text:
\section{Introduction}
\label{section.01}

\subsection{Background}
In a 2D conformal field theory, a correlation function is a sum of
(or an integral over) products of holomorphic and anti-holomorphic
conformal blocks. Methods to compute the conformal block include
\textbf{1.} making use of a null state that flows in an internal channel in
the block to derive and solve a differential equation for the block
\cite{belavin.polyakov.zamolodchikov},
\textbf{2.} representing the conformal block in terms of a Coulomb gas
of charges with a background charge and screening charges, then
evaluating the conformal block as an integral over the positions
of screening charges
\cite{nienhuis, dotsenko.fateev.01, dotsenko.fateev.02}, and
\textbf{3.} using the analytic properties of the conformal block to derive
and solve a recursion relation that can be solved for the block
\cite{zamolodchikov.01,zamolodchikov.02}.
These methods are powerful and lead to deep insights into the analytic structure
of the conformal blocks, but they are also non-algorithmic in the sense that the
answer cannot (in general) be written explicitly and directly, and become complicated to apply in
the presence of vertex operators of a sufficiently-large highest-weight charge
\footnote{\,
Vertex operators that require multiple screening charges in the Coulomb gas
approach.
},
and for 5- and higher-point conformal blocks
\footnote{\,
The more powerful (elliptic) version of the recursion relation is available
only for 4-point conformal blocks \cite{zamolodchikov.02, poghossian}.
}.

\subsection{Conformal blocks as products of normalized matrix elements}
An algorithmic approach to computing the conformal blocks is to regard them
as products of matrix elements
$\mathcal{M}^{\, 2D}$ of primary-state vertex operators between arbitrary descendant states,
these being normalized by Shapovalov matrix elements.
However, no closed-form expressions for these matrix elements are known
\footnote{\,
In $\mathcal{W}_2$ conformal field theories,
the Virasoro algebra $\mathcal{W}_2$ provides sufficiently-many constraints to
allow us to compute any Virasoro matrix element on an element-by-element
basis, but in $\mathcal{W}_N$ theories, $N= 3, 4, \cdots$, more conditions than
those provided by the $\mathcal{W}_N$ algebra are required \cite{kanno.matsuo.zhang},
and the computation of the matrix elements, while still on an element-by-element
basis, is more complicated than in the $\mathcal{W}_2$ case.
}.

\subsection{From 2D conformal blocks to 4D instanton partition functions}
An approach to compute the normalized matrix elements
$\mathcal{M}^{\, 2D}$ in closed form is the AGT correspondence
\cite{alday.gaiotto.tachikawa, mironov.morozov, wyllard.01}, which applies
to  $\mathcal{W}_N \times \mathcal{H}$ conformal field theories, where the $\mathcal{W}_N$ algebra,
generated by chiral spin-2, spin-3, $\cdots$, spin-$N$ currents, is augmented
by a Heisenberg algebra $\mathcal{H}$ generated by a chiral spin-1 current
\footnote{\,
The contributions of the $\mathcal{H}$ algebra factor out in the result
of conformal block computations in $\mathcal{W}_N \times \mathcal{H}$ theories, and
one obtains conformal blocks in $\mathcal{W}_N$ theories.
}.
The AGT correspondence identifies matrix element $\mathcal{M}^{\, 2D}$
in 2D $\mathcal{W}_N \times \mathcal{H}$ conformal field theories and instanton
partition functions $\mathcal{Z}^{\, 4D}_{\, N}$, in 4D $\mathcal{N} = 2$ supersymmetric
Yang-Mills theories with matter in bifundemental
$SU \left\lgroup N \right\rgroup$ representations.

\subsection{From 4D instanton partition functions to 5D strip partition
functions} The 4D instanton partition function $\mathcal{Z}^{\, 4D}_{\, N}$ is
engineered by taking the 4D limit of a 5D topological string strip partition
function $\mathcal{S}^{\, 5D}_{\, N}$ \cite{katz.klemm.vafa, katz.mayr.vafa},
obtained by gluing topological vertices
\cite{
iqbal, aganagic.klemm.marino.vafa,
awata.kanno.01, awata.kanno.02,
iqbal.kozcaz.vafa, iqbal.kashani-poor}.
In this sense, a topological vertex is the most fundamental building
block of the correlation functions in a 2D conformal field theory.

\subsection{2D parafermion conformal field theories}
In \cite{
alfimov.belavin.tarnopolsky,
alfimov.tarnopolsky,
belavin.belavin.bershtein,
belavin.bershtein.feigin.litvinov.tarnopolsky,
belavin.bershtein.tarnopolsky,
belavin.feigin,
belavin.mukhametzhanov,
belavin.wyllard,
bonelli.maruyoshi.tanzini.01,
bonelli.maruyoshi.tanzini.02,
itoyama.oota.yoshioka,
nishioka.tachikawa,
spodyneiko,
wyllard.02},
and other works, 4D instanton partition functions on $\mathbb{C}^{\, 2} / \mathbb{Z}_{\, n}$
were related \footnote{\,
The action of $\mathbb{Z}_{\, n}$ on $\mathbb{C}^{\, 2} = \mathbb{C}_{\, 1} \times \mathbb{C}_{\, 2}$ is
$ \left\lgroup z_{\, 1}, \, z_{\, 2} \right\rgroup
\mapsto
 \left\lgroup \omega \, z_{\, 1}, \, \omega^{\, -1} \, z_{\, 2} \right\rgroup$,
where
$z_{\, 1} \in \mathbb{C}_{\, 1},
\,
z_{\, 2} \in \mathbb{C}_{\, 2},
\,
\omega^{\, n} \, = \, 1$
\cite{belavin.bershtein.feigin.litvinov.tarnopolsky}
},
using an extension of the AGT correspondence, to matrix elements in non-rational
2D conformal field theories based on the algebra
$
\mathcal{A}  \left\lgroup N, n \right\rgroup =
 \left\lgroup
\widehat{sl}  \left\lgroup N \right\rgroup_n
\times
\widehat{sl}  \left\lgroup N \right\rgroup_p
\, / \,
\widehat{sl}  \left\lgroup N \right\rgroup_{n + p}
\right\rgroup
\times
\widehat{sl}  \left\lgroup n \right\rgroup_N
\times
\mathcal{H}
$
\footnote{\,
Our notation for $\mathcal{A}  \left\lgroup N, n \right\rgroup$ is adapted to that used in the present work
and as such it differs, in an obvious way, from that used in the original papers.
Further, we stress that the discussion of the conformal field theories in the original
papers on the subject, and definitely in the present work, is restricted to non-rational
conformal field theories. The space of states in rational theories based on
$\mathcal{A}  \left\lgroup N, n \right\rgroup$ contains degenerate representations with null states that
require special treatment.
}.
However, no connection with 5D topological string partition functions and topological
vertices was made.

\subsection{In this work}
\textbf{1.} We propose an extension of the refined topological vertex $\mathcal{C}$ of
\cite{iqbal.kozcaz.vafa},
constructed using a single free boson, to a refined topological
vertex $\mathcal{C}_n$ constructed using $n$ free bosons, and a conjugate
vertex $\mathcal{C}^{\, \star}_n$.
\textbf{2.} We compute $\mathcal{S}^{\, 5D}_{\, N, \, n}$, the 5D topological string strip
partition function that consists of $N$ pairs of vertices where
each pair consists of a single $\mathcal{C}_n$ vertex and a single $\mathcal{C}^{\, \star}_n$.
\textbf{3.} We take the 4D limit of
$\mathcal{S}^{\, 5D}_{\, N,  \,  n}$ of to obtain
$\mathcal{S}^{\, 4D}_{\, N,  \,  n}$, and identify the result, using geometric
engineering \cite{katz.klemm.vafa, katz.mayr.vafa}, with
$\mathcal{Z}^{\, 4D}_{\, N, \, n}$, the 4D instanton
partition function of matter in a bifundamental representation of
$\textit{SU}  \left\lgroup N \right\rgroup$ in $\mathbb{C}^{\, 4} / \mathbb{Z}_{\, n}$.
\textbf{4.} We use the AGT correspondence, as defined in
\cite{
alfimov.belavin.tarnopolsky,
alfimov.tarnopolsky,
belavin.belavin.bershtein,
belavin.bershtein.feigin.litvinov.tarnopolsky,
belavin.bershtein.tarnopolsky,
belavin.feigin,
belavin.mukhametzhanov,
belavin.wyllard,
bonelli.maruyoshi.tanzini.01,
bonelli.maruyoshi.tanzini.02,
spodyneiko,
wyllard.02},
to identify
$\mathcal{Z}^{\, 4D}_{\, N, \, n}$ with
a matrix element
$\mathcal{M}^{\, 2D}_{
\, \mathcal{A}  \left\lgroup  N, \, n \right\rgroup
}
 \left\lgroup \balpha_{\, L}, \, \balpha, \, \balpha_{\, R} \right\rgroup$
of a primary vertex operator that carries a highest-weight charge
$\balpha$ between left and right states,
$\langle \, \balpha_{\, L} \, |$ and $| \, \balpha_{\, R} \, \rangle$,
where
$\langle \, \balpha_{\, L} \, |$ and $| \, \balpha_{\, R} \, \rangle$
are arbitrary descendant states.
\textbf{5.} We obtain the linear relation between the K\"ahler parameters
of the 5D instanton partition function and the parameters of the
corresponding $\mathcal{A}  \left\lgroup N, \, n \right\rgroup$ matrix element,
\textbf{6.} We discuss in detail the differences in normalizations of these
objects, and show that these differences cancel out when gluing
matrix elements to compute conformal blocks.

\subsection{Outline of contents}
In section \textbf{2}, we recall the combinatorics of partitions and
Young diagrams that is used in the sequel,
and then in Section \textbf{3}, we do the same for
symmetric functions in infinitely-many
variables, with emphasis on the Schur
functions, Heisenberg
algebras, and correspondences between
them.
In \textbf{4}, we introduce the $n$-coloured refined topological vertex
$\mathcal{C}_{\, n}$, and the conjugate vertex $\mathcal{C}^{\, \star}_{\, n}$.
In \textbf{5}, we compute $\mathcal{S}^{\, 5D}_{\, N, \, n}$,
the 5D \textit{SU(N)} topological string strip partition function
made of $N$ pairs of vertices.
In \textbf{6}, we compute $\mathcal{S}^{\, 4D}_{\, N, \, n}$, the 4D limit of
$\mathcal{S}^{\, 5D}_{\, N, \, n}$, and identify it with the 4D instanton partition
function $\mathcal{Z}^{\, 4D}_{\, N, \, n}$.
In \textbf{7}, we use the 5D strips to compute 5D web diagrams, and then
in \textbf{8}, we take the 4D limit of the 5D web diagrams.
In \textbf{9}, we reproduce a 4-point conformal block computed in
\cite{alfimov.tarnopolsky}, and in \textbf{10}, we make a number of comments.

\section{Young and Maya diagrams}
\label{section.02.refined.topological.vertex}

\subsection{Young diagrams}
\label{partitions.young.diagrams}
A partition $Y =  \left\lgroup y_1, y_2, \cdots \right\rgroup$, of a non-negative integer
$|\,  Y \,|$, is a set of non-negative, non-increasing integers
$y_i \geqslant y_{i+1} \geqslant 0$, $\sum_{i = 1} y_i= | \, Y \, |$, and can
be represented as a Young diagram, that consists of rows such that
row $i$ has $y_i$ cells (see Figure \ref{young.diagram.01}).
We use $y_i$ for the $i$-th row as well as for the number of cells in
that row, and $Y^{\intercal}$ for the transpose of $Y$.
In this work, a Young diagram has infinitely-many rows.
By a finite Young diagram, we mean a Young diagram that
has fintely many non-null rows.
The null Young diagram
$Y = \emptyset$ is such that all rows are null.
Given a set of $n$ Young diagrams $\pmb{Y} =  \left\lgroup Y_1, \cdots, Y_n \right\rgroup$,
define

\begin{equation}
| \, \pmb{Y} \, | =
\sum_{i = 1}^n
| \, Y_i \, |,
\quad
\pmb{Y}^\intercal =
 \left\lgroup
Y^{\, \intercal}_1, \cdots,
Y^{\, \intercal}_n
\right\rgroup
\end{equation}

\subsubsection{The (infinite) borderline of a (finite) Young diagram}
The union of the positive $x$-axis and the negative $y$-axis, that
is the borderline of the south-east quadrant, is the borderline of
the null Young diagram $Y = \emptyset$.
The (infinite) borderline of a (finite) Young diagram is the union
of
the right vertical   boundaries of the right-most cells of each row,
the lower horizontal boundaries of the bottom     cells of each column,
the semi-infinite segment of the positive $x$-axis to the right, and
the semi-infinite segment of the negative $y$-axis below the Young
diagram, as in Figure \ref{young.diagram.01}.

\subsubsection{Cells}

We use $\square$ for a cell (a square) in the south-east quadrant
of the plane, and refer to the coordinates of $\square$ as $ \left\lgroup i, j \right\rgroup$.
If $\square \in Y$, then
$i$ is the $Y$-row-number,    counted from top to bottom, and
$j$ is the $Y$-column-number, counted from left to right,
that $\square$ lies in.
If $\square \notin Y$,
we still regard $i$ ($j$) as a $Y$-row-number ($Y$-column-number)
albeit that row (column) is null.
In other words, the coordinates
$ \left\lgroup i, j \right\rgroup$ of a cell are measured with respect to the (original)
boundaries of the south-east quadrant, rather than with respect to
the borderline of any specific Young diagram.

\subsubsection{Arms, legs, and hooks}
\label{arms.legs} Consider a cell $\square$ with coordinates
$ \left\lgroup i, j \right\rgroup$. We define the lengths of
the           arm $A^{  }_{\square, Y}$,
half-extended arm $A^{+ }_{\square, Y}$,
     extended arm $A^{++}_{\square, Y}$,
the           leg $L^{  }_{\square, Y}$,
half-extended leg $L^{+ }_{\square, Y}$,
     extended leg $L^{++}_{\square, Y}$, of $\square$
with respect to the Young diagram $Y$,

\begin{eqnarray}
A^{  }_{\square, Y} & = & y_i               - \, j, \quad
A^{+ }_{\square, Y}   =   A^{  }_{\square, Y} + \frac12,            \quad
A^{++}_{\square, Y}   =   A^{  }_{\square, Y} +      1,
\\
L^{  }_{\square, Y} & = & y_j^{\intercal} - \, i,   \quad
L^{+ }_{\square, Y}   =   L^{  }_{\square, Y} + \frac12,            \quad
L^{++}_{\square, Y}   =   L^{  }_{\square, Y} +      1
\end{eqnarray}

Note that $A_{\square,Y}$ and $L_{\square,Y}$ are negative when
$\square \notin Y$. The hook of a cell $\square$, with respect
to the borderline of a Young diagram $Y$ is,

\begin{equation}
H_{\square} = A_{\square, Y} + L_{\square, Y} + 1
\end{equation}

\begin{figure}
\begin{tikzpicture}[scale=.6]

\node at (- 1.0,   0.5) {$i = 1$};
\node at (- 1.0, - 0.5) {$i = 2$};
\node at (- 1.0, - 1.5) {$i = 3$};
\node at (- 1.0, - 2.5) {$i = 4$};
\node at (- 1.0, - 3.5) {$i = 5$};

\node at ( -0.2,   1.4) {$j =  $};
\node at (  0.5,   1.5) {$    1$};
\node at (  1.5,   1.5) {$    2$};
\node at (  2.5,   1.5) {$    3$};
\node at (  3.5,   1.5) {$    4$};
\node at (  4.5,   1.5) {$    5$};
\node at (  5.5,   1.5) {$    6$};
\node at (  6.5,   1.5) {$    7$};

\draw [thick] (0, 0) rectangle (1,1);
\draw [very thick] (5, 1.00)--(7, 1.00);
\draw [very thick] (5, 0.95)--(7, 0.95);
\draw [very thick] (5.00, 1)--(5.00, 0);
\draw [very thick] (5.05, 1)--(5.05, 0);
\draw [very thick] (4,  0.00)--(5.05,  0.00);
\draw [very thick] (4, -0.05)--(5.05, -0.05);
\draw [very thick] (4.00, 0)--(4.00, -1);
\draw [very thick] (4.05, 0)--(4.05, -1);
\draw [very thick] (2, -1.00)--(4.05, -1.00);
\draw [very thick] (2, -1.05)--(4.05, -1.05);
\draw [very thick] (2.00, -1)--(2.00, -2);
\draw [very thick] (2.05, -1)--(2.05, -2);
\draw [very thick] (0, -2.00)--(2.05, -2.00);
\draw [very thick] (0, -2.05)--(2.05, -2.05);
\draw [very thick] (0.00, -2)--(0.00, -4);
\draw [very thick] (0.05, -2)--(0.05, -4);

\draw [thick] (1, 0) rectangle (1,1);
\draw [thick] (2, 0) rectangle (1,1);
\draw [thick] (3, 0) rectangle (1,1);
\draw [thick] (4, 0) rectangle (1,1);
\draw [thick] (5, 0) rectangle (1,1);
\draw [thick] (0,-1) rectangle (1,1);
\draw [thick] (1,-1) rectangle (1,1);
\draw [thick] (2,-1) rectangle (1,1);
\draw [thick] (3,-1) rectangle (1,1);
\draw [thick] (4,-1) rectangle (1,1);
\draw [thick] (0,-2) rectangle (1,1);
\draw [thick] (1,-2) rectangle (1,1);
\draw [thick] (2,-2) rectangle (1,1);

\node at (1.5,-0.5) {$\text{\ding{51}}$};
\node at (4.5,-1.5) {$\text{\ding{55}}$};

\draw [thick] (0,-3)--(0.2,-3);
\draw [thick] (6,1)--(6,0.8);

\end{tikzpicture}
\caption{
{\it
The Young diagram $Y =  \left\lgroup 5, 4, 2 \right\rgroup$, and the corresponding
transpose Young diagram
$Y^{\, \intercal} =  \left\lgroup 3, 3, 2, 2, 1 \right\rgroup$.
The infinite borderline is shown in thick
lines.
The rows are numbered from top to bottom,
the columns are from left to right, with
null-rows and null-columns included.
The arm-length and leg-length of the box with $\text{\ding{51}}$ are $A_{\text{\ding{51}}, Y} = 2, A^+_{\text{\ding{51}}, Y} = 5/2, A^{++}_{\text{\ding{51}}, Y} = 3, L_{\text{\ding{51}}, Y} = 1, L^+_{\text{\ding{51}}, Y} = 3/2, L^{++}_{\text{\ding{51}}, Y} = 2$. Similarly for the box with $\text{\ding{55}}$, we have $A_{\text{\ding{55}}, Y} = -3, A^+_{\text{\ding{55}}, Y} = -5/2, A^{++}_{\text{\ding{55}}, Y} = -2, L_{\text{\ding{55}}, Y} = -2, L^+_{\text{\ding{55}}, Y} = -3/2, L^{++}_{\text{\ding{55}}, Y} = -1$.
}
}
\label{young.diagram.01}
\end{figure}

\subsubsection{Charged Young diagrams}
We define a charged Young diagram as a pair $ \left\lgroup Y, c_{\, Y} \right\rgroup$, where $Y$ is a finite Young
diagram and $c_{\, Y} \in \mathbb{Z}$ is the charge.

\subsection{Maya diagrams}\label{Maya.diagram.construction}
A Maya diagram is an infinite 1-dimensional lattice with a black stone
or a white stone on each segment, the segments are labelled by a position
coordinate $c_{\, p}$, such that
sufficiently far to the left, all stones are black, and
sufficiently far to the right, all stones are white \cite{miwa.jimbo.date.book}.

\subsubsection{The ground-state, 0-charge Maya diagram}
The simplest Maya diagram is the ground-state,  0-charge Maya diagram
$M_{\, 0}$, where
all segments from
position $ \left\lgroup c_{\, p} = - \infty \right\rgroup$, to
position $ \left\lgroup c_{\, p} = - 1 \right\rgroup$,
inclusive, carry black stones,
and
all segments from
position $ \left\lgroup c_{\, p} = 0 \right\rgroup$, to
position $ \left\lgroup c_{\, p} = \infty \right\rgroup$, inclusive,
carry white stones, see Figure \ref{young.maya.01}.
From $M_{\, 0}$, we can generate all other diagrams by applying
a charged Young diagram $ \left\lgroup Y, c_{\, Y} \right\rgroup$
to $M_{\, 0}$, such that, $Y$ shuffles the black and the white stones
at finite distances from the origin, and
$c_{\, Y}$ shifts
the positions of \textit{all} stones to the right by the same distance
$c_{\, Y}$.
The result is an excited state, charge-$c_{\, Y}$ Maya diagram.

\begin{figure}
\begin{center}
\begin{scriptsize}
\begin{tikzpicture}[scale = 0.6]

\draw [very thick] (-1.5,1.0)--(5.5,8.0)--(12.5,1.0);
\draw [very thick] (-1.55,1.05)--(5.5,8.1)--(12.55,1.05);

\draw [very thick] ( 6.3,6.8)--(6.5,7.0);
\draw [very thick] ( 7.3,5.8)--(7.5,6.0);
\draw [very thick] ( 8.3,4.8)--(8.5,5.0);
\draw [very thick] ( 9.3,3.8)--(9.5,4.0);
\draw [very thick] ( 10.3,2.8)--(10.5,3.0);
\draw [very thick] ( 11.3,1.8)--(11.5,2.0);
\draw [very thick] ( -0.5,2.0)--(-0.3,1.8);
\draw [very thick] ( 0.5,3.0)--(0.7,2.8);
\draw [very thick] ( 1.5,4.0)--(1.7,3.8);
\draw [very thick] ( 2.5,5.0)--(2.7,4.8);
\draw [very thick] ( 3.5,6.0)--(3.7,5.8);
\draw [very thick] ( 4.5,7.0)--(4.7,6.8);

\foreach \iota in {0,...,6}
{
\draw [thick, dashed, red] (\iota - .5, \iota + 1.5)--(\iota - .5, 0);
}
\foreach \j in {7,...,12}
{
\draw [thin, dashed, red](\j -.5, 13.5 -\j)--(\j - .5, 0);
}
\node [left]  at (-1.3,0) {$\cdots$};
\node [right] at (12.3,0) {$\cdots$};
\foreach x^{(1)} in {0,...,2}
{
\draw [fill=black!100] (x^{(1)}- 1,0) circle (0.3);
\draw (x^{(1)}+10,0) circle (0.3);
}

\draw [fill=black!100] (2,0) circle (0.3);
\draw [fill=black!100] (3,0) circle (0.3);
\draw [fill=black!100] (4,0) circle (0.3);
\draw [fill=black!100] (5,0) circle (0.3);
\draw (6,0) circle (0.3);
\draw (7,0) circle (0.3);
\draw (8,0) circle (0.3);
\draw (9,0) circle (0.3);

\foreach \a in {7, 6, ..., 1}
{
\node [below] at (- \a + 5.8, -.3) {$- \a$};
}
\foreach \a in {0, 1, ..., 6}
{
\node [below] at (  \a + 6.0, -.3) {$  \a$};
}
\node [below] at (-2.5, -.3) {$c_{\, p} = $};
\end{tikzpicture}
\end{scriptsize}
\end{center}
\caption{ \it The ground-state, 0-charge Maya diagram.
That no stones are shuffled corresponds to
introducing a null Young diagram.
That the charge is zero corresponds to
positioning the apex between $ \left\lgroup c_{\, p} = -1 \right\rgroup$, and
$ \left\lgroup c_{\, p} = 0 \right\rgroup$.}
\label{young.maya.01}
\end{figure}

\begin{figure}
\begin{center}
\begin{scriptsize}
\begin{tikzpicture}[scale = 0.6]

\draw [thin] (-1.5,1.0)--(5.5,8.0)--(12.5,1.0);
\draw [thin] ( 2.5,3.0)--(6.5,7.0);
\draw [thin] ( 3.5,2.0)--(7.5,6.0);
\draw [thin] ( 5.5,2.0)--(8.5,5.0);
\draw [thin] ( 8.5,3.0)--(9.5,4.0);
\draw [thin] ( 1.5,4.0)--(3.5,2.0);
\draw [thin] ( 2.5,5.0)--(5.5,2.0);
\draw [thin] ( 3.5,6.0)--(6.5,3.0);
\draw [thin] ( 4.5,7.0)--(8.5,3.0);

\draw [very thick] (-1.5, 1.0)--(1.5, 4.0);
\draw [very thick] (-1.45,0.95)--(1.5,3.9);

\draw [very thick] ( 1.5,4.0)--(3.5,2.0);
\draw [very thick] ( 1.5,3.9)--(3.5,1.9);

\draw [very thick] ( 3.5,2.0)--(4.5,3.0);
\draw [very thick] ( 3.5,1.9)--(4.5,2.9);

\draw [very thick] ( 4.5,3.0)--(5.5,2.0);
\draw [very thick] ( 4.5,2.9)--(5.5,1.9);

\draw [very thick] ( 5.5,2.0)--(7.5,4.0);
\draw [very thick] ( 5.5,1.9)--(7.5,3.9);

\draw [very thick] ( 7.5,4.0)--(8.5,3.0);
\draw [very thick] ( 7.5,3.9)--(8.5,2.9);

\draw [very thick] ( 8.5,3.0)--(9.5,4.0);
\draw [very thick] ( 8.5,2.9)--(9.5,3.9);

\draw [very thick] ( 9.5, 4.0)--(12.50, 1.00);
\draw [very thick] ( 9.5, 3.9)--(12.45, 0.95);

\draw [very thick] ( 10.3,2.8)--(10.5,3.0);
\draw [very thick] ( 11.3,1.8)--(11.5,2.0);
\draw [very thick] ( -0.5,2.0)--(-0.3,1.8);
\draw [very thick] ( 0.5,3.0)--(0.7,2.8);

\foreach \iota in {0,...,6}
{
\draw [thin, dashed, red] (\iota - .5, \iota + 1.5)--(\iota - .5, 0);
}
\foreach \j in {7,...,12}
{
\draw [thin, dashed, red](\j -.5, 13.5 -\j)--(\j - .5, 0);
}
\node [left]  at (-1.3,0) {$\cdots$};
\node [right] at (12.3,0) {$\cdots$};
\foreach x^{(1)} in {0,...,2}
{
\draw [fill=black!100] (x^{(1)}- 1,0) circle (0.3);
\draw                 (x^{(1)}+10,0) circle (0.3);
}

\draw                  (2,0) circle (0.3);
\draw                  (3,0) circle (0.3);
\draw [fill=black!100]  (4,0) circle (0.3);
\draw                  (5,0) circle (0.3);
\draw [fill=black!100]  (6,0) circle (0.3);
\draw [fill=black!100]  (7,0) circle (0.3);
\draw                  (8,0) circle (0.3);
\draw [fill=black!100]  (9,0) circle (0.3);

\foreach \a in {7, 6, ..., 1}
{
\node [below] at (- \a + 5.8, -.3) {$- \a$};
}
\foreach \a in {0, 1, ..., 6}
{
\node [below] at (  \a + 6.0, -.3) {$  \a$};
}

\node [below] at (-2.5, -.35) {$c_{p} = $};

\end{tikzpicture}
\end{scriptsize}
\end{center}
\caption{ \it An excited-state, 0-charge Maya diagram with a shuffling
that corresponds to $Y =  \left\lgroup 4,3,3,2 \right\rgroup$.
The apex of the Young diagram is between
position $ \left\lgroup c_{\, p} = -1 \right\rgroup$, and
position $ \left\lgroup c_{\, p} = 0 \right\rgroup$.
}
\label{young.maya.02}
\end{figure}

\subsubsection{Introducing a Young diagram}
Starting from the ground-state, 0-charge Maya diagram $M_{\, 0}$, we can
use a finite Young diagram $Y$ to shuffle the black and white stones
at finite distances from the origin and produce an excited-state Maya diagram
as follows.
\textbf{1.} We position the diagram $Y$ as in Figure \ref{young.maya.02}, so that
the apex of $Y$ projects on the point between positions $ \left\lgroup -1 \right\rgroup$
and $ \left\lgroup 0 \right\rgroup$ on the Maya diagram.
The infinite borderline of $Y$ then consists of upward and downward segments
$ \left\lgroup \diagup, \diagdown \right\rgroup$, where
all segments are $\diagup$   sufficiently far to the left, and
all segments are $\diagdown$ sufficiently far to the right.
\textbf{2.} We map the infinite borderline to a Maya diagram according to,

\begin{equation}
\diagup   \rightleftharpoons \CIRCLE,
\quad
\quad
\diagdown \rightleftharpoons \Circle
\end{equation}

The result is an excited-state, 0-charge Maya diagram, where the configuration
of the black and white stones is
in bijection with a finite Young diagram $Y$.
Note that the position coordinates
$c_{\, p}$ are the same as in the ground-state Maya diagram.

\subsubsection{Introducing a charge}
\label{charged.Maya}

\begin{figure}
\begin{center}
\begin{scriptsize}
\begin{tikzpicture}[scale = 0.6]

\draw [thin] (-1.5,1.0)--(5.5,8.0)--(12.5,1.0);
\draw [thin] ( 2.5,3.0)--(6.5,7.0);
\draw [thin] ( 3.5,2.0)--(7.5,6.0);
\draw [thin] ( 5.5,2.0)--(8.5,5.0);
\draw [thin] ( 8.5,3.0)--(9.5,4.0);
\draw [thin] ( 1.5,4.0)--(3.5,2.0);
\draw [thin] ( 2.5,5.0)--(5.5,2.0);
\draw [thin] ( 3.5,6.0)--(6.5,3.0);
\draw [thin] ( 4.5,7.0)--(8.5,3.0);

\draw [very thick] (-1.50, 1.00)--(1.5, 4.0);
\draw [very thick] (-1.45, 0.95)--(1.5, 3.9);

\draw [very thick] ( 1.5,4.0)--(3.5,2.0);
\draw [very thick] ( 1.5,3.9)--(3.5,1.9);

\draw [very thick] ( 3.5,2.0)--(4.5,3.0);
\draw [very thick] ( 3.5,1.9)--(4.5,2.9);

\draw [very thick] ( 4.5,3.0)--(5.5,2.0);
\draw [very thick] ( 4.5,2.9)--(5.5,1.9);

\draw [very thick] ( 5.5,2.0)--(7.5,4.0);
\draw [very thick] ( 5.5,1.9)--(7.5,3.9);

\draw [very thick] ( 7.5,4.0)--(8.5,3.0);
\draw [very thick] ( 7.5,3.9)--(8.5,2.9);

\draw [very thick] ( 8.5,3.0)--(9.5,4.0);
\draw [very thick] ( 8.5,2.9)--(9.5,3.9);

\draw [very thick] ( 9.5, 4.00)--(12.50, 1.00);
\draw [very thick] ( 9.5, 3.90)--(12.45, 0.95);

\draw [very thick] ( 10.3,2.8)--(10.5,3.0);
\draw [very thick] ( 11.3,1.8)--(11.5,2.0);
\draw [very thick] ( -0.5,2.0)--(-0.3,1.8);
\draw [very thick] ( 0.5,3.0)--(0.7,2.8);

\foreach \iota in {0,...,6}
{
\draw [thin, dashed, red] (\iota - .5, \iota + 1.5)--(\iota - .5, 0);
}
\foreach \j in {7,...,12}
{
\draw [thin, dashed, red](\j -.5, 13.5 -\j)--(\j - .5, 0);
}
\node [left]  at (-1.3,0) {$\cdots$};
\node [right] at (12.3,0) {$\cdots$};
\foreach x^{(1)} in {0,...,2}
{
\draw [fill=black!100] (x^{(1)}- 1,0) circle (0.3);
\draw                 (x^{(1)}+10,0) circle (0.3);
}

\draw                  (2,0) circle (0.3);
\draw                  (3,0) circle (0.3);
\draw [fill=black!100]  (4,0) circle (0.3);
\draw                  (5,0) circle (0.3);
\draw [fill=black!100]  (6,0) circle (0.3);
\draw [fill=black!100]  (7,0) circle (0.3);
\draw                  (8,0) circle (0.3);
\draw [fill=black!100]  (9,0) circle (0.3);

\foreach \a in {5, 4, ..., 1}
{
\node [below] at (- \a + 3.8, -.3) {$- \a$};
}
\foreach \a in {0, 1, ..., 8}
{
\node [below] at (  \a + 4.0, -.3) {$  \a$};
}

\node [below] at (-3, -.35) {$c_{p} + c_Y = $};

\end{tikzpicture}
\end{scriptsize}
\end{center}
\caption{ \it An excited-state charge-2 Maya diagram, with a shuffling
that corresponds to $Y =  \left\lgroup 4, 3, 3, 2 \right\rgroup$.
The apex of the Young diagram is between
position $ \left\lgroup c_{\, p} = 1 \right\rgroup$, and
position $ \left\lgroup c_{\, p} = 2 \right\rgroup$.
}
\label{young.maya.04}
\end{figure}

We can introduce a charge $c_{\, Y}$ into any 0-charge Maya diagram
by shifting all stones globally by $c_{\, Y}$ segments to the right, as
in Figure \ref{young.maya.04}.
A stone that is at position $c_{p}$ before introducing the charge is
shifted to position $ \left\lgroup c_{\, p} + c_{\, Y} \right\rgroup$.

\section{Symmetric functions and Heisenberg algebras}
\label{section.03}
\textit{We recall basic definitions related to symmetric functions,
Heisenberg algebras and relations between them.
}

\subsection{Exponentiated sequences}
Given any two sequences of integers $ \pmb{a} =  \left\lgroup a_1,a_2,\cdots \right\rgroup,  \pmb{b} =  \left\lgroup b_1,b_2,\cdots \right\rgroup$ and two variables $x, y$ we define

\begin{equation}
    x^{\,  \pmb{a}} \, y^{\,  \pmb{b}} =
 \left\lgroup x^{a_1} y^{b_1}, x^{a_2} y^{b_2}, \cdots \right\rgroup,
\end{equation}

In particular, given a Young diagram
$Y =  \left\lgroup y_1,y_2,\cdots \right\rgroup$,
and an infinite sequence of integers $\pmb{\iota } =  \left\lgroup 1,2,\cdots \right\rgroup$, we have

\begin{equation}
x^{\, \pmb{\iota }}\, y^{\, \pm Y}  =  \left\lgroup x  \, y^{\, \pm y_1}, x^2\, y^{\, \pm y_2} \cdots \right\rgroup,
\quad
x^{\, \pmb{\iota } - 1}\, y^{\, \pm Y}  =  \left\lgroup y^{\, \pm y_1}, x\, y^{\, \pm y_2} \cdots \right\rgroup,
\quad
\cdots
\end{equation}

For the purposes of section \ref{section.04},
we define the sub-sequence,

\begin{equation}
\label{subsequence.symbol}
[x^{\,  \pmb{a}} y^{\,  \pmb{b}}]_c =
 \left\lgroup x^{\, a_i} y^{\, b_i} \, | \, i =
1, 2, \cdots,
a_i + b_i = c \, \mod{\, n} \right\rgroup
\end{equation}

For example, for $n = 3$ and
$Y =  \left\lgroup 4, 3, 3, 2 \right\rgroup$,
$x^{\, \pmb{\iota }} \, y^{\, -Y} =
 \left\lgroup x y^{-4}, x^2 y^{-3}, x^3 y^{-3}, x^4 y^{-2}, x^5, x^6 \cdots \right\rgroup$,
we have
$[x^{\, \pmb{\iota }} \, y^{\, -Y}]_0 =
 \left\lgroup
x\, y^{-4}, x^3 \, y^{-3}, x^6, \cdots
\right\rgroup$.

\subsection{The power-sum symmetric functions}
Given
$ \pmb{x} =  \left\lgroup x_1, x_2, \cdots \right\rgroup$,
the power-sum symmetric function
$p_n  \left\lgroup  \pmb{x} \right\rgroup$,
$n \in  \left\lgroup 0, 1, \cdots \right\rgroup$,
is
\footnote{\,
Ch. I, p. 23, in \cite{macdonald.book}
},

\begin{equation}
p_{\, 0}  \left\lgroup  \pmb{x} \right\rgroup = 1,
\quad
p_{\, n}  \left\lgroup  \pmb{x} \right\rgroup = \sum_{\, i} x_i^{\, n}
,
\quad
n = 1, 2, \cdots,
\end{equation}

and $p_{\, Y}  \left\lgroup  \pmb{x} \right\rgroup$, indexed by a Young diagram $Y =  \left\lgroup y_1, y_2, \cdots \right\rgroup$, is
\footnote{\,
Ch. I, p. 24, in \cite{macdonald.book}
},

\begin{equation}
p_{\, Y}    \left\lgroup  \pmb{x} \right\rgroup \, = \,
p_{\, y_1}  \left\lgroup  \pmb{x} \right\rgroup\,
p_{\, y_2}  \left\lgroup  \pmb{x} \right\rgroup\,
\cdots
\end{equation}

\subsection{The inner product of power-sum functions}
\label{power.sum.inner.product.schur.basis}
Consider the ring of symmetric functions in a set of variables $ \pmb{x} =  \left\lgroup x_1, x_2 \cdots \right\rgroup$,
with constant coefficients. In this case, the power-sum functions are
defined to be orthogonal with inner product
\cite{macdonald.book}

\begin{equation}
\langle\, p_{\, Y_1}  \left\lgroup  \pmb{x} \right\rgroup\, |\, p_{\, Y_2}  \left\lgroup  \pmb{x} \right\rgroup \rangle_{\, q \, t} =
z_{\, Y_1}\, \delta_{Y_1 Y_2},
\quad
z_Y
=
1^{n_1}  \left\lgroup n_1 ! \right\rgroup
2^{n_2}  \left\lgroup n_2 ! \right\rgroup
\cdots,
\label{young.diagram.power.sum.inner.product}
\end{equation}

\subsection{The Schur function}
Given a Young diagram
$Y =  \left\lgroup y_1, y_2, \cdots \right\rgroup$,
$Y^{\, \intercal} =
 \left\lgroup
y^{\, \intercal}_1,
y^{\, \intercal}_2, \cdots \right\rgroup$,
and a set of variables
$ \pmb{x} =  \left\lgroup x_1, x_2, \cdots, x_n \right\rgroup$,
such that
$| \,  \pmb{x} \, | \geqslant y^{\, \intercal}_1$,
the Schur function
$s_{\, Y_1}  \left\lgroup\,  \pmb{x} \, \right\rgroup$
is
\footnote{\,
Ch. I, p. 40, in \cite{macdonald.book}
},

\begin{equation}
    s_Y \left\lgroup  \pmb{x} \right\rgroup = \frac{\operatorname{det} \left\lgroup x_i^{y_j + n - j} \right\rgroup_{1\leqslant i,j \leqslant n}}{\prod_{1\leqslant i,j \leqslant n} \left\lgroup x_i - x_j \right\rgroup}
\end{equation}

\subsection{The skew Schur function}
For two partitions $Y_1$ and $Y_2$, the product of the Schur
functions,

\begin{equation}
s_{\, Y_1} \, s_{\, Y_2} =
\sum_{\, Y} c^{\, Y}_{\, Y_1 \, Y_2} \, s_{\, Y}
\end{equation}

defines the integers $c^{\, Y}_{\, Y_1 \, Y_2}$, and
the skew Schur symmetric functions are then defined as
\footnote{\,
Ch. I, p. 69, in \cite{macdonald.book}
},

\begin{equation}
s_{\, Y / Y_1} =
\sum_{\, Y_2} c^{\, Y}_{\, Y_1 \, Y_2} s_{\, Y_2}
\end{equation}

By definition,

\begin{equation}\label{schur.id.4}
s_{\emptyset / \emptyset}  \left\lgroup  \pmb{x} \right\rgroup = 1,
\quad
s_{\emptyset/\eta} \left\lgroup  \pmb{x} \right\rgroup = 0, \textup{\, if $\eta \neq \emptyset$},
\quad
Q^{|\lambda| - |\eta|} \,
s_{\lambda/\eta}  \left\lgroup  \pmb{x} \right\rgroup =
\, s_{\lambda/\eta}  \left\lgroup Q  \pmb{x} \right\rgroup
\end{equation}

\subsection{Cauchy identities}
The (skew) Schur functions satisfy the Cauchy identities
\footnote{\,
Ch. I, p. 93, in \cite{macdonald.book}
},

\begin{equation}
\sum_Y
s_{Y    /   Y_2}  \left\lgroup\, \pmb{x}\, \right\rgroup \,
s_{Y    /   Y_1}  \left\lgroup\, \pmb{y}\, \right\rgroup
=
\prod_{i,\, j \, = \, 1}^\infty
\frac{
1}{
1\, -\, x_i\, y_j
}
\sum_Y
 \,
s_{Y_1 / Y}  \left\lgroup\, \pmb{x}\, \right\rgroup\,
s_{Y_2 / Y}  \left\lgroup\, \pmb{y}\, \right\rgroup
\label{schur.id.1}
\end{equation}

\begin{equation}
\sum_Y
s_{Y^{\, \intercal} / Y_2^{\, \intercal}}  \left\lgroup\, \pmb{x}\, \right\rgroup
s_{Y        / Y_1  }  \left\lgroup\, \pmb{y}\, \right\rgroup
=
\prod_{i, j = 1}^\infty  \left\lgroup 1 + x_i\, y_j \right\rgroup
\sum_Y
s_{Y_1^{\, \intercal} / Y^{\, \intercal}}  \left\lgroup\, \pmb{x}\,\right\rgroup\,
s_{Y_2        / Y  }  \left\lgroup\, \pmb{y}\,\right\rgroup
\label{schur.id.2}
\end{equation}

\subsection{The Heisenberg algebra}

In this work, a Heisenberg algebra $\mathcal{H}$ is the infinite-dimensional algebra generated
by
$ \pmb{a}_+ =  \left\lgroup a_{+1}, a_{+2},\cdots \right\rgroup$ and
$ \pmb{a}_- =  \left\lgroup a_{-1}, a_{-2},\cdots \right\rgroup$,
that satisfy the commutation relations,

\begin{equation}
[a_m, a_n] = m \, \delta_{n + m, 0}
\label{heisenberg.commutation.relations}
\end{equation}

act on the left-state as creation and annihilation
operators, respectively, and on the  right-state as
annihilation and creation operators, respectively.

\subsection{The left- and right-Heisenberg states}
The left-Heisenberg state  $ \langle \,  \pmb{a}_{ Y} \, |$, and
the right-Heisenberg state $ |       \,  \pmb{a}_{ Y} \, \rangle$,
$ Y =  \left\lgroup y_1, y_2, \cdots \right\rgroup$,
are generated from the left- and the right-vacuum states,

\begin{equation}
\langle\,  \pmb{a}_Y\, | =  \langle\, 0\, | \,
a_{\, y_1}\,
a_{\, y_2}\,
\cdots,
\quad
|\,  \pmb{a}_Y\, \rangle =
\cdots
a_{ - y_2}\,
a_{ - y_1}\,
|\, 0\, \rangle
\end{equation}

Using the Heisenberg commutation relations,
the inner product of
$ \langle\,  \pmb{a}_{\, Y_1}\, |$
and
$ |\,  \pmb{a}_{\, Y_2}\,\rangle$
is,

\begin{equation}
\langle\, a_{\, Y_1}\, |\, a_{\, Y_2}\,\rangle =
z_{ Y_1}\, \delta_{ Y_1 Y_2},
\quad
z_Y
=
1^{n_1}  \left\lgroup n_1 ! \right\rgroup
2^{n_2}  \left\lgroup n_2 ! \right\rgroup
\cdots.
\label{inner.product.right.left.states}
\end{equation}

\subsection{The power-sum/Heisenberg correspondence}

From the inner products
\ref{young.diagram.power.sum.inner.product}
and
\ref{inner.product.right.left.states},
we deduce that the power-sum basis
spanned by $p  \left\lgroup  \pmb{x} \right\rgroup$ is isomorphic
to the left-state Heisenberg basis spanned
by $\langle\,a_Y\, |$, as well as
the right-state Heisenberg basis
spanned by $|\, a_Y\, \rangle$,

\begin{equation}
p_n  \left\lgroup  \pmb{x} \right\rgroup \rightleftharpoons - \, a_{ n},
\quad
p_n  \left\lgroup  \pmb{x} \right\rgroup \rightleftharpoons - \, a_{-n}, \quad
n \geqslant 1.
\label{power.sum.heisenberg.correspondence.dual}
\end{equation}

\subsection{The left- and right-Schur states}

Expanding the Schur functions in terms of the power-sum functions,
then formally replacing the latter with Heisenberg generators, we
obtain operator-valued Schur functions that act on left- and right-vacuum states to produce left- and right-Schur states,

\begin{equation}
\langle \, s_{\, Y} \, | =  \langle\, 0\, | \, s_{\, Y} \,  \left\lgroup  \pmb{a}_{\, +} \right\rgroup,
\quad
| \, s_{\, Y} \, \rangle = \, s_{\, Y}  \left\lgroup  \pmb{a}_{\, -} \right\rgroup  |\, 0 \, \rangle.
\end{equation}

The left- and the right-Schur states satisfy the orthogonality condition,

\begin{equation}
\langle\,
s_{\, Y_1}   \left\lgroup\,  \pmb{x}\, \right\rgroup
\, |\,
s_{\, Y_2}   \left\lgroup\,  \pmb{x}\, \right\rgroup
\rangle =
\delta_{Y_1 Y_2}.
\label{schur.inner.product}
\end{equation}

\subsection{The $\Gamma$-operators}
To the Heisenberg algebra
$\mathcal{H}$,
we associate the operators $\Gamma_{\pm}$,

\begin{equation}\label{q.t.vertex.operators}
\Gamma_{\pm}  \left\lgroup x \right\rgroup =
\exp
 \left\lgroup
- \sum_{n=1}^\infty
\frac{
x^{\mp n}
}{
n
}
\,
a_{\pm\, n}
\right\rgroup
\end{equation}

Using the Heisenberg commutation relations Equation
\ref{heisenberg.commutation.relations}, we obtain

\begin{equation}
\label{q.t.vertex.operator.commutation.relation}
\Gamma_{+}  \left\lgroup x^{-1} \right\rgroup
\Gamma_{-}  \left\lgroup y      \right\rgroup =
 \left\lgroup
\frac{
1
}{
1 - xy
}
\right\rgroup
\Gamma_{-}  \left\lgroup y      \right\rgroup
\Gamma_{+}  \left\lgroup x^{-1} \right\rgroup.
\end{equation}

\subsection{The action of the $\Gamma_{\pm}$ operators on Schur states}
In \cite{foda.wu.02, foda.zhu}, identities that describe the action of
the $\Gamma_{\pm}$ operators on states labelled by Macdonald and
$q$-Whittaker symmetric functions, respectively, were derived. In the
Schur limit, these identities reduce to,

\begin{equation}
\langle\, s_{\, Y_1}\, |\,
\prod_{i \, =\, 1}^\infty
\Gamma_{-}  \left\lgroup y_j \right\rgroup  =
\sum_Y
\langle\, s_Y  \, |\,
s_{Y_1 / Y}  \left\lgroup  \pmb{y} \right\rgroup,
\quad
\prod_{i \, =\, 1}^\infty
\Gamma_+   \left\lgroup x^{\, -1}_i \right\rgroup\, |\, s_{\, Y_1}\, \rangle =
\sum_Y
s_{Y_1 / Y}  \left\lgroup\,  \pmb{x} \,\right\rgroup\,
|\, s_Y \rangle .
\label{two.identities}
\end{equation}

\section{The $n$-coloured refined topological vertex}
\label{section.04}

\textit{We construct the $n$-coloured refined topological
vertex using $n$ free bosons.}

The $n$-coloured refined topological vertex is
a trivalent vertex with
an incoming leg labelled by $n$ Young diagrams
$\pmb{Y}_1 =  \left\lgroup Y_{\, 1, 0},\cdots,Y_{\, 1, \, n-1} \right\rgroup$,
an outgoing leg labelled by
$n$ Young diagrams
$\pmb{Y}_2 =  \left\lgroup Y_{\, 2, 0},\cdots,Y_{\, 2, \, n-1} \right\rgroup$,
and a preferred leg labelled by a single
charged Young diagram $ \left\lgroup Y, c_Y \right\rgroup$,
$c_{\, Y} \in  \left\lgroup 0, 1, \cdots, n - 1 \right\rgroup$.
We construct the $n$-coloured vertex in six steps
\footnote{\,
The construction of the $n$-coloured vertex and the conjugate
vertex, in this section, was guided by the form of
the instanton partition function on
$\mathbb{C}^{\, 2} / \mathbb{Z}_{\, n}$ of the Landau School
\cite{
alfimov.belavin.tarnopolsky,
alfimov.tarnopolsky,
belavin.belavin.bershtein,
belavin.bershtein.feigin.litvinov.tarnopolsky,
belavin.bershtein.tarnopolsky,
belavin.feigin,
belavin.mukhametzhanov}
}.

\subsection{\textbf{1.} Introduce $n$ species of free bosons}
Instead of a single Heisenberg algebra, as in the case of the refined
topological vertex, we work in terms of $n$ commuting Heisenberg algebras
$\mathcal{H}_{\, m}$,
$m \in  \left\lgroup 0, 1, \cdots, n-1 \right\rgroup$. From now on, all operators that belong
to $\mathcal{H}_{\, m}$ will carry a subscript $m$. In particular, we
have $\Gamma_{\, m \, \pm}$, $m \in  \left\lgroup 0, 1, \cdots, n -1 \right\rgroup$.

\subsection{\textbf{2.} Introduce an excited-state, charged Maya diagram}

We start from a ground-state, 0-charge Maya diagram and use a charged
Young diagram $ \left\lgroup Y, c_Y \right\rgroup$ as in Section \ref{Maya.diagram.construction}
to produce an excited-state, charge-$c_{\, Y}$ Maya diagram.
The position of each stone is shifted from $c_{\, p}$ to
$ \left\lgroup c_{\, p} + c_{\, Y} \right\rgroup$.

\subsection{\textbf{3.} Introduce the $\Gamma$-operators}
We map the excited-state, charge-$c_{\, Y}$ Maya diagram to an infinite
sequence of $\Gamma$-vertex operators $\prod_{Maya  \left\lgroup Y \right\rgroup} \Gamma_{\pm}$,
using the bijections

\begin{equation}
\diagup   \rightleftharpoons \Circle \rightleftharpoons \Gamma_{\, c_H \, +},
\quad
\quad
\diagdown \rightleftharpoons \CIRCLE \rightleftharpoons \Gamma_{\, c_H \, -},
\quad
\quad
c_{\, H} =  \left\lgroup c_{\, p} + c_Y \right\rgroup \, \mod{\, n},
\end{equation}

for each stone at position $ \left\lgroup c_{\, p} + c_Y \right\rgroup$. We call $c_H$ the Heisenberg
charge.

\subsection{\textbf{4.} Assign arguments to the vertex operators}
We take the arguments of the $\Gamma$-operators to be,

\begin{equation}
\Gamma_{\, c_H \, +}  \left\lgroup x^{  - i}\, y^{  Y_{ i}            }   \right\rgroup,
\quad
\Gamma_{\, c_H \, -}  \left\lgroup y^{j - 1}\, x^{- Y^{\intercal}_{j}}   \right\rgroup,
\label{full.arguments}
\end{equation}

where
$Y_{\, i}$ is the length of the $i$-row of the Young diagram $Y$ that labels the
preferred leg of the vertex, and
$Y^{\intercal}_{j}$ is the length of the $j$-column of the Young diagram
$Y$. The sum of the exponents of the arguments of $\Gamma_{\pm}$ is related to $c_{p}$ as follows.
For $\Gamma_{\, c_H \, +}  \left\lgroup x^{  - i}\, y^{  Y_{i}            }   \right\rgroup$ at position $ \left\lgroup c_{\, p} + c_{\, Y} \right\rgroup$,

\begin{equation}
c_{\, p} = Y_{i} - i,
\end{equation}

and for $\Gamma_{\, c_H \, -}  \left\lgroup y^{j - 1}\, x^{- Y^{\, \intercal}_j}   \right\rgroup$ at
position $ \left\lgroup c_{p} + c_{\, Y} \right\rgroup$,

\begin{equation}
    c_{p} = -Y^{\, \intercal}_{j} + j - 1
\end{equation}

\subsection{\textbf{5.} From the infinite sequence of $\Gamma$-operators to an expectation value}
We evaluate the sequence
$\prod_{Maya  \left\lgroup Y \right\rgroup} \Gamma_{\pm}$
between a left-state,

\begin{equation}
\bra{s_{\mathbf{Y_1}}} =
\bra{s_{Y_{1, \, 0}}} \otimes \cdots \otimes \bra{s_{Y_{1, \, (n-1)}}} =
\bra{0}
s_{Y_{1, \, 0}}
 \left\lgroup  \pmb{a}_{1+} \right\rgroup
\cdots
s_{Y_{1, \, (n-1)}}
 \left\lgroup
 \pmb{a}_{n+}
\right\rgroup
\end{equation}

and a right-state,

\begin{equation}
\ket{s_{\mathbf{Y_2}}} =
\ket{s_{Y_{2, \, 0}}} \otimes \cdots \otimes \ket{s_{Y_{2, \, (n-1)}}} =
s_{Y_{2, \, 0}}  \left\lgroup  \pmb{a}_{1-} \right\rgroup \cdots
s_{Y_{2, \, (n-1)}}
 \left\lgroup  \pmb{a}_{n-} \right\rgroup
\ket{0}
\end{equation}

to get the unnormalized $n$-coloured refined topological vertex

\begin{equation}\label{unnormalized.vertex}
\mathcal{C}_{\pmb{Y}_1 \, \pmb{Y}_2 \, Y}^{\, p, \,  c_{\, Y}, \, unnorm}
 \left\lgroup x, y \right\rgroup =
\bra{s_{\pmb{Y}_1}}
 \left\lgroup
\prod_{
Maya  \left\lgroup Y \right\rgroup
}
\Gamma_{\pm}
\right\rgroup
\ket{s_{\pmb{Y}_2}}
\end{equation}

For $n = 3$ and  $Y =  \left\lgroup 4, 3, 3, 2 \right\rgroup$
(see Figure  \ref{young.maya.04}),
the sequences for the possible
$c_{\, Y}$ are
\footnote{\,
We list all three possibilities, then we choose one.
},

\begin{multline}
\mathcal{C}^{\, 3, \, c_{\, Y} = 0, \, unnorm}_{\pmb{Y_1} \pmb{Y_2} Y}  \left\lgroup x, y \right\rgroup =
\bra{s_{\pmb{Y_1}}}
\cdots
\Gamma_{1+}  \left\lgroup x^{-5}        \right\rgroup
\Gamma_{2-}  \left\lgroup x^{-4}        \right\rgroup
\Gamma_{0-}  \left\lgroup y      x^{-4} \right\rgroup
\Gamma_{1+}  \left\lgroup x^{-4} y^2    \right\rgroup
\Gamma_{2-}  \left\lgroup y^2    x^{-3} \right\rgroup
\\
\Gamma_{0+}  \left\lgroup x^{-3} y^{ 3} \right\rgroup
\Gamma_{1+} \left\lgroup  x^{-2} y^{ 3} \right\rgroup
\Gamma_{2-} \left\lgroup  y^{3}  x^{-1} \right\rgroup
\Gamma_{0+} \left\lgroup  x^{-1} y^{ 4} \right\rgroup
\Gamma_{1-} \left\lgroup  y^4           \right\rgroup
\cdots
\ket{s_{\pmb{Y_2}}}
\end{multline}

\begin{multline}
\mathcal{C}^{\, 3, \, c_{\, Y} = 1, \, unnorm}_{\pmb{Y_1} \pmb{Y_2} Y}  \left\lgroup x,y \right\rgroup =
\bra{s_{\pmb{Y_1}}}
\cdots
\Gamma_{2+} \left\lgroup x^{-5}         \right\rgroup
\Gamma_{0-} \left\lgroup x^{-4}         \right\rgroup
\Gamma_{1-} \left\lgroup y       x^{-4} \right\rgroup
\Gamma_{2+} \left\lgroup x^{-4}  y^2    \right\rgroup
\Gamma_{0-} \left\lgroup y^2     x^{-3} \right\rgroup
\\
\Gamma_{1+}  \left\lgroup x^{-3}y^{3} \right\rgroup
\Gamma_{2+}  \left\lgroup x^{-2}y^{3} \right\rgroup
\Gamma_{0-}  \left\lgroup y^{3}x^{-1} \right\rgroup
\Gamma_{1+}  \left\lgroup x^{-1}y^{4} \right\rgroup
\Gamma_{2-}  \left\lgroup y^4 \right\rgroup
\cdots
\ket{s_{\pmb{Y_2}}}
\end{multline}

\begin{multline}
\mathcal{C}^{\, 3, \, c_{\, Y} = 2, \, unnorm}_{\pmb{Y_1} \pmb{Y_2} Y}  \left\lgroup x,y \right\rgroup =
\bra{s_{\pmb{Y_1}}}
\cdots
\Gamma_{0+} \left\lgroup x^{-5}         \right\rgroup
\Gamma_{1-} \left\lgroup x^{-4}         \right\rgroup
\Gamma_{2-} \left\lgroup y       x^{-4} \right\rgroup
\Gamma_{0+} \left\lgroup x^{-4}  y^2    \right\rgroup
\Gamma_{1-} \left\lgroup y^2     x^{-3} \right\rgroup
    \\
\Gamma_{2+}  \left\lgroup x^{-3} y^{3} \right\rgroup
\Gamma_{0+} \left\lgroup x^{-2}  y^{3} \right\rgroup
\Gamma_{1-} \left\lgroup y^{3}  x^{-1} \right\rgroup
\Gamma_{2+} \left\lgroup x^{-1} y^{4}  \right\rgroup
\Gamma_{0-} \left\lgroup y^4           \right\rgroup
\cdots
\ket{s_{\pmb{Y_2}}}
\end{multline}

To evaluate the expectation value in Equation \ref{unnormalized.vertex}, we
normal-order the sequence
$ \left\lgroup \prod_{Maya  \left\lgroup Y \right\rgroup} \Gamma_{\pm} \right\rgroup$
to put
all $\Gamma_{\, c_H \, +}$ vertex operators on the right, and
all $\Gamma_{\, c_H \, -}$ vertex operators on the left.
Since $\Gamma$-operators with different Heisenberg charges commute, the $n$-coloured vertex is a product of $n$ components labelled by $c_H \in \mathbb{Z}_{\, n}$,

\begin{equation}\label{unnormalized.vertex.factored}
\mathcal{C}_{\pmb{Y}_1 \, \pmb{Y}_2 Y}^{\, n, \,  c_{\, Y}, \, unnorm}
 \left\lgroup x, y \right\rgroup =
\prod_{c_H = 0}^{n-1}
\bra{s_{Y_{1c_H}}}
 \left\lgroup
\prod_{
\substack{
Maya \left\lgroup Y \right\rgroup
\\
c_{\, p} + c_Y = c_H\, \mod{\, n}
}
}
\Gamma_{\pm c_H}
\right\rgroup
\ket{s_{Y_{2c_H}}},
\end{equation}

where
$ \left\lgroup
\prod_{
\substack{
Maya \left\lgroup Y \right\rgroup
\\
c_{\, p} + c_Y = c_H\, \mod{\, n}
}
}
\Gamma_{\pm c_H} \right\rgroup$
is restricted to the sub-Maya diagram of $Maya \left\lgroup Y \right\rgroup$ with stones at positions $ \left\lgroup c_{\, p} + c_{\, Y} = c_H \, \mod{\, n} \right\rgroup$. Within each factor, we commute $\Gamma$-operators (all of which are built from the same Heisenberg algebra) using,

\begin{multline}
\Gamma_{\, c_H \, +}  \left\lgroup x^{  - i}\, y^{  Y_{\, i}} \right\rgroup  \,
\Gamma_{\, c_H \, -}  \left\lgroup y^{j - 1}\, x^{- Y^{\intercal}_{\, j}} \right\rgroup =
\\
 \left\lgroup
\frac{
1
}{
1 -
x^{\, - \, Y^{\intercal}_{\, 3, \, j} + i} \, y^{- Y_{\, 3, \, i} + j - 1}
}
\right\rgroup
\Gamma_{\, c_H \, -}  \left\lgroup  y^{\, j \, - \, 1} \, x^{\, - \, Y^{\intercal}_{\, 3, \, j}} \right\rgroup\,
\Gamma_{\, c_H \, +}  \left\lgroup  x^{\,      - \, i} \, y^{\,      Y_{\, 3, \, i}}
\right\rgroup
\label{q.t.vertex.operator.commutation.relation.with.arguments}
\end{multline}

Since $\Gamma_{\, c_H \, +}$ is attached to a segment
$\diagup$ in the borderline of $Y$, and
$\Gamma_{\, c_H \, -}$ is attached to an adjacent segment
$\diagdown$ to the right of the former,
the commutation relation,
Equation \ref{q.t.vertex.operator.commutation.relation.with.arguments}, is the same as adding a strip of length $n$ to $Y$. Repeating this process,

\begin{multline}
\mathcal{C}_{
\mathbf{Y_1}
\mathbf{Y_2} Y}^{\, n, \, c_{\, Y}, \, unnorm} \left\lgroup x,y \right\rgroup =
\prod_{c_H = 0}^{n-1}
\prod_{
\substack{
(i,j) \notin Y
\\
Y_{\, i} - i + c_{\, Y} = c_H\, \mod{\, n}
\\
-Y^{\, \intercal}_{\, j}+j-1+c_{\, Y} = c_H\, \mod{\, n}}}
\frac{
1
}{
1 - x^{-Y^{\, \intercal}_{\, j} + i} y^{-Y_{\, i} + j - 1}}
\\
\bra{s_{\mathbf{Y_1}}}
\prod_{c_H = 0}^{n-1}
 \left\lgroup
\prod_{
\substack{
j=1
\\
-Y^{\, \intercal}_{\, j} + j - 1 + c_{\, Y} = c_H \, \mod{\, n}}}^\infty
\Gamma_{c_H-}  \left\lgroup y^{\, j - 1}x^{-Y^{\, \intercal}_{\, j}} \right\rgroup
\prod_{
\substack{
i=1
\\
Y_{\, i} - i + c_{\, Y} = c_H\, \mod{\, n}}}^\infty
\Gamma_{c_H+}  \left\lgroup x^{-i} y^{Y_{\, i}} \right\rgroup
\right\rgroup
\ket{s_{\mathbf{Y_2}}}
\\
=
 \left\lgroup
\prod_{
\substack{
\square \notin Y
\\
A_{\square,Y} + L_{\square, Y}+1 = 0\, \mod{\, n}}
}
\frac{
1
}{
1 - x^{-L_{\square,Y}} y^{-A^{++}_{\square,Y}}} \right\rgroup
\prod_{c_H = 0}^{n-1}
\sum_{Y} s_{Y_{1c_H}/Y}
 \left\lgroup [y^{\, \bj - 1} x^{\, - Y^{\, \intercal}}]_{c_H - c_{\, Y}} \right\rgroup
s_{Y_{2 c_H}/Y}
 \left\lgroup [x^{\, \pmb{\iota }} y^{ - Y}]_{-c_H+c_{\, Y}} \right\rgroup
\end{multline}

where, to obtain the last equality, we used $\pmb{\iota } = \bj =  \left\lgroup 1, 2, \cdots \right\rgroup$,
Equations \ref{two.identities} and \ref{schur.inner.product}, and the definition of
$[x^{\,  \pmb{a}}y^{\,  \pmb{b}}]_c$, for any two sequences of integers $ \pmb{a},  \pmb{b}$, in Equation
\ref{subsequence.symbol}.

\subsection{6. Normalization of the vertex}

Using the unnormalized vertex
$\mathcal{C}^{\, n, \, c_{\, Y}, \, unnorm}_{\mathbf{Y_1} \mathbf{Y_2} Y}  \left\lgroup x, y \right\rgroup$,
we obtain the normalization
$\mathcal{C}^{\, n, \, c_{\, Y}, \, unnorm}_{\emptyset\emptyset\emptyset}  \left\lgroup x, y \right\rgroup$,
from which we then obtain the normalized vertex:

\begin{multline}
\mathcal{C}_{\mathbf{Y_1}\mathbf{Y_2}Y}^{\, n, \, c_{\, Y}} \left\lgroup x,y \right\rgroup =
\frac{
\mathcal{C}^{\, n, \, c_{\, Y}, \, unnorm}_{\mathbf{Y_1} \mathbf{Y_2} Y}  \left\lgroup x, y \right\rgroup
}{
\mathcal{C}^{\, n, \, c_{\, Y}, \, unnorm}_{\emptyset\emptyset\emptyset}  \left\lgroup x, y \right\rgroup}
\\
=
\prod_{
\substack{
i, j = 1
\\
1 - i - j = 0 \, \mod{\, n}
}
}^\infty
 \left\lgroup 1 - x^{i} y^{\, j - 1} \right\rgroup
 \left\lgroup
\prod_{
\substack{
\square \notin Y
\\
A_{\square,Y} + L_{\square, Y}+1 = 0\, \mod{\, n}}
}
\frac{
1
}{
1 - x^{-L_{\square,Y}} y^{-A_{\square,Y}^{++}}} \right\rgroup
\\
\prod_{c_H = 0}^{n-1}
\sum_{Y} s_{Y_{1c_H}/Y}
 \left\lgroup [y^{\, \bj-1} x^{-Y^{\, \intercal}}]_{c_H-c_{\, Y}} \right\rgroup
    s_{Y_{2c_H}/Y}  \left\lgroup [x^{\, \pmb{\iota }} y^{-Y}]_{-c_H+c_{\, Y}} \right\rgroup
\\
=
 \left\lgroup
\prod_{
\substack{
\square \in Y
\\
A_{\square,Y} + L_{\square, Y}+1 = 0\, \mod{\, n}}
}
\frac{
1
}{
1 - x^{L_{\square,Y}^{++}} y^{A_{\square,Y}}} \right\rgroup
\prod_{c_H = 0}^{n-1}
\sum_{Y} s_{Y_{1c_H}/Y}
 \left\lgroup [y^{\, \bj-1} x^{-Y^{\, \intercal}}]_{c_H-c_{\, Y}} \right\rgroup
s_{Y_{2c_H}/Y}  \left\lgroup [x^{\, \pmb{\iota }}
y^{-Y}]_{-c_H+c_{\, Y}} \right\rgroup.
\end{multline}

Then, on defining

\begin{equation}
Z^{\, n}_{\, Y}  \left\lgroup x, y \right\rgroup =
\prod_{c_H = 0}^{n-1}
\prod_{
\substack{
(i,j) \in Y
\\
Y_{\,i} - i + c_{\, Y} = c_H\, \mod{\, n}
\\
-Y^{\, \intercal}_{\,j}+j-1+c_{\, Y} = c_H\, \mod{\, n}}}
\frac{
1
}{
1 - x^{Y^{\, \intercal}_{\,j}-i+1}y^{Y_{\,i}-j}}
=
\prod_{
\substack{
\square \in Y
\\
A_{\square, Y}+L_{\square, Y} + 1 = 0\, \mod{\, n}
}}
\frac{
1
}{
1 - x^{L^{++}_{\square, Y}}y^{A_{\square, Y}}
},
\end{equation}

we obtain

\begin{equation}
\boxed{
\mathcal{C}_{\mathbf{Y_1} \mathbf{Y_2}Y}^{\, n, \, c_{\, Y}}  \left\lgroup x,y \right\rgroup =
Z^{\, n}_{\, Y} \left\lgroup x,y \right\rgroup
\prod_{c_H = 0}^{n-1}
\sum_{Y} s_{Y_{1c_H}/Y}
 \left\lgroup [y^{\, \bj-1} x^{-Y^{\, \intercal}}]_{c_H-c_{\, Y}} \right\rgroup
s_{Y_{2c_H}/Y}  \left\lgroup [x^{\, \pmb{\iota }} y^{-Y}]_{-c_H+c_{\, Y}} \right\rgroup
}
\end{equation}

\begin{figure}
\begin{center}
\begin{scriptsize}
\begin{tikzpicture}[scale=.6]
\draw [thin] (-1.5,1.0)--(5.5,8.0)--(12.5,1.0);
\draw [thin] ( 2.5,3.0)--(6.5,7.0);
\draw [thin] ( 3.5,2.0)--(7.5,6.0);
\draw [thin] ( 5.5,2.0)--(8.5,5.0);
\draw [thin] ( 8.5,3.0)--(9.5,4.0);
\draw [thin] ( 1.5,4.0)--(3.5,2.0);
\draw [thin] ( 2.5,5.0)--(5.5,2.0);
\draw [thin] ( 3.5,6.0)--(6.5,3.0);
\draw [thin] ( 4.5,7.0)--(8.5,3.0);

\draw [very thick] (-1.50, 1.00)--(1.5, 4.0);
\draw [very thick] (-1.45, 0.95)--(1.5, 3.9);

\draw [very thick] ( 1.5,4.0)--(3.5,2.0);
\draw [very thick] ( 1.5,3.9)--(3.5,1.9);

\draw [very thick] ( 3.5,2.0)--(4.5,3.0);
\draw [very thick] ( 3.5,1.9)--(4.5,2.9);

\draw [very thick] ( 4.5,3.0)--(5.5,2.0);
\draw [very thick] ( 4.5,2.9)--(5.5,1.9);

\draw [very thick] ( 5.5,2.0)--(7.5,4.0);
\draw [very thick] ( 5.5,1.9)--(7.5,3.9);

\draw [very thick] ( 7.5,4.0)--(8.5,3.0);
\draw [very thick] ( 7.5,3.9)--(8.5,2.9);

\draw [very thick] ( 8.5,3.0)--(9.5,4.0);
\draw [very thick] ( 8.5,2.9)--(9.5,3.9);

\draw [very thick] ( 9.5, 4.00)--(12.50, 1.00);
\draw [very thick] ( 9.5, 3.90)--(12.45, 0.95);

\draw [very thick] ( 10.3,2.8)--(10.5,3.0);
\draw [very thick] ( 11.3,1.8)--(11.5,2.0);
\draw [very thick] ( -0.5,2.0)--(-0.3,1.8);
\draw [very thick] ( 0.5,3.0)--(0.7,2.8);

\foreach \iota in {0,...,6}
{
\draw [dashed, red] (\iota - .5, \iota + 1.5)--(\iota - .5, 0);
}
\foreach \j in {7,...,12}
{
\draw [dashed, red](\j -.5, 13.5 -\j)--(\j - .5, 0);
}
\node [left]  at (-1.3,0) {$\cdots$};
\node [right] at (12.3,0) {$\cdots$};
\foreach x^{(1)} in {0,...,2}
{
\draw                 (x^{(1)}- 1,0) circle (0.3);
\draw [fill=black!100] (x^{(1)}+10,0) circle (0.3);
}

\draw [fill=black!100] (2,0) circle (0.3);
\draw [fill=black!100] (3,0) circle (0.3);
\draw (4,0) circle (0.3);
\draw [fill=black!100] (5,0) circle (0.3);
\draw (6,0) circle (0.3);
\draw  (7,0) circle (0.3);
\draw [fill=black!100] (8,0) circle (0.3);
\draw (9,0) circle (0.3);

\foreach \a in {6, 5, ..., 0}
{
\node [below] at (- \a + 5.0, -.3) {$\a$};
}
\foreach \a in {1, 2, ..., 7}
{
\node [below] at (  \a + 5.0, -.3) {$ - \a$};
}

\node [below] at (-2.5, -.3) {$c_{\, p} = $};

\end{tikzpicture}
\end{scriptsize}
\end{center}
\caption{\it An $n$-coloured Maya diagram for conjugate vertex that corresponds to a partition $Y =  \left\lgroup 4,3,3,2 \right\rgroup$. Note that the position coordinates are reversed with respect to those in Figure \ref{young.maya.02}.
}
\label{conjugate.p.coloured.maya.diagram}
\end{figure}

\subsection{The conjugate vertex}
To form strips, we need the conjugate vertex which is obtained by reversing the positions of the Maya diagram as shown in Figure \ref{conjugate.p.coloured.maya.diagram},
that is, $c_{p} \mapsto - \, c_{p} - 1$,
and repeat the construction of the vertex
outlined above, but now the sum of the exponents of the argument of $\Gamma_{\pm}$ is related to the initial position $c_{p}$ in a different way.
For
$\Gamma_{\, c_H \, +}
 \left\lgroup x^{\, - i}\, y^{\, Y_{\, i}} \right\rgroup$,
at position
$ \left\lgroup c_{\, p} + c_{\, Y} \right\rgroup$,

\begin{equation}
    c_{  p} = -Y_{i} + i - 1,
\end{equation}

and for $\Gamma_{\, c_H \, -}  \left\lgroup y^{j - 1}\, x^{- Y^{\, \intercal}_{j}}   \right\rgroup$, at position $ \left\lgroup c_{\, p} + c_{\, Y} \right\rgroup$,

\begin{equation}
c_{\, p} = Y^{\, \intercal}_{j} - j,
\end{equation}

so that,

\begin{multline}
\mathcal{C}_{
\mathbf{Y_1}
\mathbf{Y_2}Y}^{\, n, \, c_{\, Y}, \, unnorm \, *}
 \left\lgroup x, y \right\rgroup =
\prod_{c_H = 0}^{n-1}
\bra{s_{\mathbf{Y_1}}}
 \left\lgroup
\prod_{
\substack{
Maya \left\lgroup Y \right\rgroup
\\
c_{\, p}+c_Y = c_{\, H} \, \mod{\, n}
}
}
\Gamma_{c_H \pm}
\right\rgroup
\ket{s_{\mathbf{Y_2}}}
\\
=
\prod_{c_{\, H} = 0}^{\, n - 1}
\prod_{
\substack{
(i,j) \notin Y
\\
-Y_{\,i} + i - 1 + c_{\, Y} =
c_{\, H} \, \mod{\, n}
\\
Y^{\, \intercal}_{\, j} - j + c_{\, Y} = c_{\, H} \, \mod{\, n}
}
}
\frac{
1
}{
1 - x^{-Y^{\, \intercal}_{\,j}+i}y^{-Y_{\,i}+j-1}}
\\
\bra{s_{\mathbf{Y_1}}}
\prod_{c_{\, H} = 0}^{n-1}
 \left\lgroup
\prod_{
\substack{
j=1
\\
Y^{\, \intercal}_{\,j}-j+c_{\, Y} = c_H \, \mod{\, n}}}^\infty
\Gamma_{\, c_H \, -}
 \left\lgroup
y^{\, j - 1}
x^{\, - Y^{\, \intercal}_{\,j}}
\right\rgroup
\prod_{
\substack{
i=1
\\
-Y_{\, i} + i - 1 + c_{\, Y} = c_H
\, \mod{\, n}}}^\infty
\Gamma_{\, c_H \, +}
 \left\lgroup
x^{-i} y^{Y_{\,i}}
\right\rgroup
\right\rgroup
\ket{s_{\mathbf{Y_2}}}
\\
=
 \left\lgroup
\prod_{
\substack{
\square \notin Y
\\
A_{\square,Y} + L_{\square, Y}+1 = 0\, \mod{\, n}}
}
\frac{
1
}{
1 - x^{-L_{\square,Y}} y^{-A^{++}_{\square,Y}}} \right\rgroup
\prod_{c_H = 0}^{n-1}\sum_{Y} s_{Y_{1c_H}/Y}
 \left\lgroup [y^{\, \bj-1} x^{-Y^{\, \intercal}}]_{-c_H+c_{\, Y}-1} \right\rgroup
s_{Y_{2c_H}/Y}  \left\lgroup [x^{\, \pmb{\iota }} y^{-Y}]_{c_H-c_{\, Y}+1} \right\rgroup, \end{multline}

and after normalization,

\begin{equation}
\boxed{
\mathcal{C}_{
\mathbf{Y_1}
\mathbf{Y_2} Y}^{\, n, \, c_{\, Y}*}
 \left\lgroup x, y \right\rgroup =
Z^{\, n}_{\, Y}
 \left\lgroup x,y \right\rgroup
\prod_{c_H = 0}^{n-1}
\sum_{Y}
s_{Y_{1c_H} / Y}
 \left\lgroup [y^{\, \bj-1} x^{-Y}]_{-c_H + c_{\, Y}-1} \right\rgroup
s_{Y_{2c_H} / Y}
 \left\lgroup [x^{\, \pmb{\iota }}  \, y^{-Y^{\, \intercal}}]_{c_H - c_{\, Y} + 1} \right\rgroup
}
\end{equation}

\begin{scriptsize}
\begin{figure}
\begin{tikzpicture} [scale=.8]

\draw [ultra thick] (0,0)--(0,1);
\node [below] at (0,0) {$\pmb{\emptyset}$};
\draw [thick] (0,1)--(2,1);
\node [right, blue] at (2,1) {$W_N$};
\draw [ultra thick] (0,1)--(-1,2);
\node [right] at (-0.25,1.5) {$\pmb{\xi_N}, \Delta_N$};
\draw [thick] (-1,2)--(-3,2);
\node [left, red] at (-3,2) {$V_N$};
\draw [ultra thick] (-1,2)--(-1,3);
\node [right] at (-0.75,2.5) {$\pmb{\xi_{M_{N-1}}}, \Delta_{M_{N-1}}$};
\draw [ultra thick, dashed] (-1,3)--(-2,4);
\draw [ultra thick] (-2,4)--(-2,5);
\node [right] at (-1.75,4.5) {$\pmb{\xi_{M_2}}, \Delta_{M_2}$};
\draw [thick] (-2,5)--(0,5);
\node [right, blue] at (0,5) {$W_2$};
\draw [ultra thick] (-2,5)--(-3,6);
\node [right] at (-2.25,5.5) {$\pmb{\xi_2}, \Delta_2$};
\draw [thick] (-3,6)--(-5,6);
\node [left, red] at (-5,6) {$V_2$};
\draw [ultra thick] (-3,6)--(-3,7);
\node [right] at (-2.75,6.5) {$\pmb{\xi_{M_1}}, \Delta_{M_1}$};
\draw [thick] (-3,7)--(-1,7);
\node [right, blue] at (-1,7) {$W_1$};
\draw [ultra thick] (-3,7)--(-4,8);
\node [right] at (-3.25,7.5) {$\pmb{\xi_1}, \Delta_1$};
\draw [thick] (-4,8)--(-6,8);
\node [left, red] at (-6,8) {$V_1$};
\draw [ultra thick] (-4,8)--(-4,9);
\node [above] at (-4,9) {$\pmb{\emptyset}$};

\end{tikzpicture}
\caption{
\textit{
The strip diagram made of $N$ pairs of $n$-coloured vertices
and conjugate vertices. Each internal line is labeled by
a partition and a K{\"a}hler parameter. Each external
(horizontal) line is labeled by a partition. The external
legs are preferred and labelled by Young's diagrams
$\pmb{V} =
 \left\lgroup
 \left\lgroup
V_1, \, c_{V_1}
\right\rgroup, \cdots,
 \left\lgroup
V_N,\, c_{V_N}
\right\rgroup
\right\rgroup$ and
$\pmb{W} =
 \left\lgroup
 \left\lgroup W_1,\, c_{W_1}
\right\rgroup, \cdots,
 \left\lgroup
W_N, \, c_{W_N}
\right\rgroup
\right\rgroup$.
}
}
\label{strip.diagram}
\end{figure}
\end{scriptsize}

\begin{figure}
\begin{tikzpicture} [scale=.7]
\draw [thick] (-6,4)--(-4,4);
\draw [ultra thick] (-4,6)--(-4,4);
\draw [ultra thick] (-4,4)--(-3,3);
\node [left] at (-6,4) {$Y$} ;
\node [right] at (-3,3) {$\pmb{Y_2}$};
\node [above] at (-4,6) {$\pmb{Y_1}$};

\draw [ultra thick] (1,6)--(2,5);
\draw [ultra thick] (2,5)--(2,3);
\draw [thick] (2,5)--(4,5);
\node [left] at (1,6) {$\pmb{Y_2}$} ;
\node [below] at (2,3) {$\pmb{Y_1}$};
\node [right] at (4,5) {$Y$};
\end{tikzpicture}
\caption{
\textit{
The vertex $\mathcal{C}^{\, p, \, c_{\nu}}_{\pmb{Y_1Y_2}Y}  \left\lgroup x, y \right\rgroup$ is on the left and
the conjugate vertex
$\mathcal{C}^{\, p, \, c_{\nu}*}_{\pmb{Y_1}^{\intercal} \pmb{Y_2}^{\intercal} Y^{\intercal}}  \left\lgroup y, x \right\rgroup $
is on the right.
}
}
\label{topological.vertices}
\end{figure}

\section{From $n$-coloured vertices to $n$-coloured $N$-strip partition functions}
\label{section.05}
\textit{We construct 5D strip partition functions from
$N$ pairs of $n$-coloured refined topological vertices,
where a pair of vertices consists of a vertex and a conjugate vertex.}

We consider an $N$-strip partition function on
$\mathbb{C}^{\, 2}/\mathbb{Z}_{\, n}$ shown in Figure \ref{strip.diagram}.
The horizontal external legs are labelled by Young diagrams
with $\mathbb{Z}_n$-charges. Denote a Young diagram
$Y$ carrying $\mathbb{Z}_n$-charge $c_Y \in \mathbb{Z}_n$ by
$ \left\lgroup Y,c_Y \right\rgroup$. There are $N$ horizontal external legs to the left
labelled by,

\begin{equation}
     \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}}\right\rgroup =  \left\lgroup  \left\lgroup V_1,\, c_{V_1} \right\rgroup,\cdots,  \left\lgroup V_N,\, c_{V_N} \right\rgroup \right\rgroup
\end{equation}

There are $N$ horizontal external legs to the right labelled by,

\begin{equation}
 \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}}\right\rgroup =  \left\lgroup  \left\lgroup W_1,\, c_{W_1} \right\rgroup,\cdots,  \left\lgroup W_N,\, c_{W_N} \right\rgroup \right\rgroup
\end{equation}

Each internal edge is labelled by uncharged
$n$-Young diagrams
$\pmb{\xi} =  \left\lgroup \xi_0,\cdots, \xi_{n-1} \right\rgroup$.
For an internal edge with diagram $\pmb{\xi}_i$ we define,

\begin{equation}
\text{Edge factor} =  \left\lgroup - Q_{\, i} \right\rgroup^{|\, \pmb{\xi}_i \, |},
\quad
Q_{\, i} = e^{- R \, \Delta_i},
\quad i = 1, M_1, 2, M_2, \cdots, M_{N-1}, N
\end{equation}

The external vertical legs are labelled by $n$ null Young
diagrams $\pmb{\emptyset}$.

\subsection{The unnormalized strip partition function}
The unnormalized $N$-strip $\mathbb{C}^{\, 2}/\mathbb{Z}_{\, n}$ partition
function can be written directly from Figure
\ref{strip.diagram},

\begin{multline}
\mathcal{S}^{\, n, \, unnorm
}_{
 \left\lgroup \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup
\bDelta
}
 \left\lgroup x, y, R \right\rgroup
=
\\
\sum_{
\pmb{\xi_{\, 1}}, \, \cdots, \, \pmb{\xi_{\, N}}
}
\sum_{
\pmb{\xi_{\, M_1}}, \, \cdots, \, \pmb{\xi_{\, M_{\, N - 1}
}
}
}
\prod_{I = 1}^N
 \left\lgroup
 \left\lgroup -Q_{\, i}        \right\rgroup^{\, | \, \pmb{\xi_I}     \, |}
 \left\lgroup -Q_{\, M_{\, I}} \right\rgroup^{\, | \, \pmb{\xi_{M_I}} \, |}
\mathcal{C}^{\, n, \,  V_I
}_{
\pmb{\xi_{\, M_{I-1}}}^{\, \intercal}
\pmb{\xi_{\, I      }}
V^\intercal_I
}
 \left\lgroup x, y \right\rgroup
\mathcal{C}^{\, n, \, W_I \, *
}_{
\pmb{\xi_{\, M_I}}
\pmb{\xi_{\, I}}^{\, \intercal}
W_I
}
 \left\lgroup y, x \right\rgroup
\right\rgroup
\end{multline}

where the summations are over all possible internal $n$-Young diagrams.
Expanding the definition of the vertices and  the conjugate vertices gives

\begin{multline}
\label{uneval.s.N}
\mathcal{S}^{\, n, \, unnorm}_{ \left\lgroup \, \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta
}
 \left\lgroup x, y, R \right\rgroup
\\
=
\prod_{I=1}^N
 \left\lgroup
Z^{\, n}_{V_I}  \left\lgroup x, y \right\rgroup
Z^{\, n}_{W^{\, \intercal}_I}
 \left\lgroup y, x \right\rgroup
\right\rgroup
\sum_{
\pmb{\xi_{1  }}, \, \cdots, \, \pmb{\xi_{N}    }
}
\sum_{
\pmb{\xi_{M_1}}, \, \cdots, \, \pmb{\xi_{M_{N-1}
}
 }
}
\prod_{I=1}^N
 \left\lgroup
 \left\lgroup - Q_{\,   i} \right\rgroup^{\, | \, \pmb{\xi_I}     \, |}
 \left\lgroup - Q_{\, M_I} \right\rgroup^{\, | \, \pmb{\xi_{M_I}} \, |}
\right\rgroup
\\
\prod_{c_H= 0 }^{n-1}
\sum_{\eta'_I}
s_{\, \xi^{\, \intercal}_{\, M_{I-1} c_H} /\eta'_I}
 \left\lgroup
[x^{-V_I} y^{\, \bj-1}]_{c_H - c_{V_I}}
\right\rgroup
s_{\, \xi_{I c_H} / \eta'_I}
 \left\lgroup
[x^{\, \pmb{\iota }} y^{-V^{\, \intercal}_I}]_{- c_H + c_{V_I}}
\right\rgroup
\\
\prod_{c_H= 0 }^{n-1}
\sum_{\eta''_I}
s_{\xi_{\, M_I c_H}/\eta''_I}
 \left\lgroup
[y^{- W^{\, \intercal}_I} x^{\, \pmb{\iota } - 1}]_{- c_H + c_{W_I} - 1}
\right\rgroup
s_{\xi^{\, \intercal}_{I c_H} / \eta''_I}
 \left\lgroup
[y^{\, \bj} x^{- W_I}]_{c_H - c_{W_I} + 1}
\right\rgroup
\\
=
\prod_{I=1}^N
 \left\lgroup Z^{\, n}_{V_I} \left\lgroup x,y \right\rgroup
Z^{\, n}_{W^{\, \intercal}_I}
 \left\lgroup y, x \right\rgroup
\right\rgroup
\prod_{c_H= 0 }^{n-1}
\sum_{ \left\lgroup \xi_{1  }, \cdots, \xi_{N      } \right\rgroup}
\sum_{ \left\lgroup \xi_{M_1}, \cdots, \xi_{M_{N-1}} \right\rgroup}
\prod_{I=1}^N
 \left\lgroup
 \left\lgroup -Q_{\, i} \right\rgroup^{|\xi_I|}
 \left\lgroup -Q_{M_I} \right\rgroup^{|\xi_{M_I}|}
\right\rgroup
\\
\sum_{\eta'_I}
s_{\xi^{\, \intercal}_{M_{I-1}}/\eta'_I}
 \left\lgroup [x^{-V_I}y^{\, \bj-1}]_{c_H-c_{V_I}} \right\rgroup
s_{\xi_{I}/\eta'_I}
 \left\lgroup
[x^{\, \pmb{\iota }}
y^{-V^{\, \intercal}_I}]_{-c_H+c_{V_I}} \right\rgroup
\\
\sum_{\eta''_I}
s_{\xi_{M_I}/\eta''_I}
 \left\lgroup
[ y^{-W^{\, \intercal}_I} x^{\, \pmb{\iota }-1}]_{-c_H+c_{W_I}-1}
\right\rgroup
s_{\xi^{\, \intercal}_{I}/\eta''_I}
 \left\lgroup
[ y^{\, \bj} x^{-W_I}]_{c_H-c_{W_I}+1}
\right\rgroup
\end{multline}

where
$\xi_{M_0} = \xi_{M_N} = \emptyset$.
Using the result of Appendix \ref{appendix.B} then gives

\begin{multline}
\label{5d.N.strip.partition.function}
\mathcal{S}^{\, n, \, unnorm}_{ \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}}\right\rgroup \bDelta}
 \left\lgroup x, y, R \right\rgroup =
\prod_{I=1}^N
 \left\lgroup
Z^{\, n}_{V_I}
 \left\lgroup x,y \right\rgroup
Z^{\, n}_{W^{\, \intercal}_I}
 \left\lgroup y,x \right\rgroup
\right\rgroup
\\
\prod_{c_H= 0 }^{n-1}
\prod_{J=1}^N \prod_{I=1}^J
\prod_{
\substack{
i=1
\\
-V^{\, \intercal}_{I, i} + i = - c_H + c_{V_I}\, \mod{\, n}
}
}^\infty
\prod_{
\substack{
j=1
\\
-W_{J,j}+j-1= c_H - c_{W_J} \, \mod{\, n}
}
}^\infty
 \left\lgroup 1 - \prod_{K=I}^{J-1}Q_{M_K}\prod_{K=I}^J Q_K x^{-W_{J,j}+i}y^{-V^{\, \intercal}_{I,i}+j} \right\rgroup
\\
\prod_{c_H= 0 }^{n-1}\prod_{J=1}^N\prod_{I=1}^{J-1}
\prod_{
\substack{
i=1
\\
-W^{\, \intercal}_{I,i}+i-1 = -c_H+c_{W_I}-1 \, \mod{\, n}
}
}^\infty
\prod_{
\substack{
j=1
\\
-V_{J,j}+j-1 = c_H-c_{V_J}\, \mod{\, n}
}
}^\infty
 \left\lgroup 1 - \prod_{K=I}^{J-1} Q_{M_K} \prod_{K=I+1}^{J-1} Q_K x^{-V_{J,j}+i-1}y^{-W^{\, \intercal}_{I,i}+j-1} \right\rgroup
\\
\prod_{c_N= 0 }^{n-1}\prod_{J=1}^N\prod_{I=1}^{J-1}
\prod_{
\substack{
i=1
\\
-V^{\, \intercal}_{I,i}+i = -c_H + c_{V_I}\, \mod{\, n}
}
}^\infty
\prod_{
\substack{
j=1
\\
-V_{J,j}+j-1 = c_H - c_{V_J} \, \mod{\, n}
}
}^\infty
 \left\lgroup 1 - \prod_{K=I}^{J-1} Q_{M_K}
\prod_{K=I}^{J-1} Q_K x^{-V_{J,j}+i}y^{-V^{\, \intercal}_{I,i}+j-1} \right\rgroup^{-1}
\\
\prod_{c_H= 0 }^{n-1}
\prod_{J=1}^N
\prod_{I=1}^{J-1}
\prod_{
\substack{
i=1
\\
-W^{\, \intercal}_{I,i}+i-1 = -c_H + c_{W_I}-1\, \mod{\, n}
}
}^\infty
\prod_{
\substack{j=1\\
    -W_{J,j}+j-1 = c_H - c_{W_J} \, \mod{\, n}
    }
    }^\infty
     \left\lgroup 1 - \prod_{K=I}^{J-1} Q_{M_K} \prod_{K=I+1}^J Q_K x^{-W_{J,j}+i-1}y^{-W^{\, \intercal}_{I,i}+j} \right\rgroup^{-1}
    \\
    = \prod_{I=1}^N  \left\lgroup Z^{\, n}_{V_I} \left\lgroup x,y \right\rgroup Z^{\, n}_{W^{\, \intercal}_I} \left\lgroup y,x \right\rgroup \right\rgroup
    \\
    \prod_{J=1}^N\prod_{I=1}^J
    \prod_{\substack{i,j=1\\
    (-V^{\, \intercal}_{I,i}+j) + (-W_{J,j}+i) - 1 + c_{W_J}-c_{V_I} = 0\, \mod{\, n}
    }}^\infty
     \left\lgroup 1 - \prod_{K=I}^{J-1}Q_{M_K}\prod_{K=I}^J Q_K x^{-W_{J,j}+i}y^{-V^{\, \intercal}_{I,i}+j} \right\rgroup
    \\
    \prod_{J=1}^N\prod_{I=1}^{J-1}
    \prod_{\substack{i,j=1\\
    (-W^{\, \intercal}_{I,i}+j) + (-V_{J,j}+i) - 1 + c_{V_J}-c_{W_I} = 0\, \mod{\, n}
    }}^\infty
     \left\lgroup 1 - \prod_{K=I}^{J-1} Q_{M_K} \prod_{K=I+1}^{J-1} Q_K x^{-V_{J,j}+i-1}y^{-W^{\, \intercal}_{I,i}+j-1} \right\rgroup
    \\
    \prod_{J=1}^N\prod_{I=1}^{J-1}
    \prod_{\substack{i,j=1\\
    (-V^{\, \intercal}_{I,i}+j) + (-V_{J,j}+i) - 1 + c_{V_J}-c_{V_I} = 0\, \mod{\, n}
    }}^\infty
     \left\lgroup 1 - \prod_{K=I}^{J-1} Q_{M_K} \prod_{K=I}^{J-1} Q_K x^{-V_{J,j}+i}y^{-V^{\, \intercal}_{I,i}+j-1} \right\rgroup^{-1}
    \\
    \prod_{J=1}^N\prod_{I=1}^{J-1}
    \prod_{\substack{i,j=1\\
    (-W^{\, \intercal}_{I,i}+j) + (-W_{J,j}+i) - 1 + c_{W_J}-c_{W_I} = 0\, \mod{\, n}
    }}^\infty
     \left\lgroup 1 - \prod_{K=I}^{J-1} Q_{M_K} \prod_{K=I+1}^J Q_K x^{-W_{J,j}+i-1}y^{-W^{\, \intercal}_{I,i}+j} \right\rgroup^{-1}
\end{multline}

\subsection{The normalization of the strip}

For $n=1$, the normalized $N$-strip partition
function on $\mathbb{R}^4$ is obtained by normalizing
$\mathcal{S}_{\, \pmb{V}  \, \pmb{W}  \, \bDelta} \left\lgroup x,y,R \right\rgroup$ by
$\mathcal{S}_{\, \pmb{\emptyset}  \, \pmb{\emptyset}  \, \bDelta} \left\lgroup x, y, R \right\rgroup$,
so that $\mathcal{S}^{\, norm}_{\, \pmb{\emptyset}  \, \pmb{\emptyset}  \, \bDelta}  \left\lgroup x, y, R \right\rgroup = 1$. Similarly, for $n > 1$, we write the normalized strip partition
function on $\mathbb{R}^4/\mathbb{Z}_n$ as,

\begin{equation}
\label{normalized.5d.strip.partition.function}
\mathcal{S}_{
\,   \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup \,  \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \, \bDelta
}^{
\, n, \, norm}  \left\lgroup x, y, R \right\rgroup =
\frac{
\mathcal{S}_{\,  \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}}\right\rgroup \,  \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}}\right\rgroup \, \bDelta}^{\, n}  \left\lgroup x, y, R \right\rgroup
}{
\mathcal{S}_{
\,
 \left\lgroup \bemptyset,\,  \pmb{c}_{\pmb{V}} \right\rgroup \,
 \left\lgroup \bemptyset,\,  \pmb{c}_{\pmb{W}} \right\rgroup \,
\bDelta}^{\, n}  \left\lgroup x, y, R \right\rgroup
}
\end{equation}

From Equation \ref{5d.N.strip.partition.function}
and identity \ref{R.0.normalized.product.formula},

\begin{multline}
\label{after.identity}
\mathcal{S}^{\, n, \,norm}_{
 \left\lgroup \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup
\bDelta}
 \left\lgroup x, y, R \right\rgroup =
\prod_{I = 1}^N
 \left\lgroup
Z^{\, n}_{V_I}             \left\lgroup x, y \right\rgroup
Z^{\, n}_{W^{\, \intercal}_I}   \left\lgroup y, x \right\rgroup
\right\rgroup
\\
\prod_{I = 1}^N
\prod_{J = 1}^I
\prod_{
\substack{
\blacksquare \in W_I
\\
A_{\blacksquare,W_I} +
L_{\blacksquare,V_J} + 1 - c_{W_I} +c_{V_J} = 0  \, \mod{\, n}
}
}
 \left\lgroup
1 - \prod_{K = J}^{I - 1} Q_{M_K}
\prod_{K=J}^I
Q_K
x^{- A_{\blacksquare,W_I}} \, y^{- L_{\blacksquare,V_J}}
\right\rgroup
\\
\prod_{
\substack{
\square \in V_J
\\
A_{\square, V_J} +
L_{\square, W_I} + 1 + c_{W_I} - c_{V_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup 1 - \prod_{K = J}^{I - 1}
Q_{M_K}
\prod_{K=J}^I
Q_K
x^{A^{++}_{\square, V_J}} \, y^{L^{++}_{\square, W_I}}
\right\rgroup
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\square \in  V_I
\\
A_{\square, V_I} +
L_{\square, W_J} + 1 - c_{V_I} + c_{W_J} = 0  \, \mod{\, n}
}
}
 \left\lgroup 1 -
\prod_{K=J}^{I-1} Q_{M_K}
\prod_{K=J+1}^{I-1}
Q_K
x^{-A_{\square, V_I}^{++}} \, y^{-L_{\square, W_J}^{++}}
\right\rgroup
\\
\prod_{
\substack{
\blacksquare \in W_J
\\
A_{\blacksquare, W_J} +
L_{\blacksquare, V_I}  + 1 + c_{V_I} - c_{W_J} = 0 \, \mod{\, n}
}}
 \left\lgroup 1 -
\prod_{K=J}^{I-1} Q_{M_K}
\prod_{K=J+1}^{I-1}
Q_K
x^{A_{\blacksquare,  W_J}} \, y^{L_{\blacksquare,   V_I}}
\right\rgroup
\\
\prod_{I=1}^N
\prod_{J = 1}^{I-1}
\prod_{
\substack{
\square \in V_I \\
A_{\square,V_I} +
L_{\square,V_J} + 1 - c_{V_I} + c_{V_J} = 0 \, \mod{\, n}
}}
 \left\lgroup 1 -
\prod_{K=J}^{I-1} Q_{M_K}
\prod_{K=J}^{I-1} Q_K
x^{-A_{\square,   V_I}^{++}} \, y^{-L_{\square,   V_J}}
\right\rgroup^{-1}
\\
\prod_{
\substack{
\square \in V_J
\\
A_{\square,V_J} +
L_{\square,V_I} + 1 + c_{V_I} - c_{V_J} = 0  \, \mod{\, n}
}}
 \left\lgroup 1 -
\prod_{K=J}^{I-1}
Q_{M_K}
\prod_{K=J}^{I-1}
Q_K x^{A_{\square,  V_J}} \, y^{L_{\square,   V_I}^{++}}
\right\rgroup^{-1}
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\blacksquare \in  W_I
\\
A_{\blacksquare, W_I} +
L_{\blacksquare, W_J} + 1 - c_{W_I} + c_{W_J} = 0 \, \mod{\, n}
}}
 \left\lgroup
1 -
\prod_{K=J}^{I-1}
Q_{M_K}
\prod_{K=J+1}^I
Q_K x^{-A_{\blacksquare,   W_I}} \, y^{-L_{\blacksquare, W_J}^{++}}
\right\rgroup^{-1}
\\
\prod_{
\substack{
\blacksquare \in  W_J
\\
A_{\blacksquare, W_J} +
L_{\blacksquare, W_I} + 1 + c_{W_I} - c_{W_J} = 0\, \mod{\, n}
}}
 \left\lgroup
1 -
\prod_{K=J}^{I-1} Q_{M_K}
\prod_{K=J+1}^I
Q_K x^{A^{++}_{\blacksquare,   W_J}} \, y^{L_{\blacksquare, W_I}}
\right\rgroup^{-1}
\end{multline}

\section{From $n$-coloured 5D strips to  $n$-coloured 4D instanton
partition functions}
\label{section.06}
\textit{We take the 4D limit of the 5D strip partition function
to obtain the corresponding 4D instanton partition function.
}

\subsection{Parameters}
The parameters $x, y$ of the $n$-coloured topological
vertex and the parameters $\epsilon_1, \epsilon_2$ of the instanton partition
functions are related by

\begin{equation}
x = e^{\, + R \, \epsilon_2},
\quad
y = e^{\, - R \, \epsilon_1}.
\end{equation}

Further, recall that
$Q_{\, i} = e^{- R \, \Delta_{\, i}}, i = 1, M_1, 2, M_2, \cdots, M_{N-1}, N$.

\subsection{The 4D limit}
For $n = 1$, taking the 4D limit is straightforward because both the numerator
and the denominator of
$\mathcal{S}^{
\, n, \, norm
}_{
 \left\lgroup \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta}
 \left\lgroup x, y, R \right\rgroup$
approach zero as
$R^{\, N \sum_{I=1}^N  \left\lgroup \, | \, V_I \, | + \, | \, W_I \, | \, \right\rgroup}$, in the limit $R \rightarrow 0$.
For $n > 1$, this is no longer the case and
$\mathcal{S}^{\, n, \, norm
}_{
 \left\lgroup \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta}
 \left\lgroup x, y, R \right\rgroup$
approaches zero as $R^{\, K}$, in the limit $R \rightarrow 0$,
for some integer $K$ which depends on
$ \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup$ and $ \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup$.
So, to take the 4D limit, we must cancel the excess factors of $R$ by

\begin{equation}
\mathcal{S}^{\, n, \, norm}_{ \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta} \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup =
\lim_{R \, \rightarrow \, 0}
 \left\lgroup
R^{-K} \,
\mathcal{S}^{\, n, \, norm
}_{
 \left\lgroup \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta}
 \left\lgroup x, y, R \right\rgroup
\right\rgroup,
\end{equation}

to obtain
\footnote{\,
We do not need the closed form expression of the factor
$K$, and it suffices for our purposes to compute it on
a case by case basis.},

\begin{multline}\label{sn.norm}
\mathcal{S}^{\, n, \,norm
}_{
 \left\lgroup \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta}
 \left\lgroup \epsilon_1, \, \epsilon_2       \right\rgroup
=
\\
\prod_{I = 1}^N
\prod_{J = 1}^I
\prod_{
\substack{
\blacksquare \in W_I
\\
A_{\blacksquare, W_I} + L_{\blacksquare, V_J} + 1 - c_{W_I} + c_{V_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup
\sum_{K=J}^{I-1}
\Delta_{M_K} +
\sum_{K=J}^I
\Delta_K + A_{\blacksquare, W_I} \epsilon_2 - L_{\blacksquare, V_J} \epsilon_1 \right\rgroup
\\
\prod_{
\substack{
\square, V_J
\\
A_{\square, V_J} + L_{\square, W_I} + 1 + c_{W_I} - c_{V_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup
\sum_{K=J}^{I-1}
\Delta_{M_K} + \sum_{K=J}^I \Delta_K - A^{++}_{\square, V_J} \epsilon_2 + L^{++}_{\square, W_I} \epsilon_1
\right\rgroup
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\square \in V_I
\\
A_{\square, V_I} + L_{\square, W_J} + 1 - c_{V_I} + c_{W_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup
\sum_{K=J}^{I-1}
\Delta_{M_K} +
\sum_{K=J+1}^{I-1}
\Delta_K + A^{++}_{\square, V_I} \epsilon_2 - L^{++}_{\square, W_J} \epsilon_1
\right\rgroup
\\
\prod_{
\substack{
\blacksquare \in W_J
\\
A_{\blacksquare, W_J}+L_{\blacksquare, V_I}+1+ c_{V_I}-c_{W_J} = 0\, \mod{\, n}
}
}
 \left\lgroup \sum_{K=J}^{I-1}\Delta_{M_K} + \sum_{K=J+1}^{I-1}\Delta_K - A_{\blacksquare, W_J}\epsilon_2 + L_{\blacksquare, V_I}\epsilon_1 \right\rgroup
    \\
    \prod_{I=1}^N
     \left\lgroup
    \prod_{\substack{\square \in V_I\\
    A_{\square, V_I}+L_{\square, V_I}+1 = 0\, \mod{\, n}}} \left\lgroup -A^{++}_{\square, V_I}\epsilon_2 + L_{\square, V_I}\epsilon_1 \right\rgroup^{-1}
    \prod_{\substack{\blacksquare \in W_I\\
    A_{\blacksquare, W_I}+L_{\blacksquare, W_I}+1 = 0\, \mod{\, n}}} \left\lgroup -A_{\blacksquare, W_I}\epsilon_2 + L^{++}_{\blacksquare, W_I}\epsilon_1 \right\rgroup^{-1}
    \right\rgroup
    \\
    \prod_{I=1}^N\prod_{J=1}^{I-1}\prod_{\substack{\square \in V_I\\
    A_{\square, V_I}+L_{\square, V_J} + 1 -c_{V_I}+c_{V_J} = 0\, \mod{\, n}}} \left\lgroup \sum_{K=J}^{I-1}\Delta_{M_K} + \sum_{K=J}^{I-1}\Delta_K + A^{++}_{\square, V_I}\epsilon_2 - L_{\square, V_J}\epsilon_1 \right\rgroup^{-1}
    \\
    \prod_{\substack{\square \in V_J\\
    A_{\square, V_J}+L_{\square, V_I}+1 + c_{V_I} - c_{V_J} = 0\, \mod{\, n}}} \left\lgroup \sum_{K=J}^{I-1}\Delta_{M_K} + \sum_{K=J}^{I-1}\Delta_K - A_{\square, V_J}\epsilon_2 + L^{++}_{\square, V_I} \right\rgroup^{-1}
    \\
    \prod_{I=1}^N
    \prod_{J=1}^{I-1}
    \prod_{\substack{\blacksquare \in  W_I\\
    A_{\blacksquare, W_I}+L_{\blacksquare, W_J}+1-c_{W_I}+c_{W_J} = 0\, \mod{\, n}}}
     \left\lgroup
    \sum_{K=J}^{I-1} \Delta_{M_K} +
    \sum_{K=J+1}^I \Delta_K + A_{\blacksquare,   W_I} \epsilon_2 - L_{\blacksquare,    W_J}^{++} \epsilon_1
    \right\rgroup^{-1}
    \\
    \prod_{\substack{\blacksquare \in  W_J\\
    A_{\blacksquare, W_J}+L_{\blacksquare, W_I} + 1 + c_{W_I} - c_{W_J} = 0\, \mod{\, n}}}
     \left\lgroup
    \sum_{K=J}^{I-1} \Delta_{M_K} +
    \sum_{K=J+1}^I \Delta_K - A_{\blacksquare,   W_J}^{++} \epsilon_2 + L_{\blacksquare,    W_I} \epsilon_1
    \right\rgroup^{-1}
\end{multline}

\subsection{Comparison with the $\mathbb{R}^4/\mathbb{Z}_n$ 4D instanton partition function}

The $\mathbb{R}^4/\mathbb{Z}_n$ 4D instanton
partition function is,

\begin{equation}
\label{zp.z}
\mathcal{Z}^{\, n}       \left\lgroup  \pmb{a}, \pmb{V}, \pmb{c}_{\pmb{V}} \, | \, \mu \, | \,   \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
=
\frac{
\mathcal{Z}^{\, n,  \, num}  \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \mu \, | \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
}{
\mathcal{Z}_{\, n,  \, den}  \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, |             \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
}
\end{equation}

where

\begin{multline}
\label{zp.z.num}
\mathcal{Z}^{\, n, \, num}
 \left\lgroup  \pmb{a},\pmb{V}, \pmb{c}_{\pmb{V}} \,  | \, \mu \, | \,   \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup =
\prod_{I,J = 1}^N
\prod_{
\substack{
\square \in V_I
\\
A_{\square,V_I} + L_{\square,W_J} + 1 - c_{V_I} + c_{W_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup a_I - b_J - \mu + A^{++}_{\square, V_I} \epsilon_2 - L_{\square, W_J} \epsilon_1 \right\rgroup
\\
\prod_{
\substack{
\blacksquare \in W_J
\\
A_{\blacksquare,W_J} + L_{\blacksquare,V_I} + 1 + c_{V_I} - c_{W_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup a_I - b_J - \mu - A_{\blacksquare, W_J} \epsilon_2 + L^{++}_{\blacksquare, V_I} \epsilon_1 \right\rgroup
\end{multline}

and,

\begin{multline}\label{zp.z.den}
\mathcal{Z}_{den}^{\, n}  \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}}\, | \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup =
\\
 \left\lgroup
\mathcal{Z}^{\, n, \, num} \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, 0 \, | \,   \pmb{a},\pmb{V}, \pmb{c}_{\pmb{V}} \right\rgroup
\mathcal{Z}^{\, n, \, num} \left\lgroup  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \, | \, 0 \, | \,   \pmb{b},\pmb{W}, \pmb{c}_{\pmb{W}} \right\rgroup
\right\rgroup^{\frac12}
\\
=
\prod_{I,J=1}^N
\prod_{
\substack{
\square \in V_I
\\
A_{\square, V_I} + L_{\square, V_J} + 1 - c_{V_I} + c_{V_J} = 0\, \mod{\, n}}
}
 \left\lgroup
a_I - a_J + A^{++}_{\square,V_I} \epsilon_2 - L_{\square,V_J} \epsilon_1
\right\rgroup^{\frac12}
\\
\prod_{
\substack{
\square \in V_J
\\
A_{\square, V_J} + L_{\square, V_I} + 1 + c_{V_I} - c_{V_J} = 0 \, \mod{\, n}}
}
 \left\lgroup a_I - a_J - A_{\square, V_J} \epsilon_2 + L^{++}_{\square,V_I}\epsilon_1 \right\rgroup^{\frac12}
\\
\prod_{I,J = 1}^N
\prod_{
\substack{
\blacksquare \in W_I
\\
A_{\blacksquare, W_I} + L_{\blacksquare, W_J} + 1 - c_{W_I} + c_{W_J} = 0\, \mod{\, n}}
}
 \left\lgroup
b_I - b_J + A^{++}_{\blacksquare, W_I} \epsilon_2 - L_{\blacksquare,W_J}\epsilon_1
\right\rgroup^{\frac12}
\\
\prod_{
\substack{
\blacksquare \in W_J
\\
A_{\blacksquare, W_J} + L_{\blacksquare, W_I} + 1 + c_{W_I} - c_{W_J} = 0\, \mod{\, n}}
}
 \left\lgroup
b_I - b_J -A_{\blacksquare, W_J} \epsilon_2 + L^{++}_{\blacksquare, W_I} \epsilon_1
\right\rgroup^{\frac12},
\end{multline}

where,
$ \pmb{a} =  \left\lgroup a_1,...,a_N \right\rgroup$ and $ \pmb{b} =  \left\lgroup b_1,...,b_N \right\rgroup$ are Coulomb parameters, $\mu$ is a mass parameter and $\epsilon_1, \epsilon_2$ are deformation parameters.

\subsubsection{Parameter identification}

To compare the 4D strip partition function to the 4D instanton partition functions, we substitute

\begin{equation}
\label{delta.K}
\boxed{
\Delta_i = -a_i + b_i + \mu - \epsilon_1,
\quad
\Delta_{M_i}  = a_{i+1} - b_i - \mu + \epsilon_1,
\quad
i = 1, \cdots, N}
\end{equation}

where
$a_i = a_{i+N}$, and
$b_I = b_{i+N}$,
into Equation \ref{sn.norm}, to obtain,

\begin{multline}\label{s.n}
\mathcal{S}^{\, n, \,norm}_{ \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}}\right\rgroup  \left\lgroup\pmb{W},\,  \pmb{c}_{\pmb{W}}\right\rgroup \bDelta} \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup
=
\\
\prod_{I=1}^N
\prod_{J=1}^I
\prod_{
\substack{
\square \in
V_J
\\
A_{\square, V_J} + L_{\square, W_I} + 1 + c_{W_I} - c_{V_J} = 0  \,
\mod{\, n}}}
-
 \left\lgroup
a_J - b_I - \mu + A^{++}_{\square, V_J} \epsilon_2 -
                  L^{  }_{\square, W_I} \epsilon_1
\right\rgroup
\\
\prod_{
\substack{
\blacksquare \in W_I
\\
A_{\blacksquare, W_I} + L_{\blacksquare, V_J} + 1 - c_{W_I} + c_{V_J} = 0  \,
\mod{\, n}}}
- \left\lgroup a_J - b_I - \mu - A_{\blacksquare, W_I}\epsilon_2 + L^{++}_{\blacksquare, V_J}\epsilon_1 \right\rgroup
    \\
    \prod_{I=1}^N\prod_{J=1}^{I-1}\prod_{\substack{\square \in V_I\\
    A_{\square, V_I}+L_{\square, W_J}+1 -c_{V_I}+c_{W_J} = 0\, \mod{\, n}}} \left\lgroup a_I - b_J - \mu + A^{++}_{\square, V_I}\epsilon_2 - L_{\square, W_J}\epsilon_1 \right\rgroup
    \\
    \prod_{\substack{\blacksquare \in W_J\\
    A_{\blacksquare, W_J}+L_{\blacksquare, V_I}+1 + c_{V_I}-c_{W_J} = 0\, \mod{\, n}}} \left\lgroup a_I - b_J - \mu - A_{\blacksquare, W_J}\epsilon_2 + L^{++}_{\blacksquare, V_I}\epsilon_1 \right\rgroup
    \\
    \prod_{I=1}^N
     \left\lgroup
    \prod_{\substack{\square \in V_I\\
    A_{\square, V_I}+L_{\square, V_I}+1 = 0\, \mod{\, n}}}- \left\lgroup A^{++}_{\square, V_I}\epsilon_2 - L_{\square, V_I}\epsilon_1 \right\rgroup^{-1}
    \prod_{\substack{\blacksquare \in W_I\\
    A_{\blacksquare, W_I}+L_{\blacksquare, W_I}+1 = 0\, \mod{\, n}}} \left\lgroup -A_{\blacksquare, W_I}\epsilon_2 + L^{++}_{\blacksquare, W_I}\epsilon_1 \right\rgroup^{-1}
    \right\rgroup
    \\
    \prod_{I=1}^N\prod_{J=1}^{I-1}\prod_{\substack{\square \in V_I\\
    A_{\square, V_I}+L_{\square, V_J} + 1 -c_{V_I}+c_{V_J} = 0\, \mod{\, n}}} \left\lgroup a_I-a_J + A^{++}_{\square, V_I}\epsilon_2 - L_{\square, V_J}\epsilon_1 \right\rgroup^{-1}
    \\
    \prod_{\substack{\square \in V_J\\
    A_{\square, V_J}+L_{\square, V_I}+1 + c_{V_I} - c_{V_J} = 0\, \mod{\, n}}} \left\lgroup a_I-a_J - A_{\square, V_J}\epsilon_2 + L^{++}_{\square, V_I} \epsilon_1 \right\rgroup^{-1}
    \\
    \prod_{I=1}^N
    \prod_{J=1}^{I-1}
    \prod_{\substack{\blacksquare \in  W_I\\
    A_{\blacksquare, W_I}+L_{\blacksquare, W_J}+1-c_{W_I}+c_{W_J} = 0\, \mod{\, n}}}
    - \left\lgroup
    b_J-b_I - A_{\blacksquare,   W_I} \epsilon_2 + L_{\blacksquare,    W_J}^{++} \epsilon_1
    \right\rgroup^{-1}
    \\
    \prod_{\substack{\blacksquare \in  W_J\\
    A_{\blacksquare, W_J}+L_{\blacksquare, W_I} + 1 + c_{W_I} - c_{W_J} = 0\, \mod{\, n}}}
    - \left\lgroup
    b_J-b_I + A_{\blacksquare,   W_J}^{++} \epsilon_2 - L_{\blacksquare,    W_I} \epsilon_1
    \right\rgroup^{-1}
    \\
    = \frac{\mathcal{S}^{\, n, \,num}_{ \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup\bDelta} \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup}{\mathcal{S}^{\, n}_{den \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup\bDelta} \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup}
\end{multline}

\subsubsection{Sign factors and abbreviations}
We define,

\begin{equation}
F  \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \, |\, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup =
\frac{
F^{num}  \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}}\, |\, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
}{
F_{den}  \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}}\, |\, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
},
\end{equation}

\begin{multline}
F^{\, num}
 \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \, |\, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup =
\prod_{I=1}^N
\prod_{J=1}^I
\prod_{
\substack{
\square \in V_{J}
\\
A_{\square, V_J} +
L_{\square, W_I} + 1 + c_{W_I} - c_{V_J} = 0\, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\prod_{
\substack{
\blacksquare \in W_I
\\
A_{\blacksquare, W_I} +
L_{\blacksquare, V_J} + 1 - c_{W_I} + c_{V_J} = 0\, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\end{multline}

\begin{multline}
F_{den}
 \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup =
\prod_{I=1}^N
\prod_{
\substack{
\square \in V_I
\\
A_{\square, V_I} +
L_{\square, V_J} + 1 - c_{V_I} - c_{V_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\blacksquare \in W_I
\\
A_{\blacksquare, W_I} +
L_{\blacksquare, W_J} + 1 - c_{W_I} + c_{W_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\prod_{
\substack{
\blacksquare \in W_J
\\
A_{\blacksquare, W_J} +
L_{\blacksquare, W_I} + 1 + c_{W_I} - c_{W_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\end{multline}

and introduce the abbreviations,

\begin{equation}
\mathcal{H}_{Y_{IJ}}   \left\lgroup  \pmb{x},  \pmb{c}_{\pmb{Y}} \right\rgroup
=
\mathcal{A}_{Y_{IJ}}   \left\lgroup  \pmb{x},  \pmb{c}_{\pmb{Y}} \right\rgroup \, \mathcal{L}_{Y_{IJ}}
 \left\lgroup  \pmb{x},  \pmb{c}_{\pmb{Y}} \right\rgroup,
\end{equation}

where,

\begin{equation}
\mathcal{A}_{V_{IJ}}  \left\lgroup  \pmb{x},  \pmb{c}_{\pmb{V}} \right\rgroup =
\prod_{
\substack{
\square \in V_I
\\
A_{\square, V_I} +
L_{\square, V_J} + 1 - c_{V_I} + c_{V_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup
x_I - x_J
+ A^{++}_{\square,  V_I} \epsilon_2
- L^{  }_{\square,  V_J} \epsilon_1
\right\rgroup,
\quad
\mathcal{A}_{\emptyset, \emptyset}   \left\lgroup  \pmb{x}, \pmb{c}_{\pmb{V}} \right\rgroup = 1,
\end{equation}

\begin{equation}
\mathcal{L}_{W_{IJ}}   \left\lgroup  \pmb{x},  \pmb{c}_{\pmb{W}} \right\rgroup =
\prod_{
\substack{
\blacksquare \in  W_J
\\
A_{\blacksquare, W_J} +
L_{\blacksquare, W_I} + 1 + c_{W_I} - c_{W_J} = 0 \, \mod{\, n}
}
}
 \left\lgroup x_I - x_J
- A^{  }_{\blacksquare,    W_J} \epsilon_2
+ L^{++}_{\blacksquare,    W_I} \epsilon_1
\right\rgroup,
\quad
\mathcal{L}_{\emptyset, \emptyset}   \left\lgroup  \pmb{x},  \pmb{c}_{\pmb{W}} \right\rgroup = 1,
\end{equation}

\subsubsection{Comparing the numerators}
The numerators
$\mathcal{S}^{
\, n, \,num
}_{
 \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta
}
 \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup
$
and
$\mathcal{Z}^{\, n, \, num}
 \left\lgroup  \pmb{a},\pmb{V}, \pmb{c}_{\pmb{V}} \,  | \, \mu \, | \,   \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup$
are identical up to possible overall signs,

\begin{equation}
\label{abbrev.s.num}
\mathcal{S}^{\, n, \,num}_{ \left\lgroup \, \pmb{V}, \,  \pmb{c}_{\pmb{V}} \right\rgroup
                    \left\lgroup \, \pmb{W}, \,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta}
                    \left\lgroup \, \epsilon_1, \, \epsilon_2 \right\rgroup =
F^{num}  \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
\mathcal{Z}^{\, n, \, num}
 \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \mu \, | \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
\end{equation}

\subsubsection{Comparing the denominators}
The denominators are,

\begin{multline}
\label{abbrev.s.den}
\mathcal{S}^{\, n
}_{
\, den\,  \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta
}
 \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup =
\\
F_{den} \left\lgroup \pmb{V}, \pmb{c}_{\pmb{V}} \, | \,  \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
\prod_{I=1}^N
\mathcal{A}_{V_{II}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup \mathcal{L}_{W_{II}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
\prod_{J=1}^N
\prod_{I=1}^{J-1}
\mathcal{H}_{V_{JI}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\mathcal{H}_{W_{IJ}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
\end{multline}

and

\begin{equation}
\label{abbrev.z.den}
\mathcal{Z}^{\, n}_{\, den}
 \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, | \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup  =
\prod_{I=1}^N
\prod_{J=1}^N
 \left\lgroup
\mathcal{H}_{V_{IJ}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\mathcal{H}_{W_{IJ}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
\right\rgroup^{1/2}
\end{equation}

In other words,
$\mathcal{S}^{\, n, \, norm
}_{
 \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup \bDelta
}
 \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup$ and
$\mathcal{Z}^{\, n, \, norm
}
 \left\lgroup
 \pmb{a}, \pmb{V}, \pmb{c}_{\pmb{V}}
\, | \,
\mu
\, | \,
 \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup$
have different denominators.
However, we show in the sequel that in computing
linear as well as cyclic 4D instanton partition
functions, we get exactly the same result
by using $\mathcal{S}^{\, n, \, norm}$ (with an appropriate
choice of framing factors) and by using $\mathcal{Z}^{\, n, \, norm}$.

\section{From $n$-coloured 5D strips to $n$-coloured 5D webs}
\label{section.07}
\textit{We glue 5D $n$-coloured strip diagrams to form 5D $n$-coloured web diagrams.}

\begin{scriptsize}
\begin{figure}
\begin{tikzpicture} [scale=.8]

\draw [ultra thick] (0,0)--(0,1);
\node [below] at (0,0) {$\pmb{\emptyset}$};
\draw [thick] (0,1)--(2,1);
\node [above] at (1,1) {$V_{1N},\Delta_{C_{1,N}}$};
\draw [ultra thick] (0,1)--(-1,2);
\draw [thick] (-1,2)--(-2,2);
\node [left] at (-2,2) {$\emptyset$};
\draw [ultra thick] (-1,2)--(-1,3);
\draw [ultra thick, dashed] (-1,3)--(-2,4);
\draw [ultra thick] (-2,4)--(-2,5);
\draw [thick] (-2,5)--(0,5);
\node [above] at (-1,5) {$V_{12},\Delta_{C_{1,2}}$};
\draw [ultra thick] (-2,5)--(-3,6);
\draw [thick] (-3,6)--(-4,6);
\node [left] at (-4,6) {$\emptyset$};
\draw [ultra thick] (-3,6)--(-3,7);
\draw [thick] (-3,7)--(-1,7);
\node [above] at (-2,7) {$V_{11},\Delta_{C_{1,1}}$};
\draw [ultra thick] (-3,7)--(-4,8);
\draw [thick] (-4,8)--(-5,8);
\node [left] at (-5,8) {$\emptyset$};
\draw [ultra thick] (-4,8)--(-4,9);
\node [above] at (-4,9) {$\pmb{\emptyset}$};

\draw [ultra thick] (2,1)--(2,2);
\draw [ultra thick] (2,1)--(3,0);
\draw [ultra thick] (3,0)--(3,-1);
\node [below] at (3,-1) {$\pmb{\emptyset}$};
\draw [thick] (3,0)--(5,0);
\node [above] at (4,0) {$V_{2N}, \Delta_{C_{2,N}}$};
\draw [ultra thick,dashed] (2,2)--(1,3);
\draw [ultra thick] (1,3)--(1,4);
\draw [thick] (1,4)--(3,4);
\node [above] at (2,4) {$V_{22}, \Delta_{C_{2,2}}$};
\draw [ultra thick] (1,4)--(0,5);
\draw [ultra thick] (0,5)--(0,6);
\draw [thick] (0,6)--(2,6);
\node [above] at (1,6) {$V_{21}, \Delta_{C_{2,1}}$};
\draw [ultra thick] (0,6)--(-1,7);
\draw [ultra thick] (-1,7)--(-1,8);
\node [above] at (-1,8) {$\pmb{\emptyset}$};

\draw [thick,dashed] (5,0)--(6,0);
\draw [thick,dashed] (3,4)--(4,4);
\draw [thick,dashed] (2,6)--(3,6);

\draw [thick] (6,0)--(8,0);
\node [above] at (7,0) {$V_{m1}, \Delta_{C_{m,N}}$};
\draw [thick] (4,4)--(6,4);
\node [above] at (5,4) {$V_{m2}, \Delta_{C_{m,2}}$};
\draw [thick] (3,6)--(5,6);
\node [above] at (4,6) {$V_{mN}, \Delta_{C_{m,1}}$};

\draw [ultra thick] (5,6)--(5,7);
\node [above] at (5,7) {$\pmb{\emptyset}$};
\draw [ultra thick] (5,6)--(6,5);
\draw [ultra thick] (6,5)--(6,4);
\draw [thick] (6,5)--(7,5);
\node [right] at (7,5) {$\emptyset$};
\draw [ultra thick] (6,4)--(7,3);
\draw [ultra thick] (7,3)--(7,2);
\draw [thick] (7,3)--(8,3);
\node [right] at (8,3) {$\emptyset$};
\draw [ultra thick,dashed] (7,2)--(8,1);
\draw [ultra thick] (8,1)--(8,0);
\draw [ultra thick] (8,0)--(9,-1);
\draw [ultra thick] (9,-1)--(9,-2);
\node [below] at (9,-2) {$\pmb{\emptyset}$};
\draw [thick] (9,-1)--(10,-1);
\node [right] at (10,-1) {$\emptyset$};

\end{tikzpicture}
\caption{
\textit{
The linear web diagram constructed by gluing $(m+1)$ $N$-strip
diagrams. Each $N$-strip is linked to the next by a series
of internal horizontal edges assigned with K\"ahler parameters
$\Delta_{C_{iI}}$ and sum over all possible associated Young
diagrams $\pmb{V}_i =  \left\lgroup V_{i1}, \cdots, V_{iN} \right\rgroup$ in a given
series.
}
}
\label{linear.web.diagram}
\end{figure}
\end{scriptsize}

\begin{scriptsize}
\begin{figure}
\begin{tikzpicture} [scale=.8]

\draw [ultra thick] (0,0)--(0,1);
\node [below] at (0,0) {$\pmb{\emptyset}$};
\draw [thick] (0,1)--(2,1);
\node [above] at (1,1) {$V_{1N},\Delta_{C_{1,N}}$};
\draw [ultra thick] (0,1)--(-1,2);
\draw [thick] (-1,2)--(-2,2);
\draw [ultra thick] (-1,2)--(-1,3);
\draw [ultra thick, dashed] (-1,3)--(-2,4);
\draw [ultra thick] (-2,4)--(-2,5);
\draw [thick] (-2,5)--(0,5);
\node [above] at (-1,5) {$V_{12},\Delta_{C_{1,2}}$};
\draw [ultra thick] (-2,5)--(-3,6);
\draw [thick] (-3,6)--(-4,6);
\draw [ultra thick] (-3,6)--(-3,7);
\draw [thick] (-3,7)--(-1,7);
\node [above] at (-2,7) {$V_{11},\Delta_{C_{1,1}}$};
\draw [ultra thick] (-3,7)--(-4,8);
\draw [thick] (-4,8)--(-5,8);
\draw [ultra thick] (-4,8)--(-4,9);
\node [above] at (-4,9) {$\pmb{\emptyset}$};

\draw [thick,dashed] (-5,8)--(-6,8);
\node [above] at (-5,8) {$V_{01}, \Delta_{C_{0,1}}$};
\draw [thick,dashed] (-4,6)--(-5,6);
\node [above] at (-4,6) {$V_{02}, \Delta_{C_{0,2}}$};
\draw [thick,dashed] (-2,2)--(-3,2);
\node [above] at (-2,2) {$V_{0N}, \Delta_{C_{0,N}}$};

\draw [ultra thick] (2,1)--(2,2);
\draw [ultra thick] (2,1)--(3,0);
\draw [ultra thick] (3,0)--(3,-1);
\node [below] at (3,-1) {$\pmb{\emptyset}$};
\draw [thick] (3,0)--(5,0);
\node [above] at (4,0) {$V_{2N}, \Delta_{C_{2,N}}$};
\draw [ultra thick,dashed] (2,2)--(1,3);
\draw [ultra thick] (1,3)--(1,4);
\draw [thick] (1,4)--(3,4);
\node [above] at (2,4) {$V_{22}, \Delta_{C_{2,2}}$};
\draw [ultra thick] (1,4)--(0,5);
\draw [ultra thick] (0,5)--(0,6);
\draw [thick] (0,6)--(2,6);
\node [above] at (1,6) {$V_{21}, \Delta_{C_{2,1}}$};
\draw [ultra thick] (0,6)--(-1,7);
\draw [ultra thick] (-1,7)--(-1,8);
\node [above] at (-1,8) {$\pmb{\emptyset}$};

\draw [thick,dashed] (5,0)--(6,0);
\draw [thick,dashed] (3,4)--(4,4);
\draw [thick,dashed] (2,6)--(3,6);

\draw [thick] (6,0)--(8,0);
\node [above] at (7,0) {$V_{m1}, \Delta_{C_{m,N}}$};
\draw [thick] (4,4)--(6,4);
\node [above] at (5,4) {$V_{m2}, \Delta_{C_{m,2}}$};
\draw [thick] (3,6)--(5,6);
\node [above] at (4,6) {$V_{mN}, \Delta_{C_{m,1}}$};

\draw [ultra thick] (5,6)--(5,7);
\node [above] at (5,7) {$\pmb{\emptyset}$};
\draw [ultra thick] (5,6)--(6,5);
\draw [ultra thick] (6,5)--(6,4);
\draw [thick] (6,5)--(7,5);
\draw [ultra thick] (6,4)--(7,3);
\draw [ultra thick] (7,3)--(7,2);
\draw [thick] (7,3)--(8,3);
\draw [ultra thick,dashed] (7,2)--(8,1);
\draw [ultra thick] (8,1)--(8,0);
\draw [ultra thick] (8,0)--(9,-1);
\draw [ultra thick] (9,-1)--(9,-2);
\node [below] at (9,-2) {$\pmb{\emptyset}$};
\draw [thick] (9,-1)--(10,-1);

\draw [thick,dashed] (7,5)--(8,5);
\node [above] at (7,5) {$V_{01}, \Delta_{C_{0,1}}$};
\draw [thick,dashed] (8,3)--(9,3);
\node [above] at (9,3) {$V_{02}, \Delta_{C_{0,2}}$};
\draw [thick,dashed] (10,-1)--(11,-1);
\node [above] at (11,-1) {$V_{0N}, \Delta_{C_{0,N}}$};

\end{tikzpicture}
\caption{
\textit{
The cyclic web diagram constructed by gluing the first
and the last $N$-strips in the linear web diagram.
}
}
\label{cyclic.web.diagram}
\end{figure}
\end{scriptsize}

The linear and cyclic web diagram
topological partition functions are obtained from Figures
\ref{linear.web.diagram} and \ref{cyclic.web.diagram}, once
the horizontal edge factors are specified. The answer takes
the general form,

\begin{equation}
\label{generic.web.partition}
\mathcal{S}^{\, n} \left\lgroup x,y,R \right\rgroup =
\sum_{ \left\lgroup \pmb{V}_1, \,
 \pmb{c}_{\pmb{V}_1} \right\rgroup, \cdots,  \left\lgroup \pmb{V}_p, \,
 \pmb{c}_{\pmb{V}_p} \right\rgroup}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1} \right\rgroup,  \left\lgroup \pmb{V}_2,\,
 \pmb{c}_{\pmb{V}_2} \right\rgroup, \cdots,  \left\lgroup \pmb{V}_m,\,
 \pmb{c}_{\pmb{V}_m} \right\rgroup
}
 \left\lgroup x, y, R \right\rgroup
\end{equation}

where
$\mathcal{S}^{\, n, \, block}_{
 \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1} \right\rgroup,
 \left\lgroup \pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2} \right\rgroup, \cdots,
 \left\lgroup \pmb{V}_p,\,  \pmb{c}_{\pmb{V}_m} \right\rgroup}  \left\lgroup x, y, R \right\rgroup$
is a product of factors of type
$\mathcal{S}^{\, p, \,norm}_{
 \left\lgroup \pmb{V}_i,  \pmb{c}_{\pmb{V}_i} \right\rgroup  \left\lgroup \pmb{V}_{i+1},\,
 \pmb{c}_{\pmb{V}_{i+1}}\right\rgroup \bDelta_{\, i, i+1}}$
for every strip, and
an edge factor for every internal
horizontal edge.

\subsection{The summation in Equation
\ref{generic.web.partition}}

For $n=1$, the summation is over all $N$-Young diagrams because the moduli space
of instantons on $\mathbb{C}^{\, 2}$ is connected, and the localization theorem requires that we sum over
the contributions of all fixed points, each of which
is labelled by $N$-Young diagrams.
For $n = 2, 3, \cdots$, the moduli space of instantons on
$\mathbb{C}^{\, 2}/\mathbb{Z}_{\, n}$ is a union of disjoint smaller
spaces \cite{fucito.morales.poghossian},
and the summation in Equation \ref{generic.web.partition}
is not over all possible charged $N$-Young diagrams, but
restricted to certain series of charged Young diagrams.

\subsection{
$ \left\lgroup \pmb{N}, \pmb{k} \right\rgroup$-type
Young diagrams
}
Following \cite{alfimov.tarnopolsky,ito.maruyoshi.okuda}, given
$ \left\lgroup \pmb{Y},\,  \pmb{c}_{\pmb{Y}}\right\rgroup =
 \left\lgroup  \left\lgroup Y_1,\, c_{Y_1} \right\rgroup, \cdots,  \left\lgroup Y_N, \, c_{Y_N} \right\rgroup \right\rgroup$,
we define $\pmb{N} =  \left\lgroup N_{\, 0}, \cdots, N_{\, n-1} \right\rgroup$, where
$N_c$ is the number of Young diagrams of charge $c$. Further,
we define $ \pmb{k} =  \left\lgroup k_{\, 0}, \cdots, k_{\, n-1} \right\rgroup$, where
$k_c$ is the number of cells with coordinates
$ \left\lgroup \iota, \jmath \right\rgroup \in Y_i$, $i = 1, \cdots, N$ (that is, in all Young diagrams), such that,

\begin{equation}
c_{\, Y_i} + (\iota - 1) - (\jmath -  1) \, =  \, c,
\quad
c \in  \left\lgroup 0, \cdots, n-1 \right\rgroup
\end{equation}

and say that $ \left\lgroup \pmb{Y}, \pmb{c}_{\pmb{Y}} \right\rgroup$ is of $ \left\lgroup \pmb{N}, \pmb{k} \right\rgroup$-type.
Clearly,
$\sum_{c = 0}^{n-1} N_c = N$, and
$\sum_{c = 0}^{n-1} k_c = \sum_{I=1}^N \, | \, Y_I\, | \, $.

\subsection{$ \left\lgroup \pmb{N},  \pmb{u} \right\rgroup$-series of solutions}
Let $ \pmb{u} =  \left\lgroup u_1,\cdots, u_{n-1} \right\rgroup$ be given by
\footnote{\,
$ \pmb{u}$ is related to the first Chern class
of the instanton gauge bundle
$c_1 ( E ) =
\sum_{i=1}^{n-1}
u_i \,
c_1  \left\lgroup \mathcal{T}_i \right\rgroup$,
\cite{fucito.morales.poghossian, alfimov.tarnopolsky}.
},

\begin{equation}
\label{first.chern.class}
u_{\, i} =
N_{\, i}
+
 \left\lgroup
k_{\, i-1}
- 2 k_{\, i}
+   k_{\, i+1}
\right\rgroup,
\end{equation}

then, for fixed $ \pmb{u}$, solve
Equation \ref{first.chern.class} to
find all possible values of $ \left\lgroup \pmb{N}, \,  \pmb{k} \right\rgroup$.
For fixed $\pmb{N}$, there may be no solutions,
or
there may be infinitely-many solutions
$ \pmb{k}$ that satisfy
Equation \ref{first.chern.class}.
In the latter (relevant) case, for a corresponding fixed $\pmb{N}$,
$ \pmb{k}$ is a sequence in $\mathbb{Z}^{\, n}_{\, \geqslant \, 0}$ indexed
by $k_{\, 0} = r \in \mathbb{Z}_{\, \geqslant \, 0}$  which we denote
by $ \pmb{k} \, (r)$. We define,

\begin{align}
\text{$ \left\lgroup \pmb{N},  \pmb{u} \right\rgroup$-series}
&
=
 \left\lgroup
 \left\lgroup \pmb{Y},  \pmb{c}_{\pmb{Y}}\right\rgroup \, | \,
\textrm{
$ \left\lgroup \pmb{Y},  \pmb{c}_{\pmb{Y}}\right\rgroup$ is of $ \left\lgroup \pmb{N},  \pmb{k} \right\rgroup$-type
for $ \pmb{k}$ such that $\pmb{N}$ and $ \pmb{k}$
satisfy \eqref{first.chern.class}
}
\right\rgroup
\\
&
=
 \left\lgroup
 \left\lgroup \pmb{Y},  \pmb{c}_{\pmb{Y}}\right\rgroup \, | \,
\textrm{
$ \left\lgroup \pmb{Y},  \pmb{c}_{\pmb{Y}}\right\rgroup$ is of $ \left\lgroup \pmb{N},  \pmb{k} \, (r) \right\rgroup$-type
for some $r \in \mathbb{Z}_{\, \geqslant \, 0}$
}
\right\rgroup,
\end{align}

and restrict the summation in
$ \left\lgroup \pmb{Y},  \pmb{c}_{\pmb{Y}}\right\rgroup$ to one or more of
these series.

\subsection{Example}
For
$N = 2$, $n = 4$, and
$ \pmb{u} =  \left\lgroup 0, 0, 0 \right\rgroup$,
there are three
$ \left\lgroup \pmb{N}, \,  \pmb{k}\right\rgroup$-series
\footnote{\,
We use this example, which can be found in
\cite{alfimov.tarnopolsky},
to reproduce the same result.
},

\begin{align}
\text{Series 1:} & \quad \pmb{N} =  \left\lgroup 2, 0, 0, 0 \right\rgroup,
\quad  \pmb{k} =  \left\lgroup r, r, r, r \right\rgroup
\label{u.0.series.1}
\\
\text{Series 2:} & \quad \pmb{N} =  \left\lgroup 0, 1, 0, 1 \right\rgroup,
\quad  \pmb{k} =  \left\lgroup r, r+1, r+1, r+1 \right\rgroup
\label{u.0.series.2}
\\
\text{Series 3:} & \quad \pmb{N} =  \left\lgroup 0, 0, 2, 0 \right\rgroup,
\quad  \pmb{k} =  \left\lgroup r, r+1, r+2, r+1 \right\rgroup
\label{u.0.series.3}
\end{align}

for $r \in \mathbb{Z}_{\, \geqslant \, 0}$.
This gives rise to three series of charged
Young diagrams
$ \left\lgroup \pmb{Y},\,  \pmb{c}_{\pmb{Y}} \right\rgroup$. The charges $ \pmb{c}_{\pmb{Y}}$ are fixed within the series by $\pmb{N}$,

\begin{align*}
\text{Series 1 : } &   \quad  \pmb{c}_{\pmb{Y}} =  \left\lgroup 0, 0 \right\rgroup
\\
\text{Series 2 : } &   \quad  \pmb{c}_{\pmb{Y}} =  \left\lgroup 1, 3 \right\rgroup
\quad
\text{or equivalently} \quad  \pmb{c}_{\pmb{Y}} =  \left\lgroup 3, 1 \right\rgroup
\\
\text{Series 3 : } &   \quad  \pmb{c}_{\pmb{Y}} =  \left\lgroup 2, 2 \right\rgroup
\end{align*}

Switching $ \pmb{c}_{\pmb{Y}} =  \left\lgroup 1, 3 \right\rgroup$ and
$ \pmb{c}_{\pmb{Y}} =  \left\lgroup 3,1 \right\rgroup$ is the same as switching the labels
of $Y_1$ and $Y_2$ in the summation, hence they are equivalent.
The summation in Equation \ref{generic.web.partition} is
restricted to one or more of these series.

\subsection{The edge factors}
For the $I$-th line between
$\mathcal{S}^{\, n, \, norm}_{ \left\lgroup \pmb{V}_{i-1}, \,  \pmb{c}_{\pmb{V}_{i-1}} \right\rgroup  \left\lgroup
\pmb{V}_i,     \,  \pmb{c}_{\pmb{V}_i}     \right\rgroup \bDelta_{\, i-1,\, i}}$
and
$\mathcal{S}^{\, n, \, norm}_{ \left\lgroup \pmb{V}_i,     \,  \pmb{c}_{\pmb{V}_i}     \right\rgroup  \left\lgroup \pmb{V}_{i+1}, \,     \pmb{c}_{\pmb{V}_{i+1}} \right\rgroup \bDelta_{\, i,\, i+1}}$,
we define,

\begin{equation}
\text{Edge factor} =
 \left\lgroup
- Q_{C_{iI}} \right\rgroup^{\, | \, V_{iI}\, | \, } \,
f^{\, n}_{i-1, i, i+1, I},
\end{equation}

with $Q_{C_{i I}} = z_i \, e^{- R \, \Delta_{\, C_{i I}}}$,
where $z_i$ is an instanton
expansion parameter, and the framing
factor $f^{\, n}_{i-1,i,i+1,I}$ is,

\begin{equation}
f^{\, n}_{i-1,i,i+1,I} =
 \left\lgroup -1 \right\rgroup^{\, | \, V_{iI}\, | \, }
f^{\, n, \,left}_{i - 1, i, I} f^{\, p, \,right}_{i,i+1,I}
\end{equation}

\begin{equation}
f^{\, n, \,right}_{i,i+1,I} =
\prod_{J=I}^N
\prod_{
\substack{
\square \in V_{iI}
\\
A_{\square, V_{iI}} + L_{\square, V_{i+1,J}} + 1
\\
+ c_{V_{i+1,J}} - c_{V_{iI}} = \, 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\prod_{J=1}^I
\prod_{
\substack{
\square \in V_{iI}
\\
A_{\square, V_{iI}} + L_{\square, V_{iJ}} + 1
\\
- c_{V_{iI}} - c_{V_{iJ}} = \, 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\end{equation}

\begin{equation}
f^{\, n, \,left}_{i-1,i,I} =
\prod_{J=1}^I
\prod_{
\substack{
\square \in V_{iI}
\\
A_{\square, V_{iI}} + L_{\square, V_{i-1,J}} + 1
\\
- c_{V_{iI}} + c_{V_{i-1,J}} = \, 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\prod_{J=1}^{I-1}
\prod_{
\substack{
\square \in V_{iI}
\\
A_{\square, V_{iI}} + L_{\square, V_{iJ}} + 1
\\
- c_{V_{iI}} + c_{V_{iJ}} = \, 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\prod_{J=I+1}^N
\prod_{
\substack{
\square \in V_{iI}
\\
A_{\square, V_{iI}} + L_{\square, V_{iJ}} + 1
\\
+ c_{V_{iJ}} - c_{V_{iI}} = \, 0 \, \mod{\, n}
}
}
 \left\lgroup -1 \right\rgroup
\end{equation}

The framing factor is designed to take care of the sign difference between the conformal blocks calculated from $\mathcal{Z}^{\, n}$ and $\mathcal{S}^{\, n, \, norm}$.
When $n = 1$, the framing factor is,

\begin{multline}
f^{p=1}_{i-1, i, i+1, I} =
 \left\lgroup -1 \right\rgroup^{\, | \, V_{iI}\, | \, }
\prod_{J=I}^N
\prod_{
\square \in V_{iI}}
 \left\lgroup -1 \right\rgroup
\prod_{J=1}^I
\prod_{
\square \in V_{iI}}
 \left\lgroup -1 \right\rgroup
\prod_{
\square \in V_{iI}}
 \left\lgroup -1 \right\rgroup
\prod_{J=1}^{I-1}
\prod_{\square \in V_{iI}}
 \left\lgroup -1 \right\rgroup \prod_{J=I+1}^N
\prod_{
\square \in V_{iI}}
 \left\lgroup -1 \right\rgroup
\\
=
 \left\lgroup -1 \right\rgroup^{\, | \,  V_{iI} \, | \, }
 \left\lgroup -1 \right\rgroup^{(N-I+1) \, | \, V_{iI}\, | \, }
 \left\lgroup -1 \right\rgroup^{I\, | \, V_{iI}\, | \, }
 \left\lgroup -1 \right\rgroup^{\, | \, V_{iI}\, | \, }
 \left\lgroup -1 \right\rgroup^{(I-1)\, | \, V_{iI}\, | \, }
 \left\lgroup -1 \right\rgroup^{(N-I)\, | \, V_{iI}\, | \, } = 1
\end{multline}

$\mathcal{S}^{\, 5D}_{\, N}$ is given in terms of $N$-strip partition functions
for linear and cyclic cases as follows,

\subsubsection{Linear conformal blocks}
(See Figure \ref{linear.web.diagram})
\begin{multline}
\mathcal{S}^{
\, n, \,linear\, block
}_{
 \left\lgroup
\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}
\right\rgroup,
 \left\lgroup
\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}
\right\rgroup, \cdots,
 \left\lgroup
\pmb{V}_m, \,  \pmb{c}_{\pmb{V}_m}\right\rgroup}
 \left\lgroup x, y, R
\right\rgroup
\\
=
\mathcal{S}_{
 \left\lgroup
\pmb{\emptyset}, \,  \pmb{c}_{\pmb{\emptyset}_0}
\right\rgroup
 \left\lgroup
\pmb{V}_1, \,  \pmb{c}_{\pmb{V}_1}
\right\rgroup
\bDelta_{\, 0 1}}^{\, n, \, norm}
 \left\lgroup x, y, R
\right\rgroup
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup -Q_{C_{1I}} \right\rgroup^{\, | \, V_{1I}\, | \, }f^{\, n}_{012,I}
\right\rgroup
\mathcal{S}_{
 \left\lgroup
\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}
\right\rgroup
 \left\lgroup
\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}
\right\rgroup
\bDelta_{\, 1 2}}^{\, n, \, norm}
 \left\lgroup x, y, R \right\rgroup
\cdots
\\
\cdots
\mathcal{S}_{
 \left\lgroup
\pmb{V}_{m-1},\,  \pmb{c}_{\pmb{V}_{m-1}}
\right\rgroup
 \left\lgroup
\pmb{V}_{m},\,  \pmb{c}_{\pmb{V}_{m}}
\right\rgroup
\bDelta_{\, m-1, m}
}^{
\, n, \, norm
}
 \left\lgroup x, y, R \right\rgroup
 \left\lgroup
\prod_{I = 1}^N
 \left\lgroup
- Q_{C_{m,I}} \right\rgroup^{\, | \, V_{m,I}\, | \, } f^{\, n}_{m-1, m, m+1, I}
\right\rgroup
\mathcal{S}_{
 \left\lgroup \pmb{V}_{p},\,  \pmb{c}_{\pmb{V}_{p}} \right\rgroup
 \left\lgroup
\pmb{\emptyset}, \,  \pmb{c}_{\pmb{\emptyset}_{m+1}}
\right\rgroup
\bDelta_{\, m, m+1}}^{\, n, \, norm}
 \left\lgroup
x, y, R
\right\rgroup
\end{multline}

\subsubsection{Cyclic conformal blocks}
(See Figure \ref{cyclic.web.diagram})

\begin{multline}
    \mathcal{S}^{\, n, \,cyclic\,  block}_{ \left\lgroup \pmb{V}_0,\,  \pmb{c}_{\pmb{V}_0}\right\rgroup, \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup, \left\lgroup \pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}\right\rgroup,\cdots, \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m}\right\rgroup} \left\lgroup x,y,R \right\rgroup\\
    =
    \mathcal{S}_{ \left\lgroup\pmb{V}_0,\,  \pmb{c}_{\pmb{V}_0}\right\rgroup  \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup\bDelta_{\, 0 1}}^{\, n, \, norm} \left\lgroup x,y,R \right\rgroup
     \left\lgroup\prod_{I=1}^N \left\lgroup -Q_{C_{1I}} \right\rgroup^{\, | \, V_{1I}\, | \, }f^{\, n}_{012,I}\right\rgroup
    \mathcal{S}_{ \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\, \pmb{V}_1} \right\rgroup  \left\lgroup \pmb{V}_2,\,  \pmb{c}_{\pmb{V}_{2}}\right\rgroup\bDelta_{\, 1 2}}^{\, n, \, norm} \left\lgroup x,y,R \right\rgroup\cdots
    \\
    \cdots \mathcal{S}_{ \left\lgroup\pmb{V}_{m-1},\,  \pmb{c}_{\pmb{V}_{m-1}} \right\rgroup  \left\lgroup\pmb{V}_{m},\,  \pmb{c}_{\pmb{V}_{m}}\right\rgroup\bDelta_{\, m-1, m}}^{\, n, \, norm} \left\lgroup x,y,R \right\rgroup
     \left\lgroup \prod_{I=1}^N \left\lgroup -Q_{C_{m,I}} \right\rgroup^{\, | \, V_{m,I}\, | \, }f^{\, n}_{m-1,m,0,I}\right\rgroup
    \mathcal{S}_{ \left\lgroup\pmb{V}_{m},\,  \pmb{c}_{\pmb{V}_{m}} \right\rgroup  \left\lgroup \pmb{V}_0,\,  \pmb{c}_{\pmb{V}_{0}}\right\rgroup \bDelta_{\, m 0}}^{\, n, \, norm} \left\lgroup x,y,R \right\rgroup
    \\
     \left\lgroup\prod_{I=1}^N \left\lgroup -Q_{C_{0,I}} \right\rgroup^{\, | \, V_{0,I}\, | \, }f^{\, n}_{m01,I}\right\rgroup
\end{multline}

\section{The 4D limit of the 5D $n$-coloured web diagrams}
\label{section.08}
\textit{We take the 4D limit of the 5D $n$-coloured web diagrams, and show that
$\mathcal{S}^{\, n,\, norm}$ and
$\mathcal{Z}^{\, n,\, 4D}_{\, N}$
lead to the same
4D instanton partition functions.
}

Given a pair of charged $N$-Young diagrams
$ \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \right\rgroup$ and $ \left\lgroup \pmb{W},  \pmb{c}_{\pmb{W}}\right\rgroup$ of
$ \left\lgroup \pmb{N}_{\pmb{V}},  \pmb{k}_{\pmb{V}} \right\rgroup$-type and $ \left\lgroup \pmb{N}_{\pmb{W}},  \pmb{k}_{\pmb{W}} \right\rgroup$-type respectively,
the number of factors that appear in
$\mathcal{Z}^{\, n, \, num} \left\lgroup  \pmb{a},\pmb{V}, \pmb{c}_{\pmb{V}}\, |\, \mu\, |\,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup$ (as given in Equation \ref{zp.z.num}) is,

\begin{equation}
K_n \left\lgroup \pmb{N}_{\pmb{V}},\,  \pmb{k}_{\pmb{V}}\, |\, \pmb{N}_{\pmb{W}},  \pmb{k}_{\pmb{W}} \right\rgroup =
\sum_{c = 0}^{n-1}
 \left\lgroup
   k_{V,c+1} \, k_{W,c  }
- 2k_{V,c  } \, k_{W,c  }
+  k_{V,c  } \, k_{W,c+1}
+ N_{V,c} \, k_{W,c} + N_{W,c} \,     k_{V,c}
\right\rgroup
\end{equation}

For the web diagram partition function in Equation
\ref{generic.web.partition}, where
$ \left\lgroup \pmb{V}_i,\,  \pmb{c}_{\pmb{V}_i}\right\rgroup$ is a $ \left\lgroup \pmb{N}_i,\,  \pmb{u}_i \right\rgroup$-series,

\begin{multline}
\mathcal{S}^{\, n} \left\lgroup x,y,R \right\rgroup =
\sum_{ \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1} \right\rgroup \in  \left\lgroup \pmb{N}_1,\,  \pmb{u}_1 \right\rgroup-series}
\cdots
\sum_{ \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m} \right\rgroup \in  \left\lgroup \pmb{N}_m,\,  \pmb{u}_m \right\rgroup-series}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup, \left\lgroup\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2} \right\rgroup, \cdots,
 \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m} \right\rgroup}
 \left\lgroup x, y, R \right\rgroup
\\
=
\sum_{m_1,\cdots, m_n = 0}^\infty
\sum_{\text{ $ \left\lgroup \pmb{V}_1, \pmb{c}_{\pmb{V}_1}\right\rgroup$ is $ \left\lgroup\pmb{N}_1,  \pmb{k}_1(m_1)\right\rgroup$-type } }
\cdots
\sum_{\text{ $ \left\lgroup \pmb{V}_n, \pmb{c}_{\pmb{V}_n}\right\rgroup$ is $ \left\lgroup\pmb{N}_m,  \pmb{k}_m(r_m)\right\rgroup$-type }}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup, \left\lgroup\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}\right\rgroup,
\cdots,
 \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m}\right\rgroup} \left\lgroup x,y,R \right\rgroup
\end{multline}

we multiply each summand of $\mathcal{S}^{\, n} \left\lgroup x,y,R \right\rgroup$ with
$ \left\lgroup r_1, \cdots, r_m \right\rgroup$ by a factor of,

\begin{equation}
R^{\sum_{i= 0 }^{p}
 \left\lgroup
K_n
 \left\lgroup \pmb{N}_i, \,  \pmb{k}_i(r_i) \, | \, \pmb{N}_i,     \,  \pmb{k}_i (r_i)         \right\rgroup
- K_n
 \left\lgroup \pmb{N}_i, \,  \pmb{k}_i(r_i) \, | \, \pmb{N}_{i+1}, \,  \pmb{k}_{i+1} (r_{i+1}) \right\rgroup
\right\rgroup
}
\end{equation}

with either linear or cyclic boundary conditions for $ \left\lgroup \pmb{N}_i,  \pmb{k}_i \right\rgroup$
for a linear or a cyclic web diagram, respectively.
For linear conformal blocks,

\begin{equation}
 \left\lgroup\pmb{N}_0,\,  \pmb{k}_0\right\rgroup =
 \left\lgroup
\pmb{N}_{\pmb{\emptyset}_0}, \pmb{0}
\right\rgroup,
\quad
 \left\lgroup \pmb{N}_{m+1},\,  \pmb{k}_0 \right\rgroup =
 \left\lgroup \pmb{N}_{\pmb{\emptyset}_{m+1}}, \pmb{0} \right\rgroup,
\end{equation}

where $N_{\emptyset_i,c}$ is the number of $\mathbb{Z}_n$-charges in $ \pmb{c}_{\pmb{\emptyset}_i}$ that are equal to $c$, $i = 0, m+1$.
For cyclic conformal blocks,

\begin{equation}
 \left\lgroup \pmb{N}_{0}, \,  \pmb{k}_{0} \right\rgroup =
 \left\lgroup \pmb{0}, \, \pmb{0} \right\rgroup,
\quad
 \left\lgroup
\pmb{N}_{m+1}, \,
 \pmb{k}_{m+1}
\right\rgroup =
 \left\lgroup
\pmb{N}_1,  \pmb{k}_1
\right\rgroup
\end{equation}

In the limit $R \rightarrow 0$,

\begin{multline}
\mathcal{S}^{\, n}
 \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup =
\lim_{R\rightarrow 0}
\sum_{m_1,\cdots,m_p = 0}^\infty
R^{\sum_{i= 0 }^{n}
 \left\lgroup K_n
 \left\lgroup \pmb{N}_i, \pmb{k}_i(r_i)\, |\, \pmb{N}_i, \pmb{k}_i(r_i)             \right\rgroup - K_n
 \left\lgroup \pmb{N}_i, \pmb{k}_i(r_i)\, |\, \pmb{N}_{i+1}, \pmb{k}_{i+1}(r_{i+1}) \right\rgroup
\right\rgroup
}
\\
\sum_{\text{ $ \left\lgroup \pmb{V}_1, \pmb{c}_{\pmb{V}_1}\right\rgroup$ is $ \left\lgroup\pmb{N}_1,  \pmb{k}_1(m_1)\right\rgroup$-type } }
\cdots
\sum_{\text{ $ \left\lgroup \pmb{V}_m, \pmb{c}_{\pmb{V}_m}\right\rgroup$ is $ \left\lgroup\pmb{N}_n,  \pmb{k}_p(r_m)\right\rgroup$-type }}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup, \left\lgroup\pmb{V}_2,\,
 \pmb{c}_{\pmb{V}_2}\right\rgroup,\cdots, \left\lgroup\pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m}\right\rgroup} \left\lgroup x,y,R \right\rgroup
\\
=
\sum_{\text{ $ \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1} \right\rgroup \in  \left\lgroup \pmb{N}_1,\,  \pmb{u}_1 \right\rgroup$-series }}
\cdots
\sum_{\text{ $ \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m} \right\rgroup \in  \left\lgroup \pmb{N}_m,\,  \pmb{u}_m \right\rgroup$-series }}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1} \right\rgroup,
 \left\lgroup \pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2} \right\rgroup,
\cdots,
 \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m} \right\rgroup
}
 \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup.
\end{multline}

Since the 4D limit of the edge factors are,

\begin{equation}
\lim_{R\rightarrow 0}
 \left\lgroup - Q_{C_{iI}} \right\rgroup^{\, | \, V_{iI}\, | \, } f^{\, n}_{i-1,i,i+1,I} =
 \left\lgroup -1 \right\rgroup^{\, | \, V_{iI}\, | \, } f^{\, n}_{i-1,i,i+1,I},
\end{equation}

$\mathcal{S}^{\, n, \, block
}_{
 \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup, \left\lgroup\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}\right\rgroup,\cdots, \left\lgroup
\pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m}\right\rgroup} \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup$ for linear and cyclic
building blocks is

\begin{multline}
\label{4d.bb.linear}
\mathcal{S}^{\, n, \,linear\, block}_{ \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup,
 \left\lgroup \pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}\right\rgroup, \cdots,  \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m}\right\rgroup}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\\
=
\mathcal{S}_{
 \left\lgroup \pmb{\emptyset}, \,  \pmb{c}_{\pmb{\emptyset}_0} \right\rgroup
 \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_{1}} \right\rgroup \bDelta_{\, 0 1}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
 \left\lgroup -1 \right\rgroup^{\sum_{I=1}^N \, | \, V_{1I}\, | \, }
\prod_{I=1}^N f^{\, n}_{0 1 2, I}
\mathcal{S}_{
 \left\lgroup \pmb{V}_1, \,  \pmb{c}_{\pmb{V}_1} \right\rgroup
 \left\lgroup \pmb{V}_2, \,  \pmb{c}_{\pmb{V}_2} \right\rgroup \bDelta_{\, 1 2}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\cdots
\\
\cdots
\mathcal{S}_{
 \left\lgroup \pmb{V}_{m-1},\,  \pmb{c}_{\pmb{V}_{m-1}} \right\rgroup
 \left\lgroup \pmb{V}_{m},  \,  \pmb{c}_{\pmb{V}_{m}}   \right\rgroup \bDelta_{\, m-1, m}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
 \left\lgroup -1 \right\rgroup^{\sum_{I = 1}^N \, | \, V_{mI}\, | \, }
\prod_{I=1}^N f^{\, n}_{m-1, m, m+1, I}
\mathcal{S}_{
 \left\lgroup \pmb{V}_{m},\,  \pmb{c}_{\pmb{V}_{m}} \right\rgroup
 \left\lgroup \pmb{\emptyset},\,  \pmb{c}_{\pmb{\emptyset}_{m+1}} \right\rgroup
\bDelta_{\, m, m+1}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\end{multline}

\begin{multline}
\label{4d.bb.cyclic}
\mathcal{S}^{
\, n, \,cyclic\,  block
}_{
 \left\lgroup \pmb{V}_0, \,  \pmb{c}_{\pmb{V}_0} \right\rgroup,
 \left\lgroup \pmb{V}_1, \,  \pmb{c}_{\pmb{V}_1} \right\rgroup,
 \left\lgroup \pmb{V}_2, \,  \pmb{c}_{\pmb{V}_2} \right\rgroup,
\cdots,
 \left\lgroup \pmb{V}_m, \,  \pmb{c}_{\pmb{V}_m} \right\rgroup}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\\
=
\mathcal{S}_{
 \left\lgroup \pmb{V}_0, \,  \pmb{c}_{\pmb{V}_0} \right\rgroup
 \left\lgroup \pmb{V}_1, \,  \pmb{c}_{\pmb{V}_1} \right\rgroup
\bDelta_{\, 0 1}
}^{
\, n, \, norm
}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
 \left\lgroup -1 \right\rgroup^{\sum_{I=1}^N \, | \, V_{1I} \, | \, }
\prod_{I=1}^N
f^{\, n}_{012,I}
\mathcal{S}_{
 \left\lgroup
\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}
\right\rgroup
 \left\lgroup
\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}
\right\rgroup
\bDelta_{\, 1 2}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\cdots
\\
\cdots
\mathcal{S}_{
 \left\lgroup \pmb{V}_{m-1},\,  \pmb{c}_{\pmb{V}_{m-1}} \right\rgroup
 \left\lgroup \pmb{V}_{m},  \,  \pmb{c}_{\pmb{V}_m}     \right\rgroup
\bDelta_{\, m-1, m}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
 \left\lgroup -1 \right\rgroup^{\sum_{I=1}^N \, | \, V_{mI}\, | \, }
\prod_{I=1}^N
f^{\, n}_{m-1,m,0,I}
\\
\mathcal{S}_{
 \left\lgroup \pmb{V}_m, \,  \pmb{c}_{\pmb{V}_m} \right\rgroup
 \left\lgroup \pmb{V}_0, \,  \pmb{c}_{\pmb{V}_0} \right\rgroup
\bDelta_{\, n 0}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
 \left\lgroup -1 \right\rgroup^{\sum_{I=1}^N \, | \, V_{0I}\, | \, }
\prod_{I=1}^N f^n_{m01,I}
\end{multline}

Observing that,

\begin{equation}
F  \left\lgroup \pmb{V}_i,  \pmb{c}_{\pmb{V}_i} \, | \,  \pmb{V}_{i+1},  \pmb{c}_{\pmb{V}_{i+1}} \right\rgroup =
\prod_{I = 1}^N
f^{\, n, \, right}_{i,i+1,I} f^{\, n, \,left}_{i,i+1,I},
\end{equation}

the relation between
$F \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup$ and
the framing factors $f^{\, p}_{\, i-1, i, i+1, I}$ for
the linear conformal blocks is,

\begin{multline}
 \left\lgroup
\prod_{I = 1}^N  \left\lgroup -1 \right\rgroup^{\, | \, V_{1I}\, | \, }
f^{\, n}_{\, 012, I} \right\rgroup
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup -1 \right\rgroup^{\, | \, V_{2I}\, | \, } f^{\, n}_{\, 123, I}
\right\rgroup
\cdots
\\
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup - 1 \right\rgroup^{\, | \, V_{m-1,I}\, | \, }
f^{\, n}_{\, m-2, m-1, m, I} \right\rgroup
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup -1 \right\rgroup^{\, | \, V_{mI} \, | \, } f^{\, n}_{m-1,m,m+1, I} \right\rgroup
\\
=
\prod_{I=1}^N
 \left\lgroup
f^{\, p, \,  left}_{01, I}
f^{\, p, \, right}_{12, I}
f^{\, p, \,  left}_{12, I}
f^{\, p, \, right}_{23, I}
\cdots
f^{\, p,  \, left}_{m-2, m-1, I}
f^{\, p, \, right}_{m-1, m,   I}
f^{\, p, \,  left}_{m-1, m,   I}
f^{\, p, \, right}_{m,   m+1, I}
\right\rgroup
\\
=
\prod_{I=1}^N
 \left\lgroup
f^{\, n, \,right}_{01, I}
f^{\, n, \,left}_{01,I}
f^{\, n, \, right}_{12,I}
f^{\, n, \,left}_{12,I}
f^{\, n, \,right}_{23,I}
\cdots
f^{\, n, \,left}_{m-2,m-1,I}
f^{\, n, \,right}_{m-1,m,I}
f^{\, n, \, left}_{m-1,m,I}
f^{\, n, \, right}_{m,m+1,I}
f^{\, n, \,left}_{m,m+1,I}
\right\rgroup
\\
=
\prod_{i = 0}^m
F  \left\lgroup \pmb{V}_i,  \pmb{c}_{\pmb{V}_i} \, | \,  \pmb{V}_{i+1},  \pmb{c}_{\pmb{V}_{i+1}} \right\rgroup
\end{multline}

Since
$\pmb{V}_{0} = \pmb{V}_{m+1} = \emptyset$,
we have
$f^{\, n, \,right}_{01,I} = f^{\, n, \,left}_{m,m+1,I} = 1$,
and it is justified to multiply the right hand side in
the second last equality by
$f^{\, n, \,right}_{01,I}f^{\, n, \,left}_{m,m+1,I}$.
For the cyclic conformal blocks, the relation is,

\begin{multline}
 \left\lgroup \prod_{I=1}^N
 \left\lgroup -1 \right\rgroup^{\, | \, V_{1I}\, | \, } f^{\, n}_{012, I} \right\rgroup
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup -1 \right\rgroup^{\, | \, V_{1I}\, | \, } f^{\, n}_{123, I} \right\rgroup
\cdots
\\
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup -1 \right\rgroup^{\, | \, V_{mI}\, | \, } f^{\, n}_{m-1,m,0, I}
\right\rgroup
 \left\lgroup
\prod_{I=1}^N
 \left\lgroup -1 \right\rgroup^{\, | \, V_{0I}\, | \, } f^{\, n}_{m01, I} \right\rgroup
\\
= \prod_{I=1}^N
 \left\lgroup
f^{\, n,  left}_{01,I}
f^{\, n, right}_{12,I}
f^{\, n,  left}_{12,I}
f^{\, n, right}_{23,I}
\cdots
f^{\, n,  left}_{m-2,m-1,I}
f^{\, n, right}_{m-1,m,I}
f^{\, n,  left}_{m-1,m,I}
f^{\, n, right}_{m,0,I}
f^{\, n,  left}_{m,0,I}
f^{\, n, right}_{0 1,I}
\right\rgroup
\\
=
\prod_{i = 0}^m
F  \left\lgroup \pmb{V}_i,  \pmb{c}_{\pmb{V}_i} \, | \,  \pmb{V}_{i+1},  \pmb{c}_{\pmb{V}_{i+1}} \right\rgroup
\end{multline}

and we can write the linear and cyclic conformal blocks in Equation
\ref{4d.bb.linear} and Equation \ref{4d.bb.cyclic} as,

\begin{multline}
\label{4d.bb.linear.v2}
\mathcal{S}^{\, n, \,linear\, block}_{ \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup, \left\lgroup\pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2}\right\rgroup,\cdots, \left\lgroup\pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m}\right\rgroup} \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\\
=
F  \left\lgroup \pmb{\emptyset},  \pmb{c}_{\pmb{\emptyset}_0} \, | \,  \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} \right\rgroup
\mathcal{S}_{\pmb{\emptyset} \pmb{V}_1 \bDelta_{\, 0 1}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
F  \left\lgroup \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} | \pmb{V}_{2},  \pmb{c}_{\pmb{V}_{2}} \right\rgroup
\mathcal{S}_{ \left\lgroup\pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1}\right\rgroup
 \left\lgroup\pmb{V}_2, \,  \pmb{c}_{\pmb{V}_2} \right\rgroup
\bDelta_{\, 12}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\cdots
\\
\cdots
F
 \left\lgroup \pmb{V}_{m-1},  \pmb{c}_{\pmb{V}_{m-1}} \, | \,  \pmb{V}_{m},  \pmb{c}_{\pmb{V}_{m}} \right\rgroup
\mathcal{S}_{ \left\lgroup \pmb{V}_{m-1},\,  \pmb{c}_{\pmb{V}_{m-1}} \right\rgroup
 \left\lgroup \pmb{V}_{m}, \,  \pmb{c}_{\pmb{V}_{m}} \right\rgroup
\bDelta_{\, m-1,m}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
F
 \left\lgroup
\pmb{V}_m,  \pmb{c}_{\pmb{V}_m} \, | \,  \pmb{\emptyset},  \pmb{c}_{\pmb{\emptyset}_{m+1}}
\right\rgroup
\\
\mathcal{S}_{
 \left\lgroup \pmb{V}_{m},\,  \pmb{c}_{\pmb{V}_{m}} \right\rgroup
 \left\lgroup \pmb{\emptyset},\,  \pmb{c}_{\pmb{\emptyset}_{n+1}} \right\rgroup \bDelta_{\, m,m+1}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\end{multline}

\begin{multline}\label{4d.bb.cyclic.v2}
\mathcal{S}^{
\, n, \,cyclic\,  block
}_{
 \left\lgroup \pmb{V}_0,\,  \pmb{c}_{\pmb{V}_0} \right\rgroup,
 \left\lgroup \pmb{V}_1,\,  \pmb{c}_{\pmb{V}_1} \right\rgroup,
 \left\lgroup \pmb{V}_2,\,  \pmb{c}_{\pmb{V}_2} \right\rgroup,
\cdots,
 \left\lgroup \pmb{V}_m,\,  \pmb{c}_{\pmb{V}_m} \right\rgroup}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\\
=
F
 \left\lgroup \pmb{V}_0,  \pmb{c}_{\pmb{V}_0} \, | \,  \pmb{V}_1,  \pmb{c}_{\pmb{V}_{1}} \right\rgroup
\mathcal{S}_{
 \left\lgroup \pmb{V}_0, \,  \pmb{c}_{\pmb{V}_0} \right\rgroup
 \left\lgroup \pmb{V}_1, \,  \pmb{c}_{\pmb{V}_1} \right\rgroup
\bDelta_{\, 0 1}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
F
 \left\lgroup \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} \, | \,  \pmb{V}_2,  \pmb{c}_{\pmb{V}_2} \right\rgroup
\mathcal{S}_{
 \left\lgroup \pmb{V}_1, \,  \pmb{c}_{\pmb{V}_1} \right\rgroup
 \left\lgroup \pmb{V}_2, \,  \pmb{c}_{\pmb{V}_2} \right\rgroup
\bDelta_{\, 12}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\cdots
\\
\cdots
F
 \left\lgroup
\pmb{V}_{m-1},  \pmb{c}_{\pmb{V}_{m-1}} \, | \,  \pmb{V}_{m},  \pmb{c}_{\pmb{V}_{m}}
\right\rgroup
\mathcal{S}_{
 \left\lgroup
\pmb{V}_{m-1}, \,  \pmb{c}_{\pmb{V}_{m-1}}
\right\rgroup
 \left\lgroup
\pmb{V}_{m},   \,  \pmb{c}_{\pmb{V}_{m}}
\right\rgroup
\bDelta_{\, m-1,m}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
F
 \left\lgroup \pmb{V}_m,  \pmb{c}_{\pmb{V}_m} \, | \,  \pmb{V}_0,  \pmb{c}_{\pmb{V}_0} \right\rgroup
\\
\mathcal{S}_{
 \left\lgroup
\pmb{V}_m, \,  \pmb{c}_{\pmb{V}_m} \right\rgroup
 \left\lgroup
\pmb{V}_0, \,  \pmb{c}_{\pmb{V}_0} \right\rgroup
\bDelta_{\, m 0}}^{\, n, \, norm}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\end{multline}

\subsection{
$\mathcal{S}^{\, n,\, norm}$ and
$\mathcal{Z}^{\, n,\, 4D}_{\, N}$
lead to the same
4D instanton partition functions}
Let,

\begin{multline}
\mathcal{F}
 \left\lgroup  \pmb{a}, \pmb{V} \, | \,   \pmb{b}, \pmb{W} \right\rgroup =
F
 \left\lgroup \pmb{V},  \pmb{c}_{\pmb{V}} \, | \,  \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
\frac{
\mathcal{S}^{\, n,\, norm}_{\,  \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup
 \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup\bDelta}  \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
}{
\mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup   \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \mu \, | \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup
}
\\
= \frac{
\prod_{I=1}^N
\prod_{J=1}^N
 \left\lgroup
\mathcal{H}_{V_{IJ}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\mathcal{H}_{W_{IJ}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
\right\rgroup^{1/2}
}{
\prod_{I=1}^N
\mathcal{A}_{V_{II}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup \mathcal{L}_{W_{II}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
\prod_{J=1}^N
\prod_{I=1}^{J-1}
\mathcal{H}_{V_{JI}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\mathcal{H}_{W_{IJ}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
}
\\
=
\mathcal{F}_{left}   \left\lgroup  \pmb{a},\pmb{V} \right\rgroup
\mathcal{F}_{right}  \left\lgroup  \pmb{b},\pmb{W} \right\rgroup
\end{multline}

where we have used Equations \ref{abbrev.s.num}, \ref{abbrev.s.den} and \ref{abbrev.z.den} for the second equality. We also define,

\begin{equation}
    \mathcal{F}_{left}  \left\lgroup  \pmb{a}, \pmb{V} \right\rgroup =
\frac{
\prod_{I=1}^N
\prod_{J=1}^N
\mathcal{H}_{V_{IJ}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup^{\frac12}
}{
\prod_{I=1}^N
\mathcal{A}_{V_{II}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\prod_{J=1}^N
\prod_{I = 1}^{J-1}
\mathcal{H}_{V_{JI}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup}
\end{equation}

\begin{equation}
    \mathcal{F}_{right}  \left\lgroup \pmb{b}, \pmb{W}\right\rgroup =
\frac{
\prod_{I=1}^N
\prod_{J=1}^N
\mathcal{H}_{W_{IJ}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup^{\frac12}
}{
\prod_{I=1}^N \mathcal{L}_{W_{II}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
\prod_{J=1}^N
\prod_{I = 1}^{J-1} \mathcal{H}_{W_{IJ}}  \left\lgroup  \pmb{b},  \pmb{c}_{\pmb{W}} \right\rgroup
}
\end{equation}

The linear and cyclic conformal blocks take the form,

\begin{multline}
\label{linear.quiver}
\mathcal{Z}^{\, n,\, linear\, block}
= \mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup  \pmb{a}_0, \emptyset,  \pmb{c}_{\pmb{\emptyset}} \, | \, \mu_{0 1} \, | \,  \pmb{a}_1, \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} \right\rgroup
\mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup  \pmb{a}_1, \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} \, | \, \mu_{12} \, | \,  \pmb{a}_2, \pmb{V}_2, \pmb{c}_{\pmb{V}_2} \right\rgroup
\cdots
\\
\mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup  \pmb{a}_m, \pmb{V}_m,  \pmb{c}_{\pmb{V}_m} \, | \, \mu_{m,m+1} \, | \,  \pmb{a}_{m+1},
\emptyset,  \pmb{c}_{\pmb{\emptyset}} \right\rgroup
\end{multline}

\begin{multline}
\label{cyclic.quiver}
\mathcal{Z}^{\, n,\, cyclic\, block}
= \mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup  \pmb{a}_0, \pmb{V}_0,  \pmb{c}_{\pmb{V}_0} \, | \, \mu_{0 1} \, | \,  \pmb{a}_1, \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} \right\rgroup
\mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup  \pmb{a}_1, \pmb{V}_1,  \pmb{c}_{\pmb{V}_1} \, | \, \mu_{12} \, | \,  \pmb{a}_2, \pmb{V}_2,  \pmb{c}_{\pmb{V}_2} \right\rgroup
\cdots
\\
\mathcal{Z}^{\, n,\, 4D}_{\, N}
 \left\lgroup  \pmb{a}_m,\pmb{V}_m,  \pmb{c}_{\pmb{V}_m} \,| \, \mu_{m0} \, |\,  \pmb{a}_0, \pmb{V}_0,  \pmb{c}_{\pmb{V}_0} \right\rgroup
\end{multline}

We claim that
\begin{equation}
    \boxed{\mathcal{Z}^{\, n,\, linear\, block} = \mathcal{S}^{\, n,\, linear\, block}\qquad \text{and} \qquad \mathcal{Z}^{\, n,\, cyclic\, block} = \mathcal{S}^{\, n,\, cyclic\, block}}
\end{equation}
Comparing Equation
\ref{4d.bb.linear.v2} to \ref{linear.quiver} and
\ref{4d.bb.cyclic.v2} to \ref{cyclic.quiver},
the claim is equivalent to,

\begin{equation}
\mathcal{F}_{left}  \left\lgroup \pmb{a}_0, \mathbf{\emptyset}\right\rgroup
\mathcal{F}_{right}  \left\lgroup \pmb{a}_1, \pmb{V}_1\right\rgroup
\cdots
\mathcal{F}_{left}  \left\lgroup \pmb{a}_m, \pmb{V}_m\right\rgroup
\mathcal{F}_{right}  \left\lgroup \pmb{a}_{m+1}, \mathbf{\emptyset}\right\rgroup = 1,
\end{equation}

\begin{equation}
\mathcal{F}_{left}   \left\lgroup  \pmb{a}_0,\pmb{V}_0\right\rgroup
\mathcal{F}_{right}  \left\lgroup  \pmb{a}_1,\pmb{V}_1\right\rgroup
\cdots
\mathcal{F}_{left}   \left\lgroup  \pmb{a}_m,\pmb{V}_m\right\rgroup
\mathcal{F}_{right}  \left\lgroup  \pmb{a}_0, \pmb{V}_0\right\rgroup = 1
\end{equation}

Since
$\mathcal{F} \left\lgroup  \pmb{a}, \pmb{\emptyset} \, | \,   \pmb{b}, \pmb{\emptyset} \right\rgroup = 1$,
it follows that
$\mathcal{F}_{left}  \left\lgroup  \pmb{a}, \emptyset \right\rgroup =
\mathcal{F}_{right}  \left\lgroup  \pmb{a}, \emptyset \right\rgroup = 1$,

\begin{multline}
\mathcal{F}_{left}  \left\lgroup  \pmb{a}, \pmb{V} \right\rgroup \mathcal{F}_{right}  \left\lgroup  \pmb{a}, \pmb{V} \right\rgroup =
\\
\frac{
\prod_{I=1}^N
\prod_{J=1}^N \mathcal{H}_{V_{IJ}}  \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
}
{
\prod_{I=1}^N  \left\lgroup \mathcal{A}_{V_{II}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup \mathcal{L}_{V_{II}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup \right\rgroup
\prod_{J=1}^N
\prod_{I=1}^{J-1}\mathcal{H}_{V_{IJ}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\prod_{I=1}^N
\prod_{J=1}^{I-1}\mathcal{H}_{V_{IJ}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
}
=\\
\frac{
 \left\lgroup -1 \right\rgroup^{\sum_{I=1}^N\, | \, V_I\, | \, }
\prod_{I=1}^N
\prod_{J=1}^N \mathcal{H}_{V_{IJ}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
}
{
\prod_{I=1}^N\mathcal{H}_{V_{II}}(0)
\prod_{I<J}\mathcal{H}_{V_{IJ}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
\prod_{I>J}\mathcal{H}_{V_{IJ}} \left\lgroup  \pmb{a},  \pmb{c}_{\pmb{V}} \right\rgroup
} = 1
\end{multline}

proving the claim.

\section{A 4-point conformal block}
\label{section.09}
\textit{We recover the 4-point conformal block on $\mathbb{R}^4/\mathbb{Z}_n$
in Equation (2.10) of \cite{alfimov.tarnopolsky}
by gluing two strip partition functions. To help with the comparison, in this section, we use the notation of \cite{alfimov.tarnopolsky}.
}

\subsection{Change of parameters and notation}
In previous sections, the case of computing
4-point conformal blocks is the same as linear blocks with $\pmb{V}_0 = \pmb{\emptyset}, \pmb{V}_1 = \pmb{Y}, \pmb{V}_2 = \pmb{\emptyset}$. We used the K\"{a}hler
parameters
$\Delta_{\, 0 1,   K}, \cdots$,
the Coulomb parameters
$ \pmb{a}_0,  \pmb{a}_1,  \pmb{a}_2$,
the mass parameters $\mu_{0 1}$ and $\mu_{12}$,
the deformation parameters $\epsilon_1$ and $\epsilon_2$,
and $c_I$ for the Young diagram charges.

In \cite{alfimov.tarnopolsky}, Alfimov and Tarnopolsky
use the Coulomb parameters
$\pmb{P}_i =
 \left\lgroup P_{i,1}, \cdots, P_{i,N} \right\rgroup,
\pmb{P} =
 \left\lgroup P_{1},   \cdots, P_{N} \right\rgroup$,
$i = 1, 2$,
the mass parameters $\alpha_1, \alpha_2$,
the deformation parameters
$b^{-1}, b$, and $\pmb{q} =  \left\lgroup q_1,\cdots q_N \right\rgroup$ for the Young diagram charges.
These two sets of parameters are related as,

\begin{equation}
 \pmb{a}_0 = \pmb{P}_1, \quad
 \pmb{a}_1 = \pmb{P}, \quad
 \pmb{a}_2 = \pmb{P}_2, \quad
\mu_{0 1} = \alpha_1, \quad
\mu_{12} = \alpha_2, \quad
\epsilon_1 = b^{-1}, \quad
\epsilon_2 = b, \quad
c_{Y_I} = -q_I,
\end{equation}

and the K\"{a}hler parameters
$ \left\lgroup \Delta_{\, 0 1,   K}, \cdots \right\rgroup$,
in terms of the parameters of
\cite{alfimov.tarnopolsky} are,

\begin{multline}
\Delta_{\, 0 1,   K} = -P_{\, 1, K} + P_{\, K} + \alpha_{\, 1} - b^{\, -1},
\quad
\Delta_{\, 0 1, M_K} = P_{\, 1, K + 1} - P_{\, K} + \alpha_{\, 1} - b^{\, -1},
\\
\Delta_{\, 12, K} = - P_{\, K} + P_{\, 2, K} + \alpha_{\, 2} - b^{\, -1},
\quad
\Delta_{\, 12, M_K} = P_{\, K + 1} - P_{\, 2, K} + \alpha_{\, 2} - b^{\, -1}
\end{multline}

We will also use,

\begin{equation}
 \left\lgroup \pmb{Y},\,  \pmb{c}_{\pmb{Y}}\right\rgroup =  \left\lgroup \pmb{Y},\, - \, \pmb{q} \right\rgroup
\end{equation}

\subsection{The 4-point conformal block}
In the notation of \cite{alfimov.tarnopolsky}, introduced
above, the 4-point $n$-coloured conformal block constructed from two $n$-coloured strip partition functions is,

\begin{equation}
\mathcal{S}^{\, n, \, 4-point}  \left\lgroup b \right\rgroup =
\sum_{ \left\lgroup \pmb{Y},\, -\pmb{q} \right\rgroup}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup \pmb{Y}, \, \pmb{q} \right\rgroup}
 \left\lgroup b \right\rgroup z^{\, |\, \pmb{Y} \, | \, / \, n},
\end{equation}

where $z$ is an instanton expansion parameter (a position parameter in 2D conformal field theory).
The summation is over each of the
$ \left\lgroup \pmb{N},  \pmb{u} =
 \left\lgroup 0, 0, 0 \right\rgroup \right\rgroup$-series in
Equations  \ref{u.0.series.1}--\ref{u.0.series.3}.
For example, if we choose Series 2 in Equation \ref{u.0.series.2} then
$\pmb{N} =  \left\lgroup 0, 1, 0, 1 \right\rgroup$ and
$ \pmb{k} (r) =  \left\lgroup r, r + 1, r + 1, r + 1 \right\rgroup$
and the summation is over all
$ \left\lgroup \pmb{Y},  \pmb{c}_{\pmb{Y}} \right\rgroup =
 \left\lgroup \pmb{Y}, - \pmb{q} \right\rgroup$ of
$ \left\lgroup \pmb{N},  \pmb{k} (r) \right\rgroup$-type. We obtain the expression for the 4-point conformal block $\mathcal{S}^{\, n, \, block}_{ \left\lgroup \pmb{Y}, \, \pmb{q} \right\rgroup} \left\lgroup b \right\rgroup$ from the expression of linear conformal block in Equation \ref{4d.bb.linear.v2} by substituting $\pmb{V}_0 = \pmb{\emptyset}, \pmb{V}_1 = \pmb{Y}, \pmb{V}_2 = \pmb{\emptyset}$.
In the notation of \cite{alfimov.tarnopolsky},

\begin{multline}
\mathcal{S}^{\, n, \, block}_{ \left\lgroup \pmb{Y}, \, \pmb{q} \right\rgroup} \left\lgroup b \right\rgroup
=
F  \left\lgroup \pmb{\emptyset}, \pmb{0} \, | \,  \pmb{Y}, \pmb{c}_{\pmb{Y}} \right\rgroup
\mathcal{S}^{
\, n, \,norm
}_{
 \left\lgroup \pmb{\emptyset}, \,  \pmb{0} \right\rgroup
 \left\lgroup \pmb{Y},             \, \pmb{c}_{\pmb{Y}}   \right\rgroup
\bDelta_{\, 0 1}}
 \left\lgroup \epsilon_1, \, \epsilon_2 \right\rgroup
F  \left\lgroup \pmb{Y}, \pmb{c}_{\pmb{Y}} \, | \,  \pmb{\emptyset}, \pmb{0} \right\rgroup
\mathcal{S}^{
\, n, \,norm
}_{
 \left\lgroup \pmb{Y},            \, \pmb{c}_{\pmb{Y}}    \right\rgroup
 \left\lgroup \pmb{\emptyset},\,  \pmb{0} \right\rgroup
\bDelta_{\, 1 2}
}
 \left\lgroup \epsilon_1, \epsilon_2 \right\rgroup
\\
=
F  \left\lgroup \pmb{\emptyset}, \pmb{0} \, | \,  \pmb{Y}, -\pmb{q} \right\rgroup
\mathcal{S}^{
\, n, \,norm
}_{
 \left\lgroup \pmb{\emptyset}, \,  \pmb{0} \right\rgroup
 \left\lgroup \pmb{Y},             \, -\pmb{q}     \right\rgroup
\bDelta_{\, 0 1}}
 \left\lgroup b^{-1}, \, b \right\rgroup
F  \left\lgroup \pmb{Y}, - \pmb{q} \, | \,  \pmb{\emptyset}, \pmb{0} \right\rgroup
\mathcal{S}^{
\, n, \,norm
}_{
 \left\lgroup \pmb{Y},            \, -\pmb{q}     \right\rgroup
 \left\lgroup \pmb{\emptyset},\,  \pmb{0} \right\rgroup
\bDelta_{\, 1 2}
}
 \left\lgroup b^{-1}, b \right\rgroup
\\
=
\prod_{I=1}^N
\prod_{J=1}^I
\prod_{
\substack{
\square \in Y_I
\\
A_{\square, Y_I} +
L_{\square, \emptyset} + 1 + q_I \, = \,  0 \, \mod{\, n}
}
}
 \left\lgroup
  P_{1, J} - P_{\, I} - \alpha_1
- A^{  }_{\square,       Y_I} b
+ L^{++}_{\square, \emptyset} b^{-1}
\right\rgroup
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\square \in Y_J
\\
A_{\square,       Y_J} +
L_{\square, \emptyset} + 1 + q_J \, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
P_{1,I} - P_J - \alpha_1
- A^{  }_{\square,       Y_J} b
+ L^{++}_{\square, \emptyset} b^{-1}
\right\rgroup
\\
\prod_{I\, = 1}^N
\prod_{
\substack{
\square \in Y_I
\\
A_{\square, Y_I} +
L_{\square, Y_I} + 1 \, =  \, 0 \, \mod{\, n}
}
}
 \left\lgroup
- A^{  }_{\square,       Y_I} b
+ L^{++}_{\square, \emptyset} b^{-1}
\right\rgroup^{-1}
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\square \in  Y_I
\\
A_{\square, Y_I} +
L_{\square, Y_J} + 1 + q_I - q_J \, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
P_J - P_{\, I}
- A_{\square,   Y_I}      b
+ L_{\square,   Y_J}^{++} b^{-1}
\right\rgroup^{-1}
\\
\prod_{
\substack{
\square \in  Y_J
\\
  A_{\square, Y_J}
+ L_{\square, Y_I} + 1 - q_I + q_J\, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
P_J - P_I
+ A_{\square,   Y_J}^{++} b
- L_{\square,   Y_I}      b^{-1}
\right\rgroup^{-1}
\\
\prod_{I=1}^N
\prod_{J=1}^I
\prod_{
\substack{
\square \in
Y_J
\\
A_{\square,       Y_J} +
L_{\square, \emptyset} + 1 + q_J \, = \, 0  \, \mod{\, n}
}
}
 \left\lgroup
P_J - P_{2,I} - \alpha_2
+ A^{++}_{\square,       Y_J} b
- L^{  }_{\square, \emptyset} b^{-1}
\right\rgroup
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
\substack{
\square \in Y_I
\\
A_{\square,       Y_I} +
L_{\square, \emptyset} + 1 + q_I \, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
P_I - P_{2,J} - \alpha_2
+ A^{++}_{\square,       Y_I} b
- L^{  }_{\square, \emptyset} b^{-1}
\right\rgroup
\\
\prod_{I=1}^N
\prod_{
\substack{
\square \in Y_I
\\
A_{\square, Y_I} + L_{\square, Y_I} + 1 \, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
  A^{++}_{\square, Y_I} b
- L^{  }_{\square, Y_I} b^{-1}
\right\rgroup^{-1}
\\
\prod_{I=1}^N
\prod_{J=1}^{I-1}
\prod_{
 \substack{
\square \in Y_I
\\
  A_{\square, Y_I}
+ L_{\square, Y_J} + 1 + q_I - q_J \, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
P_I - P_J
+ A^{++}_{\square, Y_I} b
- L^{  }_{\square, Y_J} b^{-1}
\right\rgroup^{-1}
\\
\prod_{
\substack{
\square \in Y_J
\\
  A_{\square, Y_J}
+ L_{\square, Y_I} + 1 - q_I + q_J \, = \, 0 \, \mod{\, n}
}
}
 \left\lgroup
P_I - P_J
- A^{  }_{\square, Y_J} b
+ L^{++}_{\square, Y_I} b^{-1}
\right\rgroup^{-1}
\\
=
\prod_{I,J = 1}^N
\prod_{
\substack{
\square \in Y_I
\\
  A_{\square,       Y_I}
+ L_{\square, \emptyset} + 1 \, = \,  - q_I \, \mod{\, n}
}
}
 \left\lgroup
P_{1,J} - P_I - \alpha_1
- A^{  }_{\square,       Y_I} b
+ L^{++}_{\square, \emptyset} b^{-1}
\right\rgroup
\\
\prod_{I, J \, = \,  1}^N
\prod_{
\substack{
\square \in Y_J
\\
  A_{\square,       Y_J}
+ L_{\square, \emptyset} + 1 \, = \,  - q_J \, \mod{\, n}
}
}
 \left\lgroup
P_J - P_{2,I} - \alpha_2
+ A^{++}_{\square,       Y_J} b
- L^{  }_{\square, \emptyset} b^{-1}
\right\rgroup
\\
\prod_{I,J}^N
\prod_{
\substack{
\square \in Y_I
\\
A_{\square,Y_I} + L_{\square, Y_J} + 1 \, = \,  - q_I + q_J \, \mod{\, n}
}
}
 \left\lgroup
P_I - P_J + A^{++}_{\square, Y_I} b - L_{\square, Y_J} b^{-1} \right\rgroup^{-1}
\\
\prod_{
\substack{
\square \in Y_J \\
A_{\square,Y_J} + L_{\square, Y_I} + 1 \, = \, q_I - q_J \, \mod{\, n}
}
}
 \left\lgroup P_I - P_J - A_{\square, Y_J} b + L^{++}_{\square, Y_I} b^{-1} \right\rgroup^{-1}
\\
\, = \,
\frac{
\mathcal{Z}^{\, num}
 \left\lgroup  \pmb{s}_1, \pmb{\emptyset}, \pmb{0}
\, | \,
\alpha_1 \, | \, \pmb{P}, \pmb{Y},-\pmb{q} \right\rgroup
\mathcal{Z}^{\, num}
 \left\lgroup \pmb{P}, \pmb{Y}, -\pmb{q} \, | \, \alpha_2 \, | \,  \pmb{s}_2, \pmb{\emptyset}, \pmb{0} \right\rgroup
}{
\mathcal{Z}^{\, num}
 \left\lgroup \pmb{P}, \pmb{Y},-\pmb{q}  \, | \, 0 \, | \, \pmb{P}, \pmb{Y},-\pmb{q} \right\rgroup
}
\end{multline}

which is Equation (2.10) in \cite{alfimov.tarnopolsky},
for $N = 2, n = 4$.

\section{Comments and remarks}
\label{section.10}

\subsection{A formulation in terms of $n$-coloured Young diagrams}
We have refrained from discussing the $n$-coloured vertex in terms
of $n$-coloured checkerboard Young diagrams, and in particular, we
did not consider working in terms of $n$-cores and $n$-quotients
\cite{james.kerber, loehr, olsson}.
This is because, in this work,
working in terms of
$n$-cores and $n$-quotients was not needed and would have taken us far afield from our goal
which is to reproduce the 2D parafermion matrix elements in
\cite{alfimov.tarnopolsky}.

\subsection{The $n$-coloured vertex is not a product of $n$ arbitrary
refined topological vertices}
The $n$-coloured vertex factorizes into a product of $n$ refined vertices.
However, it is not a product of $n$ arbitrary refined topological vertices
unless the Young diagrams that label the preferred legs of the component
vertices mesh together to form the Young diagram that labels the preferred
leg of the $n$-coloured vertex.

\subsection{Counting parameters}
While the 5D strip partition function has $ \left\lgroup 2 N - 1 \right\rgroup$ K\"ahler parameters,
and the 4D instanton partition function also has $ \left\lgroup 2 N - 1\right\rgroup$ parameters,
there are $N^2$ relations among them which arise from the factors in
$\mathcal{S}^{\, n, \,num}_{ \left\lgroup \pmb{V},\,  \pmb{c}_{\pmb{V}} \right\rgroup  \left\lgroup \pmb{W},\,  \pmb{c}_{\pmb{W}} \right\rgroup\bDelta} \left\lgroup \epsilon_1,\epsilon_2 \right\rgroup$ (Equation \ref{s.n}) and
$\mathcal{Z}^{\, n,  \, num}  \left\lgroup  \pmb{a}, \pmb{V},  \pmb{c}_{\pmb{V}} \, | \, \mu \, | \,  \pmb{b}, \pmb{W},  \pmb{c}_{\pmb{W}} \right\rgroup$ (Equation \ref{zp.z.num}).
The significance of the fact that $ \left\lgroup 2 N - 1 \right\rgroup$ parameters satisfy
$N^2$ relations is not clear at this stage.

\subsection{Rational models}
In this work, we restricted our attention to generic, that is non-rational
parameters and the corresponding conformal field theories. The relations
between the parameters of the 5D strip partition function and the 2D matrix
elements obtained in this work are expected to extend to the rational model,
after
\textbf{1.} re-writing all parameters in terms of the screening charges $\alpha_+$
and $\alpha_-$ of the minimal 2D conformal field theory, and
\textbf{2.} restricting the Young diagrams that label the preferred legs to those
that satisfy Burge-type conditions (thereby eliminating the null states),
as described in \cite{alkalaev.belavin, bershtein.foda, belavin.foda.santachiara}.

\subsection{The orbifold vertex \cite{bryan.cadman.young}}
In \cite{bryan.cadman.young}, Bryan, Cadman and Young define Donaldson-Thomas
invariants of Calabi-Yau orbifolds and introduce \textit{an orbifold topological
vertex} to compute them. The orbifold vertex is $n$-coloured in the sense that
it is the generating function of plane partitions that are made of interlacing
checkerboard $n$-coloured Young diagrams.
However, the orbifold vertex differs (at least on a superficial level) from
the $n$-coloured vertex introduced in this work in several technical respects.
\textbf{1.} The orbifold vertex is constructed in terms of one type of
$\Gamma_{\, \pm}$-operators, which is equivalent to using one Heisenberg algebra,
rather than $n$ Heisenberg algebras as in the $n$-coloured vertex, and consequently,
the orbifold vertex is in the form of a single sum of a bilinear of Schur functions
(as in the refined topological vertex), while the $n$-coloured vertex is in the form
of a product of $n$ sums of bilinears of Schur functions, and
\textbf{2.} the cells of the plane partitions generated by the orbifold vertex are assigned
$n$ colours, and cells of different colours are assigned different weights,
while in the case of the $n$-coloured vertex, all cells are assigned the same
(trivial) weight
\footnote{\,
There can be an orbifold vertex that is formulated in terms of
$n$ free bosons that assign different colours to different cells, but this is
not clear at this stage.},
\textbf{3.} the orbifold vertex is not refined, while the $n$-coloured vertex is
\footnote{\,
Because of \textbf{1} and \textbf{2}, the orbifold vertex is
parameterized in terms of $n$ parameters $ \left\lgroup q_0, q_1, \cdots, q_{\, n-1} \right\rgroup$,
while the $n$-coloured vertex is parameterized in terms of the equivariant
deformation parameters $ \left\lgroup \epsilon_2, \epsilon_1 \right\rgroup$, and the radius $R$. However, this
is probably not a deep difference, and it is entirely possible that there is
a refined version of the orbifold vertex.
}.
More generally,
\textbf{4.} The orbifold vertex computes topological string partition functions on
orbifold Calabi-Yau (internal) spaces such as $\mathbb{C}^{\, 3} / \mathbb{Z}_{\, n}$,
while the $n$-coloured vertex computes topological string partition functions
on the (Euclidean version of space-time) orbifolds $\mathbb{C}^{\, 2} / \mathbb{Z}_{\, n}$.
It is possible that there is a more general operator formalism that produces
a more general vertex such that the orbifold vertex and the $n$-coloured vertex
are special cases of the same object, but this is not clear at this stage, and
is unlikely given that these two vertices compute different objects.

\subsection{The intertwining operator of the Fock representations
of the quantum toroidal algebras of type $A_n$
\cite{awata.kanno.mironov.morozov.suetake.zenkevich}}

Following the completion of this work, we learned that in
\cite{awata.kanno.mironov.morozov.suetake.zenkevich},
Awata, Kanno, Mironov, Morozov, Suetake and Zenkevich
introduced an intertwining operator of the Fock representations
of the quantum toroidal algebras of type $A_n$, and used these
to obtain the instanton partition functions on
$\mathbb{C}^{\, 2} / \mathbb{Z}_{\, n}$
\footnote{\,
We thank M Bershtein and J-E Bourgine for bringing
\cite{awata.kanno.mironov.morozov.suetake.zenkevich}
to our attention.
}.
Using AGT, these instanton partition functions are identified
with the matrix elements discussed in this work, and as such,
the intertwiner of
\cite{awata.kanno.mironov.morozov.suetake.zenkevich}
is identified with the $n$-coloured vertex introduced
in this work.
The differences between the two works are that
\textbf{1.} The intertwiner of \cite{awata.kanno.mironov.morozov.suetake.zenkevich}
is in Awata-Feigin-Shiraishi operator form \cite{awata.feigin.shiraishi},
which bypasses working in terms of symmetric functions, while
the $n$-coloured vertex in this work is in the (more conventional)
Iqbal-Kozcaz-Vafa symmetric-function form \cite{iqbal.kozcaz.vafa},
\textbf{2.} The focus of \cite{awata.kanno.mironov.morozov.suetake.zenkevich}
is on the representation theory of quantum toroidal algebras, while
that of the present work is on making contact with the 2D matrix elements
that the $\mathbb{C}^{\, 2} / \mathbb{Z}_{\, n}$ topological vertex formalism computes,
with emphasis on the details, including the framing factors, the matching
of the various normalizations, \textit{etc.}

\subsection{More general orbifolds}
Following the completion of this work,
Bourgine and Jeong \cite{bourgine.jeong}
introduced an $n$-coloured topological vertex
of the Awata-Feigin-Shiraishi type
\cite{awata.feigin.shiraishi},
that allowed them to discuss new quantum toroidal algebras that
are related to 5D supersymmetric gauge theories on
$ \left\lgroup \mathbb{C}^{\, 2} / \, Z_n \right\rgroup \times S^1$ orbifolds, which were first discussed
in \cite{bonelli.maruyoshi.tanzini.01, bonelli.maruyoshi.tanzini.02}.
These orbifolds are more general than those discussed in this work,
and so far the corresponding 2D conformal field theories are not
known.

\subsection{Extracting the affine $A_n$, integral level-$N$,
WZW model algebra}
In \cite{foda.macleod}, Macleod and the second author show
that there is a choice of the Nekrasov deformation parameters
such that the coset component in the full $\mathcal{A}  \left\lgroup N, n \right\rgroup$ algebra trivializes and one obtains a formalism that involves only
the affine $A_n$ WZW models, at integral level $N$, times
a Heisenberg factor.

\section*{Acknowledgements}
We wish to thank V Belavin, M Bershtein, J-E Bourgine, J Bryan,
H Kanno, N Macleod, M Manabe, A Morozov, R Santachiara, F Yagi
and R-D Zhu for discussions on the subject of this work and related
topics, and Trevor Welsh for a careful reading of the manuscript.
WC is supported by an Australian Postgraduate Fellowship, and OF is supported by the Australian Research Council.

redpajama_set_name: RedPajamaArXiv