end{tikzpicture}
% }
% \qquad.
\end{equation}

§.§.§ The D2 affine Grassmannian

There are four components:

Component $(0,0) \in \mathbb{Z}_2\times\mathbb{Z}_2$. This component is studied in Figure <ref>. The origin is

\begin{equation}
\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\end{tikzpicture}
\end{equation}

Component $(0,1) \in \mathbb{Z}_2\times\mathbb{Z}_2$. This component is studied in Figure <ref>. The origin is

\begin{equation}
\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (4,1.5) {};
\end{tikzpicture}
\end{equation}

Component $(1,0) \in \mathbb{Z}_2\times\mathbb{Z}_2$. This component is studied in Figure <ref>. The origin is

\begin{equation}
\raisebox{-.5\height}{ \begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (4,1.5) {};
\draw (2,1.5)--(1)--(6,1.5);
\end{tikzpicture}}
\end{equation}

Component $(1,1) \in \mathbb{Z}_2\times\mathbb{Z}_2$. This component is studied in Figure <ref>. The origin is

\begin{equation}
\raisebox{-.5\height}{ \begin{adjustbox}{center}
\scalebox{0.7}{
\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (4,2) {};
\node[D5] (1) at (4,1.5) {};
\draw (2,1.5)--(1)--(6,1.5);
\node at (9.5,2) {$HW$};
\node at (9.5,1.5) {$\longleftrightarrow$};
\begin{scope}[shift={(11,0)}]
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (1,1.5) {};
\node[D5] at (7,1.5) {};
\end{scope}
\end{tikzpicture}}
\end{adjustbox}}
\end{equation}

Transition. While not strictly a Hanany-Witten transition, the following two brane webs are equivalent in that they depict the same leaf in the Coulomb branch:

\begin{equation}
\vcenter{\hbox{\scalebox{1}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1.3)--(2,1.3);
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\end{tikzpicture}}}}\sim \qquad
\vcenter{\hbox{\scalebox{1}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (0.5,2) {};
\node[D5] at (1.5,2) {};
\draw (0,1.3)--(2,1.3);
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\end{tikzpicture}}}}\qquad
\vcenter{\hbox{\scalebox{1}{\begin{tikzpicture}
\node[gauge,label=left:{$1$}] (g1) at (0,0) {};
\node[gauge,label=left:{$1$}] (g2) at (0,1) {};
\node[flavour,label=right:{$2$}] (f1) at (1,0) {};
\node[flavour,label=right:{$2$}] (f2) at (1,1) {};
\draw (g1)--(f1) (g2)--(f2);
\end{tikzpicture}}}}
\label{eq:D2choices}
\end{equation}

In Figure <ref> we use the representation on the left. Furthermore the following two transverse slices are equivalent and correspond to an $A_1$ transition:

\begin{equation}
\begin{adjustbox}{center}\raisebox{-0.5\height}{\scalebox{0.8}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,2) {};
\node[D5o] (2r) at (8.5,2) {};
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=-1pt}] (2,1)--(8,1);
\draw[transform canvas={yshift=1pt}] (0,1)--(6,1);
\draw[olive,thick,transform canvas={yshift=-1pt}] (1l)--(0,2);
\draw[olive,thick,transform canvas={yshift=1pt}] (1l)--(6,2);
\draw[olive,thick,transform canvas={yshift=1pt}] (2r)--(8,2);
\draw[olive,thick,transform canvas={yshift=-1pt}] (2r)--(2,2);
\draw[olive,thick] (0,1.5)--(2,1.5) (6,1.5)--(8,1.5);
\end{tikzpicture}}}
$\quad \sim \quad$
\raisebox{-0.5\height}{{\scalebox{0.8}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1l) at (-0.5,2) {};
\node[D5o] (2r) at (8.5,2) {};
\node[D5] (1) at (3,2) {};
\node[D5] (2) at (5,2) {};
\draw[transform canvas={yshift=-1pt}] (2,1)--(8,1);
\draw[transform canvas={yshift=1pt}] (0,1)--(6,1);
\draw[olive,thick] (1l)--(0,2) (2,2)--(1) (2)--(6,2) (8,2)--(2r);
\draw[olive,thick] (0,1.5)--(2,1.5) (6,1.5)--(8,1.5);
\end{tikzpicture}}}}
\end{adjustbox}
\label{eq:D2choices2}
\end{equation}

The only difference between the two brane systems in (<ref>) and (<ref>) lies in their Higgs branch.

(00) at (0,0) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\end{tikzpicture}}$;
(20) at (-1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\end{tikzpicture}}$;
(02) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1)--(2,1);
\end{tikzpicture}}$;
(40) at (-2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (2.5,2.5) {};
\node[D5] at (3.5,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\draw (2,1.4) .. controls (4.9,1.3) .. (0,1.1);
\end{tikzpicture}}$;
(22) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1.2)--(2,1.2);
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\end{tikzpicture}}$;
(04) at (2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\node[D5] (2) at (3,1.6) {};
\draw (2,1.6)--(2);
\draw (2) .. controls (4.45,1.5) .. (2,1.4);
\draw (0,1)--(2,1) (0,0.7)--(2,0.7);
\end{tikzpicture}}$;

[draw,label=left:$A_1$] (0020) at ($(00)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (0002) at ($(00)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick] (0,1)--(2,1) (6,1)--(8,1);
\end{tikzpicture}}$;
[draw,label=left:$A_3$] (2040) at ($(20)!0.5!(40)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1.2)--(6,1.2);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (2022) at ($(20)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1.2)--(2,1.2);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (0222) at ($(02)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\end{tikzpicture}}$;
[draw,label=left:$A_3$] (0204) at ($(02)!0.5!(04)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1)--(2,1);
\end{tikzpicture}}$;
(00)–(0020)–(20)–(2040)–(40) (20)–(2022)–(22) (00)–(0002)–(02)–(0204)–(04) (02)–(0222)–(22);
(40)–(-2*-0.4*,2*+0.4*) (40)–(-2*+0.4*,2*+0.4*) (22)–(-0.4*,2*+0.4*) (22)–(0.4*,2*+0.4*) (04)–(2*-0.4*,2*+0.4*) (04)–(2*+0.4*,2*+0.4*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{PSO}(4,\mathbb{C})}{}{[0,0]}$.

(00) at (0,0) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
(20) at (-1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
(02) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1)--(2,1);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
(40) at (-2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (2.5,2.5) {};
\node[D5] at (3.5,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\draw (2,1.4) .. controls (4.9,1.3) .. (0,1.1);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
(22) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1.2)--(2,1.2);
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
(04) at (2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\node[D5] (2) at (3,1.6) {};
\draw (2,1.6)--(2);
\draw (2) .. controls (4.45,1.5) .. (2,1.4);
\draw (0,1)--(2,1) (0,0.7)--(2,0.7);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;

[draw,label=left:$A_2$] (0020) at ($(00)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (0002) at ($(00)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick] (0,1)--(2,1) (6,1)--(8,1);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_4$] (2040) at ($(20)!0.5!(40)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1.2)--(6,1.2);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (2022) at ($(20)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1.2)--(2,1.2);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (0222) at ($(02)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
[draw,label=left:$A_3$] (0204) at ($(02)!0.5!(04)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1)--(2,1);
\node[D5] at (4,2.5) {};
\end{tikzpicture}}$;
(00)–(0020)–(20)–(2040)–(40) (20)–(2022)–(22) (00)–(0002)–(02)–(0204)–(04) (02)–(0222)–(22);
(40)–(-2*-0.4*,2*+0.4*) (40)–(-2*+0.4*,2*+0.4*) (22)–(-0.4*,2*+0.4*) (22)–(0.4*,2*+0.4*) (04)–(2*-0.4*,2*+0.4*) (04)–(2*+0.4*,2*+0.4*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{PSO}(4,\mathbb{C})}{}{[0,1]}$.

(00) at (0,0) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (m) at (4,2.2) {};
\draw (2,2.2)--(m);
\end{tikzpicture}}$;
(20) at (-1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(02) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1)--(2,1);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(40) at (-2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (2.5,2.5) {};
\node[D5] at (3.5,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\draw (2,1.4) .. controls (4.9,1.3) .. (0,1.1);
\end{tikzpicture}}$;
(22) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1.2)--(2,1.2);
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(04) at (2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\node[D5] (2) at (3,1.6) {};
\draw (2,1.6)--(2);
\draw (2) .. controls (4.45,1.5) .. (2,1.4);
\draw (0,1)--(2,1) (0,0.7)--(2,0.7);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;

[draw,label=left:$A_1$] (0020) at ($(00)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (0002) at ($(00)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick] (0,1)--(2,1) (6,1)--(8,1);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_3$] (2040) at ($(20)!0.5!(40)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1.2)--(6,1.2);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (2022) at ($(20)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1.2)--(2,1.2);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_1$] (0222) at ($(02)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_4$] (0204) at ($(02)!0.5!(04)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1)--(2,1);
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
(00)–(0020)–(20)–(2040)–(40) (20)–(2022)–(22) (00)–(0002)–(02)–(0204)–(04) (02)–(0222)–(22);
(40)–(-2*-0.4*,2*+0.4*) (40)–(-2*+0.4*,2*+0.4*) (22)–(-0.4*,2*+0.4*) (22)–(0.4*,2*+0.4*) (04)–(2*-0.4*,2*+0.4*) (04)–(2*+0.4*,2*+0.4*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{PSO}(4,\mathbb{C})}{}{[1,0]}$.

(00) at (0,0) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (4,1.5) {};
\node[D5] (m) at (4,2) {};
\draw (2,2)--(m);
\end{tikzpicture}}$;
(20) at (-1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(02) at (1*,1*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1)--(2,1);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(40) at (-2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (2.5,2.5) {};
\node[D5] at (3.5,2.5) {};
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\draw (2,1.4) .. controls (4.9,1.3) .. (0,1.1);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(22) at (0*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\draw (0,1.2)--(2,1.2);
\draw (2,0.9) .. controls (4.9,0.8) .. (0,0.6);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;
(04) at (2*,2*) $\scalebox{0.7}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (1) at (3,2) {};
\draw (2,2)--(1);
\draw (1) .. controls (4.45,1.9) .. (2,1.8);
\node[D5] (2) at (3,1.6) {};
\draw (2,1.6)--(2);
\draw (2) .. controls (4.45,1.5) .. (2,1.4);
\draw (0,1)--(2,1) (0,0.7)--(2,0.7);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m);
\end{tikzpicture}}$;

[draw,label=left:$A_2$] (0020) at ($(00)!0.5!(20)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (0002) at ($(00)!0.5!(02)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick] (0,1)--(2,1) (6,1)--(8,1);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_4$] (2040) at ($(20)!0.5!(40)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1.2)--(6,1.2);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (2022) at ($(20)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] at (3,2.5) {};
\node[D5] at (5,2.5) {};
\draw[transform canvas={yshift=2pt}] (2,0.7)--(8,0.7);
\draw[transform canvas={yshift=-2pt}] (0,0.7)--(6,0.7);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1.2)--(8,1.2);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1.2)--(2,1.2);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_2$] (0222) at ($(02)!0.5!(22)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(2,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(6,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=-2pt}] (0,1)--(6,1);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
[draw,label=left:$A_4$] (0204) at ($(02)!0.5!(04)$) $\scalebox{0.4}{\begin{tikzpicture}
\draw[red] (0,0)--(0,3) (2,0)--(2,3) (6,0)--(6,3) (8,0)--(8,3);
\draw[cyan] (4,0)--(4,3);
\node[D5] (l) at (3,1.5) {};
\node[D5] (r) at (5,1.5) {};
\draw (0,0.5)--(2,0.5) (6,0.5)--(8,0.5);
\draw[transform canvas={yshift=2pt}] (2,1.5)--(l)--(6,1.5);
\draw[transform canvas={yshift=-2pt}] (2,1.5)--(r)--(6,1.5);
\node[D5o] (1) at (-0.5,2) {};
\node[D5o] (2) at (8.5,2) {};
\draw[olive,thick,transform canvas={yshift=2pt}] (1)--(6,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (1)--(0,2);
\draw[olive,thick,transform canvas={yshift=-2pt}] (2)--(2,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (2)--(8,2);
\draw[olive,thick,transform canvas={yshift=2pt}] (6,1)--(8,1);
\draw[olive,thick,transform canvas={yshift=2pt}] (0,1)--(2,1);
\node[D5] at (4,2.5) {};
\node[D5] (m) at (4,2.8) {};
\draw (2,2.8)--(m)--(6,2.8);
\end{tikzpicture}}$;
(00)–(0020)–(20)–(2040)–(40) (20)–(2022)–(22) (00)–(0002)–(02)–(0204)–(04) (02)–(0222)–(22);
(40)–(-2*-0.4*,2*+0.4*) (40)–(-2*+0.4*,2*+0.4*) (22)–(-0.4*,2*+0.4*) (22)–(0.4*,2*+0.4*) (04)–(2*-0.4*,2*+0.4*) (04)–(2*+0.4*,2*+0.4*);

Brane depiction of the low dimensional leaves [and elementary slices between them] of $\w{\mathrm{PSO}(4,\mathbb{C})}{}{[1,1]}$.

§ QUIVER ADDITION

§.§ Algorithm

In recent papers <cit.>, the notion of quiver subtraction was developed to obtain the Hasse diagram for symplectic singularities, and has been used in various scenarios <cit.>. In this section, we consider the reverse of this process: quiver addition. Aspects of this notion have been discussed in <cit.> for $D$-type Dynkin quivers. In this work, we show that the quiver addition algorithm presented below precisely reproduces the Hasse diagrams and quivers for the slices in the affine Grassmannian of any finite dimensional simple complex Lie group. For classical groups, the quiver addition algorithm can be viewed as the quiver version of the brane constructions of Section <ref>, expressed in a way more suitable e.g. to program. For exceptional groups, however, no brane systems are known, and the quiver addition algorithm provides a simple, but powerful way to obtain the Hasse diagram of any transverse slice in the affine Grassmannian. Before proceeding to quiver addition, let us review the procedure of quiver subtraction.

One way to obtain the Coulomb branch Hasse diagram of a quiver $\mathsf{Q}$ is to use quiver subtraction. In quiver subtraction one identifies an elementary slice with quiver $\mathsf{D}$ from Table <ref> and subtracts it from $\mathsf{Q}$ following the algorithm given in <cit.> to produce a new quiver $\mathsf{Q}'=\mathsf{Q}-\mathsf{D}$. We restrict to the case where a given elementary slice cannot be subtracted more than once consecutively.[If the same slice can be subtracted more than once, we are in the realm of affine Grassmannians of affine type, rather than finite type.] In this case, the Coulomb branch of $\mathsf{Q}'$ is the closure of a minimal degeneration of the highest leaf in the Coulomb branch of $\mathsf{Q}$.
One key point in the algorithm is that the balance of all the gauge nodes remains invariant during subtraction. If more than one quiver can be subtracted, the Hasse diagram bifurcates. As introduced above, we call the number of elementary slices which can be subtracted at any given step the disposition $\mathfrak{D}$ of the leaf.

Conversely, to a given quiver one can generically add an infinite number of elementary quivers. However, if one imposes the restriction that the Dynkin diagram formed by the gauge nodes remains invariant then the number of possible slices allowed to be added becomes finite. We now discuss the algorithm.

Quiver addition algorithm.
Let $\mathsf{Q}$ be a quiver of unitary gauge nodes forming a rank $r$ finite Dynkin diagram with Cartan matrix $(C_{\mathfrak{g}})_{i,j=1 , \dots , r}$. Let the ranks of the gauge nodes be $k_i \geq 0$, and the ranks of the flavors nodes be $N_i \geq 0$ for $i=1 , \dots , r$.[We emphasize that $k_i = 0$ is allowed, even when $N_i >0$. ]

* Identification of the elementary quivers $\mathsf{S}$ to add. Three kinds of elementary quivers can be added to $\mathsf{Q}$.

* Define the subset of gauge nodes without any flavors

\begin{equation}
J_0 = \left\{ j \in \{1 , \dots , r\} \mid N_j =0 \right\} \,.
\end{equation}

The nodes labeled by $J_0$ form a union of connected Dynkin diagrams. For each connected Dynkin diagram, one can add the quiver $\mathsf{S}$ for the closure of the minimal nilpotent orbit (see Table <ref>).

* Consider the set of gauge nodes

\begin{equation}
J_1 = \left\{ j \in \{1 , \dots , r\} \mid N_j \geq 1  \quad \textrm{and} \quad \forall \; j'\in \{1 , \dots , r\} ,  \; N_{j'} \geq - (C_{\mathfrak{g}})_{j'j} \right\}
\end{equation}

such that gauge node itself has flavours and also its connected neighbours have enough flavour. For each $j \in J_1$, one can add on that node the quiver for a Kleinian singularity $A_{N_j +1}$.

* If there is a subset of the gauge nodes which realize the quivers on the first column of Table <ref> with  the flavors exactly as indicated, and with no flavorless gauge node connected to that subset, then one can add the $\mathsf{S}$ in the second column.

* Addition of the elementary quivers.
The result of the addition of the elementary quiver $\mathsf{S}$ to $\mathsf{Q}$ is a quiver $\mathsf{Q}'$ with gauge nodes given by the Dynkin diagram specified by $(C_{\mathfrak{g}})_{i,j=1 , \dots , r}$, where

* The ranks of the gauge groups $k'_i$ are the sums of the ranks in $\mathsf{Q}$ and $\mathsf{S}$.

* The flavors $N'_i$ are given by the corresponding flavor in $\mathsf{S}$ if $k_i \neq k'_i$ and by rebalancing otherwise.

To this section of $\mathsf{Q}$     One can add the following $\mathsf{S}$
[gauge] (4) at (-3,0) ;
(3) at (-2,0) $\cdots$;
[gauge] (2) at (-1,0) ;
[gauge] (1) at (0,0) ;
[flavour,label=above:1] (1a) at (0,1) ;
(1a)–(1) (2)–(3)–(4) (-0.4,0.2)–(-0.6,0)–(-0.4,-0.2);
[transform canvas=yshift=-1pt] (1)–(2);
[transform canvas=yshift=1pt] (1)–(2);
[flavour,label=above:1] (10) at (4,1) ;
[gauge,label=below:1] (0) at (0,0) ;
[gauge,label=below:1] (1) at (1,0) ;
(2) at (2,0) $\cdots$;
[gauge,label=below:1] (3) at (3,0) ;
[gauge,label=below:1] (4) at (4,0) ;
[flavour,label=above:1] (0a) at (0,1) ;
(0)–(1)–(2)–(3) (3.6,0.2)–(3.4,0)–(3.6,-0.2) (4)–(10);
[transform canvas=yshift=-1pt] (3)–(4);
[transform canvas=yshift=1pt] (3)–(4);

[gauge] (2) at (-1,0) ;
[gauge] (1) at (0,0) ;
[flavour,label=above:1] (1a) at (0,1) ;
(1a)–(1)–(2) (-0.4,0.2)–(-0.6,0)–(-0.4,-0.2);
[transform canvas=yshift=-2pt] (1)–(2);
[transform canvas=yshift=0pt] (1)–(2);
[transform canvas=yshift=2pt] (1)–(2);
[gauge,label=below:1] (3) at (3,0) ;
[gauge,label=below:1] (4) at (4,0) ;
[flavour,label=above:1] (2) at (3,1) ;
[transform canvas=yshift=-2pt] (3)–(4);
[transform canvas=yshift=0pt] (3)–(4);
[transform canvas=yshift=2pt] (3)–(4);
(3.6,0.2)–(3.4,0)–(3.6,-0.2) (2)–(3);

[gauge] (2) at (-1,0) ;
[gauge] (1) at (0,0) ;
[flavour,label=above:2] (1a) at (0,1) ;
(1a)–(1)–(2) (-0.4,0.2)–(-0.6,0)–(-0.4,-0.2);
[transform canvas=yshift=-2pt] (1)–(2);
[transform canvas=yshift=0pt] (1)–(2);
[transform canvas=yshift=2pt] (1)–(2);
[gauge,label=below:1] (3) at (3,0) ;
[gauge,label=below:1] (4) at (4,0) ;
[flavour,label=above:1] (5) at (4,1) ;
[flavour,label=above:1] (2) at (3,1) ;
[transform canvas=yshift=-2pt] (3)–(4);
[transform canvas=yshift=0pt] (3)–(4);
[transform canvas=yshift=2pt] (3)–(4);
(3.6,0.2)–(3.4,0)–(3.6,-0.2) (4)–(5) (2)–(3);

Identification of the situations where the quivers for $ac_n$, $cg_2$ and $ag_2$ can be added. We insist on the fact that the flavor nodes must be exactly as indicated in the first column, and that there should be no flavorless gauge node connected to the portions of quiver drawn in the first column, so that quiver of the second column can be added.

Note that the total number of flavors either stays the same or increases when a slice is added.[Just like the number of D5 branes between NS5 branes stays the same or increases when moving upwards in the Hasse diagrams in Section <ref>] As we grow the Hasse diagram further, we reach a stage where all gauge groups have non-zero flavor nodes. Once this happens, the only slices that can be added are Kleinian $A_k$ singularities. Hence, as the Hasse diagram grows, more and more Kleinian singularities populate the Hasse diagram.
When all the nodes have flavors, the disposition is equal to the rank of the group, and can not decrease upon quiver addition.

§.§ Examples with simply laced quivers

Let us exemplify how the algorithm works in the case of good linear quivers.
The set $J_0$ of unflavored nodes decomposes into smaller $A$-type diagrams, and for each such diagram, an $a_n$-type slice can be added.[The transition in the corresponding brane system is (<ref>).] For instance, consider the following scenario:

\begin{equation}
\raisebox{-.5\height}{
\begin{tikzpicture}[scale=0.8]
\begin{pgfonlayer}{nodelayer}
\node [style=gauge3] (0) at (-4.5, 0.25) {};
\node [style=gauge3] (1) at (-3.5, 0.25) {};
\node [style=gauge3] (2) at (-2.5, 0.25) {};
\node [style=gauge3] (3) at (-1, 0.25) {};
\node [style=gauge3] (4) at (0, 0.25) {};
\node [style=gauge3] (5) at (1, 0.25) {};
\node [style=none] (6) at (-1.75, 0.25) {$\dots$};
\node [style=gauge3] (7) at (4, 0.25) {};
\node [style=none] (10) at (2, 0.25) {$\dots$};
\node [style=none] (14) at (-5.25, 0.25) {$\dots$};
\node [style=flavour2] (16) at (4, 1.25) {};
\node [style=flavour2] (17) at (-3.5, 1.25) {};
\node [style=flavour2] (18) at (0, 1.25) {};
\node [style=gauge3] (19) at (-6, 0.25) {};
\node [style=gauge3] (20) at (3, 0.25) {};
\node [style=none] (21) at (-6, -0.75) {};
\node [style=none] (22) at (-4.5, -0.75) {};
\node [style=none] (24) at (3, -0.75) {};
\node [style=none] (25) at (-5.25, -1.5) {$m_1$};
\node [style=none] (26) at (-1.75, -1.5) {$m_2$};
\node [style=none] (27) at (2, -1.5) {$m_3$};
\node [style=none] (28) at (-5.5, 2) {};
\node [style=none] (29) at (-5.5, 1) {};
\node [style=none] (30) at (-5.5, 6) {Can add $a_{m_1}$};
\node [style=none] (31) at (-1.75, 2) {};
\node [style=none] (32) at (-1.75, 1) {};
\node [style=none] (34) at (2, 2) {};
\node [style=none] (35) at (2, 1) {};
\node [style=gauge3] (38) at (-4.75, 4) {};
\node [style=gauge3] (39) at (-6.25, 4) {};
\node [style=none] (40) at (-5.5, 4) {$\dots$};
\node [style=flavour2] (41) at (-6.25, 5) {};
\node [style=flavour2] (42) at (-4.75, 5) {};
\node [style=none] (44) at (-6.25, 3.25) {};
\node [style=none] (45) at (-4.75, 3.25) {};
\node [style=none] (46) at (-5.5, 2.5) {$m_1$};
\node [style=none] (47) at (-1.75, 6) {Can add $a_{m_2}$};
\node [style=gauge3] (48) at (-1, 4) {};
\node [style=gauge3] (49) at (-2.5, 4) {};
\node [style=none] (50) at (-1.75, 4) {$\dots$};
\node [style=flavour2] (51) at (-2.5, 5) {};
\node [style=flavour2] (52) at (-1, 5) {};
\node [style=none] (53) at (-2.5, 3.25) {};
\node [style=none] (54) at (-1, 3.25) {};
\node [style=none] (55) at (-1.75, 2.5) {$m_2$};
\node [style=none] (59) at (2, 6) {Can add $a_{m_3}$};
\node [style=gauge3] (60) at (2.75, 4) {};
\node [style=gauge3] (61) at (1.25, 4) {};
\node [style=none] (62) at (2, 4) {$\dots$};
\node [style=flavour2] (63) at (1.25, 5) {};
\node [style=flavour2] (64) at (2.75, 5) {};
\node [style=none] (65) at (1.25, 3.25) {};
\node [style=none] (66) at (2.75, 3.25) {};
\node [style=none] (67) at (2, 2.5) {$m_3$};
\node [style=none] (68) at (-6.25, 3.5) {1};
\node [style=none] (69) at (-4.75, 3.5) {1};
\node [style=none] (70) at (-2.5, 3.5) {1};
\node [style=none] (71) at (-1, 3.5) {1};
\node [style=none] (72) at (1.25, 3.5) {1};
\node [style=none] (73) at (2.75, 3.5) {1};
\node [style=none] (74) at (-6.25, 5.5) {1};
\node [style=none] (75) at (-4.75, 5.5) {1};
\node [style=none] (76) at (-2.5, 5.5) {1};
\node [style=none] (77) at (-1, 5.5) {1};
\node [style=none] (78) at (1.25, 5.5) {1};
\node [style=none] (79) at (2.75, 5.5) {1};
\node [style=none] (80) at (-2.5, -0.75) {};
\node [style=none] (81) at (-1, -0.75) {};
\node [style=none] (82) at (1, -0.75) {};
\node [style=none] (83) at (-6, -0.25) {\small $k_a$};
\node [style=none] (84) at (-4.5, -0.25) {\small $k_b$};
\node [style=none] (85) at (-3.5, -0.25) {\small $k_c$};
\node [style=none] (86) at (-2.5, -0.25) {\small $k_d$};
\node [style=none] (87) at (-1, -0.25) {\small $k_f$};
\node [style=none] (88) at (0, -0.25) {\small $k_g$};
\node [style=none] (89) at (1, -0.25) {\small $k_h$};
\node [style=none] (90) at (3, -0.25) {\small $k_i$};
\node [style=none] (91) at (4, -0.25) {\small $k_j$};
\node [style=none] (92) at (-3.5, 1.75) {$N_1$};
\node [style=none] (93) at (0, 1.75) {$N_2$};
\node [style=none] (94) at (4, 1.75) {$N_3$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\draw (1) to (2);
\draw (5) to (4);
\draw (4) to (3);
\draw (17) to (1);
\draw (18) to (4);
\draw (16) to (7);
\draw (20) to (7);
\draw [style=->] (28.center) to (29.center);
\draw [style=->] (31.center) to (32.center);
\draw [style=->] (34.center) to (35.center);
\draw (41) to (39);
\draw (42) to (38);
\draw (51) to (49);
\draw (52) to (48);
\draw (63) to (61);
\draw (64) to (60);
\draw [style=brace] (66.center) to (65.center);
\draw [style=brace] (54.center) to (53.center);
\draw [style=brace] (45.center) to (44.center);
\draw [style=brace] (22.center) to (21.center);
\draw [style=brace] (81.center) to (80.center);
\draw [style=brace] (24.center) to (82.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

There are three possible slices one can add $a_{m_1}$, $a_{m_2}$, and $a_{m_3}$. For example, adding the $a_{m_2}$ slice produces

\begin{equation}
\raisebox{-.5\height}{
\begin{tikzpicture}[scale=0.8]
\begin{pgfonlayer}{nodelayer}
\node [style=gauge3] (0) at (-4.5, 0.25) {};
\node [style=gauge3] (1) at (-3.5, 0.25) {};
\node [style=gauge3] (2) at (-2.5, 0.25) {};
\node [style=gauge3] (3) at (-1, 0.25) {};
\node [style=gauge3] (4) at (0, 0.25) {};
\node [style=gauge3] (5) at (1, 0.25) {};
\node [style=none] (6) at (-1.75, 0.25) {$\dots$};
\node [style=gauge3] (7) at (4, 0.25) {};
\node [style=none] (10) at (2, 0.25) {$\dots$};
\node [style=none] (14) at (-5.25, 0.25) {$\dots$};
\node [style=flavour2] (16) at (4, 1.25) {};
\node [style=flavour2] (17) at (-3.5, 1.25) {};
\node [style=flavour2] (18) at (0, 1.25) {};
\node [style=gauge3] (19) at (-6, 0.25) {};
\node [style=gauge3] (20) at (3, 0.25) {};
\node [style=none] (21) at (-6, -0.75) {};
\node [style=none] (22) at (-4.5, -0.75) {};
\node [style=none] (24) at (3, -0.75) {};
\node [style=none] (25) at (-5.25, -1.5) {$m_1$};
\node [style=none] (26) at (-1.75, -1.5) {$m_2$};
\node [style=none] (27) at (2, -1.5) {$m_3$};
\node [style=none] (80) at (-2.5, -0.75) {};
\node [style=none] (81) at (-1, -0.75) {};
\node [style=none] (82) at (1, -0.75) {};
\node [style=none] (83) at (-6, -0.25) {\small $k_a$};
\node [style=none] (84) at (-4.5, -0.25) {\small $k_b$};
\node [style=none] (85) at (-3.5, -0.25) {\small $k_c$};
\node [style=none] (86) at (-2.5, -0.25) {\small $k_d{+}1$};
\node [style=none] (87) at (-1, -0.25) {\small $k_f{+}1$};
\node [style=none] (88) at (0, -0.25) {\small $k_g$};
\node [style=none] (89) at (1, -0.25) {\small $k_h$};
\node [style=none] (90) at (3, -0.25) {\small $k_i$};
\node [style=none] (91) at (4, -0.25) {\small $k_j$};
\node [style=none] (92) at (-3.5, 1.75) {$N_1{-}1$};
\node [style=none] (93) at (0, 1.75) {$N_2{-}1$};
\node [style=none] (94) at (4, 1.75) {$N_3$};
\node [style=flavour2] (95) at (-2.5, 1.25) {};
\node [style=flavour2] (96) at (-1, 1.25) {};
\node [style=none] (97) at (-2.5, 1.75) {1};
\node [style=none] (98) at (-1, 1.75) {1};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\draw (1) to (2);
\draw (5) to (4);
\draw (4) to (3);
\draw (17) to (1);
\draw (18) to (4);
\draw (16) to (7);
\draw (20) to (7);
\draw [style=brace] (22.center) to (21.center);
\draw [style=brace] (81.center) to (80.center);
\draw [style=brace] (24.center) to (82.center);
\draw (95) to (2);
\draw (96) to (3);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

On the other hand, the $A$-type Kleinian singularities can be added in the following situation:[The transition in the corresponding brane system is (<ref>).]

\begin{equation}
\raisebox{-.5\height}{
\begin{tikzpicture}[scale=0.8]
\begin{pgfonlayer}{nodelayer}
\node [style=gauge3] (36) at (-4, 0) {};
\node [style=gauge3] (37) at (-3, 0) {};
\node [style=gauge3] (38) at (-2, 0) {};
\node [style=flavour2] (39) at (-4, 1) {};
\node [style=flavour2] (40) at (-3, 1) {};
\node [style=flavour2] (41) at (-2, 1) {};
\node [style=none] (42) at (-4, 1.5) {$N_1$};
\node [style=none] (43) at (-3, 1.5) {$N_2$};
\node [style=none] (44) at (-2, 1.5) {$N_3$};
\node [style=none] (45) at (-5.75, 0) {$\dots$};
\node [style=none] (46) at (-0.25, 0) {$\dots$};
\node [style=none] (47) at (-3, 3) {};
\node [style=none] (48) at (-3, 2) {};
\node [style=none] (49) at (0, 4.5) {Can add $A_{N_2{+}1}$};
\node [style=gauge3] (50) at (-5, 0) {};
\node [style=gauge3] (51) at (-1, 0) {};
\node [style=gauge3] (52) at (-3, 4) {};
\node [style=flavour2] (53) at (-3, 5) {};
\node [style=none] (55) at (-3, 3.5) {1};
\node [style=none] (56) at (1, 1) {};
\node [style=none] (57) at (3, 1) {};
\node [style=gauge3] (58) at (5.75, 0) {};
\node [style=gauge3] (59) at (6.75, 0) {};
\node [style=gauge3] (60) at (7.75, 0) {};
\node [style=flavour2] (61) at (5.75, 1) {};
\node [style=flavour2] (62) at (6.75, 1) {};
\node [style=flavour2] (63) at (7.75, 1) {};
\node [style=none] (64) at (5.25, 1.5) {\small $N_1{-}1$};
\node [style=none] (65) at (6.75, 1.5) {\small $N_2{+}2$};
\node [style=none] (66) at (8.25, 1.5) {\small $N_3{-}1$};
\node [style=none] (67) at (4, 0) {$\dots$};
\node [style=none] (68) at (9.5, 0) {$\dots$};
\node [style=gauge3] (69) at (4.75, 0) {};
\node [style=gauge3] (70) at (8.75, 0) {};
\node [style=none] (71) at (-5, -0.5) {$k_a$};
\node [style=none] (72) at (-4, -0.5) {$k_b$};
\node [style=none] (73) at (-3, -0.5) {$k_c$};
\node [style=none] (74) at (-2, -0.5) {$k_d$};
\node [style=none] (75) at (-1, -0.5) {$k_e$};
\node [style=none] (76) at (-3, 5.5) {$N_2{+}2$};
\node [style=none] (77) at (4.75, -0.5) {$k_a$};
\node [style=none] (78) at (5.75, -0.5) {$k_b$};
\node [style=none] (79) at (6.75, -0.5) {$k_c+1$};
\node [style=none] (80) at (7.75, -0.5) {$k_d$};
\node [style=none] (81) at (8.75, -0.5) {$k_e$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (36) to (37);
\draw (37) to (38);
\draw (39) to (36);
\draw (40) to (37);
\draw (41) to (38);
\draw [style=->] (47.center) to (48.center);
\draw (36) to (50);
\draw (38) to (51);
\draw (53) to (52);
\draw [style=->] (56.center) to (57.center);
\draw (58) to (59);
\draw (59) to (60);
\draw (61) to (58);
\draw (62) to (59);
\draw (63) to (60);
\draw (58) to (69);
\draw (60) to (70);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

which corresponds to nodes in the set $J_1$.

Single unbalanced gauge group.
As an illustration, we consider linear quivers with one unbalanced node, with imbalance $+1$. These quivers cover all cases of transverse slice to the lowest leaf in the components of $\mathrm{Gr}_{\mathrm{PSL}(N , \mathbb{C})}$ not connected to the trivial lattice. Quiver addition then reproduces the Hasse diagram and quivers for the transverse slices in these components of $\mathrm{Gr}_{\mathrm{PSL}(N , \mathbb{C})}$. Specifically, Figure <ref> shows the first few leaves in the affine Grassmannian Hasse diagram of $\mathrm{Gr}_{\mathrm{PSL}(m_1 + m_2 + 2 , \mathbb{C})}$ for $m_1,m_2 \geq 6$ and lowest dominant coweight (see Table <ref>)

\begin{equation}
\Omega_{1+m_1}=[\underbrace{0\dots0}_{m_1}1\underbrace{0\dots 0}_{m_2}] \, .
\end{equation}

It is clear that for quivers where all gauge groups have non-zero ranks, the global symmetry is $A_{m_1}\times A_{m_2} \times \mathfrak{u}(1)$. As a feature of Hasse diagrams, the non-abelian part of the global symmetry is given by the two slices directly connected to the origin. Note, since $m_1$ and $m_2$ can be different, the Hasse diagram becomes less symmetric as one moves away from the bottom.

We observe that in the Hasse diagram of $\mathrm{Gr}_{\mathrm{PSL}(m_1 + m_2 + 2 , \mathbb{C})}$, if we keep the leaves where only the ranks of the first $m_1$ gauge nodes are non-zero, then the resulting diagram is finite and is the Hasse diagram of the nilpotent orbits of $\mathfrak{sl}(m_1 +1, \mathbb{C})$.

[style=gauge3] (611) at (0, -37.5) ;
[style=none] (613) at (-1.75, -37.5) $\dots$;
[style=flavour2] (614) at (0, -36.5) ;
[style=gauge3] (615) at (-2.5, -37.5) ;
[style=none] (616) at (-2.5, -38.75) ;
[style=none] (618) at (-1.75, -39.75) $m_1$;
[style=none] (619) at (1.75, -39.75) $m_2$;
[style=none] (621) at (-2.5, -38.25) 0;
[style=none] (623) at (0, -38.25) 0;
[style=none] (624) at (0, -35.75) 1;
[style=gauge3] (626) at (-1, -37.5) ;
[style=none] (627) at (-1, -38.25) 0;
[style=miniBlue] (629) at (0, -37.5) ;
[style=gauge3] (630) at (2.5, -37.5) ;
[style=gauge3] (631) at (1, -37.5) ;
[style=none] (632) at (1.75, -37.5) $\dots$;
[style=none] (634) at (1, -38.25) 0;
[style=none] (636) at (2.5, -38.25) 0;
[style=none] (641) at (2.5, -38.75) ;
[style=gauge3] (642) at (-8.75, -31.5) ;
[style=none] (643) at (-10.5, -31.5) $\dots$;
[style=gauge3] (645) at (-12.25, -31.5) ;
[style=none] (650) at (-12.25, -32.25) 1;
[style=none] (651) at (-8.75, -32.25) 1;
[style=gauge3] (653) at (-11.25, -31.5) ;
[style=gauge3] (654) at (-9.75, -31.5) ;
[style=none] (655) at (-9.75, -32.25) 1;
[style=none] (656) at (-11.25, -32.25) 1;
[style=gauge3] (658) at (-5.25, -31.5) ;
[style=gauge3] (659) at (-7.75, -31.5) ;
[style=none] (660) at (-6, -31.5) $\dots$;
[style=none] (661) at (-7.75, -32.25) 0;
[style=none] (662) at (-5.25, -32.25) 0;
[style=flavour2] (666) at (-12.25, -30.5) ;
[style=flavour2] (667) at (-8.75, -30.5) ;
[style=none] (668) at (-12.25, -29.75) 1;
[style=none] (669) at (-8.75, -29.75) 1;
[style=gauge3] (670) at (-12.5, -25.5) ;
[style=none] (671) at (-14.25, -25.5) $\dots$;
[style=gauge3] (672) at (-16, -25.5) ;
[style=none] (677) at (-16, -26.25) 1;
[style=none] (678) at (-12.5, -26.25) 1;
[style=gauge3] (679) at (-15, -25.5) ;
[style=gauge3] (680) at (-13.5, -25.5) ;
[style=none] (681) at (-13.5, -26.25) 2;
[style=none] (682) at (-15, -26.25) 2;
[style=gauge3] (684) at (-9, -25.5) ;
[style=gauge3] (685) at (-11.5, -25.5) ;
[style=none] (686) at (-9.75, -25.5) $\dots$;
[style=none] (687) at (-11.5, -26.25) 0;
[style=none] (688) at (-9, -26.25) 0;
[style=flavour2] (692) at (-15, -24.5) ;
[style=flavour2] (693) at (-13.5, -24.5) ;
[style=none] (694) at (-15, -23.75) 1;
[style=none] (695) at (-13.5, -23.75) 1;
[style=gauge3] (697) at (-13.25, -15.25) ;
[style=none] (698) at (-16.25, -15.25) $\dots$;
[style=gauge3] (699) at (-19.25, -15.25) ;
[style=none] (704) at (-19.25, -16) 1;
[style=none] (705) at (-13.25, -16) 1;
[style=gauge3] (706) at (-18.25, -15.25) ;
[style=gauge3] (707) at (-14.25, -15.25) ;
[style=none] (708) at (-14.25, -16) 2;
[style=none] (709) at (-18.25, -16) 2;
[style=gauge3] (711) at (-10, -15.25) ;
[style=gauge3] (712) at (-12.25, -15.25) ;
[style=none] (713) at (-10.75, -15.25) $\dots$;
[style=none] (714) at (-12.25, -16) 0;
[style=none] (715) at (-10, -16) 0;
[style=gauge3] (724) at (-19, -19.5) ;
[style=none] (725) at (-20.75, -19.5) $\dots$;
[style=gauge3] (726) at (-22.5, -19.5) ;
[style=none] (731) at (-22.5, -20.25) 2;
[style=none] (732) at (-19, -20.25) 1;
[style=gauge3] (733) at (-21.5, -19.5) ;
[style=gauge3] (734) at (-20, -19.5) ;
[style=none] (735) at (-20, -20.25) 2;
[style=none] (736) at (-21.5, -20.25) 2;
[style=gauge3] (738) at (-15.5, -19.5) ;
[style=none] (740) at (-16.25, -19.5) $\dots$;
[style=none] (742) at (-15.5, -20.25) 0;
[style=flavour2] (747) at (-20, -18.5) ;
[style=none] (749) at (-20, -17.75) 1;
[style=gauge3] (751) at (-17.25, -15.25) ;
[style=gauge3] (752) at (-15.25, -15.25) ;
[style=none] (753) at (-15.25, -16) 3;
[style=none] (754) at (-17.25, -16) 3;
[style=flavour2] (755) at (-15.25, -14.25) ;
[style=flavour2] (756) at (-17.25, -14.25) ;
[style=none] (757) at (-15.25, -13.5) 1;
[style=none] (758) at (-17.25, -13.5) 1;
[style=flavour2] (759) at (-22.5, -18.5) ;
[style=none] (760) at (-22.5, -17.75) 2;
[style=none] (761) at (-4.5, -36.75) ;
[style=none] (762) at (-6.75, -34.75) ;
[style=none] (763) at (-10.5, -29.75) ;
[style=none] (764) at (-12.5, -27.75) ;
[style=none] (765) at (-15, -23) ;
[style=none] (766) at (-15, -17.5) ;
[style=none] (767) at (-16.25, -24.25) ;
[style=none] (768) at (-18, -21.75) ;
[style=gauge3] (888) at (1, -25.5) ;
[style=gauge3] (889) at (0, -25.5) ;
[style=none] (891) at (1.75, -25.5) $\dots$;
[style=flavour2] (892) at (0, -24.5) ;
[style=gauge3] (893) at (2.5, -25.5) ;
[style=none] (899) at (2.5, -26.25) 1;
[style=none] (900) at (1, -26.25) 1;
[style=none] (901) at (0, -26.25) 1;
[style=none] (904) at (0, -23.75) 1;
[style=gauge3] (910) at (-2.5, -25.5) ;
[style=none] (911) at (-1.75, -25.5) $\dots$;
[style=gauge3] (912) at (-1, -25.5) ;
[style=none] (913) at (-1, -26.25) 1;
[style=none] (914) at (-2.5, -26.25) 1;
[style=miniBlue] (923) at (0, -25.5) ;
[style=flavour2] (924) at (-2.5, -24.5) ;
[style=flavour2] (925) at (2.5, -24.5) ;
[style=none] (926) at (2.5, -23.75) 1;
[style=none] (927) at (-2.5, -23.75) 1;
[style=none] (928) at (-5.75, -30) ;
[style=none] (929) at (-4.25, -28) ;
[style=gauge3] (932) at (-18, -19.5) ;
[style=none] (933) at (-18, -20.25) 0;
[style=gauge3] (934) at (8.75, -31.5) ;
[style=none] (935) at (10.5, -31.5) $\dots$;
[style=gauge3] (936) at (12.25, -31.5) ;
[style=none] (941) at (12.25, -32.25) 1;
[style=none] (942) at (8.75, -32.25) 1;
[style=gauge3] (943) at (11.25, -31.5) ;
[style=gauge3] (944) at (9.75, -31.5) ;
[style=none] (945) at (9.75, -32.25) 1;
[style=none] (946) at (11.25, -32.25) 1;
[style=gauge3] (948) at (5.25, -31.5) ;
[style=gauge3] (949) at (7.75, -31.5) ;
[style=none] (950) at (6, -31.5) $\dots$;
[style=none] (951) at (7.75, -32.25) 0;
[style=none] (952) at (5.25, -32.25) 0;
[style=flavour2] (954) at (12.25, -30.5) ;
[style=flavour2] (955) at (8.75, -30.5) ;
[style=none] (956) at (12.25, -29.75) 1;
[style=none] (957) at (8.75, -29.75) 1;
[style=gauge3] (958) at (12.5, -25.5) ;
[style=none] (959) at (14.25, -25.5) $\dots$;
[style=gauge3] (960) at (16, -25.5) ;
[style=none] (965) at (16, -26.25) 1;
[style=none] (966) at (12.5, -26.25) 1;
[style=gauge3] (967) at (15, -25.5) ;
[style=gauge3] (968) at (13.5, -25.5) ;
[style=none] (969) at (13.5, -26.25) 2;
[style=none] (970) at (15, -26.25) 2;
[style=gauge3] (972) at (9, -25.5) ;
[style=gauge3] (973) at (11.5, -25.5) ;
[style=none] (974) at (9.75, -25.5) $\dots$;
[style=none] (975) at (11.5, -26.25) 0;
[style=none] (976) at (9, -26.25) 0;
[style=flavour2] (978) at (15, -24.5) ;
[style=flavour2] (979) at (13.5, -24.5) ;
[style=none] (980) at (15, -23.75) 1;
[style=none] (981) at (13.5, -23.75) 1;
[style=gauge3] (983) at (13.75, -16) ;
[style=none] (984) at (16.75, -16) $\dots$;
[style=gauge3] (985) at (19.75, -16) ;
[style=none] (990) at (19.75, -16.75) 1;
[style=none] (991) at (13.75, -16.75) 1;
[style=gauge3] (992) at (18.75, -16) ;
[style=gauge3] (993) at (14.75, -16) ;
[style=none] (994) at (14.75, -16.75) 2;
[style=none] (995) at (18.75, -16.75) 2;
[style=gauge3] (997) at (10.25, -16) ;
[style=gauge3] (998) at (12.75, -16) ;
[style=none] (999) at (11, -16) $\dots$;
[style=none] (1000) at (12.75, -16.75) 0;
[style=none] (1001) at (10.25, -16.75) 0;
[style=gauge3] (1004) at (20, -19.5) ;
[style=none] (1005) at (21.75, -19.5) $\dots$;
[style=gauge3] (1006) at (23.5, -19.5) ;
[style=none] (1011) at (23.5, -20.25) 2;
[style=none] (1012) at (20, -20.25) 1;
[style=gauge3] (1013) at (22.5, -19.5) ;
[style=gauge3] (1014) at (21, -19.5) ;
[style=none] (1015) at (21, -20.25) 2;
[style=none] (1016) at (22.5, -20.25) 2;
[style=gauge3] (1018) at (16.5, -19.5) ;
[style=none] (1019) at (17.25, -19.5) $\dots$;
[style=none] (1020) at (16.5, -20.25) 0;
[style=flavour2] (1022) at (21, -18.5) ;
[style=none] (1023) at (21, -17.75) 1;
[style=gauge3] (1025) at (17.75, -16) ;
[style=gauge3] (1026) at (15.75, -16) ;
[style=none] (1027) at (15.75, -16.75) 3;
[style=none] (1028) at (17.75, -16.75) 3;
[style=flavour2] (1029) at (15.75, -15) ;
[style=flavour2] (1030) at (17.75, -15) ;
[style=none] (1031) at (15.75, -14.25) 1;
[style=none] (1032) at (17.75, -14.25) 1;
[style=flavour2] (1033) at (23.5, -18.5) ;
[style=none] (1034) at (23.5, -17.75) 2;
[style=none] (1035) at (4.75, -36.75) ;
[style=none] (1036) at (7, -34.75) ;
[style=none] (1037) at (10.5, -29.75) ;
[style=none] (1038) at (12.5, -27.75) ;
[style=none] (1039) at (14.75, -23) ;
[style=none] (1040) at (14.75, -18.25) ;
[style=none] (1041) at (16.25, -24.25) ;
[style=none] (1042) at (19.5, -21.75) ;
[style=none] (1043) at (5.5, -30) ;
[style=none] (1044) at (4, -28) ;
[style=gauge3] (1045) at (18, -19.5) ;
[style=none] (1046) at (18, -20.25) 0;
[style=gauge3] (1047) at (-5.25, -19.75) ;
[style=none] (1048) at (-8.25, -19.75) $\dots$;
[style=none] (1054) at (-5.25, -20.5) 1;
[style=gauge3] (1055) at (-10.25, -19.75) ;
[style=gauge3] (1056) at (-6.25, -19.75) ;
[style=none] (1057) at (-6.25, -20.5) 2;
[style=none] (1058) at (-10.25, -20.5) 1;
[style=miniBlue] (1059) at (-5.25, -19.75) ;
[style=gauge3] (1060) at (-2.75, -19.75) ;
[style=gauge3] (1061) at (-4.25, -19.75) ;
[style=none] (1062) at (-3.5, -19.75) $\dots$;
[style=none] (1063) at (-4.25, -20.5) 1;
[style=none] (1064) at (-2.75, -20.5) 1;
[style=gauge3] (1067) at (-9.25, -19.75) ;
[style=gauge3] (1068) at (-7.25, -19.75) ;
[style=none] (1069) at (-7.25, -20.5) 2;
[style=none] (1070) at (-9.25, -20.5) 2;
[style=flavour2] (1072) at (-9.25, -18.75) ;
[style=none] (1074) at (-9.25, -18) 1;
[style=flavour2] (1075) at (-2.75, -18.75) ;
[style=none] (1077) at (-2.75, -18) 1;
[style=none] (1081) at (-7.25, -19.75) ;
[style=flavour2] (1082) at (-6.25, -18.75) ;
[style=none] (1083) at (-6.25, -18) 1;
[style=miniBlue] (1084) at (-7.75, -31.5) ;
[style=miniBlue] (1085) at (-11.5, -25.5) ;
[style=miniBlue] (1086) at (-12.25, -15.25) ;
[style=miniBlue] (1087) at (-18, -19.5) ;
[style=none] (1088) at (-11.5, -24) ;
[style=none] (1089) at (-9.75, -22) ;
[style=gauge3] (1093) at (5.5, -20) ;
[style=none] (1094) at (8.5, -20) $\dots$;
[style=none] (1098) at (5.5, -20.75) 1;
[style=gauge3] (1099) at (10.5, -20) ;
[style=gauge3] (1100) at (6.5, -20) ;
[style=none] (1101) at (6.5, -20.75) 2;
[style=none] (1102) at (10.5, -20.75) 1;
[style=miniBlue] (1103) at (5.5, -20) ;
[style=gauge3] (1104) at (3, -20) ;
[style=gauge3] (1105) at (4.5, -20) ;
[style=none] (1106) at (3.75, -20) $\dots$;
[style=none] (1107) at (4.5, -20.75) 1;
[style=none] (1108) at (3, -20.75) 1;
[style=gauge3] (1111) at (9.5, -20) ;
[style=gauge3] (1112) at (7.5, -20) ;
[style=none] (1113) at (7.5, -20.75) 2;
[style=none] (1114) at (9.5, -20.75) 2;
[style=flavour2] (1115) at (9.5, -19) ;
[style=none] (1116) at (9.5, -18.25) 1;
[style=flavour2] (1117) at (3, -19) ;
[style=none] (1118) at (3, -18.25) 1;
[style=none] (1119) at (7.5, -20) ;
[style=flavour2] (1120) at (6.5, -19) ;
[style=none] (1121) at (6.5, -18.25) 1;
[style=none] (1122) at (12, -24.25) ;
[style=none] (1123) at (10.25, -22.25) ;
[style=none] (1125) at (-3.25, -21.5) ;
[style=none] (1126) at (-1.25, -23.25) ;
[style=none] (1127) at (3, -21.75) ;
[style=none] (1128) at (1.25, -23.25) ;
[style=gauge3] (1129) at (0.5, -13.25) ;
[style=none] (1130) at (-2.5, -13.25) $\dots$;
[style=none] (1134) at (0.5, -14) 2;
[style=gauge3] (1135) at (-4.5, -13.25) ;
[style=gauge3] (1136) at (-0.5, -13.25) ;
[style=none] (1137) at (-0.5, -14) 2;
[style=none] (1138) at (-4.5, -14) 1;
[style=miniBlue] (1139) at (0.5, -13.25) ;
[style=gauge3] (1147) at (-3.5, -13.25) ;
[style=gauge3] (1148) at (-1.5, -13.25) ;
[style=none] (1149) at (-1.5, -14) 2;
[style=none] (1150) at (-3.5, -14) 2;
[style=flavour2] (1151) at (-3.5, -12.25) ;
[style=none] (1152) at (-3.5, -11.5) 1;
[style=none] (1155) at (-1.5, -13.25) ;
[style=none] (1159) at (-4.75, -18.25) ;
[style=none] (1160) at (-3.25, -16) ;
[style=none] (1161) at (2.75, -16) ;
[style=none] (1162) at (5, -18.5) ;
[style=none] (1163) at (3.5, -13.25) $\dots$;
[style=gauge3] (1164) at (5.5, -13.25) ;
[style=gauge3] (1165) at (1.5, -13.25) ;
[style=none] (1166) at (1.5, -14) 2;
[style=none] (1167) at (5.5, -14) 1;
[style=gauge3] (1168) at (4.5, -13.25) ;
[style=gauge3] (1169) at (2.5, -13.25) ;
[style=none] (1170) at (2.5, -14) 2;
[style=none] (1171) at (4.5, -14) 2;
[style=flavour2] (1172) at (4.5, -12.25) ;
[style=none] (1173) at (4.5, -11.5) 1;
[style=none] (1174) at (2.5, -13.25) ;
[style=flavour2] (1175) at (0.5, -12.25) ;
[style=none] (1176) at (0.5, -11.5) 1;
[style=none] (1177) at (-5.25, -35.25) $a_{m_1}$;
[style=none] (1178) at (-10.5, -28.5) $a_{m_1-2}$;
[style=none] (1179) at (-17.5, -23.25) $A_1$;
[style=none] (1180) at (-13.75, -20.25) $a_{m_1-4}$;
[style=none] (1181) at (-11.25, -22.5) $a_{m_2+2}$;
[style=none] (1182) at (-5.75, -28.75) $a_{m_2+1}$;
[style=none] (1183) at (5.25, -28.5) $a_{m_1+1}$;
[style=none] (1184) at (5.25, -35.5) $a_{m_2}$;
[style=none] (1185) at (10.75, -28.5) $a_{m_2-2}$;
[style=none] (1186) at (11.75, -22.5) $a_{m_1+2}$;
[style=none] (1187) at (17.5, -22.75) $A_1$;
[style=none] (1188) at (13.5, -20.5) $a_{m_2-4}$;
[style=none] (1189) at (1.5, -22) $a_{m_2-1}$;
[style=none] (1190) at (-1.5, -22) $a_{m_1-1}$;
[style=none] (1191) at (-5, -16.75) $a_{m_2}$;
[style=none] (1192) at (4.75, -16.75) $a_{m_1}$;
[style=none] (1193) at (-23.25, -16.5) ⋮;
[style=none] (1194) at (-15.75, -12.25) ⋮;
[style=none] (1195) at (-7.25, -17) ⋮;
[style=none] (1196) at (0.25, -10.25) ⋮;
[style=none] (1197) at (7, -17) ⋮;
[style=none] (1198) at (15.5, -13) ⋮;
[style=none] (1199) at (21.75, -16) ⋮;
[style=none] (1205) at (-1, -38.75) ;
[style=none] (1206) at (1, -38.75) ;
[style=gauge3] (1207) at (-11.25, -15.25) ;
[style=gauge3] (1208) at (-17, -19.5) ;
[style=gauge3] (1209) at (-10.5, -25.5) ;
[style=none] (1210) at (-11.25, -16) 0;
[style=none] (1211) at (-17, -20.25) 0;
[style=none] (1212) at (-10.5, -26.25) 0;
[style=gauge3] (1213) at (-6.75, -31.5) ;
[style=miniBlue] (1214) at (11.5, -25.5) ;
[style=miniBlue] (1215) at (7.75, -31.5) ;
[style=gauge3] (1216) at (10.5, -25.5) ;
[style=gauge3] (1217) at (6.75, -31.5) ;
[style=gauge3] (1220) at (19, -19.5) ;
[style=none] (1221) at (19, -20.25) 0;
[style=miniBlue] (1222) at (19, -19.5) ;
[style=miniBlue] (1223) at (12.75, -16) ;
[style=gauge3] (1224) at (11.75, -16) ;
[style=none] (1225) at (11.75, -16.75) 0;
[style=none] (1226) at (6.75, -32.25) 0;
[style=none] (1227) at (10.5, -26.25) 0;
[style=none] (1228) at (-6.75, -32.25) 0;

(614) to (611);
(629) to (631);
(626) to (629);
(653) to (645);
(666) to (645);
(679) to (672);
(693) to (680);
(679) to (692);
(733) to (726);
(747) to (734);
(699) to (706);
(706) to (751);
(752) to (707);
(755) to (752);
(756) to (751);
(759) to (726);
[style=->] (761.center) to (762.center);
[style=->] (763.center) to (764.center);
[style=->] (765.center) to (766.center);
[style=->] (767.center) to (768.center);
(888) to (889);
(892) to (889);
(912) to (889);
(925) to (893);
(924) to (910);
[style=->] (928.center) to (929.center);
(943) to (936);
(954) to (936);
(967) to (960);
(979) to (968);
(967) to (978);
(1013) to (1006);
(1022) to (1014);
(985) to (992);
(992) to (1025);
(1026) to (993);
(1029) to (1026);
(1030) to (1025);
(1033) to (1006);
[style=->] (1035.center) to (1036.center);
[style=->] (1037.center) to (1038.center);
[style=->] (1039.center) to (1040.center);
[style=->] (1041.center) to (1042.center);
[style=->] (1043.center) to (1044.center);
(1059) to (1061);
(1056) to (1059);
(1055) to (1067);
(1068) to (1056);
(1072) to (1067);
(1075) to (1060);
(1082) to (1056);
(680) to (670);
(670) to (685);
(654) to (642);
(642) to (667);
(642) to (659);
(697) to (707);
(697) to (1086);
(734) to (724);
(724) to (1087);
[style=->] (1088.center) to (1089.center);
(1103) to (1105);
(1100) to (1103);
(1099) to (1111);
(1112) to (1100);
(1115) to (1111);
(1117) to (1104);
(1120) to (1100);
[style=->] (1122.center) to (1123.center);
[style=->] (1126.center) to (1125.center);
[style=->] (1128.center) to (1127.center);
(1136) to (1139);
(1135) to (1147);
(1148) to (1136);
(1151) to (1147);
[style=->] (1159.center) to (1160.center);
[style=->] (1162.center) to (1161.center);
(1164) to (1168);
(1169) to (1165);
(1172) to (1168);
(1139) to (1165);
(1175) to (1139);
[style=brace] (641.center) to (1206.center);
[style=brace] (1205.center) to (616.center);
(1207) to (1086);
(1208) to (1087);
(1209) to (1085);
(1084) to (1213);
(1216) to (1214);
(958) to (968);
(958) to (1214);
(944) to (934);
(934) to (1215);
(1215) to (1217);
(955) to (934);
(1045) to (1222);
(1222) to (1004);
(1004) to (1014);
(1224) to (1223);
(1223) to (983);
(983) to (993);

Hasse diagram for the first orbits in the component of $\mathrm{Gr}_{\mathrm{PSL}(m_1 + m_2 + 2 , \mathbb{C})}$ corresponding to the lowest coweight $\Omega_{1+m_1}=[\underbrace{0\dots0}_{m_1}1\underbrace{0\dots 0}_{m_2}]$ for $m_1,m_2 \geq 6$. Blue nodes represent unbalanced gauge groups.

Non-linear $D_N$, $E_{6,7,8}$ quivers.
The treatment is almost identical to the $A$-type affine Grassmannians except for the novelty that one may add $d_n$, $e_6$, $e_7$, $e_8$ elementary slices. For $A$-type affine Grassmannian, we mentioned how $n$ unflavored gauge groups between two flavored gauge groups allow the addition of an $a_n$ slice. We can now generalize this by considering the subset of $n$ flavorless gauge nodes that are bounded by flavored gauge nodes. Then, one is allowed to add the elementary slice that is the closure of the minimal nilpotent orbit of $\mathfrak{g}_n$, where $\mathfrak{g}_n$ is the finite Dynkin diagram in the shape of the unflavored gauge nodes.  Let us demonstrate this with some slices of $\mathrm{E}_6$ affine Grassmannian:

\begin{equation}
\raisebox{-.5\height}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=gauge3] (0) at (0.5, 4) {};
\node [style=gauge3] (1) at (-0.5, 4) {};
\node [style=gauge3] (2) at (-1.5, 4) {};
\node [style=gauge3] (3) at (1.5, 4) {};
\node [style=gauge3] (4) at (2.5, 4) {};
\node [style=gauge3] (5) at (0.5, 5) {};
\node [style=flavour2] (18) at (0.5, 6) {};
\node [style=oliveX] (23) at (2.75, 7.25) {};
\node [style=oliveX] (24) at (3.75, 7.25) {};
\node [style=oliveX] (25) at (4.75, 7.25) {};
\node [style=oliveX] (26) at (5.75, 7.25) {};
\node [style=oliveX] (27) at (6.75, 7.25) {};
\node [style=olivesquare] (28) at (2.75, 8.25) {};
\node [style=olivesquare] (29) at (6.75, 8.25) {};
\node [style=none] (30) at (2.75, 8.75) {1};
\node [style=none] (31) at (6.75, 8.75) {1};
\node [style=none] (32) at (2.75, 6.75) {1};
\node [style=none] (33) at (3.75, 6.75) {1};
\node [style=none] (34) at (4.75, 6.75) {1};
\node [style=none] (35) at (5.75, 6.75) {1};
\node [style=none] (36) at (6.75, 6.75) {1};
\node [style=none] (37) at (4.75, 9) {add $a_5$};
\node [style=none] (38) at (-2, 4.5) {};
\node [style=none] (39) at (-2, 3.25) {};
\node [style=none] (40) at (3, 4.5) {};
\node [style=none] (41) at (3, 3.25) {};
\node [style=none] (94) at (-1.5, 3.5) {$k_a$};
\node [style=none] (95) at (-0.5, 3.5) {$k_b$};
\node [style=none] (96) at (0.5, 3.5) {$k_c$};
\node [style=none] (97) at (1.5, 3.5) {$k_d$};
\node [style=none] (98) at (2.5, 3.5) {$k_e$};
\node [style=none] (109) at (1, 5) {$k_f$};
\node [style=gauge3] (112) at (5.75, 7.25) {};
\node [style=gauge3] (113) at (6.75, 7.25) {};
\node [style=gauge3] (114) at (4.75, 7.25) {};
\node [style=gauge3] (115) at (3.75, 7.25) {};
\node [style=gauge3] (116) at (2.75, 7.25) {};
\node [style=flavour2] (128) at (2.75, 8.25) {};
\node [style=flavour2] (129) at (6.75, 8.25) {};
\node [style=none] (130) at (3.5, 4) {};
\node [style=none] (131) at (6.25, 4) {};
\node [style=gauge3] (136) at (9.25, 4) {};
\node [style=gauge3] (137) at (8.25, 4) {};
\node [style=gauge3] (138) at (7.25, 4) {};
\node [style=gauge3] (139) at (10.25, 4) {};
\node [style=gauge3] (140) at (11.25, 4) {};
\node [style=gauge3] (141) at (9.25, 5) {};
\node [style=flavour2] (142) at (9.25, 6) {};
\node [style=none] (147) at (7.25, 3.5) {$k_a{+}1$};
\node [style=none] (148) at (8.25, 3) {$k_b{+}1$};
\node [style=none] (149) at (9.25, 3.5) {$k_c{+}1$};
\node [style=none] (150) at (10.25, 3) {$k_d{+}1$};
\node [style=none] (151) at (11.25, 3.5) {$k_e{+}1$};
\node [style=none] (152) at (9.75, 5) {$k_f$};
\node [style=none] (153) at (1, 6) {$N$};
\node [style=none] (154) at (9.25, 6.5) {$N{-}1$};
\node [style=olivesquare] (199) at (7.25, 5) {};
\node [style=none] (200) at (7.25, 5.5) {1};
\node [style=flavour2] (201) at (7.25, 5) {};
\node [style=olivesquare] (202) at (11.25, 5) {};
\node [style=none] (203) at (11.25, 5.5) {1};
\node [style=flavour2] (204) at (11.25, 5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (1);
\draw (1) to (0);
\draw (0) to (5);
\draw (0) to (3);
\draw (3) to (4);
\draw (18) to (5);
\draw (23) to (24);
\draw (24) to (25);
\draw (26) to (25);
\draw (26) to (27);
\draw (28) to (23);
\draw (29) to (27);
\draw [style=Reddotted] (40.center) to (38.center);
\draw [style=Reddotted] (38.center) to (39.center);
\draw [style=Reddotted] (39.center) to (41.center);
\draw [style=Reddotted] (41.center) to (40.center);
\draw [style=->] (130.center) to (131.center);
\draw (138) to (137);
\draw (137) to (136);
\draw (136) to (141);
\draw (136) to (139);
\draw (139) to (140);
\draw (142) to (141);
\draw (201) to (138);
\draw (204) to (140);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

Here, we can add a $a_5$ slice to the nodes that make up a $a_5$ finite Dynkin diagram in the red box.

\begin{equation}
\raisebox{-.5\height}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=gauge3] (6) at (0.5, -3.25) {};
\node [style=gauge3] (7) at (-0.5, -3.25) {};
\node [style=gauge3] (8) at (-1.5, -3.25) {};
\node [style=gauge3] (9) at (1.5, -3.25) {};
\node [style=gauge3] (10) at (2.5, -3.25) {};
\node [style=gauge3] (11) at (0.5, -2.25) {};
\node [style=flavour2] (42) at (-1.5, -2.25) {};
\node [style=none] (44) at (-1, -1.75) {};
\node [style=none] (45) at (-1, -4) {};
\node [style=none] (46) at (3, -1.75) {};
\node [style=none] (47) at (3, -4) {};
\node [style=oliveX] (49) at (3.25, -0.25) {};
\node [style=oliveX] (50) at (4.25, -0.25) {};
\node [style=oliveX] (51) at (5.25, -0.25) {};
\node [style=oliveX] (52) at (6.25, -0.25) {};
\node [style=none] (58) at (3.25, -0.75) {1};
\node [style=none] (59) at (4.25, -0.75) {2};
\node [style=none] (60) at (5.25, -0.75) {2};
\node [style=none] (61) at (6.25, -0.75) {1};
\node [style=none] (62) at (4.75, 2) {add $d_5$};
\node [style=olivesquare] (63) at (5.25, 0.75) {};
\node [style=none] (64) at (5.25, 1.25) {1};
\node [style=oliveX] (65) at (4.25, 0.75) {};
\node [style=none] (66) at (4.25, 1.25) {1};
\node [style=none] (99) at (-1.5, -3.75) {$k_a$};
\node [style=none] (100) at (-0.5, -3.75) {$k_b$};
\node [style=none] (101) at (0.5, -3.75) {$k_c$};
\node [style=none] (102) at (1.5, -3.75) {$k_d$};
\node [style=none] (103) at (2.5, -3.75) {$k_e$};
\node [style=none] (110) at (1, -2.25) {$k_f$};
\node [style=gauge3] (117) at (4.25, -0.25) {};
\node [style=gauge3] (118) at (4.25, 0.75) {};
\node [style=gauge3] (119) at (3.25, -0.25) {};
\node [style=gauge3] (120) at (5.25, -0.25) {};
\node [style=gauge3] (121) at (6.25, -0.25) {};
\node [style=flavour2] (127) at (5.25, 0.75) {};
\node [style=none] (132) at (3.5, -3.25) {};
\node [style=none] (133) at (6.25, -3.25) {};
\node [style=none] (155) at (-1.5, -1.75) {$N$};
\node [style=gauge3] (156) at (9, -3.25) {};
\node [style=gauge3] (157) at (8, -3.25) {};
\node [style=gauge3] (158) at (7, -3.25) {};
\node [style=gauge3] (159) at (10, -3.25) {};
\node [style=gauge3] (160) at (11, -3.25) {};
\node [style=gauge3] (161) at (9, -2.25) {};
\node [style=flavour2] (162) at (7, -2.25) {};
\node [style=none] (167) at (7, -3.75) {$k_a$};
\node [style=none] (168) at (8, -3.75) {$k_b{+}1$};
\node [style=none] (169) at (9, -4.25) {$k_c{+}2$};
\node [style=none] (170) at (10, -3.75) {$k_d{+}2$};
\node [style=none] (171) at (11, -4.25) {$k_e{+}1$};
\node [style=none] (172) at (9, -1.75) {$k_f{+}1$};
\node [style=none] (173) at (7, -1.75) {$N{-}1$};
\node [style=olivesquare] (196) at (10, -2.25) {};
\node [style=none] (197) at (10, -1.75) {1};
\node [style=flavour2] (198) at (10, -2.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (8) to (7);
\draw (7) to (6);
\draw (6) to (11);
\draw (6) to (9);
\draw (9) to (10);
\draw (42) to (8);
\draw [style=Reddotted] (46.center) to (44.center);
\draw [style=Reddotted] (44.center) to (45.center);
\draw [style=Reddotted] (45.center) to (47.center);
\draw [style=Reddotted] (47.center) to (46.center);
\draw (49) to (50);
\draw (51) to (50);
\draw (51) to (52);
\draw (63) to (51);
\draw (65) to (50);
\draw [style=->] (132.center) to (133.center);
\draw (158) to (157);
\draw (157) to (156);
\draw (156) to (161);
\draw (156) to (159);
\draw (159) to (160);
\draw (162) to (158);
\draw (198) to (159);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

Here, we can add a $d_5$ slice to the nodes that make up a $d_5$ finite Dynkin diagram in the red box.

\begin{equation}
\raisebox{-.5\height}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=gauge3] (67) at (0.75, -11) {};
\node [style=gauge3] (68) at (-0.25, -11) {};
\node [style=gauge3] (69) at (-1.25, -11) {};
\node [style=gauge3] (70) at (1.75, -11) {};
\node [style=gauge3] (71) at (2.75, -11) {};
\node [style=gauge3] (72) at (0.75, -10) {};
\node [style=flavour2] (73) at (-1.25, -10) {};
\node [style=none] (74) at (-0.75, -9.5) {};
\node [style=none] (75) at (-0.75, -11.75) {};
\node [style=none] (76) at (2.25, -9.5) {};
\node [style=none] (77) at (2.25, -11.75) {};
\node [style=oliveX] (78) at (4, -7.75) {};
\node [style=oliveX] (79) at (5, -7.75) {};
\node [style=oliveX] (80) at (6, -7.75) {};
\node [style=none] (82) at (4, -8.25) {1};
\node [style=none] (83) at (5.5, -7.25) {2};
\node [style=none] (84) at (6, -8.25) {1};
\node [style=none] (86) at (5, -5.75) {add $d_4$};
\node [style=oliveX] (89) at (5, -6.75) {};
\node [style=none] (90) at (5, -6.25) {1};
\node [style=flavour2] (91) at (2.75, -10) {};
\node [style=olivesquare] (92) at (5, -8.75) {};
\node [style=none] (93) at (5, -9.25) {1};
\node [style=none] (104) at (-1.25, -11.5) {$k_a$};
\node [style=none] (105) at (-0.25, -11.5) {$k_b$};
\node [style=none] (106) at (0.75, -11.5) {$k_c$};
\node [style=none] (107) at (1.75, -11.5) {$k_d$};
\node [style=none] (108) at (2.75, -11.5) {$k_e$};
\node [style=none] (111) at (1.25, -10) {$k_f$};
\node [style=gauge3] (122) at (5, -7.75) {};
\node [style=gauge3] (123) at (6, -7.75) {};
\node [style=gauge3] (124) at (5, -6.75) {};
\node [style=gauge3] (125) at (4, -7.75) {};
\node [style=flavour2] (126) at (5, -8.75) {};
\node [style=none] (134) at (3.5, -11) {};
\node [style=none] (135) at (6.25, -11) {};
\node [style=none] (174) at (-1.25, -9.5) {$N_1$};
\node [style=none] (175) at (2.75, -9.5) {$N_2$};
\node [style=gauge3] (176) at (9.25, -11) {};
\node [style=gauge3] (177) at (8.25, -11) {};
\node [style=gauge3] (178) at (7.25, -11) {};
\node [style=gauge3] (179) at (10.25, -11) {};
\node [style=gauge3] (180) at (11.25, -11) {};
\node [style=gauge3] (181) at (9.25, -10) {};
\node [style=flavour2] (182) at (7.25, -10) {};
\node [style=flavour2] (187) at (11.25, -10) {};
\node [style=none] (188) at (7, -11.5) {$k_a$};
\node [style=none] (189) at (8.25, -11.5) {$k_b{+}1$};
\node [style=none] (190) at (8.75, -10.5) {$k_c{+}2$};
\node [style=none] (191) at (10.25, -11.5) {$k_d{+}1$};
\node [style=none] (192) at (11.5, -11.5) {$k_e$};
\node [style=none] (193) at (9.25, -9.5) {$k_f{+}1$};
\node [style=none] (194) at (7.25, -9.5) {$N_1{-}1$};
\node [style=none] (195) at (11.25, -9.5) {$N_2{-}1$};
\node [style=olivesquare] (205) at (9.25, -12) {};
\node [style=none] (206) at (9.25, -12.5) {1};
\node [style=flavour2] (207) at (9.25, -12) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (69) to (68);
\draw (68) to (67);
\draw (67) to (72);
\draw (67) to (70);
\draw (70) to (71);
\draw (73) to (69);
\draw [style=Reddotted] (76.center) to (74.center);
\draw [style=Reddotted] (74.center) to (75.center);
\draw [style=Reddotted] (75.center) to (77.center);
\draw [style=Reddotted] (77.center) to (76.center);
\draw (78) to (79);
\draw (80) to (79);
\draw (89) to (79);
\draw (91) to (71);
\draw (79) to (92);
\draw [style=->] (134.center) to (135.center);
\draw (178) to (177);
\draw (177) to (176);
\draw (176) to (181);
\draw (176) to (179);
\draw (179) to (180);
\draw (182) to (178);
\draw (187) to (180);
\draw (176) to (207);
\end{pgfonlayer}