$\frac{d\bar{X}^{\prime}}{\bar{X}^{\prime}}\wedge\frac{d\bar{Z}^{\prime}}{\bar{Z}^{\prime}}=\frac{d\bar{X}}{\bar{X}}\wedge\frac{d\bar{Z}}{\bar{Z}}$.
This $SL(2,\mathbb{Z})$ symmetry maps the holomorphic curve
$C^{\mathbf{(p,q)}}$ to $C^{\mathbf{(p^{\prime},q^{\prime})}}$, where

$\displaystyle\begin{pmatrix}\mathbf{p^{\prime}}\\\
\mathbf{q^{\prime}}\end{pmatrix}=g\begin{pmatrix}\mathbf{p}\\\
\mathbf{q}\end{pmatrix},\qquad g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL(2,\mathbb{Z}).$ (202)

Thus, the $SL(2,\mathbb{Z})$ maps the M5-branes supported on those holomorphic
curves accordingly. After the dimensional reduction and T-dualization, this
precisely matches with the $SL(2,\mathbb{Z})$ symmetry of the IIB theory,
which maps the $\mathbf{(p,q)}$-fivebrane to the
$\mathbf{(p^{\prime},q^{\prime})}$-fivebrane.

The $SL(2,\mathbb{Z})$ symmetry of the twisted M-theory descends to the 5d
non-commutative $\mathfrak{gl}(1)$ Chern-Simons theory on
$\mathbb{R}_{t}\times\mathbb{C}^{\times}_{X}\times\mathbb{C}^{\times}_{Z}$.
Thus, the $SL(2,\mathbb{Z})$ symmetry should be reflected on the universal
associative algebra as its automorphism group. Indeed, the quantum toroidal
algebra $U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$ is known
to possess an $SL(2,\mathbb{Z})$ automorphism group Miki:2007 . Then, the
$q$-boson representations associated to the M5-branes supported on different
holomorphic curves should be mapped to one another under the action of this
$SL(2,\mathbb{Z})$ automorphism group.

The $SL(2,\mathbb{Z})$ group is generated by two elements,

$\displaystyle T=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix},\qquad
S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$ (203)

Let us first consider the subgroup generated by $T$. When acted on
$C^{\mathbf{(1,0)}}=\mathbb{C}^{\times}_{X}$, the subgroup generates all the
holomorphic curves of the form $C^{\mathbf{(1,n)}}$,
$\mathbf{n}\in\mathbb{Z}$. The $q$-boson representations associated to the
M5-brane on these curves can thus be obtained by applying the
$T$-transformations to the $q$-boson representation
$\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}}$ (58),

$\displaystyle\rho_{\text{M5}_{c}^{\mathbf{(1,n)}}}:U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))\twoheadrightarrow\text{M5}_{c}^{\mathbf{(1,n)}}=\text{End}(\mathcal{F}^{\mathbf{(1,n)}}_{c}(v)).$
(204)

This turns out to be a slight modification of the $q$-boson representation
$\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}}$ (58), given by

$\displaystyle\begin{split}&E(X)\mapsto-e^{a_{0}^{(c)}\frac{\log q_{c+1}\log
q_{c-1}}{\log
q_{c}}}\frac{1-q_{c}}{\kappa_{1}}\left(\frac{q_{c}^{1/2}}{X}\right)^{\mathbf{n}}\eta_{c}(X),\\\
&F(X)\mapsto e^{-a_{0}^{(c)}\frac{\log q_{c+1}\log q_{c-1}}{\log
q_{c}}}\frac{1-q_{c}^{-1}}{\kappa_{1}}\left(\frac{X}{q_{c}^{1/2}}\right)^{\mathbf{n}}\xi_{c}(X),\\\
&K^{\pm}(X)\mapsto q_{c}^{\mp\mathbf{n}/2}\varphi^{\pm}_{c}(X),\end{split}$
(205)

Note $(C,C^{\perp})\mapsto\left(q_{c}^{1/2},q_{c}^{\mathbf{n}/2}\right)$, in
particular.

Meanwhile, the $q$-boson representation associated to the M5-brane wrapping
$C^{\mathbf{(0,1)}}=\mathbb{C}^{\times}_{Z}$ can be understood in a different
way. Recall that one way of obtaining the $q$-boson representation is reducing
the twisted M-theory along the circle of
$\mathbb{C}^{\times}_{Z}\simeq\mathbb{R}_{z}\times S^{1}$, after which this
M5-brane becomes a D4-brane. The $q$-boson representation is viewed as the
algebra of local operators on the line $\mathbb{R}_{z}$ acting on the
equivariant K-theory of the moduli space of $U(1)$ instantons. Thus, the
$q$-boson representation arising in this case is the one where the
representation constructed in 2009arXiv0904.1679F ; 2009arXiv0905.2555S
directly applies. Namely, we have

$\displaystyle\rho_{\text{M5}_{c}^{\mathbf{(0,1)}}}:U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))\twoheadrightarrow\text{M5}_{c}^{\mathbf{(0,1)}}=\text{End}(\mathcal{F}^{\mathbf{(0,1)}}_{c}(a)).$
(206)

Let us recall the basis states of the Fock representation
$\mathcal{F}_{c}^{\mathbf{(0,1)}}(a)=\bigoplus_{\\{\lambda\\}}\mathbb{C}|a,\lambda\rangle^{(c)}$
are enumerated by a single Young diagram $\lambda$. The action of the
generating currents on the state $|a,\lambda\rangle^{(c)}$ is given by

$\displaystyle\begin{split}&E(X)|a,\lambda\rangle^{(c)}=\frac{1-q_{c}}{\kappa_{1}}\sum_{\Box\in\partial_{+}\lambda}\delta(\chi_{\Box}/X)\underset{X=\chi_{\Box}}{\text{Res}}{X^{-1}\mathcal{Y}_{\lambda}^{(c)}(X,a)^{-1}}|a,\lambda+\Box\rangle^{(c)},\\\
&F(X)|a,\lambda\rangle^{(c)}=-\frac{1-q_{c}^{-1}}{\kappa_{1}}q_{c}^{-1/2}\sum_{\Box\in\partial_{-}\lambda}\delta(\chi_{\Box}/X)\underset{X=\chi_{\Box}}{\text{Res}}X^{-1}\mathcal{Y}_{\lambda}^{(c)}(Xq_{c}^{-1},a)|a,\lambda-\Box\rangle^{(c)},\\\
&K^{\pm}(X)|a,\lambda\rangle^{(c)}=q_{c}^{-1/2}\left(\frac{\mathcal{Y}_{\lambda}^{(c)}(Xq_{c}^{-1},a)}{\mathcal{Y}_{\lambda}^{(c)}(X,a)}\right)_{\pm}|a,\lambda\rangle^{(c)},\qquad
C|a,\lambda\rangle^{(c)}=|a,\lambda\rangle^{(c)},\end{split}$ (207)

where we defined $\chi_{\Box_{(i,j)}}=aq_{c+1}^{i-1}q_{c-1}^{j-1}$ and

$\displaystyle\mathcal{Y}^{(c)}_{\lambda}(X,a)=\left(1-\frac{a}{X}\right)\prod_{\Box\in\lambda}\frac{\left(1-\frac{q_{c+1}\chi_{\Box}}{X}\right)\left(1-\frac{q_{c-1}\chi_{\Box}}{X}\right)}{\left(1-\frac{\chi_{\Box}}{X}\right)\left(1-\frac{q_{c}^{-1}\chi_{\Box}}{X}\right)}=\frac{\prod_{\Box\in\partial_{+}\lambda}\left(1-\frac{\chi_{\Box}}{X}\right)}{\prod_{\Box\in\partial_{-}\lambda}\left(1-\frac{q_{c}^{-1}\chi_{\Box}}{X}\right)}.$
(208)

Here, $\partial_{+}\lambda$ (resp. $\partial_{-}\lambda$) is the set of boxes
that can be added to (resp. removed from) the Young diagram $\lambda$. Also,
$f(X)_{+}$ (resp. $f(X)_{-}$) is the expansion of $f(X)$ in the domain
$|X|>\\!\\!>1$ (resp. $|X|<\\!\\!<1$). Note that
$(C,C^{\perp})\mapsto(1,q_{c}^{1/2})$, in particular.404040In some literature,
(204) is referred to as the horizontal Fock representation, while (206) is
referred to as the vertical Fock representation. Since the terminology may
cause confusion when compared to the $\mathbf{(p,q)}$-fivebrane web, we will
avoid using this terminology.

It was indeed shown that the $q$-boson representations
$\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}}$ and
$\rho_{\text{M5}_{c}^{\mathbf{(0,1)}}}$ are exchanged with each other under
the action of the $S$-transformation, also known as the Miki’s automorphism
Miki:2007 , of the quantum toroidal algebra
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$ Bourgine:2018fjy
. This is consistent with the corresponding reparametrization swapping the
$\mathbb{C}^{\times}_{X}$-plane and the $\mathbb{C}^{\times}_{Z}$-plane.

####  5.1.2 $q$-boson intertwiners for non-transverse M5-brane intersections

The M5-branes supported on different holomorphic curves $C^{\mathbf{(p,q)}}$
can make transverse or non-transverse intersections. Since it is difficult to
visualize this web of M5-branes directly, it is helpful to pass to the IIB
dual frame where the M5-brane supported on the curve $C^{\mathbf{(p,q)}}$
becomes the $\mathbf{(p,q)}$-fivebrane (see section 1.3). Recall that
$\mathbf{(p,q)}$-fivebranes are supported on
$\mathbb{R}^{2}_{{\varepsilon}_{c+1}}\times\mathbb{R}^{2}_{{\varepsilon}_{c-1}}\times
l^{\mathbf{(p,q)}}\times S^{1}$, where
$l^{\mathbf{(p,q)}}=\\{\mathbf{q}x-\mathbf{p}z=\text{const}\\}$ is a line on
the two-dimensional plane $\mathbb{R}_{x}\times\mathbb{R}_{z}$.

When the two fivebranes only share one of the three topological planes in
their supports (namely, $c_{1}\neq c_{2}$), the two fivebranes cannot end on
each other as they pass through along the mutually transverse directions. By a
similar argument for the transverse M2-M5 intersection, the local operator at
such a transverse intersection of M5-branes would provide the R-matrix between
the associated $q$-boson representations; explicitly,
$R_{\text{M5}_{c_{1}}^{\mathbf{(p_{1},q_{1})}},\text{M5}_{c_{2}}^{\mathbf{(p_{2},q_{2})}}}=(\rho_{\text{M5}_{c_{1}}^{\mathbf{(p_{1},q_{1})}}}\otimes\rho_{\text{M5}_{c_{2}}^{\mathbf{(p_{2},q_{2})}}})\mathcal{R}$.
We will not discuss this transverse M5-brane intersection further in this
work.

When the two fivebranes share two of the three topological planes (namely,
$c_{1}=c_{2}$), they can make an intersection at a point on
$\mathbb{R}_{x}\times\mathbb{R}_{z}$ if the $\mathbf{(p,q)}$ charge is
conserved Aharony:1997ju . From now on, we will focus on non-transverse
M5-brane intersections. Let us fix $c=3$ so that all the fivebranes have
$\mathbb{R}^{2}_{{\varepsilon}_{1}}\times\mathbb{R}^{2}_{{\varepsilon}_{2}}$
as the topological part of their supports. By consecutively joining their
intersections, we can form $\mathbf{(p,q)}$-web of fivebranes. The basic
building blocks for the web are the trivalent intersections depicted in Figure
15.

$\text{M5}_{0,0,1}^{\mathbf{(1,n+1)}}$$\text{M5}_{0,0,1}^{\mathbf{(0,1)}}$$\text{M5}_{0,0,1}^{\mathbf{(1,n)}}$$\Psi^{\mathbf{(n)}}$$\mathbb{R}_{x}$$\mathbb{R}_{z}$
(a)

$\text{M5}_{0,0,1}^{\mathbf{(1,n+1)}}$$\text{M5}_{0,0,1}^{\mathbf{(1,n)}}$$\text{M5}_{0,0,1}^{\mathbf{(0,1)}}$$\Psi^{*\mathbf{(n)}}$
(b)

Figure 15: Basic non-transverse intersections of fivebranes and associated
intertwiners of $q$-boson representations. We call (a) the first kind; (b) the
second kind.

The local operators on the fivebranes are ordered in the increasing
$\mathbb{R}_{x}$-direction, which is equivalent to the radial-ordering on
$\mathbb{C}^{\times}_{X}$ in the M-theory setting. These local operators can
act on the Hilbert space attached to each fivebrane, formed as the Fock space
$\mathcal{F}^{\mathbf{(p,q)}}_{3}(v)$ for the associated $q$-boson
representation $\text{M5}_{0,0,1}^{\mathbf{(p,q)}}$. In this sense, the local
operators supported at the first kind of the trivalent intersection provides a
map,

$\displaystyle\Psi^{\mathbf{(n)}}(v,v^{\prime}):\mathcal{F}^{\mathbf{(0,1)}}_{3}(v^{\prime})\otimes\mathcal{F}^{\mathbf{(1,n)}}_{3}(v)\to\mathcal{F}^{\mathbf{(1,n+1)}}_{3}(-vv^{\prime}),$
(209)

between the Fock spaces attached to the three fivebranes. By an argument
similar to that of the non-transverse M2-M5 intersection (see section 3.2.2),
we expect that the gauge-invariance condition for this intersection at full
quantum level reads

$\displaystyle\rho_{\text{M5}_{0,0,1}^{\mathbf{(1,n+1)}}}(g)\Psi^{\mathbf{(n)}}=\Psi^{\mathbf{(n)}}\Delta_{\text{M5}_{0,0,1}^{\mathbf{(0,1)}},\text{M5}_{0,0,1}^{\mathbf{(1,n)}}}(g).$
(210)

In other words, the local operator at the trivalent intersection provides an
intertwiner between the $q$-boson representations of the quantum toroidal
algebra $U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$. In
fact, this constraint can be solved to uniquely determine the intertwiner, up
to an overall scaling Awata:2011ce . We will not review the derivation of this
intertwiner here, but only present its expression in appendix E.

Similarly, let us consider the local operator supported at the second kind of
the trivalent intersection. The local operator gives a map,

$\displaystyle\Psi^{*\mathbf{(n)}}(v,v^{\prime}):\mathcal{F}^{\mathbf{(1,n+1)}}_{3}(-vv^{\prime})\to\mathcal{F}^{\mathbf{(0,1)}}_{3}(v^{\prime})\otimes\mathcal{F}^{\mathbf{(1,n)}}_{3}(v),$
(211)

between the Fock spaces attached to the three M5-branes. The gauge-invariance
condition at the quantum level is expected to be

$\displaystyle\Delta^{\text{op}}_{\text{M5}_{0,0,1}^{\mathbf{(0,1)}},\text{M5}_{0,0,1}^{\mathbf{(1,n)}}}(g)\Psi^{*\mathbf{(n)}}=\Psi^{*\mathbf{(n)}}\rho_{\text{M5}_{0,0,1}^{\mathbf{(1,n+1)}}}(g)$
(212)

The constraint exactly determines the intertwiner up to an overall scaling
Awata:2011ce . We give the expression for this intertwiner in appendix E.

###  5.2 $qq$-characters from Miura operators on web of $q$-bosons

Having introduced the $q$-boson representations associated to M5-branes
supported on holomorphic curves $C^{\mathbf{(p,q)}}$ and the intertwiners
associated to their non-transverse intersections, now we can compose these
M5-brane intersections to form a brane web engineering 5-dimensional
$\mathcal{N}=1$ gauge theories compactified on a circle. Moreover, we can
introduce an M2-brane which is transverse to the M5-branes. We have studied
such a transverse M2-M5 intersection giving rise to the R-matrix between the
$q$-bosons and the quantum torus algebra (namely, a product of Miura
operators). Here, we will show that the insertion of the R-matrix in the web
of $q$-bosons establishes the qq-characters Nikita:I .

We will only exemplify the simplest case: the 5-dimensional $\mathcal{N}=1$
pure $SU(2)$ gauge theory. In the IIB dual frame, the associated brane web is
given as in Figure 16. Note, however, that the principle directly generalizes
to arbitrary M5-brane web. The $qq$-characters of the 5d $\mathcal{N}=1$
effective gauge theory on the worldvolume of the M5-brane web would arise by
the additional insertion of transverse M2-branes. We emphasize that the
correspondence between the transverse M2-M5 intersections and the
$qq$-characters is anticipated due to a string duality. Specifically, the
original crossed D-brane configuration for the $qq$-characters Nikita:I ;
Nikita:III  transforms into the M2-branes transversally intersecting the
M5-brane web through T-duality and uplift to M-theory (see section 1.3). The
example we establish below is confirmation of this expected correspondence.

$\otimes$NS5D5D3$\mathbf{(1,1)}$$\mathbf{(1,-1)}$$\mathbb{R}_{x}$$\mathbb{R}_{z}$$\mathbb{R}_{t}$
(a)

$\otimes$$\text{M5}_{0,0,1}^{\mathbf{(1,0)}}$$\text{M5}_{0,0,1}^{\mathbf{(0,1)}}$$\text{M2}_{0,0,1}$$\text{M5}_{0,0,1}^{\mathbf{(1,1)}}$$\text{M5}_{0,0,1}^{\mathbf{(1,-1)}}$$\Psi^{\mathbf{(0)}}$$\Psi^{*\mathbf{(-1)}}$$\Psi^{*\mathbf{(0)}}$$\Psi^{\mathbf{(-1)}}$
(b)

Figure 16: (a) $\mathbf{(p,q)}$-fivebrane web of IIB theory engineering the 5d
$\mathcal{N}=1$ pure $SU(2)$ gauge theory. The $\mathbf{(p,q)}$-fivebranes are
supported on
$\mathbb{R}^{2}_{{\varepsilon}_{1}}\times\mathbb{R}^{2}_{{\varepsilon}_{2}}\times\\{\text{line
on the figure}\\}\times S^{1}$. The $qq$-character as a codimension-4 defect
of the 5d $\mathcal{N}=1$ gauge theory is introduced by a transverse D3-brane
supported on $\mathbb{R}^{2}_{{\varepsilon}_{3}}\times\mathbb{R}_{t}\times
S^{1}$; (b) Web of $q$-bosons and intertwiners. At the transverse intersection
with the D3-brane, the R-matrix between the $q$-bosons and the quantum torus
algebra $\text{M2}_{0,0,1}$ is assigned.

In our example, there are four trivalent intersections of the fivebranes, each
of which is assigned with an intertwiner of the associated $q$-boson
representations. Let us break the $\mathbf{(p,q)}$-fivebrane web into the
lower half and the upper half, splitting the intertwiners into two groups. For
the lower half of the web, we can compose the two intertwiners through the
space $\mathcal{F}^{\mathbf{(0,1)}}_{3}$ of states of the intermediate
M5-brane as

$\displaystyle\begin{split}&\sigma\left(\Psi^{\mathbf{(-1)}}(v_{2},a_{1})\otimes\Psi^{*\mathbf{(0)}}(v_{1}a_{2},a_{1})\right)\sigma\\\
&\qquad:\mathcal{F}^{\mathbf{(1,1)}}_{3}(-v_{1}a_{1}a_{2})\otimes\mathcal{F}_{3}^{\mathbf{(1,-1)}}(v_{2})\to\mathcal{F}_{3}^{\mathbf{(1,0)}}(v_{1}a_{2})\otimes\mathcal{F}_{3}^{\mathbf{(1,0)}}(-v_{2}a_{1}),\end{split}$
(213)

where we swapped the ordering of the Fock spaces in the tensor product using
the exchange operators. Similarly, for the upper half of the web we have

$\displaystyle\begin{split}&\sigma\left(\Psi^{\mathbf{(0)}}(-v_{2}a_{1},a_{2})\otimes\Psi^{*\mathbf{(-1)}}(-v_{1},a_{2})\right)\sigma\\\
&\qquad:\mathcal{F}_{3}^{\mathbf{(1,0)}}(v_{1}a_{2})\otimes\mathcal{F}_{3}^{\mathbf{(1,0)}}(-v_{2}a_{1})\to\mathcal{F}_{3}^{\mathbf{(1,-1)}}(-v_{1})\otimes\mathcal{F}^{\mathbf{(1,1)}}_{3}(v_{2}a_{1}a_{2}),\end{split}$
(214)

Then, the web as a whole provides a correlation function of intertwiners,

$\displaystyle\begin{split}&\mathcal{T}:=\sigma\left(\Psi^{\mathbf{(0)}}(-v_{2}a_{1},a_{2})\otimes\Psi^{*\mathbf{(-1)}}(-v_{1},a_{2})\right)\circ\left(\Psi^{\mathbf{(-1)}}(v_{2},a_{1})\otimes\Psi^{*\mathbf{(0)}}(v_{1}a_{2},a_{1})\right)\sigma\\\
&\qquad:\mathcal{F}^{\mathbf{(1,1)}}_{3}(-v_{1}a_{1}a_{2})\otimes\mathcal{F}_{3}^{\mathbf{(1,-1)}}(v_{2})\to\mathcal{F}_{3}^{\mathbf{(1,-1)}}(-v_{1})\otimes\mathcal{F}^{\mathbf{(1,1)}}_{3}(v_{2}a_{1}a_{2}).\end{split}$
(215)

We compute its vacuum expectation value in the $q$-boson representations by
evaluating it at the respective vacua,

$\displaystyle\mathcal{Z}(a;\mathfrak{q}):=(\langle\varnothing|\otimes\langle\varnothing|)\mathcal{T}(|\varnothing\rangle\otimes|\varnothing\rangle).$
(216)

By normal-ordering all the vertex operators in (215), we can explicitly
compute this vacuum expectation value. The outcome is (see appendix E for
notations)

$\displaystyle\begin{split}&\mathcal{Z}(a;\mathfrak{q})=\mathcal{Z}^{\text{1-loop}}(a)\sum_{\\{\lambda_{1},\lambda_{2}\\}}\mathfrak{q}^{|\lambda_{1}|+|\lambda_{2}|}\frac{1}{\prod_{\alpha,\beta=1,2}N_{\lambda_{\alpha}\lambda_{\beta}}(a_{\alpha}/a_{\beta};q_{1},q_{2})}\\\
&\mathcal{Z}^{\text{1-loop}}(a)=\exp\left[-\sum_{r=1}^{\infty}\frac{(a_{1}/a_{2})^{r}}{r}\frac{1+q_{3}^{-r}}{(1-q_{1}^{r})(1-q_{2}^{r})}\right]\end{split},$
(217)

where we introduced $\mathfrak{q}=q_{3}^{-1}\frac{v_{2}}{v_{1}}$. This exactly
reproduces the partition function of the 5d $\mathcal{N}=1$ pure $SU(2)$ gauge
theory on
$\mathbb{R}^{2}_{{\varepsilon}_{1}}\times\mathbb{R}^{2}_{{\varepsilon}_{2}}\times
S^{1}$, with the complexified gauge coupling $\mathfrak{q}$ and the Coulomb
parameter $a=a_{1}/a_{2}$. This is precisely the algebraic reconstruction of
the 5d $\mathcal{N}=1$ gauge theory partition function obtained in
Awata:2011ce .

Now, let us introduce an M2-brane supported on
$\mathbb{R}^{2}_{{\varepsilon}_{3}}\times\mathbb{R}_{t}$ on top of the
M5-brane web. The resulting transverse M2-M5 intersection assigns the R-matrix
$R_{\text{M2}_{0,0,1},\text{M5}_{0,0,2}^{\mathbf{(1,0)}}}\in\text{M2}_{0,0,1}\otimes\text{M5}_{0,0,2}^{\mathbf{(1,0)}}$.
Recalling that
${\text{M5}_{0,0,2}^{\mathbf{(1,0)}}}\hookrightarrow{\text{M5}_{0,0,1}^{\mathbf{(1,0)}}}\otimes{\text{M5}_{0,0,1}^{\mathbf{(1,0)}}}\simeq\text{End}(\mathcal{F}_{3}^{\mathbf{(1,0)}})\otimes\text{End}(\mathcal{F}_{3}^{\mathbf{(1,0)}})$,
this R-matrix is inserted in the middle between the lower and the upper halves
of the web, producing

$\displaystyle\begin{split}\mathcal{X}&:=\left(\sigma(\Psi^{\mathbf{(0)}}\otimes\Psi^{*\mathbf{(-1)}})\sigma\right)R_{\text{M2}_{0,0,1},\text{M5}_{0,0,2}^{\mathbf{(1,0)}}}\left(\sigma(\Psi^{\mathbf{(-1)}}\otimes\Psi^{*\mathbf{(0)}})\sigma\right)\\\
&\in\text{M2}_{0,0,1}\otimes\text{Hom}\left(\mathcal{F}^{\mathbf{(1,1)}}_{3}(-v_{1}a_{1}a_{2})\otimes\mathcal{F}_{3}^{\mathbf{(1,-1)}}(v_{2}),\mathcal{F}_{3}^{\mathbf{(1,-1)}}(-v_{1})\otimes\mathcal{F}^{\mathbf{(1,1)}}_{3}(v_{2},a_{1}a_{2})\right).\end{split}$
(218)

We can also define its vacuum expectation value by evaluating it at the vacua
of the $q$-bosons and the basis elements of $\mathcal{V}_{3}(u)$,

$\displaystyle\left\langle\mathcal{X}(u)\right\rangle_{k}:=\left([uq_{3}^{-1}]_{i+k}^{*}\otimes\langle\varnothing|\otimes\langle\varnothing|\right)\mathcal{X}\left([u]_{i}\otimes|\varnothing\rangle\otimes|\varnothing\rangle\right).$
(219)

Even though $i\in\mathbb{Z}$ enumerating the states
$[u]_{i}\in\mathcal{V}_{3}(u)$ is left arbitrary, the non-trivial dependence
is only on the distance $k\in\mathbb{Z}$ between the vector $[u]_{i}$ and the
dual vector $[uq_{3}^{-1}]_{i+k}^{*}$, not on the specific choice of $i$. Let
us call the vacuum expectation value
$\left\langle\mathcal{X}\right\rangle_{k}$ the $k$-th fundamental qq-
character.

We can explicitly evaluate the vacuum expectation values. As we studied in
section 3.3.2, the R-matrix is given by the product of Miura operators as

$\displaystyle\begin{split}&R_{\text{M2}_{0,0,1},\text{M5}_{0,0,2}^{\mathbf{(1,0)}}}=R_{2}^{(3)}R_{1}^{(3)}\\\
&=\left[1-q_{3}^{-1}\left(e^{-a_{0}^{[2]}\frac{\log q_{1}\log q_{2}}{\log
q_{3}}}\xi^{[2]}(X_{1})+e^{-a_{0}^{[1]}\frac{\log q_{1}\log q_{2}}{\log
q_{3}}}\xi^{[1]}(q_{3}^{\frac{1}{2}}X_{1})\varphi^{+,[2]}(q_{3}^{\frac{1}{2}}X_{1})\right)Z_{1}\right.\\\
&\quad\left.+q_{3}^{-2}e^{-(a_{0}^{[1]}+a_{0}^{[2]})\frac{\log q_{1}\log
q_{2}}{\log
q_{3}}}\xi^{[1]}(q_{3}^{-\frac{1}{2}}X_{1})\xi^{[2]}(X_{1})\varphi^{+,[2]}(q_{3}^{-\frac{1}{2}}X_{1})Z_{1}^{2}\right]q_{3}^{d|_{\text{M2}_{0,0,1}}}V^{[1]}(X_{1})V^{[2]}(q_{3}^{-\frac{1}{2}}X_{1}).\end{split}$
(220)

Here the square bracket indicates which of the two $q$-bosons the operators
are valued in. Plugging this expression into (219), it is apparent that
$\langle\mathcal{X}\rangle_{k}\neq 0$ only if $k=0,1,2$ since the R-matrix is
a degree-two polynomial in $Z_{1}$. Also note that the module parameter
$u\in\mathbb{C}^{\times}$ for $\mathcal{V}_{3}(u)$ gets shifted by
$q_{3}^{-1}$ due to $q_{3}^{d|_{\text{M2}_{0,0,1}}}$, as appearing in (219).

The $0$-th fundamental $qq$-character turns out to be exactly identical to the
partition function $\mathcal{Z}$ itself,
$\left\langle\mathcal{X}\right\rangle_{0}=\mathcal{Z}$, since the terms
emergent from normal-ordering $V^{[1]}(X_{1})$ and
$V^{[2]}(q_{3}^{-\frac{1}{2}}X_{1})$ to the right cancel with each other. The
second fundamental $qq$-character, though seemingly more complicated, also
yields more or less trivial expression since the terms appearing from normal-
ordering $\xi^{[1]}(q_{3}^{-\frac{1}{2}}X_{1})$ and
$\xi^{[2]}(X_{1})\varphi^{+,[2]}(q_{3}^{-\frac{1}{2}}X_{1})$ also cancel with
each other:
$\left\langle\mathcal{X}\right\rangle_{2}=-q_{3}^{-2}v_{1}^{-1}v_{2}^{-1}a_{1}^{-1}a_{2}^{-1}\mathcal{Z}$.

Finally, let us compute the first fundamental $qq$-character, which is the
only one that gives a non-trivial expression in our example.414141To have more
than one non-trivial fundamental $qq$-characters, we need to consider the
M5-brane web which engineers quiver gauge theories. If the quiver has $n$
gauge nodes, the transverse M2-brane assigns the R-matrix which is a product
of $n+1$ Miura operators. It is a degree-$(n+1)$ polynomial in $Z_{1}$,
yielding $\langle\mathcal{X}\rangle_{k}\neq 0$ only if $k=0,1,\cdots,n+1$.
Moreover, $k=0$ and $k=n+1$ give trivial expressions, so that there are $n$
non-trivial fundamental $qq$-characters. This exactly matches with the number
of fundamental $qq$-characters for the linear quiver gauge theory with $n$
gauge nodes (see section 1.3, where $n=l-2$) Nikita:I ; Nikita:III . A
straightforward computation gives

$\displaystyle\begin{split}\left\langle\mathcal{X}(X_{1})\right\rangle_{1}&=\mathcal{Z}^{\text{1-loop}}(a)\sum_{\\{\lambda_{1},\lambda_{2}\\}}\mathfrak{q}^{|\lambda_{1}|+|\lambda_{2}|}\frac{1}{\prod_{\alpha,\beta=1,2}N_{\lambda_{\alpha}\lambda_{\beta}}(a_{\alpha}/a_{\beta};q_{1},q_{2})}\\\
&\qquad\qquad\qquad\quad\times\frac{q_{3}^{-1/2}}{v_{2}}\left(\frac{q_{3}^{-1}}{a_{1}a_{2}}X_{1}\mathcal{Y}(X_{1}q_{3}^{-1/2})+\mathfrak{q}\frac{1}{X_{1}\mathcal{Y}(X_{1}q_{3}^{1/2})}\right),\end{split}$
(221)

where
$\mathcal{Y}(X):=\prod_{\alpha=1,2}\mathcal{Y}_{\lambda_{\alpha}}(X,a_{\alpha})$.
We abused the notation and denoted $uq_{3}^{i}$ simply by $X_{1}$ (recall the
former is the eigenvalue of the latter on the state $[uq_{3}^{-1}]_{i+1}$). Up
to an overall constant, this precisely recovers the fundamental $qq$-character
as an observable of the 5d $\mathcal{N}=1$ pure $SU(2)$ gauge theory Nikita:I
. Thus, we establish the simplest example of our general prescription: the
Miura operators inserted on the web of $q$-bosons, realized by the transverse
M2-branes passing through the web of M5-branes, give rise to the
$qq$-characters as codimension-4 defect observables of the 5d $\mathcal{N}=1$
effective gauge theory on the worldvolume of the M5-brane web.

##### Remark

Note that the degree-one term in the R-matrix (220), which is responsible for
the non-trivial $qq$-character, is proportional to
$\Delta_{\text{M5}_{0,0,1}^{\mathbf{(1,0)}},\text{M5}_{0,0,1}^{\mathbf{(1,0)}}}(F(X_{1}))$.
It was previously observed in Bourgine:2016vsq  that the $qq$-characters are
realized by inserting such generators into the web of $q$-bosons. In this
sense, at the computational level, our findings do not introduce new
information. The emphasis, however, is on the origin of the generator
insertion. In our prescription, the generators are not simply inserted by
hand, but appear as a part of the R-matrix (i.e., product of Miura operators)
which is present due to the M2-brane transversally intersecting the M5-brane
web. As we emphasized earlier, our construction of the $qq$-characters by the
M2-M5 intersection is compatible with the original crossed D-brane
configuration in the gauge origami Nikita:I ; Nikita:III , related to each
other by a string duality (see section 1.3).

##  6 Discussion

There are many interesting directions that deserve further studies.

##### MacMahon representations

We may replace
$\mathcal{C}=\mathbb{C}^{\times}_{X}\times\mathbb{C}^{\times}_{Z}$ in the
twisted M-theory background by a (multiplicative) $A_{P}$ singularity, that is
to say a $\mathbb{C}^{\times}_{Z}$ fibration over $\mathbb{C}^{\times}_{X}$,
where the fiber degenerates over $P$ points on the base. In the Type IIB web,
this is dual to adding $P$ D7 branes supported on
$\mathbb{R}^{2}_{{\varepsilon}_{1}}\times\mathbb{R}^{2}_{{\varepsilon}_{2}}\times\mathbb{R}^{2}_{{\varepsilon}_{3}}\times
S^{1}(R)\times\mathbb{R}_{t}$ at points on $\mathbb{R}_{x}$. It has been
suggested that such branes are natural candidates to be MacMahon
representations of the quantum toroidal algebra Zenkevich:2020ufs . It would
be important to make this point precise, and ultimately further our
understanding of 7-brane physics via the representation theory of quantum
toroidal algebras. See Gaiotto:2019wcc  for the case of
$\mathcal{C}=(\mathbb{C}\times\mathbb{C})/\mathbb{Z}_{P+1}$.

##### Miki automorphism and 5-branes

In section 4, we saw that the Miura R-matrices were captured by crossings of
D3 and NS5-branes in type IIB. In an S-dual frame, we would study crossings of
D3 and D5-branes instead, related to the previous setup by action of the Miki
automorphism, see section 5.1.1. At such crossings, we would construct S-dual
Miura operators at such crossings, where quantum toroidal vector
representations are taken as before, while the Fock representations are no
longer “horizontal” $q$-boson vertex representations, but instead the
“vertical” representations (206). In a fully generic S-dual frame, the Miura
operator will be realized via crossing of a D3-brane with a $\mathbf{(p,q)}$
5-brane. We leave the derivation of such S-dual operators to future work.

##### 3d non-abelian gauge theory

We only considered only a single D3-brane in the type IIB web; for an
arbitrary number of NS5-branes, the low energy 3d theory was a quiver whose
gauge content was a product of abelian groups. We can increase the total
number of D3-branes, which increases the rank of the gauge group(s) in the
quiver. This non-abelian case brings about new subtleties with regards to the
cancellation of the gauge anomaly, which generically requires turning on bare
Chern-Simons terms, but the computation of the half-index remains
straightforward: in the simple case of $K$ semi-infinite D3-branes
intersecting a single NS5-brane (and ending on another NS5-brane), the half-
index is a product of $K$ R-matrices of the form (129); it follows from
section 2.1.2 that if we expand this product as a series in $q$-shift
operators, the coefficients can all be expressed in terms of generators of
spherical DAHA. It would be nice to work out nontrivial examples directly from
the type IIB frame.

##### Geometric Langlands Program

Let $\mathfrak{g}$ be a simple Lie algebra. The quantum $q$-Langlands
correspondence is the statement on the Riemann surface
$\mathbb{C}^{\times}=S^{1}\times\mathbb{R}_{t}$, there exists an isomorphism
of spaces of $q$-conformal blocks for two different algebras :

${\cal
W}_{q_{1},q_{2}}(\mathfrak{g})\;\leftrightarrow\;U_{\hbar}(\widehat{{}^{L}\mathfrak{g}}_{{}^{L}k})\;,$
(222)

where the parameters $q_{1}$ and $q_{2}$ on the left-hand side are related to
the quantization parameter $\hbar$ and the affine Kac-Moody level ${}^{L}k$ on
the right-hand side, all taken complex and not equal to roots of unity. At the
present time, the correspondence has been shown to hold only for blocks valued
in tensor products of fundamental representations Aganagic:2017smx  and Verma
representations Haouzi:2023doo ; Tamagni:2023wan . It is a tantalizing
possibility that the quantum $q$-Langlands correspondence could be embedded
into the quantum toroidal algebra, at least for
$\mathfrak{g}=\mathfrak{sl}(r)$. Indeed, in that case, the algebra on the
left-hand side is the one governing the Fock representations of
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$ at level $r$,
and each $q$-conformal block likewise has an elegant interpretation as the vev
of specific operators in
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$, following
section 4.1. The puzzle lies in the appearance of the algebra
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}}_{{}^{L}k})$ on the right-hand side; in
particular, the interpretation of its $q$-conformal blocks will require a
representation theoretic understanding of K-theoretic stable envelopes
Okounkov:2015spn . Taking the degeneration limit, it would also be desirable
to relate our study of M-brane intersections and R-matrices to the 4d
$\mathcal{N}=2$ gauge theory formulation of the ($\hbar$-)Langlands
correspondence Jeong:2018qpc ; Jeong:2023qdr ; Jeong:2024hwf . This is work in
progress.

##  Appendix A Quantum toroidal algebra of $\mathfrak{gl}(1)$

###  A.1 Definition

The quantum toroidal algebra of $\mathfrak{gl}(1)$, denoted by
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$, is an
associative algebra with the generators $E_{k},F_{k},K_{\pm r}^{\pm}$
$(k\in\mathbb{Z},\,r\in\mathbb{Z}_{\geq 0})$ and $C$. We introduce the
generating currents as

$\displaystyle E(X)=\sum_{k\in\mathbb{Z}}E_{k}X^{-k},\quad
F(X)=\sum_{k\in\mathbb{Z}}F_{k}X^{-k},\quad
K^{\pm}(X)=\sum_{r=0}^{\infty}K_{\pm r}^{\pm}X^{\mp r}.$ (223)

In terms of these generating currents, the defining relations are

$\displaystyle C\text{ is central}$ (224a) $\displaystyle
1=K^{+}_{0}K^{-}_{0}=K^{-}_{0}K^{+}_{0}$ (224b) $\displaystyle
K^{\pm}(X)K^{\pm}(X^{\prime})=K^{\pm}(X^{\prime})K^{\pm}(X)$ (224c)
$\displaystyle\frac{g(C^{-1}X,X^{\prime})}{g(CX,X^{\prime})}K^{-}(X)K^{+}(X^{\prime})=\frac{g(X^{\prime},C^{-1}X)}{g(X^{\prime},CX)}K^{+}(X^{\prime})K^{-}(X)$
(224d) $\displaystyle
g(X,X^{\prime})E(X)E(X^{\prime})+g(X^{\prime},X)E(X^{\prime})E(X)=0$ (224e)
$\displaystyle
g(X^{\prime},X)F(X)F(X^{\prime})+g(X,X^{\prime})F(X^{\prime})F(X)=0$ (224f)
$\displaystyle g(X,X^{\prime})K^{\pm}(C^{(1\mp
1)/2}X)E(X^{\prime})+g(X^{\prime},X)E(X^{\prime})K^{\pm}(C^{(1\mp 1)/2}X)=0$
(224g) $\displaystyle g(X^{\prime},X)K^{\pm}(C^{(1\pm
1)/2}X)F(X^{\prime})+g(X,X^{\prime})F(X^{\prime})K^{\pm}(C^{(1\pm 1)/2}X)=0$
(224h)
$\displaystyle[E(X),F(X^{\prime})]=\tilde{g}\left\\{\delta\left(C\frac{X^{\prime}}{X}\right)K^{+}(X)-\delta\left(C\frac{X}{X^{\prime}}\right)K^{-}(X^{\prime})\right\\},$
(224i)

augmented by the Serre relations

$\displaystyle
0=\text{Sym}_{i_{1},i_{2},i_{3}}[E_{i_{1}},[E_{i_{2}+1},E_{i_{3}-1}]]$ (225a)
$\displaystyle
0=\text{Sym}_{i_{1},i_{2},i_{3}}[F_{i_{1}},[F_{i_{2}+1},F_{i_{3}-1}]].$ (225b)

Here, $g(X,X^{\prime})$ is the structure function defined by

$\displaystyle g(X,X^{\prime}):=\prod_{a=1}^{3}(X-q_{a}X^{\prime}).$ (226)

Note that $\tilde{g}$ only affects the relative normalization of the currents
$E(X)$, $F(X)$, and $K^{\pm}(X)$. We will always choose it to be
$\tilde{g}=\kappa_{1}^{-1}$, where we defined

$\displaystyle\kappa_{n}=\prod_{i=1}^{3}(q_{i}^{\frac{n}{2}}-q_{i}^{-\frac{n}{2}})=\prod_{i=1}^{3}(1-q_{i}^{-n})=-\kappa_{-n}.$
(227)

The quantum toroidal algebra
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$ is parametrized
by two complex numbers,
$(q_{1},q_{2},q_{3})\in\left(\mathbb{C}^{\times}\right)^{\times 3}$
constrained by $q_{1}q_{2}q_{3}=1$. They enter in the definition through (226)
and (227). Note that there is an explicit triality of permuting $q_{1}$,
$q_{2}$, and $q_{3}$.

We reorganize $\left(K^{\pm}_{\pm r}\right)_{r=0}^{\infty}$ in terms of
$C^{\perp}$ and $\left(H_{\pm r}\right)_{r=0}^{\infty}$ by

$\displaystyle K^{\pm}(X)=(C^{\perp})^{\mp
1}\exp\left(\pm\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}H_{\pm r}X^{\mp
r}\right),$ (228)

where $K^{-}_{0}=(K^{+}_{0})^{-1}=C^{\perp}$ is a central element.

It is helpful to recast the defining relations in terms of the modes. It is
straightforward to see (224c) and (224d) are equivalent to the $q$-boson
commutation relation satisfied by $H_{r}$,

$\displaystyle[H_{r},H_{s}]=\delta_{r+s,0}\frac{r}{\kappa_{r}}(C^{r}-C^{-r}).$
(229)

Also, the relations (224e), (224f), and (224i) read

$\displaystyle[E_{k+2},E_{l-1}]-(q_{1}+q_{2}+q_{3})E_{k+1}E_{l}+(q_{1}^{-1}+q_{2}^{-1}+q_{3}^{-1})E_{l}E_{k+1}+(k\leftrightarrow
l)=0$ (230a)
$\displaystyle[F_{k+2},F_{l-1}]-(q_{1}^{-1}+q_{2}^{-1}+q_{3}^{-1})F_{k+1}F_{l}+(q_{1}+q_{2}+q_{3})F_{l}F_{k+1}+(k\leftrightarrow
l)=0$ (230b)
$\displaystyle[E_{k},F_{l}]=\tilde{g}\left(C^{-l}K^{+}_{k+l}\theta_{k+l\geq
0}-C^{-k}K^{-}_{k+l}\theta_{k+l\leq 0}\right),$ (230c)

where in the last relation we used the notation

$\displaystyle\theta_{\\{\text{condition}\\}}=\begin{cases}1\qquad\text{if the
condition is true}\\\ 0\qquad\text{if the condition is false}\end{cases}.$
(231)

Finally, the relations (224g) and (224h) are equivalent to

$\displaystyle[H_{r},E_{k}]=C^{(r-|r|)/2}E_{k+r}$ (232a)
$\displaystyle[H_{r},F_{k}]=-C^{(r+|r|)/2}F_{k+r}.$ (232b)

Again, note the explicit invariance under the triality of exchanging
$(q_{1},q_{2},q_{3})$.

###  A.2 Coproduct

The quantum toroidal algebra of $\mathfrak{gl}(1)$ is a Hopf algebra. In
particular, it has a (formal) coproduct
$\Delta:U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))\to
U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))\,\widehat{\otimes}\,U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$
defined by

$\displaystyle\Delta(E(X))=E(X)\otimes 1+K^{-}(C_{1}X)\otimes E(C_{1}X)$
(233a) $\displaystyle\Delta(F(X))=1\otimes F(X)+F(C_{2}X)\otimes
K^{+}(C_{2}X)$ (233b) $\displaystyle\Delta(K^{+}(X))=K^{+}(X)\otimes
K^{+}(C_{1}^{-1}X)$ (233c)
$\displaystyle\Delta(K^{-}(X))=K^{-}(C_{2}^{-1}X)\otimes K^{-}(X)$ (233d)
$\displaystyle\Delta(C)=C\otimes C,$ (233e)

where we used the notation $C_{1}=C\otimes 1$ and $C_{2}=1\otimes C$. The
image of the formal coproduct contains infinite summations of elements, and
thus has to be understood as lying in the completed tensor product. The
infinite summation is expected to truncate to a finite summation when
represented on appropriate modules over
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))$.

###  A.3 Grading operators

The quantum toroidal algebra of $\mathfrak{gl}(1)$ is equipped with a
$\mathbb{Z}^{2}$-grading with two grading operators $d$ and $d^{\perp}$, which
act by

$\displaystyle\begin{split}&[d,E(X)]=-X\partial_{X}E(X),\quad[d,F(X)]=-X\partial_{X}F(X),\quad[d,K^{\pm}(X)]=-X\partial_{X}K^{\pm}(X),\\\
&[d^{\perp},E(X)]=E(X),\quad[d^{\perp},F(X)]=-F(X),\quad[d^{\perp},K^{\pm}(X)]=0,\\\
&[d,C]=[d^{\perp},C]=0.\end{split}$ (234)

We define
$\text{deg}=(d^{\perp},d):U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))\to\mathbb{Z}^{2}$,
so that we have

$\displaystyle\deg E_{k}=(1,k),\quad\deg F_{k}=(-1,k),\quad\deg
H_{r}=(0,r),\quad\deg C=\deg C^{\perp}=(0,0).$ (235)

In particular, for any parameter $q\in\mathbb{C}^{\times}$ it follows from
(234) that

$\displaystyle q^{d}E(X)=E(q^{-1}X)q^{d},\quad q^{d}F(X)=F(q^{-1}X)q^{d},\quad
q^{d}K^{\pm}(X)=K^{\pm}(q^{-1}X)q^{d}.$ (236)

##  Appendix B Spherical double affine Hecke algebras

###  B.1 Double affine Hecke algebras

The double affine Hecke algebra (DAHA) of $GL(m)$, denoted by
$\mathbf{\ddot{H}}^{(m)}_{q,t^{-1}}$, is an associative algebra generated by
$T_{i}^{\pm 1}$ ($i=1,2,\cdots,m-1$), $X_{i}^{\pm 1}$, and $Y_{i}^{\pm 1}$
($i=1,2,\cdots,m$) subject to the following relations:

$\displaystyle(T_{i}-t^{\frac{1}{2}})(T_{i}+t^{-\frac{1}{2}})=0,\qquad
T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}$ (237a) $\displaystyle
T_{i}T_{j}=T_{j}T_{i}\qquad\text{if}\;\;|i-j|>1$ (237b) $\displaystyle
X_{i}X_{j}=X_{j}X_{i},\qquad Y_{i}Y_{j}=Y_{j}Y_{i}$ (237c) $\displaystyle
T_{i}X_{i}T_{i}=X_{i+1},\qquad T_{i}^{-1}Y_{i}T_{i}^{-1}=Y_{i+1}$ (237d)
$\displaystyle T_{i}X_{j}=X_{j}T_{i},\qquad
T_{i}Y_{j}=Y_{j}T_{i}\qquad\text{if}\;\;j\neq i,i+1$ (237e) $\displaystyle
Y_{1}X_{1}\cdots X_{m}=qX_{1}\cdots X_{m}Y_{1}$ (237f) $\displaystyle
X_{1}^{-1}Y_{2}=Y_{2}X_{1}^{-1}T_{1}^{-2}.$ (237g)

Here, the subscripts are understood modulo $m$. The DAHA is parameterized by
two complex numbers $q,t^{-1}\in\mathbb{C}^{\times}$ (we intentionally use
$t^{-1}$ instead of $t$, since it is $q$ and $t^{-1}$ which get identified
with two out of the triple $(q_{1},q_{2},q_{3})$, parametrizing the quantum
toroidal algebra of $\mathfrak{gl}(1)$ under the algebra homomorphism).

The DAHA is endowed with the automorphism group given by the braid group on
three strands,

$\displaystyle
B_{3}=\left\langle\left.\tau_{\pm}\;\right|\;\tau_{+}\tau_{-}^{-1}\tau_{+}=\tau_{-}^{-1}\tau_{+}\tau_{-}^{-1}\right\rangle.$
(238)

The action on the generators is given by

$\displaystyle\tau_{+}:\begin{cases}T_{i}\mapsto T_{i}\\\ X_{i}\mapsto
X_{i}Y_{i}(T_{i-1}\cdots T_{1})(T_{1}\cdots T_{i-1})\\\ Y_{i}\mapsto
Y_{i}\end{cases},\quad\tau_{-}:\begin{cases}T_{i}\mapsto T_{i}\\\ X_{i}\mapsto
X_{i}\\\ Y_{i}\mapsto Y_{i}X_{i}(T_{i-1}^{-1}\cdots
T_{1}^{-1})(T_{1}^{-1}\cdots T_{i-1}^{-1})\end{cases}.$ (239)

Note that

$\displaystyle\sigma:=\tau_{+}\tau_{-}^{-1}\tau_{+}=\tau_{-}^{-1}\tau_{+}\tau_{-}^{-1}:\begin{cases}T_{i}\mapsto
T_{i}\\\ X_{i}\mapsto Y_{i}^{-1}\\\ Y_{i}\mapsto
Y_{i}X_{i}Y_{i}^{-1}\end{cases}.$ (240)

The braid group $B_{3}$ is an extension of $SL(2,\mathbb{Z})$ by $\mathbb{Z}$,

$\displaystyle 0\to\mathbb{Z}\to B_{3}\to SL(2,\mathbb{Z})\to 1,$ (241)

where the kernel of the projection $B_{3}\to SL(2,\mathbb{Z})$ is generated by
$(\tau_{+}\tau_{-}^{-1}\tau_{+})^{4}$. Moreover, we have

$\displaystyle\tau_{+}$ $\displaystyle\mapsto\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\qquad\tau_{-}\mapsto\begin{pmatrix}1&0\\\
1&1\end{pmatrix},\qquad\sigma\mapsto\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$
(242)

There is a faithful representation $\varphi_{m}$ of
$\ddot{\mathbf{H}}_{q,t^{-1}}^{(m)}$ on the space
$\mathbb{C}(q,t^{\frac{1}{2}})[X_{1}^{\pm 1},\cdots,X_{m}^{\pm 1}]$ defined by
cherednik2005

$\displaystyle\begin{split}&\varphi_{m}(X_{i})=X_{i},\\\
&\varphi_{m}(T_{i})=t^{\frac{1}{2}}s_{i}+\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{X_{i}/X_{i+1}-1}(s_{i}-1),\\\
&\varphi_{m}(Y_{i})=\varphi_{m}(T_{i})\cdots\varphi_{m}(T_{m-1})\omega\varphi_{m}(T_{1}^{-1})\cdots\varphi_{m}(T_{i-1}^{-1}),\end{split}$
(243)

where $s_{i}$ stands for the transposition $X_{i}\leftrightarrow X_{i+1}$;
$\omega=s_{m-1}\cdots s_{1}q^{D_{X_{1}}}$; and $q^{D_{X_{i}}}$ is the
difference operator $q^{D_{X_{i}}}f(X_{j})=f(q^{\delta_{i,j}}X_{j})$.

###  B.2 Spherical double affine Hecke algebras

Let $s_{i}\in S_{m}$ be the transposition $s_{i}=(i,i+1)$, and let
$l:S_{m}\to\mathbb{Z}_{\geq 0}$ be the length function. For a reduced
decomposition $w=s_{i_{1}}\cdots s_{i_{r}}$ of $w\in S_{m}$, we define
$T_{w}=T_{i_{1}}\cdots T_{i_{r}}$. It is easy to show that

$\displaystyle S=\left(\prod_{i=1}^{m}\frac{1-t}{1-t^{i}}\right)\sum_{w\in
S_{m}}t^{\frac{l(w)}{2}}T_{w}$ (244)

is an idempotent element in $\mathbf{\ddot{H}}^{(m)}_{q,t^{-1}}$, i.e.,
$S^{2}=S$. Also, for any $i$ we have $T_{i}S=ST_{i}=t^{\frac{1}{2}}S$. The
spherical DAHA is defined to be the subalgebra

$\displaystyle\text{\bf{S}}\boldsymbol{\ddot{\text{\bf{H}}}}^{(m)}_{q,t^{-1}}=S\mathbf{\ddot{H}}^{(m)}_{q,t^{-1}}S.$
(245)

The spherical DAHA is generated by

$\displaystyle\begin{split}&P^{(m)}_{0,r}=S\sum_{i=1}^{m}Y_{i}^{r}S,\qquad
P^{(m)}_{0,-r}=q^{r}S\sum_{i=1}^{m}Y_{i}^{-r}S,\\\
&P^{(m)}_{r,0}=q^{r}S\sum_{i=1}^{m}X_{i}^{r}S,\quad
P^{(m)}_{-r,0}=S\sum_{i=1}^{m}X_{i}^{-r}S,\end{split}\qquad
r\in\mathbb{Z}_{>0},$ (246)

with the relations following from (237). The action of the braid group $B_{3}$
on $\mathbf{\ddot{H}}^{(m)}_{q,t^{-1}}$ preserves the subalgebra
$\text{\bf{S}}\boldsymbol{\ddot{\text{\bf{H}}}}^{(m)}_{q,t^{-1}}$. Moreover,
the corresponding action factors through the projection $B_{3}\to
SL(2,\mathbb{Z})$; namely, $(\tau_{+}\tau_{-}^{-1}\tau_{+})^{4}$ acts
trivially on
$\text{\bf{S}}\boldsymbol{\ddot{\text{\bf{H}}}}^{(m)}_{q,t^{-1}}$. The
remaining $SL(2,\mathbb{Z})$-action on $\mathbf{S\ddot{H}}^{(m)}_{q,t^{-1}}$
is given by

$\displaystyle P^{(m)}_{a,b}\mapsto P^{(m)}_{g(a,b)},\qquad g\in
SL(2,\mathbb{Z}).$ (247)

Some useful formulas for the other generators obtained by the action of
$SL(2,\mathbb{Z})$ are

$\displaystyle\begin{split}&P^{(m)}_{1,k}=q\frac{1-t^{m}}{1-t}SX_{1}Y_{1}^{k}S,\quad
P^{(m)}_{-1,k}=\frac{1-t^{-m}}{1-t^{-1}}SY_{1}^{k}X_{1}^{-1}S,\\\
&P^{(m)}_{k,1}=\frac{1-t^{-m}}{1-t^{-1}}SY_{1}X_{1}^{k}S,\quad
P^{(m)}_{k,-1}=q\frac{1-t^{m}}{1-t}SX_{1}^{k}Y_{1}^{-1}S\end{split}\qquad
k\in\mathbb{Z}.$ (248)

The polynomial representation (243) of DAHA, $\mathbf{H}_{q,t^{-1}}^{(m)}$,
restricts to a faithful representation of the spherical DAHA,
$\text{\bf{S}}\boldsymbol{\ddot{\text{\bf{H}}}}_{q,t^{-1}}^{(m)}$, on the
space of symmetric polynomials $\mathbb{C}(q,t)[X_{1}^{\pm
1},\cdots,X_{m}^{\pm 1}]^{S_{m}}$, given by MacDonald1979SymmetricFA  (after
redefining the difference operators by a simple function of
$(X_{i})_{i=1}^{m}$)

$\displaystyle\begin{split}&\varphi_{m}(Se_{r}(X)S)=e_{r}(X),\\\
&\varphi_{m}(Se_{r}(Y)S)=\hat{H}_{r}:=t^{\frac{r(r-1)}{2}}\sum_{\begin{subarray}{c}I\subset\\{1,2,\cdots,m\\}\\\
|I|=r\end{subarray}}\left(\prod_{\begin{subarray}{c}i\in I\\\ j\notin
I\end{subarray}}\frac{tX_{i}-X_{j}}{X_{i}-X_{j}}\right)\prod_{i\in
I}q^{D_{X_{i}}},\end{split}$ (249)

where $e_{r}(X)=\sum_{1\leq i_{1}<\cdots<i_{r}\leq m}X_{i_{1}}\cdots
X_{i_{r}}$ is the $r$-th elementary symmetric polynomial of
$(X_{i})_{i=1}^{m}$, and similarly for $e_{r}(Y)$. In particular, the
difference operators $(\hat{H}_{r})_{r=1}^{m}$ are precisely the Macdonald
operators, which provide the quantum Hamiltonians of the trigonometric
Ruijsenaars-Schneider model of type $GL(m)$.

In section 2.1.2, we used the polynomial representation
$\varphi_{m}\circ\sigma$ obtained by composing (249) with the automorphism
$\sigma$ (240). This representation can be written explicitly as

$\displaystyle\begin{split}&P_{0,r}^{(m)}\mapsto
q^{r}\sum_{i=1}^{m}X_{i}^{r},\qquad
P_{0,-r}^{(m)}\mapsto\sum_{i=1}^{m}X_{i}^{-r},\qquad r\in\mathbb{Z}_{>0}\\\
\hfil\displaystyle\begin{split}&P_{1,k}^{(m)}\mapsto
q\sum_{i=1}^{m}\left(\prod_{j\neq
i}\frac{t^{-1}X_{i}-X_{j}}{X_{i}-X_{j}}\right)X_{i}^{k}q^{-D_{X_{i}}}\\\
&P_{-1,k}^{(m)}\mapsto\sum_{i=1}^{m}\left(\prod_{j\neq
i}\frac{tX_{i}-X_{j}}{X_{i}-X_{j}}\right)q^{D_{X_{i}}}X_{i}^{k}\end{split},\qquad
k\in\mathbb{Z}.\end{split}$ (250)

###  B.3 Spherical DAHA as representation of quantum toroidal algebra

At each $m\in\mathbb{Z}_{>0}$, there is a surjective algebra homomorphism
$U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1))\twoheadrightarrow\text{\bf{S}}\boldsymbol{\ddot{\text{\bf{H}}}}^{(m)}_{q_{2},q_{1}}$
given by Schiffmann_Vasserot_2011

$\displaystyle\begin{split}&E_{k}\mapsto\frac{1}{q_{2}(1-q_{2})}P^{(m)}_{1,k},\quad
F_{k}\mapsto\frac{1}{1-q_{2}^{-1}}P^{(m)}_{-1,k},\quad H_{\pm
r}\mapsto\mp\frac{1}{1-q_{2}^{r}}P^{(m)}_{0,\pm r}\\\ &C\mapsto 1,\quad
C^{\perp}\mapsto 1,\end{split}$ (251)

with $k\in\mathbb{Z}$ and $r\in\mathbb{Z}_{>0}$.

##  Appendix C Solving the R-matrix

The constraint for the R-matrix
$R^{(c)}=(\rho_{\text{M2}_{0,0,1}}\otimes\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}})\mathcal{R}$,
$c\in\\{1,2,3\\}$, reads

$\displaystyle
R^{(c)}\Delta_{\text{M2}_{0,0,1},\text{M5}_{c}^{\mathbf{(1,0)}}}(g)=\Delta^{\text{op}}_{\text{M2}_{0,0,1},\text{M5}_{c}^{\mathbf{(1,0)}}}(g)R^{(c)},\qquad\text{for
any }g\in U_{q_{1},q_{2},q_{3}}(\widehat{\widehat{\mathfrak{gl}}}(1)).$ (252)

We solve the constraint by making an ansatz,

$\displaystyle
R^{(c)}=\bar{R}^{(c)}K^{(c)}\in\text{M2}_{0,0,1}\,\widehat{\otimes}\,\text{M5}_{c}^{\mathbf{(1,0)}},$
(253)

where

$\displaystyle\begin{split}&K^{(c)}=\left(q_{3}^{\frac{d}{2}}\otimes
1\right)\exp\left[\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{X_{1}^{-r}}{1-q_{3}^{r}}\otimes\frac{a_{r}^{(c)}}{q_{c}^{r/2}-q_{c}^{-r/2}}\right]\\\
&\bar{R}^{(c)}=\sum_{m=0}^{\infty}\alpha_{m}\bar{R}^{(c)}_{m}.\end{split}$
(254)

Here, we set

$\displaystyle\bar{R}_{m}^{(c)}=\sum_{k_{1},k_{2},\cdots,k_{m}\in\mathbb{Z}}\left(\rho_{\text{M2}_{0,0,1}}\otimes\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}}\right)\left(E_{k_{1}}E_{k_{2}}\cdots
E_{k_{m}}\otimes F_{-k_{1}}F_{-k_{2}}\cdots F_{-k_{m}}\right)$ (255)

so that $\bar{R}^{(c)}$ is expanded with respect to the $d^{\perp}$-grading,
where $\alpha_{m}\in\mathbb{C}$ are coefficients to be determined.

Let us denote
$\Delta_{\text{M2}_{0,0,1},\text{M5}_{c}^{\mathbf{(1,0)}}}=(\rho_{\text{M2}_{0,0,1}}\otimes\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}})\Delta$
simply by $\Delta$ for brevity. Let us also omit explicit representations, so
that $g_{1}\otimes g_{2}$ is to be read as
$\left(\rho_{\text{M2}_{0,0,1}}\otimes\rho_{\text{M5}_{c}^{\mathbf{(1,0)}}}\right)(g_{1}\otimes
g_{2})$. Moreover, we write the zero-mode term in the $q$-boson representation
as $v\equiv e^{a_{0}^{(c)}\frac{\log q_{c+1}\log q_{c-1}}{\log q_{c}}}$.

First, we show that, for each $m\in\mathbb{Z}_{\geq 0}$,

$\displaystyle\bar{R}^{(c)}_{m}K^{(c)}\Delta(K^{+}(X))=\Delta^{\text{op}}(K^{+}(X))\bar{R}^{(c)}_{m}K^{(c)}.$
(256)

A straightforward computation gives

$\displaystyle\begin{split}&\bar{R}^{(c)}_{m}K^{(c)}\Delta(K^{+}(X))=\bar{R}^{(c)}_{m}K^{(c)}\left(K^{+}(X)\otimes\varphi^{+}_{c}(X)\right)\\\
&=\frac{v^{-m}(1-q_{c}^{-1})^{m}}{\kappa_{1}^{m}(1-q_{3})^{m}}\left(X_{1}^{k_{1}}(q_{3}^{-1}X_{1})^{k_{2}}\cdots(q_{3}^{-m+1}X_{1})^{k_{m}}\otimes\left(\xi_{c}\right)_{k_{1}}\cdots\left(\xi_{c}\right)_{k_{m}}\right)\\\
&\qquad\qquad\qquad\qquad\times(Z_{1}^{m}\otimes
1)\left(K^{+}(q_{c}^{-\frac{1}{2}}X)\otimes\varphi^{+}_{c}(X)\right)K^{(c)}\\\
&=\frac{v^{-m}(1-q_{c}^{-1})^{m}}{\kappa_{1}^{m}(1-q_{3})^{m}}\xi_{c}(X_{1})\xi_{c}(q_{3}^{-1}X_{1})\cdots\xi_{c}(q_{3}^{-m+1}X_{1})\left(K^{+}(q_{c}^{-\frac{1}{2}+m}X)\otimes\varphi^{+}_{c}(X)\right)Z_{1}^{m}K^{(c)}\\\
&=\exp\left[\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{1-q_{3}^{-mr}}{1-q_{3}^{-r}}q_{c}^{\frac{r}{2}}\left(\frac{X_{1}}{X}\right)^{r}\right]\left(K^{+}(q_{c}^{-\frac{1}{2}+m}X)\otimes\varphi^{+}_{c}(X)\right)\bar{R}^{(c)}_{m}K^{(c)}\\\
&=\left(K(q_{c}^{-\frac{1}{2}}X)\otimes\varphi^{+}_{c}(X)\right)\bar{R}^{(c)}_{m}K^{(c)}\\\
&=\Delta^{\text{op}}(K^{+}(X))\bar{R}^{(c)}_{m}K^{(c)},\end{split}$ (257)

where we used (236) for the second equality and the normal-ordering relation

$\displaystyle\begin{split}\varphi^{+}_{c}(X)\xi(X^{\prime})&=\exp\left[-\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\left(\frac{q_{c}^{\frac{1}{2}}X^{\prime}}{X}\right)^{r}\right]:\varphi^{+}_{c}(X)\xi(X^{\prime}):\\\
&=\prod_{a=1}^{3}\frac{1-q_{a}q_{c}^{\frac{1}{2}}\frac{X^{\prime}}{X}}{1-q_{a}^{-1}q_{c}^{\frac{1}{2}}\frac{X^{\prime}}{X}}:\varphi^{+}_{c}(X)\xi(X^{\prime}):,\end{split}$
(258)

in the fourth equality.

In a similar way, we can show that the following relation holds for each
$m\in\mathbb{Z}_{\geq 0}$:

$\displaystyle\bar{R}^{(c)}_{m}K^{(c)}\Delta(K^{-}(X))=\Delta^{\text{op}}(K^{-}(X))\bar{R}^{(c)}_{m}K^{(c)}.$
(259)

A straightforward computation gives

$\displaystyle\begin{split}&\bar{R}^{(c)}_{m}K^{(c)}\Delta(K^{-}(X))=\bar{R}^{(c)}_{m}K^{(c)}\left(K^{-}(q_{c}^{-\frac{1}{2}}X)\otimes\varphi^{-}_{c}(X)\right)\\\
&=\frac{v^{-m}(1-q_{c}^{-1})^{m}}{\kappa_{1}^{m}(1-q_{3})^{m}}\xi_{c}(X_{1})\cdots\xi_{c}(q_{3}^{-m+1}X_{1})(Z_{1}^{m}\otimes
1)\left(K^{-}(X)\otimes\varphi^{-}_{c}(X)\right)K^{(c)}\\\
&=\exp\left[-\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{1-q_{3}^{mr}}{1-q_{3}^{r}}\left(\frac{X}{X_{1}}\right)^{r}\right]\left(K^{-}(q_{3}^{m}X)\otimes\varphi^{-}_{c}(X)\right)\bar{R}^{(c)}_{m}K^{(c)}\\\
&=\left(K^{-}(X)\otimes\varphi^{-}_{c}(X)\right)\bar{R}^{(c)}_{m}K^{(c)}\\\
&=\Delta^{\text{op}}(K^{-}(X))\bar{R}^{(c)}_{m}K^{(c)},\end{split}$ (260)

where we used (236) and the normal-ordering relation,

$\displaystyle\begin{split}V_{c}(X_{1})\varphi^{-}(X)&=\exp\left[-\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{1-q_{c}^{-r}}{1-q_{3}^{r}}\left(\frac{q_{c}^{\frac{1}{2}}X}{X_{1}}\right)^{r}\right]:V_{c}(X_{1})\varphi^{-}(X):\\\
&=K^{-}(q_{c}^{\frac{1}{2}}X)K^{-}(q_{c}^{-\frac{1}{2}}X)^{-1}:V_{c}(X_{1})\varphi^{-}(X):,\end{split}$
(261)

for the second equality; and the normal-ordering relation

$\displaystyle\begin{split}\xi_{c}(X)\varphi^{-}_{c}(X^{\prime})&=\exp\left[-\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\left(\frac{X^{\prime}}{X}\right)^{r}\right]:\xi_{c}(X)\varphi^{-}_{c}(X^{\prime}):\\\
&=\prod_{a=1}^{3}\frac{1-\frac{q_{c}q_{a}X}{X^{\prime}}}{1-\frac{q_{c}q_{a}^{-1}X}{X^{\prime}}}:\xi_{c}(X)\varphi^{-}_{c}(X^{\prime}):,\end{split}$
(262)

for the third equality.

Next, we turn to the relation

$\displaystyle R^{(c)}\Delta(E(X))=\Delta^{\text{op}}(E(X))R^{(c)}.$ (263)

Recall that $R^{(c)}$ is expanded with respect to the $d^{\perp}$-grading, so
that the relation has to be satisfied for each degree of the grading. The
degree-$m$ relation is found to be

$\displaystyle\begin{split}&\bar{R}^{(c)}_{m}K^{(c)}\left(E(X)\otimes
1\right)+\bar{R}^{(c)}_{m+1}K^{(c)}\left(K^{-}(X)\otimes\left(-v\frac{1-q_{c}}{\kappa_{1}}\right)\eta_{c}(X)\right)\\\
&\qquad=\left(1\otimes\left(-v\frac{1-q_{c}}{\kappa_{1}}\right)\eta_{c}(X)\right)\bar{R}^{(c)}_{m+1}K^{(c)}+\left(E(q_{c}^{\frac{1}{2}}X)\otimes\varphi^{-}_{c}(q_{c}^{\frac{1}{2}}X)\right)\bar{R}^{(c)}_{m}K^{(c)}.\end{split}$
(264)

Note that there is a recursive relation

$\displaystyle\bar{R}^{(c)}_{m}=\frac{\alpha_{m}}{\alpha_{m-1}}\frac{v^{-1}(1-q_{c}^{-1})}{\kappa_{1}(1-q_{3})}\xi_{c}(X_{1})Z_{1}\bar{R}^{(c)}_{m-1}.$
(265)

Let us assume $\gamma\equiv\frac{\alpha_{m}}{\alpha_{m-1}}$, and look for its
value in which (264) holds. We proceed by induction. Assume (264) holds for
$0\leq m\leq m^{\prime}-1$. Then, for $m=m^{\prime}$, the left hand side of
(264) gives

$\displaystyle\begin{split}&\gamma\frac{v^{-1}(1-q_{c}^{-1})}{\kappa_{1}(1-q_{3})}\xi_{c}(X_{1})Z_{1}\\\
&\times\left[\left(1\otimes\left(-v\frac{1-q_{c}}{\kappa_{1}}\right)\eta_{c}(X)\right)\bar{R}^{(c)}_{m^{\prime}}K^{(c)}+\left(E(q_{c}^{\frac{1}{2}}X)\otimes\varphi^{-}_{c}(q_{c}^{\frac{1}{2}}X)\right)\bar{R}^{(c)}_{m^{\prime}-1}K^{(c)}\right].\end{split}$
(266)

The second term in the square bracket vanishes, due to the normal-ordering
relation (262) and the delta function in
$\rho_{\text{M2}_{0,0,1}}(E(q_{c}^{\frac{1}{2}}q_{3}X))$. The remaining term
is

$\displaystyle\begin{split}&-\gamma\frac{(1-q_{c}^{-1})(1-q_{c})}{\kappa_{1}^{2}(1-q_{3})}\xi_{c}(X_{1})\eta_{c}(X)Z_{1}\bar{R}^{(c)}_{m^{\prime}}K^{(c)}\\\
&=-\gamma\frac{(1-q_{c}^{-1})(1-q_{c})}{\kappa_{1}^{2}(1-q_{3})}\left[\frac{(1-q_{c+1})(1-q_{c-1})}{1-q_{c}^{-1}}\left\\{\delta\left(\frac{q_{c}^{\frac{1}{2}}X}{X_{1}}\right)\varphi^{-}_{c}(X_{1})-\delta\left(\frac{q_{c}^{\frac{1}{2}}X_{1}}{X}\right)\varphi^{+}_{c}(X)\right\\}\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\eta_{c}(X)\xi_{c}(X_{1})\Bigg{]}Z_{1}\bar{R}^{(c)}_{m^{\prime}}K^{(c)}\\\
&=\left(1\otimes\left(-v\frac{1-q_{c}}{\kappa_{1}}\right)\eta_{c}(X)\right)\bar{R}^{(c)}_{m^{\prime}+1}K^{(c)}\\\
&\qquad+\frac{\gamma}{\kappa_{1}}\left(E(q_{c}^{\frac{1}{2}}X)\otimes\varphi^{-}(q_{c}^{\frac{1}{2}}X)-E(q_{c}^{-\frac{1}{2}}X)\otimes\varphi_{c}^{+}(X)\right)\bar{R}^{(c)}_{m^{\prime}}K^{(c)},\end{split}$
(267)

where we used the commutation relation

$\displaystyle[\xi_{c}(X_{1}),\eta_{c}(X)]=\frac{(1-q_{c+1})(1-q_{c-1})}{1-q_{c}^{-1}}\left\\{\delta\left(\frac{q_{c}^{\frac{1}{2}}X}{X_{1}}\right)\varphi^{-}_{c}(X_{1})-\delta\left(\frac{q_{c}^{\frac{1}{2}}X_{1}}{X}\right)\varphi^{+}_{c}(X)\right\\}$
(268)

for the first equality. The second term in the last line of (267) vanishes,
due to the normal-ordering relation (258) and
$\rho_{\text{M2}_{0,0,1}}(E(q_{c}^{-\frac{1}{2}}X))=\frac{1}{1-q_{3}}\delta(q_{c}^{\frac{1}{2}}X_{1}/X)Z_{1}$.
Once we pass $Z_{1}$ to the right, the numerator of (258) vanishes due to the
delta function. All in all, we have

$\displaystyle\begin{split}&\bar{R}^{(c)}_{m^{\prime}}K^{(c)}\left(E(X)\otimes
1\right)+\bar{R}^{(c)}_{m^{\prime}+1}K^{(c)}\left(K^{-}(X)\otimes\left(-v\frac{1-q_{c}}{\kappa_{1}}\right)\eta_{c}(X)\right)\\\
&=\left(1\otimes\left(-v\frac{1-q_{c}}{\kappa_{1}}\right)\eta_{c}(X)\right)\bar{R}^{(c)}_{m^{\prime}+1}K^{(c)}+\frac{\gamma}{\kappa_{1}}\left(E(q_{c}^{\frac{1}{2}}X)\otimes\varphi^{-}(q_{c}^{\frac{1}{2}}X)\right)\bar{R}^{(c)}_{m^{\prime}}K^{(c)}.\end{split}$
(269)

The desired relation (264) is exactly recovered if and only if
$\gamma=\kappa_{1}$.

Finally, we check the relation

$\displaystyle R^{(c)}\Delta(F(X))=\Delta^{\text{op}}(F(X))R^{(c)}.$ (270)

The degree-$m$ relation reads

$\displaystyle\begin{split}&\bar{R}^{(c)}_{m}K^{(c)}\left(1\otimes
v^{-1}\frac{1-q_{c}^{-1}}{\kappa_{1}}\xi_{c}(X)\right)+\bar{R}^{(c)}_{m+1}K^{(c)}\left(F(q_{c}^{\frac{1}{2}}X)\otimes\varphi^{+}_{c}(q_{c}^{\frac{1}{2}}X)\right)\\\
&\quad=(F(X)\otimes 1)\bar{R}^{(c)}_{m+1}K^{(c)}+\left(K^{+}(X)\otimes
v^{-1}\frac{1-q_{c}^{-1}}{\kappa_{1}}\xi_{c}(X)\right)\bar{R}^{(c)}_{m}K^{(c)}.\end{split}$
(271)

By the induction hypothesis, the left hand side gives

$\displaystyle\begin{split}&\gamma\frac{v^{-1}(1-q_{c}^{-1})}{\kappa_{1}(1-q_{3})}\xi_{c}(X_{1})Z_{1}\\\
&\quad\times\left[(F(X)\otimes
1)\bar{R}^{(c)}_{m}K^{(c)}+\left(K^{+}(X)\otimes
v^{-1}\frac{1-q_{c}^{-1}}{\kappa_{1}}\xi_{c}(X)\right)\bar{R}^{(c)}_{m-1}K^{(c)}\right].\end{split}$
(272)

The first term in the square bracket gives zero, due to the normal-ordering
relation (86). The remaining term is

$\displaystyle\begin{split}&\gamma\frac{v^{-2}(1-q_{c}^{-1})^{2}}{\kappa_{1}^{2}(1-q_{3})}\xi_{c}(X_{1})\left(K^{+}(q_{3}X)\otimes\xi_{c}(X)\right)Z_{1}\bar{R}^{(c)}_{m-1}K^{(c)}\\\
&=\frac{\left(1-\frac{X}{X_{1}}\right)\left(1-\frac{q_{c}X}{X_{1}}\right)}{\left(1-\frac{q_{c+1}^{-1}X}{X_{1}}\right)\left(1-\frac{q_{c-1}^{-1}X}{X_{1}}\right)}\frac{\left(1-\frac{q_{c+1}^{-1}X_{1}}{X}\right)\left(1-\frac{q_{c-1}^{-1}X_{1}}{X}\right)}{\left(1-\frac{X_{1}}{X}\right)\left(1-\frac{q_{c}X_{1}}{X}\right)}(K^{+}(q_{3}X)\otimes
F(X))\bar{R}^{(c)}_{m}K^{(c)}\\\ &=(K^{+}(X)\otimes
F(X))\bar{R}^{(c)}_{m}K^{(c)}\\\
&\qquad-\left[\frac{(1-q_{c+1}^{-1})(1-q_{c-1})}{1-q_{c-1}q_{c+1}^{-1}}\delta\left(\frac{q_{c+1}X_{1}}{X}\right)+\frac{(1-q_{c-1}^{-1})(1-q_{c+1})}{1-q_{c+1}q_{c-1}^{-1}}\delta\left(\frac{q_{c-1}X_{1}}{X}\right)\right]\\\
&\qquad\qquad\times\frac{\left(1-\frac{q_{c+1}^{-1}X_{1}}{X}\right)\left(1-\frac{q_{c-1}^{-1}X_{1}}{X}\right)}{\left(1-\frac{X_{1}}{X}\right)\left(1-\frac{q_{c}X_{1}}{X}\right)}\frac{\left(1-\frac{q_{1}X_{1}}{X}\right)\left(1-\frac{q_{2}X_{1}}{X}\right)}{\left(1-\frac{X_{1}}{X}\right)\left(1-\frac{q_{3}^{-1}X_{1}}{X}\right)}(1\otimes
F(X))\bar{R}^{(c)}_{m}K^{(c)},\end{split}$ (273)

where we used the normal-ordering relation (86) for the first equality and the
relation

$\displaystyle
f(X)_{+}-f(X)_{-}=\sum_{\alpha}\underset{X=X^{(\alpha)}}{\text{Res}}\left(\frac{f(X)}{X}\right)\delta\left(\frac{X^{(\alpha)}}{X}\right)$
(274)

for the second equality. Here, $f(X)$ is a rational function with simple poles
at $X=X^{(\alpha)}$, and $f(X)_{+}$ (resp. $f(X)_{-}$) is the expansion of
$f(X)$ in the domain $|X|>\\!\\!>1$ (resp. $|X|<\\!\\!<1$). Due to the delta
functions, only one of the two terms in the square bracket survives, yielding

$\displaystyle\begin{split}&\frac{1}{1-q_{3}^{-1}}\delta\left(\frac{q_{3}X_{1}}{X}\right)\frac{v^{-1}(1-q_{c}^{-1})}{1-q_{3}}\xi_{c}(X)\bar{R}^{(c)}_{m}K^{(c)}\\\
&\qquad=\frac{1}{1-q_{3}^{-1}}Z_{1}^{-1}\delta\left(\frac{X_{1}}{X}\right)\frac{v^{-1}(1-q_{c}^{-1})}{1-q_{3}}\xi_{c}(X_{1})Z_{1}\bar{R}_{m}^{(c)}K^{(c)}\\\
&\qquad=\frac{\kappa_{1}}{\gamma}(F(X)\otimes
1)\bar{R}_{m+1}^{(c)}K^{(c)}.\end{split}$ (275)

Therefore, we arrive at

$\displaystyle\begin{split}&\bar{R}^{(c)}_{m}K^{(c)}\left(1\otimes
v^{-1}\frac{1-q_{c}^{-1}}{\kappa_{1}}\xi_{c}(X)\right)+\bar{R}^{(c)}_{m+1}K^{(c)}\left(F(q_{c}^{\frac{1}{2}}X)\otimes\varphi^{+}_{c}(q_{c}^{\frac{1}{2}}X)\right)\\\
&\quad=\frac{\kappa_{1}}{\gamma}(F(X)\otimes
1)\bar{R}^{(c)}_{m+1}K^{(c)}+\left(K^{+}(X)\otimes
v^{-1}\frac{1-q_{c}^{-1}}{\kappa_{1}}\xi_{c}(X)\right)\bar{R}^{(c)}_{m}K^{(c)}.\end{split}$
(276)

Again, the desired relation (271) is precisely recovered if and only if
$\gamma=\kappa_{1}$. By choosing the overall normalization by $\alpha_{0}=1$,
we therefore obtain the solution $\alpha_{m}=\kappa_{1}^{m}$,
$m\in\mathbb{Z}_{\geq 0}$. This completes the derivation of the R-matrix
between $\text{M2}_{0,0,1}$ and $\text{M5}_{c}^{\mathbf{(1,0)}}$.

If $c=3$, the derivation could have been simplified in fact, since the ansatz
(255) truncates to the first two terms as a result of the normal ordering. In
any case, the derivation we presented applies to all the cases
$c\in\\{1,2,3\\}$.

##  Appendix D Maximality of set of screening charges

In this appendix, we sketch a proof of the following proposition: the only
operators in the tensor product of two $q$-bosons which commute with the
generators produced by the Miura transformation are the screening operators of
section 3.3.3.

Let us first consider a single $\text{M2}_{0,0,1}$ brane, with support on
$\mathbb{R}^{2}_{{\varepsilon}_{3}}\times\mathbb{R}_{t}$. By definition, the
Miura transformation is a product of Miura operators, which in the light of
our results is a product of R-matrices:

${R}^{(c_{L+M+N})}_{L+M+N}\,\cdots\,{R}^{(c_{2})}_{2}\,{R}^{(c_{1})}_{1}\;,\qquad\qquad\qquad\;R^{(c_{I})}_{I}=\bar{R}^{(c_{I})}_{I}K^{(c_{I})}_{I},\;\;c\in\\{1,2,3\\}$
(277)

To make this appendix self-contained, we rewrite the operator definitions
here. The Cartan part reads

$\displaystyle K^{(c_{I})}_{I}=\left(q_{c_{I}}^{\frac{d}{2}}\otimes
1\right)V^{(c_{I})}_{I}(X)=\left(q_{c_{I}}^{\frac{d}{2}}\otimes
1\right)\exp\left[\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{X^{-r}}{1-q_{3}^{r}}\otimes\frac{a_{r}^{[I]}}{q_{c_{I}}^{r/2}-q_{c}^{-r/2}}\right]$
(278)

with
$V^{(c_{I})}_{I}(X):=\exp\left[\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{X^{-r}}{1-q_{3}^{r}}\otimes\frac{a_{r}^{[I]}}{q_{c_{I}}^{r/2}-q_{c_{I}}^{-r/2}}\right]$,
while the main part of the R-matrix reads

$\displaystyle\bar{R}^{(c_{I})}_{I}=\sum_{m=0}^{\infty}q_{\bar{c_{I}}}^{m}e^{-ma_{0}^{[I]}\frac{\log
q_{c_{I}+1}\log q_{c_{I}-1}}{\log
q_{c_{I}}}}\left(\prod_{j=0}^{m-1}\frac{1-q_{c_{I}}q_{3}^{-j}}{1-q_{3}^{-j-1}}\right):\prod_{j=0}^{m-1}\xi^{(c_{I})}_{I}(q_{3}^{-j}X):q_{3}^{-mD_{X}}.$
(279)

with vertex operator

$\displaystyle\xi^{(c_{I})}_{I}(X)=\exp\left(\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{q_{c}^{r/2}}{(q_{c_{I}}^{r/2}-q_{c_{I}}^{-r/2})^{2}}a_{-r}^{[I]}X^{r}\right)\exp\left(\sum_{r=1}^{\infty}\frac{\kappa_{r}}{r}\frac{1}{(q_{c_{I}}^{r/2}-q_{c_{I}}^{-r/2})^{2}}a_{r}^{[I]}X^{-r}\right)\;.$
(280)

The main property of the Miura transformation is that it is a generating
function of the current generators for the $q$-boson algebra. The expansion
parameter of this generating function is the shift operator $q_{3}^{-D_{X}}$.
The screening charges of the algebra are defined as the commutant of these
currents.

Then, let us consider a general ansatz for an endomorphism of the tensor
product of two Fock spaces
$\text{End}\left(\mathcal{F}_{c_{1}}^{\mathbf{(1,0)}}\otimes\mathcal{F}_{c_{2}}^{\mathbf{(1,0)}}\right)$:

$\displaystyle\begin{split}{\mathcal{S}}^{(3)}(Y)&=e^{{\widetilde{\gamma}}^{[1]}_{0}\,d^{\perp}\otimes
1+{\widetilde{\gamma}}^{[2]}_{0}\,1\otimes
d^{\perp}}\;{Y}^{\gamma^{[1]}_{0}\,a^{[1]}_{0}\otimes
1+\gamma^{[2]}_{0}\,1\otimes a^{[2]}_{0}+\gamma^{[0]}_{0}1\otimes 1}\\\
&\qquad\times\exp\left(\sum_{r>0}\frac{Y^{r}}{r}(\gamma^{[1]}_{-r}\,a^{[1]}_{-r}\otimes
1+\gamma^{[2]}_{-r}\,1\otimes a^{[2]}_{-r})\right)\\\
&\qquad\times\exp\left(-\sum_{r>0}\frac{Y^{-r}}{r}(\gamma^{[1]}_{+r}\,a^{[1]}_{+r}\otimes
1+\gamma^{[2]}_{+r}\,1\otimes a^{[2]}_{+r}\right)\;,\end{split}$ (281)

with five zero mode unknowns
$\gamma^{[0]}_{0},\gamma^{[1]}_{0},\gamma^{[2]}_{0}$, and
$\widetilde{\gamma}^{[1]}_{0},\widetilde{\gamma}^{[2]}_{0}$, as well as an
infinite number of nonzero mode unknowns: $\gamma^{[1]}_{\pm
r},\gamma^{[2]}_{\pm r}$ for $r>0$.

Our strategy will be to investigate
$[{R}^{(c_{2})}_{2}\,{R}^{(c_{1})}_{1}\;,\;{\mathcal{S}}^{(3)}(Y)]$ at each
order $m$ in the shift operator $q_{3}^{-mD_{X}}$, and constrain the unknowns
in the ansatz current ${\mathcal{S}}^{(3)}(Y)$ for the commutator to be a
$q$-derivative. Note that it suffices to consider a Miura transformation for
the product of two R-matrices ${R}^{(c_{2})}_{2}\,{R}^{(c_{1})}_{1}$, since
the ansatz is defined on the tensor product of precisely two Fock spaces.

At order $m=0$, the commutator takes the form

$[V^{(c_{2})}_{2}(X\,q^{1/2}_{c_{2}})V^{(c_{1})}_{1}(X\,q^{1/2}_{c_{1}}q^{1/2}_{c_{2}})\;,{\mathcal{S}}^{(3)}(Y)]$
(282)

Imposing its vanishing (up to $q$-derivatives), we find constraints:

$\displaystyle\begin{split}&\bullet\text{For}\;c_{1}=c_{2},\;\;\gamma^{[1]}_{-r}=-q^{r/2}_{c_{1}}\,\gamma^{[2]}_{-r}\;,\\\\[8.53581pt]
&\bullet\text{For}\;c_{1}\neq
c_{2},\;\;\gamma^{[1]}_{-r}=-\left(\frac{q^{r/2}_{c_{2}}-q^{-r/2}_{c_{2}}}{q^{r/2}_{c_{1}}-q^{-r/2}_{c_{1}}}\right)^{2}\,q^{r/2}_{c_{1}}\,\gamma^{[2]}_{-r}\;.\end{split}$
(283)

Note that at this point, $c_{I}\in\\{1,2,3\\}$. At order $m=1$, the commutator
is more involved:

$\displaystyle\begin{split}&\left[q_{\bar{c_{2}}}\frac{1-q_{c_{2}}}{1-q^{-1}_{3}}\,\xi^{(c_{2})}_{2}(X)V^{(c_{2})}_{2}(X\,q^{-1}_{3}q^{1/2}_{c_{2}})V^{(c_{1})}_{1}(X\,q^{-1}_{3}q^{1/2}_{c_{1}}q^{1/2}_{c_{2}})\,e^{a^{[2]}_{0}\frac{\log
q_{3}\,\log q_{\bar{c_{2}}}}{\log q_{c_{2}}}}\right.\\\
&\;\;\;+\left.q_{\bar{c_{1}}}\frac{1-q_{c_{1}}}{1-q^{-1}_{3}}\,V^{(c_{2})}_{2}(X\,q^{1/2}_{c_{2}})\xi^{(c_{1})}_{1}(X\,q^{1/2}_{c_{1}})V^{(c_{1})}_{1}(X\,q^{1/2}_{c_{1}}q^{1/2}_{c_{2}})\,e^{a^{[1]}_{0}\frac{\log
q_{3}\,\log q_{\bar{c_{1}}}}{\log
q_{c_{1}}}}\;,\;{\mathcal{S}}^{(3)}(Y)\right]\end{split}$ (284)

After normal-ordering the currents and making use of the constraints (283) at
order $m=0$, we find that all the unknowns are determined: a necessary
condition for the commutator to be null or proportional to a $q$-derivative is
that $c_{I}\neq 3$, so $c_{I}\in\\{1,2\\}$. Another necessary condition is
that ${\mathcal{S}}^{(3)}$ must be one of the four screening currents
${\mathcal{S}}^{(3)}_{1,1}$, ${\mathcal{S}}^{(3)}_{2,2}$,
${\mathcal{S}}^{(3)}_{1,2}$, or ${\mathcal{S}}^{(3)}_{2,1}$ written in (121)
and (124). The first two currents are the “bosonic” screening endomorphisms,
on the tensor product of two identical Fock spaces, while the latter two
currents are the “fermionic” screening endomorphisms, on the tensor product of
two distinct Fock spaces. In more detail, we find that for these four
solutions, the commutator is proportional to a $q_{3}$-derivative,

${\cal
D}^{Y}_{q_{3}}\left[Y\,\delta\left(q^{j_{c_{1}}}_{1}\,q^{j_{c_{2}}}_{2}Y/X\right)\;:\ldots:\;\right]\;,$
(285)

where the derivative operator is defined as ${\cal
D}^{Y}_{q_{c}}(f(Y))=\dfrac{f(Y)-f(q_{c}\,Y)}{Y}$ for any function $f$. The
expression $:\ldots:$ stands for the normal ordering of the product of all
operators, and $j_{c_{I}}$ is a half-integer which is uniquely fixed by the
computation, for each $c_{I}\in\\{1,2\\}$.

There are again at most four solutions if the initial brane is
$\text{M2}_{1,0,0}$, supported on
$\mathbb{R}^{2}_{{\varepsilon}_{1}}\times\mathbb{R}_{t}$, and four solutions
if the initial brane is $\text{M2}_{0,1,0}$, supported on
$\mathbb{R}^{2}_{{\varepsilon}_{2}}\times\mathbb{R}_{t}$. These solutions are
readily obtained by simply permuting the parameters $(q_{1},q_{2},q_{3})$,
thanks to the triality of the quantum toroidal algebra. This is precisely the
exhaustive list of screenings written in (121) and (124), for any
$c\in\\{1,2,3\\}$. This completes the proof of the proposition.

##### Remark

The work Harada:2021xnm  defines the Miura transformation slightly
differently, as

$\bar{R}^{(c_{L+M+N})}_{L+M+N}(X)\,\cdots\,\bar{R}^{(c_{2})}_{2}(X)\,\bar{R}^{(c_{1})}_{1}(X)=\sum_{m=0}^{\infty}(-1)^{m}\,T_{m}(X)\,q^{-mD_{X}}_{3}\;,$
(286)

where $T_{m}(X)$ is the $m$-th generating current of the $qY_{L,M,N}$ algebra,
expressed as a finite series in the operators (118). In particular, the
authors find that the screening charges of the $qY_{L,M,N}$ algebra are in the
commutant of the operators $T_{m}(X)$. In fact, it is equivalent to show that
the screening charges commute with the product of two operators, say
$\bar{R}^{(c_{2})}_{2}\,\bar{R}^{(c_{1})}_{1}$. In our work, we found that the
_same_ screening charges are also in the commutant of the Miura operators
${R}^{(c_{2})}_{2}\,{R}^{(c_{1})}_{1}$, with
$R^{(c_{I})}_{I}=\bar{R}^{(c_{I})}_{I}K^{(c_{I})}_{I}$. The advantage of using
the Miura operators $R^{(c_{I})}_{I}$ is that they have a direct
interpretation as R-matrices of the quantum toroidal algebra, unlike the
operators $\bar{R}^{(c_{I})}_{I}$.

##  Appendix E $q$-boson intertwiners for non-transverse M5-brane
intersections

The intertwiner $\Psi^{(\mathbf{n})}(v,v^{\prime})$ satisfying (210) can be
expanded in the dual basis of $\mathcal{F}^{\mathbf{(0,1)}}_{3}$ as,

$\displaystyle\Psi^{(\mathbf{n})}(v,v^{\prime})=\sum_{\\{\lambda\\}}\Xi_{\lambda}\Psi^{\mathbf{(n)}}_{\lambda}(v,v^{\prime})\langle
v^{\prime},\lambda|$ (287)

where the coefficients are given by

$\displaystyle\Psi^{\mathbf{(n)}}_{\lambda}(v,v^{\prime})=t_{\mathbf{n}}(\lambda,v,v^{\prime}):\Psi_{\varnothing}(v^{\prime})\prod_{\Box\in\lambda}\eta(\chi_{\Box}):,$
(288)

with

$\displaystyle\begin{split}&t_{\mathbf{n}}(\lambda,v,v^{\prime})=(-vv^{\prime})^{|\lambda|}\prod_{\Box\in\lambda}\left(q_{3}^{1/2}/\chi_{\Box}\right)^{\mathbf{n}+1},\qquad\chi_{\Box_{(i,j)}}=v^{\prime}q_{1}^{i-1}q_{2}^{j-1},\\\
&\Psi_{\varnothing}(v^{\prime})=\exp\left[\sum_{r=1}^{\infty}\frac{1}{r}\frac{1}{1-q_{3}^{-r}}a_{-r}^{(3)}v^{\prime
r}\right]\exp\left[-\sum_{r=1}^{\infty}\frac{1}{r}\frac{q_{3}^{-r/2}}{1-q_{3}^{-r}}a_{r}^{(3)}v^{\prime-r}\right].\end{split}$
(289)

Here, we used the normalization of the dual states such that (our case is
$c=3$)

$\displaystyle{}^{(c)}\langle
u,\lambda|u,\mu\rangle^{(c)}=\left(\Xi_{\lambda}^{(c)}\right)^{-1}\delta_{\lambda,\mu},$
(290)

where we defined

$\displaystyle\Xi_{\lambda}^{(c)}=\frac{(uq_{c}^{1/2})^{-|\lambda|}\prod_{\Box\in\lambda}\chi_{\Box}}{N^{(c)}_{\lambda\lambda}(1;q_{1},q_{2})},$
(291)

with

$\displaystyle
N^{(c)}_{\lambda\mu}(u_{1}/u_{2};q_{1};q_{2})=\prod_{\Box\in\lambda}\left(1-\frac{\chi_{\Box}}{q_{c}u_{2}}\right)\prod_{\blacksquare\in\mu}\left(1-\frac{u_{1}}{\chi_{\blacksquare}}\right)\prod_{\begin{subarray}{c}\Box\in\lambda\\\
\blacksquare\in\mu\end{subarray}}\frac{\left(1-\frac{q_{c+1}\chi_{\square}}{\chi_{\blacksquare}}\right)\left(1-\frac{q_{c-1}\chi_{\square}}{\chi_{\blacksquare}}\right)}{\left(1-\frac{\chi_{\square}}{\chi_{\blacksquare}}\right)\left(1-\frac{q_{c}^{-1}\chi_{\square}}{\chi_{\blacksquare}}\right)}.$
(292)

The intertwiner $\Psi^{*\mathbf{(n)}}(v,v^{\prime})$ satisfying (212) can be
expanded in the basis of $\mathcal{F}^{\mathbf{(0,1)}}_{3}$ as

$\displaystyle\Psi^{*\mathbf{(n)}}(v,v^{\prime})=\sum_{\\{\lambda\\}}\Xi_{\lambda}|a,\lambda\rangle\Psi^{*\mathbf{(n)}}_{\lambda}(v,v^{\prime})$
(293)

where

$\displaystyle\Psi^{*\mathbf{(n)}}_{\lambda}(v,v^{\prime})=t_{\mathbf{n}}^{*}(\lambda,v,v^{\prime}):\Psi^{*}_{\varnothing}(v^{\prime})\prod_{\Box\in\lambda}\xi(\chi_{\Box}):,$
(294)

with

$\displaystyle\begin{split}&t_{\mathbf{n}}^{*}(\lambda,v,v^{\prime})=(vq_{3}^{1/2})^{-|\lambda|}\prod_{\Box\in\lambda}(q_{3}^{-1/2}\chi_{\Box})^{\mathbf{n}},\\\
&\Psi^{*}_{\varnothing}(v^{\prime})=\exp\left[-\sum_{r=1}^{\infty}\frac{1}{r}\frac{q_{3}^{r/2}}{1-q_{3}^{-r}}a_{-r}^{(3)}v^{\prime
r}\right]\exp\left[\sum_{r=1}^{\infty}\frac{1}{r}\frac{1}{1-q_{3}^{r}}a_{r}^{(3)}v^{\prime-r}\right].\end{split}$
(295)

## References

  * (1) L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. 91 (2010) 167–197, arXiv:0906.3219 [hep-th].
  * (2) N. Wyllard, “A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories,” JHEP 11 (2009) 002, arXiv:0907.2189 [hep-th].
  * (3) J. Yagi, “Compactification on the \Omega-background and the AGT correspondence,” JHEP 09 (2012) 101, arXiv:1205.6820 [hep-th].
  * (4) C. Beem, L. Rastelli, and B. C. van Rees, “$\mathcal{W}$ symmetry in six dimensions,” JHEP 05 (2015) 017, arXiv:1404.1079 [hep-th].
  * (5) K. Costello, “M-theory in the Omega-background and 5-dimensional non-commutative gauge theory,” arXiv:1610.04144 [hep-th].
  * (6) D. Gaiotto and M. Rapčák, “Vertex Algebras at the Corner,” JHEP 01 (2019) 160, arXiv:1703.00982 [hep-th].
  * (7) D. Gaiotto and M. Rapcak, “Miura operators, degenerate fields and the M2-M5 intersection,” JHEP 01 (2022) 086, arXiv:2012.04118 [hep-th].
  * (8) T. Procházka and M. Rapčák, “$\mathcal{W}$ -algebra modules, free fields, and Gukov-Witten defects,” JHEP 05 (2019) 159, arXiv:1808.08837 [hep-th].
  * (9) K. Harada, Y. Matsuo, G. Noshita, and A. Watanabe, “$q$-deformation of corner vertex operator algebras by Miura transformation,” JHEP 04 (2021) 202, arXiv:2101.03953 [hep-th].
  * (10) K. Costello, “Holography and Koszul duality: the example of the $M2$ brane,” arXiv:1705.02500 [hep-th].
  * (11) D. Gaiotto and J. Oh, “Aspects of $\Omega$-deformed M-theory,” arXiv:1907.06495 [hep-th].
  * (12) D. Gaiotto and J. Abajian, “Twisted M2 brane holography and sphere correlation functions,” arXiv:2004.13810 [hep-th].
  * (13) N. Seiberg, “New theories in six-dimensions and matrix description of M theory on T**5 and T**5 / Z(2),” Phys. Lett. B 408 (1997) 98–104, arXiv:hep-th/9705221.
  * (14) A. Losev, G. W. Moore, and S. L. Shatashvili, “M & m’s,” Nucl. Phys. B 522 (1998) 105–124, arXiv:hep-th/9707250.
  * (15) M. Aganagic and N. Haouzi, “ADE Little String Theory on a Riemann Surface (and Triality),” arXiv:1506.04183 [hep-th].
  * (16) N. Haouzi and C. Kozçaz, “The ABCDEFG of little strings,” JHEP 06 (2021) 092, arXiv:1711.11065 [hep-th].
  * (17) N. Ishtiaque, S. Jeong, and Y. Zhou, “upcoming paper,”.
  * (18) N. C. Leung and C. Vafa, “Branes and toric geometry,” Adv. Theor. Math. Phys. 2 (1998) 91–118, arXiv:hep-th/9711013.
  * (19) E. Witten, “Analytic Continuation Of Chern-Simons Theory,” AMS/IP Stud. Adv. Math. 50 (2011) 347–446, arXiv:1001.2933 [hep-th].
  * (20) E. Witten, “A New Look At The Path Integral Of Quantum Mechanics,” arXiv:1009.6032 [hep-th].
  * (21) H. Awata, B. Feigin, and J. Shiraishi, “Quantum Algebraic Approach to Refined Topological Vertex,” JHEP 03 (2012) 041, arXiv:1112.6074 [hep-th].
  * (22) A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nucl. Phys. B 492 (1997) 152–190, arXiv:hep-th/9611230.
  * (23) M. Aganagic, N. Haouzi, C. Kozcaz, and S. Shakirov, “Gauge/Liouville Triality,” arXiv:1309.1687 [hep-th].
  * (24) M. Aganagic, N. Haouzi, and S. Shakirov, “$A_{n}$-Triality,” arXiv:1403.3657 [hep-th].
  * (25) A. Mironov, A. Morozov, and Y. Zenkevich, “Ding–Iohara–Miki symmetry of network matrix models,” Phys. Lett. B762 (2016) 196–208, arXiv:1603.05467 [hep-th].
  * (26) Y. Zenkevich, “Higgsed network calculus,” JHEP 08 (2021) 149, arXiv:1812.11961 [hep-th].
  * (27) Y. Zenkevich, “Mixed network calculus,” JHEP 12 (2021) 027, arXiv:2012.15563 [hep-th].
  * (28) T. Dimofte, D. Gaiotto, and N. M. Paquette, “Dual boundary conditions in 3d SCFT’s,” JHEP 05 (2018) 060, arXiv:1712.07654 [hep-th].
  * (29) Y. Yoshida and K. Sugiyama, “Localization of three-dimensional $\mathcal{N}=2$ supersymmetric theories on $S^{1}\times D^{2}$,” PTEP 2020 no. 11, (2020) 113B02, arXiv:1409.6713 [hep-th].
  * (30) A. Gadde, S. Gukov, and P. Putrov, “Walls, Lines, and Spectral Dualities in 3d Gauge Theories,” JHEP 05 (2014) 047, arXiv:1302.0015 [hep-th].
  * (31) A. Gadde, S. Gukov, and P. Putrov, “Fivebranes and 4-manifolds,” Prog. Math. 319 (2016) 155–245, arXiv:1306.4320 [hep-th].
  * (32) I. Ciocan-Fontanine and B. Kim, “Moduli stacks of stable toric quasimaps,” arXiv e-prints (Aug., 2009) arXiv:0908.4446, arXiv:0908.4446 [math.AG].
  * (33) B. Kim, “Stable quasimaps to holomorphic symplectic quotients,” arXiv e-prints (May, 2010) arXiv:1005.4125, arXiv:1005.4125 [math.AG].
  * (34) I. Ciocan-Fontanine, B. Kim, and D. Maulik, “Stable quasimaps to GIT quotients,” arXiv e-prints (June, 2011) arXiv:1106.3724, arXiv:1106.3724 [math.AG].
  * (35) A. Okounkov, “Lectures on K-theoretic computations in enumerative geometry,” arXiv:1512.07363 [math.AG].
  * (36) M. Aganagic and A. Okounkov, “Quasimap counts and Bethe eigenfunctions,” Moscow Math. J. 17 no. 4, (2017) 565–600, arXiv:1704.08746 [math-ph].
  * (37) M. Aganagic, E. Frenkel, and A. Okounkov, “Quantum $q$-Langlands Correspondence,” Trans. Moscow Math. Soc. 79 (2018) 1–83, arXiv:1701.03146 [hep-th].
  * (38) S. Jeong, N. Lee, and N. Nekrasov, “Parallel surface defects, Hecke operators, and quantum Hitchin system,” arXiv:2304.04656 [hep-th].
  * (39) N. Nekrasov, “BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and $qq$-characters,” JHEP 1603 (2016) 181, arXiv:1512.05388 [hep-th].
  * (40) N. Nekrasov, “BPS/CFT Correspondence III: Gauge Origami partition function and $qq$-characters,” Commun. Math. Phys. 358 (2017) 863–894, arXiv:1701.00189 [hep-th].
  * (41) E. Witten, “Solutions Of Four-Dimensional Field Theories Via M Theory,” Nucl.Phys.B 500 (1997) 3–42, arXiv:hep-th/9703166 [hep-th].
  * (42) M. R. Douglas and G. W. Moore, “D-branes, quivers, and ALE instantons,” arXiv:hep-th/9603167.
  * (43) E. Frenkel and N. Reshetikhin, “Deformations of $\mathcal{W}$-algebras associated to simple Lie algebras,” Comm. Math. Phys. 197 no. 1, (1998) 1–32, arXiv:q-alg/9708006 [math.QA].
  * (44) J. Shiraishi, H. Kubo, H. Awata, and S. Odake, “A Quantum deformation of the Virasoro algebra and the Macdonald symmetric functions,” Lett. Math. Phys. 38 (1996) 33–51, arXiv:q-alg/9507034.
  * (45) B. Feigin and E. Frenkel, “Quantum W algebras and elliptic algebras,” Commun. Math. Phys. 178 (1996) 653–678, arXiv:q-alg/9508009.
  * (46) H. Awata, H. Kubo, S. Odake, and J. Shiraishi, “Quantum W(N) algebras and Macdonald polynomials,” Commun. Math. Phys. 179 (1996) 401–416, arXiv:q-alg/9508011.
  * (47) P. Bouwknegt and K. Pilch, “On deformed W algebras and quantum affine algebras,” Adv. Theor. Math. Phys. 2 (1998) 357–397, arXiv:math/9801112.
  * (48) T. Kimura and V. Pestun, “Quiver W-algebras,” Lett. Math. Phys. 108 no. 6, (2018) 1351–1381, arXiv:1512.08533 [hep-th].
  * (49) T. Kimura and V. Pestun, “Quiver elliptic W-algebras,” Lett. Math. Phys. 108 no. 6, (2018) 1383–1405, arXiv:1608.04651 [hep-th].
  * (50) T. Kimura and V. Pestun, “Fractional quiver W-algebras,” arXiv:1705.04410 [hep-th].
  * (51) T. Kojima, “Quadratic relations of the deformed w-superalgebra wq,t(sl(2|1)),” arXiv: Quantum Algebra (2019) .
  * (52) T. Kojima, “Quadratic relations of the deformed w-superalgebra wq,ta(m,n),” Journal of Physics A: Mathematical and Theoretical 54 (2021) .
  * (53) T. Kojima, “Quadratic relations of the deformed $W$-algebra,” arXiv e-prints (Dec., 2023) arXiv:2312.16856, arXiv:2312.16856 [math.QA].
  * (54) M. Bershtein, B. Feigin, and G. Merzon, “Plane partitions with a “pit”: generating functions and representation theory,” arXiv e-prints (Dec., 2015) arXiv:1512.08779, arXiv:1512.08779 [math.CO].
  * (55) J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang, and R.-D. Zhu, “Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d Super Yang-Mills,” PTEP 2016 no. 12, (2016) 123B05, arXiv:1606.08020 [hep-th].
  * (56) B. Feigin, M. Jimbo, E. Mukhin, and I. Vilkoviskiy, “Deformations of $\mathcal{W}$ algebras via quantum toroidal algebras,” arXiv e-prints (Mar., 2020) arXiv:2003.04234, arXiv:2003.04234 [math.QA].
  * (57) T. Kimura, H. Mori, and Y. Sugimoto, “Refined geometric transition and $qq$-characters,” JHEP 01 (2018) 025, arXiv:1705.03467 [hep-th].
  * (58) Y. Zenkevich, “Spiralling branes and R-matrices,” arXiv:2312.16990 [hep-th].
  * (59) N. Nekrasov and N. S. Prabhakar, “Spiked Instantons from Intersecting D-branes,” Nucl. Phys. B914 (2017) 257–300, arXiv:1611.03478 [hep-th].
  * (60) H.-C. Kim, “Line defects and 5d instanton partition functions,” JHEP 03 (2016) 199, arXiv:1601.06841 [hep-th].
  * (61) C.-M. Chang, O. Ganor, and J. Oh, “An index for ray operators in 5d $E_{n}$ SCFTs,” JHEP 02 (2017) 018, arXiv:1608.06284 [hep-th].
  * (62) B. Assel and A. Sciarappa, “Wilson loops in 5d $\mathcal{N}=1$ theories and S-duality,” JHEP 10 (2018) 082, arXiv:1806.09636 [hep-th].
  * (63) P. Agarwal, J. Kim, S. Kim, and A. Sciarappa, “Wilson surfaces in M5-branes,” JHEP 08 (2018) 119, arXiv:1804.09932 [hep-th].
  * (64) N. Haouzi and C. Kozçaz, “Supersymmetric Wilson Loops, Instantons, and Deformed $\mathcal{W}$-Algebras,” Commun. Math. Phys. 393 no. 2, (2022) 669–779, arXiv:1907.03838 [hep-th].
  * (65) N. Haouzi, “Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N}=4$ Gauge Theories,” arXiv:2009.08989 [hep-th].
  * (66) J.-E. Bourgine, Y. Matsuo, and H. Zhang, “Holomorphic field realization of SHc and quantum geometry of quiver gauge theories,” JHEP 04 (2016) 167, arXiv:1512.02492 [hep-th].
  * (67) S. Jeong, “Splitting of surface defect partition functions and integrable systems,” Nucl. Phys. B938 (2019) 775–806, arXiv:1709.04926 [hep-th].
  * (68) S. Jeong and N. Nekrasov, “Opers, surface defects, and Yang-Yang functional,” Adv. Theor. Math. Phys. 24 no. 7, (2020) 1789–1916, arXiv:1806.08270 [hep-th].
  * (69) N. Lee and N. Nekrasov, “Quantum spin systems and supersymmetric gauge theories. Part I,” JHEP 03 (2021) 093, arXiv:2009.11199 [hep-th].
  * (70) S. Jeong, N. Lee, and N. Nekrasov, “Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations,” JHEP 10 (2021) 120, arXiv:2103.17186 [hep-th].
  * (71) A. Grekov and N. Nekrasov, “Elliptic Calogero-Moser system, crossed and folded instantons, and bilinear identities,”  10, 2023.  arXiv:2310.04571 [math-ph].
  * (72) S. Jeong, N. Lee, and N. Nekrasov, “di-Langlands correspondence and extended observables,” JHEP 06 (2024) 105, arXiv:2402.13888 [hep-th].
  * (73) S. Jeong and N. Lee, “Bispectral duality and separation of variables from surface defect transition,” arXiv:2402.13889 [hep-th].
  * (74) C. Elliott and V. Pestun, “Multiplicative Hitchin Systems and Supersymmetric Gauge Theory,” arXiv:1812.05516 [math.AG].
  * (75) E. Frenkel, P. Koroteev, D. S. Sage, and A. M. Zeitlin, “q-Opers, QQ-Systems, and Bethe Ansatz,” arXiv:2002.07344 [math.AG].
  * (76) S. Jeong and X. Zhang, “BPZ equations for higher degenerate fields and non-perturbative Dyson-Schwinger equations,” arXiv:1710.06970 [hep-th].
  * (77) S. Jeong and X. Zhang, “A note on chiral trace relations from qq-characters,” JHEP 04 (2020) 026, arXiv:1910.10864 [hep-th].
  * (78) S. Jeong and N. Nekrasov, “Riemann-Hilbert correspondence and blown up surface defects,” JHEP 12 (2020) 006, arXiv:2007.03660 [hep-th].
  * (79) N. Haouzi and J. Oh, “On the Quantization of Seiberg-Witten Geometry,” arXiv:2004.00654 [hep-th].
  * (80) N. Haouzi, “Quantum geometry and $\theta$-angle in five-dimensional super Yang-Mills,” JHEP 09 (2020) 035, arXiv:2005.13565 [hep-th].
  * (81) H. Liu, “A Representation-Theoretic Approach to $qq$-Characters,” arXiv e-prints (Mar., 2022) arXiv:2203.07072, arXiv:2203.07072 [math.QA].
  * (82) N. M. Paquette and B. R. Williams, “Koszul duality in quantum field theory,” arXiv:2110.10257 [hep-th].
  * (83) K. Costello, E. Witten, and M. Yamazaki, “Gauge Theory and Integrability, I,” ICCM Not. 06 no. 1, (2018) 46–119, arXiv:1709.09993 [hep-th].
  * (84) J. Oh and Y. Zhou, “Feynman diagrams and $\Omega$-deformed M-theory,” SciPost Phys. 10 no. 2, (2021) 029, arXiv:2002.07343 [hep-th].
  * (85) A. Tsymbaliuk, “The affine Yangian of $\mathfrak{gl}_{1}$ revisited,” Adv. Math. 304 (2017) 583–645, arXiv:1404.5240 [math.RT].
  * (86) O. Schiffmann and E. Vasserot, “The elliptic hall algebra, cherednik hecke algebras and macdonald polynomials,” Compositio Mathematica 147 no. 1, (2011) 188–234.
  * (87) A. Gorsky, N. Nekrasov, and V. Rubtsov, “Hilbert schemes, separated variables, and D-branes,” Commun. Math. Phys. 222 (2001) 299–318, arXiv:hep-th/9901089.
  * (88) A. Kapustin, A. Kuznetsov, and D. Orlov, “Noncommutative instantons and twistor transform,” Commun. Math. Phys. 221 (2001) 385–432, arXiv:hep-th/0002193.
  * (89) A.Oblomkov, “Double affine hecke algebras and calogero-moser spaces,”  2003\.  https://api.semanticscholar.org/CorpusID:119579748.
  * (90) D. Jordan, “Quantized multiplicative quiver varieties,” Advances in Mathematics 250 (2014) 420–466. https://www.sciencedirect.com/science/article/pii/S0001870813003393.
  * (91) V. V. Fock and A. A. Roslyi, “Poisson structure on moduli of flat connections on Riemann surfaces and r matrix,”.
  * (92) O. Chalykh and M. Fairon, “Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models,” J. Geom. Phys. 121 (2017) 413–437, arXiv:1704.05814 [math.QA].
  * (93) L. Rozansky and E. Witten, “Hyper-kähler geometry and invariants of three-manifolds,” Selecta Mathematica 3 (1996) 401–458.
  * (94) J. Yagi, “$\Omega$-deformation and quantization,” JHEP 08 (2014) 112, arXiv:1405.6714 [hep-th].
  * (95) M. Balagovic and D. A. Jordan, “The Harish-Chandra isomorphism for quantum $GL_{2}$,” Journal of Noncommutative Geometry (2016) . https://api.semanticscholar.org/CorpusID:119163354.
  * (96) J. J. Wen, “The quantum Harish-Chandra isomorphism for $GL_{n}$,” arXiv:2309.00823.
  * (97) M. Finkelberg and V. Ginzburg, Cherednik Algebras for Algebraic Curves, pp. 121–153.  Birkhäuser Boston, Boston, 2010.
  * (98) R. Kodera and H. Nakajima, “Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras,” Proc. Symp. Pure Math. 98 (2018) 49–78, arXiv:1608.00875 [math.RT].
  * (99) K. Costello and N. M. Paquette, “Twisted Supergravity and Koszul Duality: A case study in AdS3,” Commun. Math. Phys. 384 no. 1, (2021) 279–339, arXiv:2001.02177 [hep-th].
  * (100) E. Frenkel and N. Reshetikhin, “Towards deformed chiral algebras,” in 21st International Colloquium on Group Theoretical Methods in Physics, pp. 27–42.  1996\.
  * (101) P. Etingof and D. Kazhdan, “Quantization of lie bialgebras, part v: Quantum vertex operator algebras,” Selecta Mathematica 6 (2000) 105–130.
  * (102) F. Chen, X. Huang, F. Kong, and S. Tan, “Quantum vertex algebra associated to quantum toroidal $\mathfrak{gl}_{N}$,” arXiv:2405.09386 [math.QA].
  * (103) M. B. Green, J. A. Harvey, and G. W. Moore, “I-brane inflow and anomalous couplings on d-branes,” Class. Quant. Grav. 14 (1997) 47–52, arXiv:hep-th/9605033.
  * (104) R. Dijkgraaf, L. Hollands, P. Sulkowski, and C. Vafa, “Supersymmetric gauge theories, intersecting branes and free fermions,” JHEP 02 (2008) 106, arXiv:0709.4446 [hep-th].
  * (105) J. Yagi, “3d TQFT from 6d SCFT,” JHEP 08 (2013) 017, arXiv:1305.0291 [hep-th].
  * (106) S. Lee and M. Yamazaki, “3d Chern-Simons Theory from M5-branes,” JHEP 12 (2013) 035, arXiv:1305.2429 [hep-th].
  * (107) Y. Luo, M.-C. Tan, J. Yagi, and Q. Zhao, “$\Omega$-deformation of B-twisted gauge theories and the 3d-3d correspondence,” JHEP 02 (2015) 047, arXiv:1410.1538 [hep-th].
  * (108) K. Costello and J. Yagi, “Unification of integrability in supersymmetric gauge theories,” Adv. Theor. Math. Phys. 24 no. 8, (2020) 1931–2041, arXiv:1810.01970 [hep-th].
  * (109) K. Costello and M. Yamazaki, “Gauge Theory And Integrability, III,” arXiv:1908.02289 [hep-th].
  * (110) O. Schiffmann and E. Vasserot, “The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^{2}$,” arXiv e-prints (May, 2009) arXiv:0905.2555, arXiv:0905.2555 [math.QA].
  * (111) E. Frenkel and N. Reshetikhin, “Quantum affine algebras and deformations of the virasoro and $\mathcal{W}$-algebras,” Les rencontres physiciens-mathématiciens de Strasbourg -RCP25 47 (1995) . http://www.numdam.org/item/RCP25_1995__47__275_0/. Talk no. 12, 31 p.
  * (112) B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, “A commutative algebra on degenerate $cp^{1}$ and macdonald polynomials,” arXiv: Combinatorics (2009) .
  * (113) A. Garbali and A. Neguţ, “Computing the R-matrix of the quantum toroidal algebra,” J. Math. Phys. 64 no. 1, (2023) 011702, arXiv:2112.09094 [math-ph].
  * (114) M. Fukuda, K. Harada, Y. Matsuo, and R.-D. Zhu, “The Maulik–Okounkov R-matrix from the Ding–Iohara–Miki algebra,” PTEP 2017 no. 9, (2017) 093A01, arXiv:1705.02941 [hep-th].
  * (115) A. Garbali and J. de Gier, “The R-Matrix of the Quantum Toroidal Algebra $U_{q,t}(\overset{..}{gl}_{1})$ in the Fock Module,” Commun. Math. Phys. 384 no. 3, (2021) 1971–2008, arXiv:2004.09241 [math-ph].
  * (116) A. Neguţ, “The R-matrix of the quantum toroidal algebra,” arXiv:2005.14182 [math.QA].
  * (117) J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo, and R.-D. Zhu, “(p, q)-webs of DIM representations, 5d $\mathcal{N}=1$ instanton partition functions and qq-characters,” JHEP 11 (2017) 034, arXiv:1703.10759 [hep-th].
  * (118) Y. Zenkevich, “Hanany-Witten brane crossing and Ding-Iohara-Miki algebra,” arXiv e-prints (Dec., 2022) arXiv:2212.14808, arXiv:2212.14808 [hep-th].
  * (119) H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake, and Y. Zenkevich, “$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces,” JHEP 03 (2018) 192, arXiv:1712.08016 [hep-th].
  * (120) J.-E. Bourgine and S. Jeong, “New quantum toroidal algebras from 5D $\mathcal{N}$ = 1 instantons on orbifolds,” JHEP 05 (2020) 127, arXiv:1906.01625 [hep-th].
  * (121) N. Nekrasov and E. Witten, “The Omega Deformation, Branes, Integrability, and Liouville Theory,” JHEP 09 (2010) 092, arXiv:1002.0888 [hep-th].
  * (122) B. Feigin and A. Tsymbaliuk, “Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra,” arXiv e-prints (Apr., 2009) arXiv:0904.1679, arXiv:0904.1679 [math.RT].
  * (123) O. Schiffmann and E. Vasserot, “Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2̂,” arXiv e-prints (Feb., 2012) arXiv:1202.2756, arXiv:1202.2756 [math.QA].
  * (124) D. Maulik and A. Okounkov, “Quantum Groups and Quantum Cohomology,” arXiv e-prints (Nov., 2012) arXiv:1211.1287, arXiv:1211.1287 [math.AG].
  * (125) C. Beem, T. Dimofte, and S. Pasquetti, “Holomorphic Blocks in Three Dimensions,” JHEP 12 (2014) 177, arXiv:1211.1986 [hep-th].
  * (126) S. Cecotti and C. Vafa, “Topological antitopological fusion,” Nucl. Phys. B 367 (1991) 359–461.
  * (127) S. Cecotti and C. Vafa, “On classification of N=2 supersymmetric theories,” Commun. Math. Phys. 158 (1993) 569–644, arXiv:hep-th/9211097.
  * (128) K. Hori, A. Iqbal, and C. Vafa, “D-branes and mirror symmetry,” arXiv:hep-th/0005247.
  * (129) D. Gaiotto, G. W. Moore, and E. Witten, “An Introduction To The Web-Based Formalism,” arXiv:1506.04086 [hep-th].
  * (130) D. Gaiotto, G. W. Moore, and E. Witten, “Algebra of the Infrared: String Field Theoretic Structures in Massive ${\mathcal{N}}=(2,2)$ Field Theory In Two Dimensions,” arXiv:1506.04087 [hep-th].
  * (131) M. Bullimore, S. Crew, and D. Zhang, “Boundaries, Vermas, and Factorisation,” JHEP 04 (2021) 263, arXiv:2010.09741 [hep-th].
  * (132) T. Kitao, K. Ohta, and N. Ohta, “Three-dimensional gauge dynamics from brane configurations with (p,q)-fivebrane,” Nucl. Phys. B 539 (1999) 79–106, arXiv:hep-th/9808111.
  * (133) O. Bergman, A. Hanany, A. Karch, and B. Kol, “Branes and supersymmetry breaking in three-dimensional gauge theories,” JHEP 10 (1999) 036, arXiv:hep-th/9908075.
  * (134) C. Hwang, J. Kim, S. Kim, and J. Park, “General instanton counting and 5d SCFT,” JHEP 07 (2015) 063, arXiv:1406.6793 [hep-th]. [Addendum: JHEP04,094(2016)].
  * (135) C. Hwang, P. Yi, and Y. Yoshida, “Fundamental Vortices, Wall-Crossing, and Particle-Vortex Duality,” JHEP 05 (2017) 099, arXiv:1703.00213 [hep-th].
  * (136) S. Crew, N. Dorey, and D. Zhang, “Factorisation of 3d $\mathcal{N}$ = 4 twisted indices and the geometry of vortex moduli space,” JHEP 08 no. 08, (2020) 015, arXiv:2002.04573 [hep-th].
  * (137) F. Nieri, Y. Pan, and M. Zabzine, “3d Expansions of 5d Instanton Partition Functions,” JHEP 04 (2018) 092, arXiv:1711.06150 [hep-th].
  * (138) T. Banks, N. Seiberg, and E. Silverstein, “Zero and one-dimensional probes with N=8 supersymmetry,” Phys. Lett. B 401 (1997) 30–37, arXiv:hep-th/9703052.
  * (139) C. P. Bachas, M. B. Green, and A. Schwimmer, “(8,0) quantum mechanics and symmetry enhancement in type I’ superstrings,” JHEP 01 (1998) 006, arXiv:hep-th/9712086.
  * (140) L.-Y. Hung, “Comments on I1-branes,” JHEP 05 (2007) 076, arXiv:hep-th/0612207.
  * (141) D. Tong, “The holographic dual of $AdS_{3}\times S^{3}\times S^{3}\times S^{1}$,” JHEP 04 (2014) 193, arXiv:1402.5135 [hep-th].
  * (142) T. Okazaki, “Abelian dualities of $\mathcal{N}=(0,4)$ boundary conditions,” JHEP 08 (2019) 170, arXiv:1905.07425 [hep-th].
  * (143) K. Miki, “A $(q,\gamma)$ analog of the $w_{1+\infty}$ algebra,” J. Math. Phys. 48 (2007) .
  * (144) J.-E. Bourgine, “Fiber-base duality from the algebraic perspective,” JHEP 03 (2019) 003, arXiv:1810.00301 [hep-th].
  * (145) O. Aharony and A. Hanany, “Branes, superpotentials and superconformal fixed points,” Nucl. Phys. B 504 (1997) 239–271, arXiv:hep-th/9704170.
  * (146) N. Haouzi, “A new realization of quantum algebras in gauge theory and Ramification in the Langlands program,” arXiv:2311.04367 [hep-th].
  * (147) S. Tamagni, “Stable Envelopes, Vortex Moduli Spaces, and Verma Modules,” arXiv:2311.03462 [hep-th].
  * (148) I. Cherednik, Double Affine Hecke Algebras.  Cambridge University Press, 2005.  Online publication date: December 2009, Print publication year: 2005.
  * (149) I. G. MacDonald, “Symmetric functions and hall polynomials,”  1979\.  https://api.semanticscholar.org/CorpusID:122648908.