# Gauge origami and quiver W-algebras II:
Vertex function and beyond quantum $q$-Langlands correspondence

Taro Kimura  Go Noshita

###### Abstract

We continue the study of generalized gauge theory called gauge origami, based
on the quantum algebraic approach initiated in [KN23]. In this article, we in
particular explore the D2 brane system realized by the screened vertex
operators of the corresponding W-algebra. The partition function of this
system given by the corresponding conformal block is identified with the
vertex function associated with quasimaps to Nakajima quiver varieties and
generalizations, that plays a central role in the quantum $q$-Langlands
correspondence. Based on the quantum algebraic perspective, we address three
new aspects of the correspondence: (i) Direct equivalence between the electric
and magnetic blocks by constructing stable envelopes from the chamber
structure of the vertex operators, (ii) Double affine generalization of
quantum $q$-Langlands correspondence, and (iii) Conformal block realization of
the origami vertex function associated with intersection of quasimaps, that
realizes the higher-rank multi-leg Pandharipande–Thomas vertices of 3-fold and
4-fold.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Direct equivalence between electric and magnetic blocks
    2. 1.2 Double affine quantum $q$-Langlands correspondence
    3. 1.3 Origami vertex function, PT vertex, and W-algebra
  2. 2 W${}_{q_{1,2}}(A_{1})$ conformal block
    1. 2.1 Single screening current
    2. 2.2 Double screening current
  3. 3 W${}_{q_{1,2,3,4}}(\widehat{A}_{0})$ conformal block
    1. 3.1 Single screening current
    2. 3.2 Multiple screening current
  4. A Special functions

##  1 Introduction and summary

Gauge origami, introduced and investigated in a series of works by Nekrasov
[Nek15, Nek16, Nek17a, Nek17b, Nek17c], deserves a generalization of gauge
theory, formulated as a union and intersection of gauge theories supported on
various subspaces in $\mathbb{C}^{4}$. It is formulated as intersecting
coherent sheaves on the subschemes, and the equivariant integral of the
corresponding moduli space gives rise to the gauge origami partition function,
i.e., the generating function of the enumerative invariants. In this article,
we continue the study of such a generalized gauge theory based on the quantum
algebraic structure initiated in our previous work [KN23]. We in particular
focus on the D2 brane system algebraically realized by the screening current
of the corresponding W-algebra. This system is known to be related to
quasimaps $f$ from $\mathbb{P}^{1}$, a compactification of $\mathbb{C}$, to
the corresponding target space $X$, typically taken to be a Nakajima quiver
variety. The corresponding partition function, a.k.a., the vertex function, is
given by the generating function of equivariant (K-theoretic) Euler
characteristics of the moduli space of quasimaps
$\mathsf{QM}(X)=\\{f:\mathbb{P}^{1}\to X\\}/\sim$ [Oko15], that is identified
with the vortex partition function in the physics context. See, e.g., [DN23,
CZZ23]. The vertex function constructed geometrically also has an algebraic
realization addressed by Aganagic, E. Frenkel, and Okounkov in the context of
the quantum $q$-Langlands correspondence.

######  Theorem 1.1 ([AFO17]).

The vertex function is the conformal block of the $q$-deformed W-algebra.

The quantum $q$-Langlands correspondence is the two-parameter deformation of
the original geometric Langlands correspondence, that is the equivalence
between the spaces of conformal blocks of the quantum affine algebra
$U_{q}(\widehat{\mathfrak{g}})$ (electric block) and those of the $q$-deformed
W-algebra W${}_{q_{1,2}}({}^{L}\mathfrak{g})$ (magnetic block), where
${}^{L}\mathfrak{g}$ is the Langlands dual of a simple Lie algebra
$\mathfrak{g}$. Theorem 1.1 plays a central role in the correspondence since
the vertex function is a building block of a solution to the quantum
Kniznik–Zamolodchikov ($q$-KZ) equation introduced by I. Frenkel and
Reshetikhin [FR92] for the conformal block of the quantum affine algebra. One
can construct the integral formula of the solution, a.k.a., the
$q$-hypergeometric integral solution, to the $q$-KZ equation from the vertex
function together with the stable envelope [MO12]. In the context of the
$q$-hypergeometric solution, the stable envelope is identified with the so-
called weight function [GRTV12, RTV14, FRV17, Kon17]. See, e.g., [Tam23,
Hao23, JLN24] for recent gauge theory approaches to the Langlands
correspondence.

In this article, we address three new perspectives concerning the quantum
$q$-Langlands correspondence as follows.

###  1.1 Direct equivalence between electric and magnetic blocks

The $q$-KZ equation is a $q$-difference equation for the electric conformal
block of the quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$ that takes
a value in the tensor product module of the corresponding representations.
Meanwhile, the magnetic conformal block of the $q$-deformed W-algebra
W${}_{q_{1,2}}(^{L}\mathfrak{g})$ is apparently insensitive to the
representation of $U_{q}(\widehat{\mathfrak{g}})$ by definition, so that we
need to insert the stable envelope by hand to construct the $q$-KZ solution
from the magnetic block side. In this article, we reexamine the W-algebra
conformal block, and we figure out how to impose the representation dependence
of the conformal block, i.e., the stable envelope purely from the W-algebra
side. The key idea is the radial ordering and the chamber structure of the
vertex operators. This implies that we do not need to insert the stable
envelope by hand to the W-algebra block to construct the $q$-hypergeometric
solution to the $q$-KZ equation. In fact, the W-algebra block directly solves
the $q$-KZ equation.

######  Theorem 1.2 (Theorem 2.17).

The elliptic conformal block of the screened vertex operators of the
$q$-deformed W-algebra W${}_{q_{1,2}}(A_{1})$ obeys the dynamical elliptic
$q$-KZ equation.

We emphasize that this is a reformulation of the result given by Konno [Kon17,
Kon18, Kon20], who proved the corresponding statement in the context of
elliptic quantum group $U_{q,p}(\widehat{\mathfrak{sl}}_{n})$, that is known
to be equivalent to the W-algebra W${}_{q_{1,2}}(A_{n})$222We also use the
Dynkin classification notation, e.g,
$\mathrm{W}_{q_{1,2}}(A_{n-1})=\mathrm{W}_{q_{1,2}}(\mathfrak{sl}_{n})$. , for
level-one representations. As pointed out by Konno, the quantum $q$-Langlands
correspondence becomes more direct in the elliptic setup: We show that the
elliptic stable envelope introduced by Aganagic and Okounkov [AO16] naturally
arises from the chamber structure of the vertex operators. However, it has
been known that the elliptic envelope is not sufficient to construct the
$q$-KZ solution in the trigonometric setup. We need to also insert the
trigonometric version of the stable envelope to obtain the trigonometric
$q$-KZ solution. In fact, we do not need the additional stable envelope in the
elliptic case, where the elliptic envelope is sufficient to obtain the
elliptic $q$-KZ solution. We prove this Theorem for the case
$\mathfrak{g}=\mathfrak{sl}_{2}$ associated with $A_{1}$ quiver. Its
generalization to simply-laced algebras seems straightforward. It would be
also interesting to consider the non-simply-laced cases [FR97, KP17, KP22].

###  1.2 Double affine quantum $q$-Langlands correspondence

The vertex function is constructed for a wide class of Nakajima quiver
varieties. A particularly interesting family of quiver varieties is those
associated with cyclic quivers (untwisted affine $A$-type quivers). It has
been known that affine $A$-type quiver varieties play an important role in the
context of geometric representation theory of quantum toroidal algebra of
$\mathfrak{gl}_{n}$ denoted by
$U_{q}(\widehat{\vphantom{\rule{1.0pt}{5.93056pt}}\smash{\widehat{{\mathfrak{gl}}}}}_{n})$,
which is a double affine quantum algebra. See, e.g., [Her08, Tsy22, MNNZ23]
for reviews. Meanwhile, the formalism of quiver W-algebra [KP15, KP16, KP17]
allows us to consider the W-algebra side of this story beyond the construction
of E. Frenkel and Reshetikhin for finite-type simple Lie algebras [FR97]. One
can construct the $q$-deformed W-algebra associated with cyclic quivers
denoted by W${}_{q_{1,2,3,4}}(\widehat{A}_{n-1})$ that depends on four
parameters $\\{q_{1,2,3,4}\\}$ under the condition $q_{1}q_{2}q_{3}q_{4}=1$
(Calabi–Yau 4-fold condition). This affine quiver W-algebra deserves a double
affine W-algebra.

######  Theorem 1.3 (Theorems 3.6, 3.8).

The vertex function for Nakajima quiver variety associated with the Jordan
quiver ($\widehat{A}_{0}$ quiver) is given by the conformal block of affine
quiver W-algebra W${}_{q_{1,2,3,4}}(\widehat{A}_{0})$.

This is a double affine version of Theorem 1.1, establishing the double affine
quantum $q$-Langlands correspondence between the quantum toroidal
$\mathfrak{gl}_{1}$ and the affine quiver W-algebra
W${}_{q_{1,2,3,4}}(\widehat{A}_{0})$. In fact, it has been recently shown by
Smirnov that the quantum difference equation associated with the quantum
toroidal $\mathfrak{gl}_{1}$ is solved by the vertex function of the Hilbert
scheme of points on $\mathbb{C}^{2}$ [Smi21]. We prove this Theorem for the
rank-one case, whereas a higher-rank generalization is also straightforward.

We remark that there are two types of highest-weight modules of quantum
toroidal $\mathfrak{gl}_{1}$, Fock module and MacMahon module. Geometrically,
the former one is constructed from the quiver variety of the Jordan quiver,
namely the Hilbert scheme of points on $\mathbb{C}^{2}$ for the rank-one case,
and the corresponding Quot scheme for the higher-rank case. On the other hand,
the latter one has a geometric realization by a quiver with a potential, which
gives rise to the Hilbert scheme of points and the Quot scheme on
$\mathbb{C}^{3}$. In fact, the conformal block of
W${}_{q_{1,2,3,4}}(\widehat{A}_{0})$ reproduces vertex functions associated
with both types of modules depending on “the momentum” of the vertex operator.
This is analogous to the finite-type case, where the momentum parametrizes the
highest-weight module of the quantum affine algebra.

###  1.3 Origami vertex function, PT vertex, and W-algebra

It has been known that the moduli space of quasimaps to the Hilbert scheme of
points on $\mathbb{C}^{2}$ is identified with that of the stable pairs that
gives rise to the Pandharipande–Thomas (PT) invariant, a cousin of the
Donaldson–Thomas (DT) invariant, of $\mathbb{C}^{3}$ with a boundary condition
in one specific direction. This correspondence has been recently generalized
by Cao and Zhao [CZ23] to the higher-dimensional situation $\mathbb{C}^{4}$
where the vertex function associated with the Hilbert scheme of points on
$\mathbb{C}^{3}$ plays a role of the generating function of the PT invariants
for $\mathbb{C}^{4}$. In general, one can assign boundary conditions in three
(four) directions for $\mathbb{C}^{3}$ ($\mathbb{C}^{4}$) to define the so-
called multi-leg PT vertex, for which we propose an algebraic construction.

######  Conjecture 1.4 (Conjecture 3.10).

The conformal block of multi-screened vertex operators of the W-algebra
W${}_{q_{1,2,3,4}}(\widehat{A}_{0})$ is the (higher-rank) multi-leg PT vertex
of $\mathbb{C}^{3}$ and $\mathbb{C}^{4}$.

We call this generalized version of the vertex function associated with the
multi-leg situation the origami vertex function as is naturally given by the
gauge origami partition function. We derive the contour integral formula of
the conformal block and check that its pole structure, corresponding to the
equivariant fixed points in the moduli space, is consistent with the PT count
in $\mathbb{C}^{3}$ and $\mathbb{C}^{4}$ for various boundary conditions. Our
formula for the one-point conformal block reproduces the known result for the
rank-one PT3 and PT4 vertices, and we conjecture that the $n$-point block
generates the rank-$n$ PT3 and PT4 invariants.

The PT vertex is based on the so-called curve counting, hence it corresponds
to the D2 brane system as mentioned above. Recently, its higher dimensional
analog, the surface counting, has been discussed [BKP22, BKP24]. See also
[NP23]. We expect that the surface counting is similarly formulated based on
the vertex operators by replacing the screening current with the corresponding
D4 brane vertex operators [KN23]. A further analysis is left for a future
study.

### Acknowledgements

We would like to thank Y. Cao, G. Felder, H. Konno, N. Lee, M. Kool, and S.
Monavari for valuable communications. The work of TK was supported by CNRS
through MITI interdisciplinary programs, EIPHI Graduate School (No.
ANR-17-EURE-0002) and Bourgogne-Franche-Comté region. GN is supported by JSPS
KAKENHI Grant-in-Aid for JSPS fellows Grant No. JP22J20944, JSR Fellowship,
and FoPM (WINGS Program), the University of Tokyo.

### Notations

  * •

We denote $x_{i}/x_{j}=x_{ij}$, $a_{i}/a_{j}=a_{ij}$, etc, for
$\mathbb{C}^{\times}$-variables.

  * •

We define $\underline{\mathbf{2}}=\\{1,2\\}$,
$\underline{\mathbf{3}}=\\{1,2,3\\}$, $\underline{\mathbf{4}}=\\{1,2,3,4\\}$.
For $A\subseteq\underline{\mathbf{4}}$, we denote $q_{A}=\prod_{\sigma\in
A}q_{\sigma}$.

  * •

We denote

$\displaystyle
f(q_{\sigma,\sigma^{\prime}}x)=f(q_{\sigma}x)f(q_{\sigma^{\prime}}x)\,,\qquad
f(x^{\pm 1})=f(x^{+1})f(x^{-1})\,.$ (1.1)

  * •

We define the $q$-shifted factorial for $q\in\mathbb{C}^{\times}$,

$\displaystyle(x;q)_{d}=\prod_{n=0}^{d-1}(1-xq^{n})\,.$ (1.2)

For $d\in\mathbb{Z}$ in general, we define

$\displaystyle(x;q)_{d}=\frac{(x;q)_{\infty}}{(q^{d}x;q)_{\infty}}=\begin{cases}(1-x)(1-qx)\cdots(1-q^{d-1}x)&(d>0)\\\
(1-q^{d}x)^{-1}\cdots(1-q^{-1}x)^{-1}&(d<0)\end{cases}$ (1.3)

  * •

We define the theta function for $p\in\mathbb{C}^{\times}$ with $|p|<1$,

$\displaystyle\theta(z;p)=(z,pz^{-1};p)_{\infty}=(z;p)_{\infty}(pz^{-1};p)_{\infty}.$
(1.4)

##  2 W${}_{q_{1,2}}(A_{1})$ conformal block

We consider the conformal block of the $q$-deformed W-algebra associated with
$A_{1}$ quiver denoted by W${}_{q_{1,2}}(A_{1})$. This algebra is identified
with the $q$-deformed Virasoro algebra introduced by Shiraishi, Kubo, Awata,
and Odake [SKAO95].

We denote the OPE factor associated with the vertex operators $\mathsf{V}$ and
$\mathsf{W}$ by

$\displaystyle\wick{\c{\mathsf{V}}(x)\c{\mathsf{W}}(x^{\prime})}=\frac{\mathsf{V}(x)\mathsf{W}(x^{\prime})}{:\mathsf{V}(x)\mathsf{W}(x^{\prime}):}\,.$
(2.1)

In this notation, we assume the radial ordering $x\succ x^{\prime}$, which
means $|x|>|x^{\prime}|$.

######  Definition 2.1.

Let $\sigma\in\underline{\textbf{2}}$,
$\bar{\sigma}=\underline{\textbf{2}}\backslash\sigma$, $|q_{1,2}|>1$, and
$x,x^{\prime},z,K\in\mathbb{C}^{\times}$. We define the vertex operators
$\\{\mathsf{S}_{1,2},\mathsf{Z}_{K}\\}$ associated with $A_{1}$ quiver,
obeying the following OPEs,

$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\sigma}(x^{\prime})}=\frac{(x^{\prime}/x;q_{\sigma}^{-1})_{\infty}(q_{12}^{-1}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}{(q_{\sigma}^{-1}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}(q_{\bar{\sigma}}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}\,,\qquad$
$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\bar{\sigma}}(x^{\prime})}=\frac{(x^{\prime})^{2}q_{\sigma}}{(1-q_{\sigma}x^{\prime}/x)(1-q_{\bar{\sigma}}^{-1}x^{\prime}/x)}\,,$
(2.2a)
$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{Z}}_{K}(x^{\prime})}=\frac{(Kx^{\prime}/x;q_{\sigma}^{-1})_{\infty}}{(x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}\,,\qquad$
$\displaystyle\wick{\c{\mathsf{Z}}_{K}(x^{\prime})\c{\mathsf{S}}_{\sigma}(x)}=\frac{(q_{\sigma}^{-1}x/x^{\prime};q_{\sigma}^{-1})_{\infty}}{(K^{-1}q_{\sigma}^{-1}x/x^{\prime};q_{\sigma}^{-1})_{\infty}}\,,$
(2.2b)
$\displaystyle\mathsf{S}_{\sigma}(x)z=(q_{\bar{\sigma}}^{-2}z)\mathsf{S}_{\sigma}(x)\,,\qquad$
$\displaystyle\mathsf{Z}_{K}(x)z=(Kz)\mathsf{Z}_{K}(x)\,.$ (2.2c)

See, e.g., [KP15] for explicit realization of these vertex operators in terms
of the free fields. We slightly change the zero mode notation compared to
[KP15] to simplify the discussion below. In this article, all the computations
are based on the OPEs defined above.

In the notation of [KN23, Proposition C.3], the vertex operators
$\mathsf{S}_{1,2}$ are called the screening currents associated with the plane
$\mathbb{C}_{1,2}$, that commute with each other,
$\wick{\c{\mathsf{S}}_{1}(x)\c{\mathsf{S}}_{2}(x^{\prime})}=\wick{\c{\mathsf{S}}_{2}(x^{\prime})\c{\mathsf{S}}_{1}(x)}$.
As discussed in [KN23], the screening current plays a role of the creation
operator of D2 branes supported on the plane, and the partition function of
this system is realized by the corresponding conformal block. Meanwhile, the
vertex operator $\mathsf{Z}_{K}$ is associated with the whole space
$\mathbb{C}^{4}$, corresponding to D8 ($\overline{\text{D8}}$) branes. The
screening currents commute with the generating current of the deformed
Virasoro algebra ($q$-Virasoro algebra) up to the total difference term. We
denote this algebra by W${}_{q_{1,2}}(A_{1})$ since it deserves the
$q$-deformed W-algebra associated with $A_{1}$ quiver [KP15]. As seen below,
it has an essential connection to quantum affine algebra
$U_{q}(\widehat{\mathfrak{sl}}_{2})$ and also elliptic quantum group
$U_{q,p}(\widehat{\mathfrak{sl}}_{2})$.

######  Lemma 2.2.

We may rewrite the OPEs (2.2) for $\sigma\in\underline{\mathbf{2}}$ as
follows,

$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\sigma}(x^{\prime})}$
$\displaystyle=\frac{(x^{\prime}/x;q_{\sigma}^{-1})_{\infty}(x/x^{\prime};q_{\sigma}^{-1})_{\infty}}{(q_{\bar{\sigma}}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}(q_{\bar{\sigma}}x/x^{\prime};q_{\sigma}^{-1})_{\infty}}\frac{\theta(q_{\bar{\sigma}}x^{\prime}/x;q_{\sigma}^{-1})}{\theta(x/x^{\prime};q_{\sigma}^{-1})},$
(2.3a)
$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{Z}}_{K}(x^{\prime})}$
$\displaystyle=\wick{\c{\mathsf{Z}}_{K}(x^{\prime})\c{\mathsf{S}}_{\sigma}(x)}\frac{\theta(Kx^{\prime}/x;q_{\sigma}^{-1})}{\theta(x^{\prime}/x;q_{\sigma}^{-1})}\,.$
(2.3b)

###### Proof.

It directly follows from Definition 2.1. ∎

###  2.1 Single screening current

In this subsection, we set $\mathsf{S}(x)\equiv\mathsf{S}_{2}(x)$. For
$|q_{\sigma}|>1$, define a kernel function,

$\displaystyle\varphi_{\sigma,K}(\xi,\eta)=\varphi_{K}(\xi,\eta;q_{\sigma}^{-1})=\frac{\theta(\xi\eta;q_{\sigma}^{-1})\theta(K;q_{\sigma}^{-1})}{\theta(K\xi;q_{\sigma}^{-1})\theta(\eta;q_{\sigma}^{-1})}\,,$
(2.4)

which is quasiperiodic with respect to each variable,

$\displaystyle\varphi_{\sigma,K}(q_{\sigma}^{-1}\xi,\eta)=(\eta/K)\varphi_{\sigma,K}(\xi,\eta)\,,\quad\varphi_{\sigma,K}(\xi,q_{\sigma}^{-1}\eta)=\xi^{-1}\varphi_{\sigma,K}(\xi,\eta)\,,\quad\varphi_{\sigma,q_{\sigma}^{-1}K}(\xi,\eta)=\xi\varphi_{\sigma,K}(\xi,\eta)\,.$
(2.5)

For $|q_{\sigma}|<1$, we instead use the theta function
$\theta(\cdot;q_{\sigma})$ to define the kernel function, namely
$\varphi_{K}(\xi,\eta;q_{\sigma})$.

######  Definition 2.3.

Let $z\in\mathbb{C}^{\times}$. We define the $m$-screened vertex operator,

$\displaystyle\mathsf{Z}_{K}^{(m)}(a)=\oint_{\gamma}[\differential{\underline{x}}]\left(\prod_{i=1,\ldots,m}^{\curvearrowleft}\varphi_{2,K}(a/x_{i},zK)\mathsf{S}(x_{i})\right)\mathsf{Z}_{K}(a)\,,\qquad[\differential{\underline{x}}]=\prod_{i=1}^{m}\frac{\differential{x}_{i}}{2\pi\mathrm{i}x_{i}}\,,$
(2.6)

where we apply the ordered product under the radial ordering configuration
$x_{m}\succ x_{m-1}\succ\cdots\succ x_{2}\succ x_{1}\succ a$,

$\displaystyle\prod_{i=1,\ldots,m}^{\curvearrowleft}\mathsf{V}(x_{i})=\mathsf{V}(x_{m})\mathsf{V}(x_{m-1})\cdots\mathsf{V}(x_{2})\mathsf{V}(x_{1}).$
(2.7)

The contour $\gamma$ is taken such that the integral is invariant under shift
of the variables, $x_{i}\to q_{2}^{-1}x_{i}$.

This definition is motivated by the construction of the intertwining operator
of the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{sl}}_{2})$ in
[JKOS98, §5.3]. From the OPE (2.2c), we have

$\displaystyle\mathsf{Z}_{K}^{(m)}(x)z=(Kq_{1}^{-2m}z)\mathsf{Z}_{K}^{(m)}(x)\,.$
(2.8)

The factor $Kq_{1}^{-2m}$ corresponds to the weight of the $m$-th operator.
See (2.52) for a related factor. The parameter $z$ is called the dynamical
parameter in the context of elliptic integrable systems, which is also
identified with the FI parameter of three-dimensional gauge theory.

Let us consider the conformal block of the screened vertex operators. Define a
map, $c:\\{1,\ldots,k\\}\to\\{1,\ldots,n\\}$, that we call the coloring
function. Let $\underline{k}=(k_{1},\ldots,k_{n})\in(\mathbb{Z}_{\geq
0})^{n}$, such that $k_{\alpha}=\\#\\{c(i)=\alpha,i=1,\ldots,n\\}$, obeying
$\sum_{\alpha=1}^{n}k_{\alpha}=k$. We denote

$\displaystyle K_{i}^{c}=\prod_{\alpha\leq c(i)}K_{\alpha}\,,$ (2.9)

and the Weyl vector of type $A_{k-1}$ (twice the positive root sum) by

$\displaystyle\rho_{i}=\frac{k+1}{2}-i\,,\qquad i=1,\ldots,k\,.$ (2.10)

Then, we have the conformal block of the screened vertex operators.

######  Proposition 2.4.

We have the following integral formula for the conformal block in the chamber
denoted by $\mathscr{C}_{12\ldots n}$, such that $a_{1}\succ
a_{2}\succ\cdots\succ a_{n}$ under the parameter shift $z\to zq_{1}^{k-1}$,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$
$\displaystyle=\bra{0}\mathsf{Z}^{(k_{1})}_{K_{1}}(a_{1})\mathsf{Z}^{(k_{2})}_{K_{2}}(a_{2})\cdots\mathsf{Z}^{(k_{n})}_{K_{n}}(a_{n})\ket{0}/\bra{0}\mathsf{Z}_{K_{1}}(a_{1})\mathsf{Z}_{K_{2}}(a_{2})\cdots\mathsf{Z}_{K_{n}}(a_{n})\ket{0}$
$\displaystyle=\frac{\mathsf{c}_{\underline{k}}(\underline{K})}{k!}\oint_{\gamma}[\differential{\underline{x}}]\,\Phi_{2}(\underline{x},\underline{a})\operatorname{Stab}_{2,\underline{k}}(\underline{x},\underline{a})$
(2.11)

where we define the principal part of the integrand for
$\sigma\in\underline{\mathbf{2}}$,
$\bar{\sigma}=\underline{\mathbf{2}}\backslash\sigma$ with the measure

$\displaystyle\Phi_{\sigma}(\underline{x},\underline{a})=\prod_{i\neq
j}^{k}\frac{(x_{ij};q_{\sigma}^{-1})_{\infty}}{(q_{\bar{\sigma}}x_{ij};q_{\sigma}^{-1})_{\infty}}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})_{\infty}}{(a_{\alpha}/x_{i};q_{\sigma}^{-1})_{\infty}}\,,\qquad[\differential{\underline{x}}]=\prod_{i=1}^{k}\frac{\differential{x}_{i}}{2\pi\mathrm{i}x_{i}}\,,$
(2.12)

together with the constant
$\mathsf{c}_{\underline{k}}(\underline{K})=\mathsf{c}_{2,\underline{k}}(\underline{K})$
defined by

$\displaystyle\mathsf{c}_{\sigma,\underline{k}}(\underline{K})=\prod_{i=1}^{k}\frac{\theta(K_{c(i)};q_{\sigma}^{-1})}{\theta(zq_{\bar{\sigma}}^{-2\rho_{i}}K_{i}^{c};q_{\sigma}^{-1})}\,.$
(2.13)

The off-shell elliptic stable envelope is defined for
$\sigma\in\underline{\mathbf{2}}$,
$\bar{\sigma}=\underline{\mathbf{2}}\backslash\sigma$ by

$\displaystyle\operatorname{Stab}_{\sigma,\underline{k}}(\underline{x},\underline{a})$
$\displaystyle=\operatorname*{Sym}_{\underline{x}}\prod_{i<j}^{k}\frac{\theta(q_{\bar{\sigma}}x_{ij};q_{\sigma}^{-1})}{\theta(x_{ij};q_{\sigma}^{-1})}\prod_{i=1}^{k}\frac{\prod_{\alpha<c(i)}\theta(a_{\alpha}/x_{i};q_{\sigma}^{-1})\theta(zq_{\bar{\sigma}}^{-2\rho_{i}}K_{i}^{c}a_{c(i)}/x_{i};q_{\sigma}^{-1})\prod_{\alpha>c(i)}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})}{\prod_{\alpha=1}^{k}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})}.$
(2.14)

###### Proof.

By Lemma 2.2, we have

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$
$\displaystyle=\frac{1}{k!}\oint_{\gamma}\prod_{i=1}^{k}\frac{\differential{x}_{i}}{2\pi\mathrm{i}x_{i}}\varphi_{K_{c(i)}}(x_{i},a_{c(i)},zq_{1}^{-2\rho_{i}}K_{i}^{c})\prod_{i=1,\ldots,k}^{\curvearrowleft}\mathsf{S}(x_{i})\prod_{\alpha=1,\ldots,n}^{\curvearrowright}\mathsf{Z}_{K_{\alpha}}(a_{\alpha})\prod_{i=1}^{k}\prod_{\alpha<c(i)}\frac{\theta(a_{\alpha}/x_{i};q_{2}^{-1})}{\theta(K_{\alpha}a_{\alpha}/x_{i};q_{2}^{-1})}$
(2.15)

and

$\displaystyle\prod_{i=1}^{k}\varphi_{K_{c(i)}}(x_{i},a_{c(i)},zq_{1}^{-2\rho_{i}}K_{i}^{c})\prod_{\alpha<c(i)}\frac{\theta(a_{\alpha}/x_{i};q_{2}^{-1})}{\theta(K_{\alpha}a_{\alpha}/x_{i};q_{2}^{-1})}$
$\displaystyle=\prod_{i=1}^{k}\frac{\theta(K_{c(i)};q_{2}^{-1})}{\theta(zq_{1}^{-2\rho_{i}}K_{i}^{c};q_{2}^{-1})}\frac{\prod_{\alpha<c(i)}\theta(a_{\alpha}/x_{i};q_{2}^{-1})\theta(zq_{1}^{-2\rho_{i}}K_{i}^{c}a_{c(i)}/x_{i};q_{2}^{-1})\prod_{\alpha>c(i)}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{2}^{-1})}{\prod_{\alpha=1}^{k}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{2}^{-1})}$
(2.16)

Then, using the OPEs shown in Definition 2.1, we obtain the expression above.
∎

In the following, we drop the constant factor
$\mathsf{c}_{\underline{k}}(\underline{K})$ for simplicity.

######  Remark 2.5.

The factor $K_{i}^{c}$ is different from the standard convention of the
elliptic stable envelope by the factor $K_{c(i)}$. We apply this modification
to obtain a consistent result with the known expressions of the vortex
partition function. See Proposition 2.11.

The conformal block $\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$ of W${}_{q_{1,2}}(A_{1})$
algebra agrees, under a slight modification of the notation, with the
partition function of the mass-deformed $\mathcal{N}=4$ three-dimensional
$\mathrm{U}(k)$ gauge theory supported on
$\mathbb{C}_{q_{2}}\times\mathbb{S}^{1}$ with $n$ fundamental matters [AHKS13,
AHS14, AH15] together with the additional boundary contribution written in
terms of the theta functions.333Meanwhile, the partition function of
$\mathcal{N}=1$ five-dimensional gauge theory defined on
$\mathbb{C}_{q_{1}}\times\mathbb{C}_{q_{2}}\times\mathbb{S}^{1}$ is similarly
realized by the conformal block of the (infinitely many) screening charges
[KP15].  In terms of the $\mathcal{N}=2$ multiplets, it consists of a vector
multiplet, a single adjoint and $(n+n)$ fundamental matters, which is known as
the handsaw quiver gauge theory of type $A_{1}$. The corresponding quiver
diagram is given in Fig. 1(a).

##### Pole structure

In order to evaluate the contour integral, we discuss the pole structure of
the integrand. Let $x_{\alpha,i}=x_{i+\sum_{\beta>\alpha}k_{\beta}}$ for
$i=1,\ldots,k_{\alpha}$. The poles of the contour integral are parametrized by
$n$-tuple inverse partitions,

$\displaystyle x_{\alpha,1}=a_{\alpha}q_{2}^{-n_{\alpha,1}}\,,\quad
x_{\alpha,i}=x_{\alpha,i-1}q_{1}q_{2}^{-n_{\alpha,i}}=a_{\alpha}q_{1}^{i-1}q_{2}^{-\nu_{\alpha,i}}$
(2.17)

where $n_{\alpha,i}\in\mathbb{Z}_{\geq 0}$ and
$\nu_{\alpha}=(\nu_{\alpha,1}\leq\nu_{\alpha,2}\leq\cdots\leq\nu_{\alpha,k_{\alpha}})$
is an inverse partition defined by

$\displaystyle\nu_{\alpha,i}=\sum_{j=1}^{i}n_{\alpha,j}\,.$ (2.18)

We see that the residues of other poles at
$x_{\alpha,i}/x_{\alpha,j}=q_{1}q_{2}^{\mathbb{Z}_{\leq 0}}$ for $i<j$ are
zero due to the stable envelope that plays a role of the pole subtraction
matrix [AO16].

######  Remark 2.6.

Although we can convert an inverse partition to the standard one by relabeling
the indices, $i\mapsto k_{\alpha}-i+1$, we now apply this convention, which is
naturally generalized to the situation with the double screening current (See
§2.2). This convention is also compatible with the 5d gauge theory partition
function (Proposition 2.12).

######  Proposition 2.7.

Let $K_{\alpha}=q_{1}^{\mu_{\alpha}}$, $\mu_{\alpha}\in\mathbb{Z}_{\geq 0}$
for $\alpha=1,\ldots,n$. Then, $\Psi^{\underline{k}}=0$ unless $0\leq
k_{\alpha}\leq\mu_{\alpha}$.

###### Proof.

The factor in the integrand
$(K_{\alpha}a_{\alpha}/x_{\alpha,i};q_{2}^{-1})_{\infty}=(q_{1}^{\mu_{\alpha}}a_{\alpha}/x_{\alpha,i};q_{2}^{-1})_{\infty}$
has zeros at
$x_{\alpha,i}=a_{\alpha}q_{1}^{\mu_{\alpha}}q_{2}^{\mathbb{Z}_{\leq 0}}$. ∎

This truncation is essential to realize the finite-dimensional representation
of the quantum affine algebra (the $(\mu_{\alpha}+1)$-dimensional
representation of $U_{q}(\widehat{\mathfrak{sl}}_{2})$). In this context, the
simplest case $\mu=1$ ($K=q_{1}$) corresponds to the fundamental two-
dimensional representation of $\mathfrak{sl}_{2}$, and a generic case is
obtained from the tensor product of the fundamental ones via specialization of
the parameters,
$\\{a_{1},\ldots,a_{\mu}\\}=\\{a,aq_{1},\ldots,aq_{1}^{\mu-1}\\}$. See (2.8).
Under this specialization, we have

$\displaystyle\prod_{\alpha=1}^{\mu}\frac{(q_{1}a_{\alpha}/x;q_{2}^{-1})}{(a_{\alpha}/x;q_{2}^{-1})}=\frac{\cancel{(q_{1}a/x;q_{2}^{-1})}}{(a/x;q_{2}^{-1})}\frac{\cancel{(q_{1}^{2}a/x;q_{2}^{-1})}}{\cancel{(q_{1}a/x;q_{2}^{-1})}}\cdots\frac{(q_{1}^{\mu}a/x;q_{2}^{-1})}{\cancel{(q_{1}^{\mu-1}a/x;q_{2}^{-1})}}=\frac{(q_{1}^{\mu}a/x;q_{2}^{-1})}{(a/x;q_{2}^{-1})}\,.$
(2.19)

Hence, we obtain the case $K=q_{1}^{\mu}$. This condition is known as the
$q$-segment condition for (the roots of) the Drinfeld polynomial, which
characterizes the finite-dimensional representation of the quantum affine
algebra (and also the Yangian) [CP91]. From the geometric point of view, the
assignment $K=q_{1}^{\mu}$ is realized via modification of the superpotential
in the corresponding quiver variety description,
$W=\sum_{\alpha=1}^{n}J_{\alpha}B^{\mu_{\alpha}}I_{\alpha}$ [NS09, BZJ19,
CZ23].

####  2.1.1 Elliptic $R$-matrix

Let us consider the simplest case $(n,k)=(2,1)$, where the corresponding
quiver variety is $T^{\vee}\mathbb{P}^{1}$ from the geometric point of view.
Hence, the conformal block computes the vertex function associated with
quasimaps $f:\mathbb{P}^{1}\to X=T^{\vee}\mathbb{P}^{1}$. We denote
$K_{i_{1}i_{2}\cdots i_{m}}=K_{i_{1}}K_{i_{2}}\cdots K_{i_{m}}$ and we write
$(\cdot)_{d}=(\cdot;q_{2}^{-1})_{d}$,
$\theta(\cdot)=\theta(\cdot;q_{2}^{-1})$. Then, we have two possible conformal
blocks in the chamber ${\mathscr{C}_{12}}:a_{1}\succ a_{2}$,

$\displaystyle\Psi_{\mathscr{C}_{12}}^{(1,0)}(a_{1,2};K_{1,2})$
$\displaystyle=\oint_{\gamma}\frac{\differential{x}}{2\pi\mathrm{i}x}\frac{(K_{1,2}a_{1,2}/x)_{\infty}}{(a_{1,2}/x)_{\infty}}\frac{\theta(zK_{1}a_{1}/x)\theta(K_{1})}{\theta(K_{1}a_{1}/x)\theta(zK_{1})}\,,$
(2.20a) $\displaystyle\Psi_{\mathscr{C}_{12}}^{(0,1)}(a_{1,2};K_{1,2})$
$\displaystyle=\oint_{\gamma}\frac{\differential{x}}{2\pi\mathrm{i}x}\frac{(K_{1,2}a_{1,2}/x)_{\infty}}{(a_{1,2}/x)_{\infty}}\frac{\theta(zK_{12}a_{2}/x)\theta(K_{2})}{\theta(K_{2}a_{2}/x)\theta(zK_{12})}\frac{\theta(a_{1}/x)}{\theta(K_{1}a_{1}/x)}\,.$
(2.20b)

Taking the poles at $\\{x=a_{1}q_{2}^{-d}\\}_{d\geq 0}$, we have

$\displaystyle\Psi_{\mathscr{C}_{12}}^{(1,0)}\Big{|}_{\\{a_{1}q_{2}^{-d}\\}}$
$\displaystyle=\frac{(K_{1},K_{2}a_{21})_{\infty}}{(q_{2}^{-1},a_{21})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{1}^{-1}q_{2}^{-1}&K_{2}^{-1}a_{12}q_{2}^{-1}\\\
a_{12}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\,,$ (2.21a)
$\displaystyle\Psi_{\mathscr{C}_{12}}^{(0,1)}\Big{|}_{\\{a_{1}q_{2}^{-d}\\}}$
$\displaystyle=0\,,$ (2.21b)

where ${}_{2}\phi_{1}$ is the $q$-hypergeometric series (Definition A.1), and
for the poles at $\\{x=a_{2}q_{2}^{-d}\\}_{d\geq 0}$, we have

$\displaystyle\Psi_{\mathscr{C}_{12}}^{(1,0)}\Big{|}_{\\{a_{2}q_{2}^{-d}\\}}$
$\displaystyle=\frac{(K_{2},q_{2}^{-1}a_{21})_{\infty}}{(q_{2}^{-1},q_{2}^{-1}K_{1}^{-1}a_{21})_{\infty}}\frac{\theta(zK_{1}a_{12},K_{1})}{\theta(a_{12},zK_{1})}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}^{-1}q_{2}^{-1}&K_{1}^{-1}a_{21}q_{2}^{-1}\\\
a_{21}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\,,$ (2.22a)
$\displaystyle\Psi_{\mathscr{C}_{12}}^{(0,1)}\Big{|}_{\\{a_{2}q_{2}^{-d}\\}}$
$\displaystyle=\frac{(K_{2},K_{1}a_{12})_{\infty}}{(q_{2}^{-1},a_{12})_{\infty}}\frac{\theta(a_{12})}{\theta(K_{1}a_{12})}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}^{-1}q_{2}^{-1}&K_{1}^{-1}a_{21}q_{2}^{-1}\\\
a_{21}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\,,$ (2.22b)

where we assumed $|zK_{12}|<1$. Combining them together, we write

$\displaystyle\Psi_{\mathscr{C}_{12}}$
$\displaystyle=\begin{pmatrix}\Psi_{\mathscr{C}_{12}}^{(1,0)}\Big{|}_{\\{a_{1}q_{2}^{-d}\\}}+\Psi_{\mathscr{C}_{12}}^{(1,0)}\Big{|}_{\\{a_{2}q_{2}^{-d}\\}}\\\
\Psi_{\mathscr{C}_{12}}^{(0,1)}\Big{|}_{\\{a_{1}q_{2}^{-d}\\}}+\Psi_{\mathscr{C}_{12}}^{(0,1)}\Big{|}_{\\{a_{2}q_{2}^{-d}\\}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&\displaystyle\frac{\theta(zK_{1}a_{12},K_{1})}{\theta(K_{1}a_{12},zK_{1})}\\\
0&\displaystyle\frac{\theta(a_{12})}{\theta(K_{1}a_{12})}\end{pmatrix}\begin{pmatrix}\displaystyle\frac{(K_{1},K_{2}a_{21})_{\infty}}{(q_{2}^{-1},a_{21})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{1}^{-1}q_{2}^{-1}&K_{2}^{-1}a_{12}q_{2}^{-1}\\\
a_{12}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\\\\[15.00002pt]
\displaystyle\frac{(K_{2},K_{1}a_{12})_{\infty}}{(q_{2}^{-1},a_{12})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}^{-1}q_{2}^{-1}&K_{1}^{-1}a_{21}q_{2}^{-1}\\\
a_{21}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\end{pmatrix}\,.$
(2.23)

In the other chamber $\mathscr{C}_{21}:a_{1}\prec a_{2}$, we instead have

$\displaystyle\Psi_{\mathscr{C}_{21}}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{\theta(a_{21})}{\theta(K_{2}a_{21})}&0\\\
\displaystyle\frac{\theta(zK_{2}a_{21},K_{2})}{\theta(K_{2}a_{21},zK_{2})}&1\end{pmatrix}\begin{pmatrix}\displaystyle\frac{(K_{1},K_{2}a_{21})_{\infty}}{(q_{2}^{-1},a_{21})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{1}^{-1}q_{2}^{-1}&K_{2}^{-1}a_{12}q_{2}^{-1}\\\
a_{12}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\\\\[15.00002pt]
\displaystyle\frac{(K_{2},K_{1}a_{12})_{\infty}}{(q_{2}^{-1},a_{12})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}^{-1}q_{2}^{-1}&K_{1}^{-1}a_{21}q_{2}^{-1}\\\
a_{21}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\end{pmatrix}\,.$
(2.24)

Change of the chambers is simply organized by the $R$-matrix constructed from
the stable envelope.

######  Proposition 2.8.

$\displaystyle\Psi_{\mathscr{C}_{12}}(a_{1,2};K_{1,2})=R_{21}\Psi_{\mathscr{C}_{21}}(a_{1,2};K_{1,2})$
(2.25)

where the elliptic $R$-matrix is given by the elliptic stable envelope,

$\displaystyle R_{21}=R_{21\to
12}=\begin{pmatrix}1&\displaystyle\frac{\theta(zK_{1}a_{12},K_{1})}{\theta(K_{1}a_{12},zK_{1})}\\\
0&\displaystyle\frac{\theta(a_{12})}{\theta(K_{1}a_{12})}\end{pmatrix}\begin{pmatrix}\displaystyle\frac{\theta(a_{21})}{\theta(K_{2}a_{21})}&0\\\
\displaystyle\frac{\theta(zK_{2}a_{21},K_{2})}{\theta(K_{2}a_{21},zK_{2})}&1\end{pmatrix}^{-1}\,,\quad
R_{12}=R_{12\to 21}=R_{21\to 12}^{-1}\,.$ (2.26)

We may use the addition formula of the theta functions to rewrite the elliptic
$R$-matrix into the original form by Felder [Fel94a, Fel94b] together with the
gauge transform [AO16].

##### Mirror symmetry

By Proposition A.2, we may rewrite the principal part of the conformal block
as follows,

$\displaystyle\begin{pmatrix}\displaystyle\frac{(K_{1},K_{2}a_{21})_{\infty}}{(q_{2}^{-1},a_{21})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{1}^{-1}q_{2}^{-1}&K_{2}^{-1}a_{12}q_{2}^{-1}\\\
a_{12}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\\\\[15.00002pt]
\displaystyle\frac{(K_{2},K_{1}a_{12})_{\infty}}{(q_{2}^{-1},a_{12})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}^{-1}q_{2}^{-1}&K_{1}^{-1}a_{21}q_{2}^{-1}\\\
a_{21}q_{2}^{-1}&-\end{matrix};q_{2}^{-1};zK_{12}\right]\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{\theta(K_{2}a_{21})}{\theta(a_{21})}\frac{(K_{1},zK_{2}q_{2}^{-1})_{\infty}}{(q_{2}^{-1},zK_{12})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}&zK_{12}\\\
zK_{2}q_{2}^{-1}\end{matrix};q_{2}^{-1};\frac{a_{12}}{q_{2}K_{2}}\right]\\\\[15.00002pt]
\displaystyle\frac{\theta(K_{1}a_{12})}{\theta(a_{12})}\frac{(K_{2},zK_{1}q_{2}^{-1})_{\infty}}{(q_{2}^{-1},zK_{12})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{1}&zK_{12}\\\
zK_{1}q_{2}^{-1}\end{matrix};q_{2}^{-1};\frac{a_{21}}{q_{2}K_{1}}\right]\end{pmatrix}\,.$
(2.27)

The $q$-hypergeometric series on the LHS are analytic when $|zK_{12}|<1$,
while it is not clear if they are all analytic in the chamber
${\mathscr{C}_{12}}:a_{1}\succ a_{2}$, i.e., $|a_{12}|>1$. Meanwhile, we may
also write the total conformal block (including the stable envelope
contribution),

$\displaystyle\Psi_{\mathscr{C}_{12}}$
$\displaystyle=\begin{pmatrix}1&\displaystyle\frac{\theta(zK_{1}a_{12},K_{1})}{\theta(a_{12},zK_{1})}\\\
0&1\end{pmatrix}\begin{pmatrix}\displaystyle\frac{(zq_{2}^{-1},K_{1})_{\infty}}{(q_{2}^{-1},zK_{1})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{2}&z^{-1}\\\
z^{-1}K_{1}^{-1}q_{2}^{-1}\end{matrix};q_{2}^{-1};\frac{a_{21}}{q_{2}K_{1}}\right]\\\\[15.00002pt]
\displaystyle\frac{(K_{2},zK_{1}q_{2}^{-1})_{\infty}}{(q_{2}^{-1},zK_{12})_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}K_{1}&zK_{12}\\\
zK_{1}q_{2}^{-1}&\end{matrix};q_{2}^{-1};\frac{a_{21}}{q_{2}K_{1}}\right]\end{pmatrix}\,,$
(2.28)

which is now clearly analytic in the chamber ${\mathscr{C}_{12}}$ through the
basis change organized by the stable envelope, implying the 3d mirror symmetry
that exchanges the twisted masses $a_{12}$ ($a_{21}$) and the FI parameter $z$
[BZ21, DN21]. From this point of view, the $R$-matrix organizes change of the
chamber $a_{12}$, $z\gtrless 1$, i.e., the wall-crossing.

####  2.1.2 Vertex function

Let $\underline{\ell}=(\ell_{1},\ldots,\ell_{n})\in(\mathbb{Z}_{\geq 0})^{n}$
obeying $\sum_{\alpha=1}^{n}\ell_{\alpha}=k$ and the corresponding coloring
function $c^{\prime}=c_{\underline{\ell}}$, such that
$\ell_{\alpha}=\\#\\{c^{\prime}(i)=\alpha,i=1,\ldots,n\\}$. We denote by
$\mathsf{P}_{\underline{\ell}}$ the set of $n$-tuple inverse partitions
$\underline{\nu}=(\nu_{1},\ldots,\nu_{\alpha})$ obeying
$\nu_{\alpha}=(\nu_{\alpha,1}\leq\cdots\leq\nu_{\alpha,\ell_{\alpha}})$. We
consider the residue of the following poles,

$\displaystyle
x_{\alpha,i}=x_{i+\sum_{\beta>\alpha}\ell_{\beta}}=a_{\alpha}q_{1}^{i-1}q_{2}^{-\nu_{\alpha,i}}\,.$
(2.29)

######  Lemma 2.9.

The residue of the integrand of $\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}$ at (2.29) is zero unless $c^{\prime}(i)\geq c(i)$ for
$i=1,\ldots,k$.

This implies the triangular property of the elliptic stable envelope. In order
to write down the vertex function, we need the following Lemma.

######  Lemma 2.10.

Let
$\emptyset_{\underline{k}}=\\{a_{\alpha}q_{1}^{i-1}\\}_{\alpha=1,\ldots,n,i=1,\ldots,k_{\alpha}}\in\mathsf{P}_{\underline{k}}$
and $(\cdot)_{\infty}=(\cdot;q_{2}^{-1})_{\infty}$. Denoting
$k_{\beta\alpha}=k_{\beta}-k_{\alpha}$, we have

$\displaystyle I_{\underline{k}}(\underline{a};\underline{K})$
$\displaystyle=\frac{(q_{2}^{-1})_{\infty}^{k}}{k!}\oint_{\emptyset_{\underline{k}}}[\differential{\underline{x}}]\frac{\prod_{i\neq
j}^{k}(x_{ij})_{\infty}}{\prod_{i,j}^{k}(q_{1}x_{ij})_{\infty}}\prod_{\begin{subarray}{c}\alpha=1,\ldots,n\\\
i=1,\ldots,k\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i})_{\infty}}{(a_{\alpha}/x_{i})_{\infty}}=\prod_{\begin{subarray}{c}\alpha,\beta=1,\ldots,n\\\
i=1,\ldots,k_{\alpha}\end{subarray}}\frac{(K_{\beta}a_{\beta\alpha}q_{1}^{1-i})_{\infty}}{(a_{\beta\alpha}q_{1}^{k_{\beta\alpha}+i})_{\infty}}\,.$
(2.30)

###### Proof.

Let $\mathbf{K}$, $\mathbf{N}$, and $\widetilde{\mathbf{N}}$ be vector spaces
(vector bundles whose fiber is given by these vector spaces on the moduli
space of quasimaps,
$\mathsf{QM}(T^{\vee}\mathbb{P}^{1})=\\{f:\mathbb{P}^{1}\to
X=T^{\vee}\mathbb{P}^{1}\\}/\sim$) with the characters,
$\operatorname{ch}\mathbf{K}=\sum_{i=1}^{k}x_{i}$,
$\operatorname{ch}\mathbf{N}=\sum_{\alpha=1}^{n}a_{\alpha}$, and
$\operatorname{ch}\widetilde{\mathbf{N}}=\sum_{\alpha=1}^{n}K_{\alpha}a_{\alpha}$.
We define $\mathbf{P}_{a}$ with $\operatorname{ch}\mathbf{P}_{a}=1-q_{a}$. For
the dual space, we have
$\operatorname{ch}\mathbf{K}^{\vee}=\sum_{i=1}^{k}x_{i}^{-1}$,
$\operatorname{ch}\mathbf{P}_{a}^{\vee}=1-q_{a}^{-1}$, and we denote by
$\wedge\mathbf{X}=\sum_{i\geq 0}(-1)^{i}\wedge^{i}\mathbf{X}$ the alternating
sum. Then, the integral is identified with the contribution of the fixed point
$\emptyset_{\underline{k}}$ of the equivariant index (equivariant K-theoretic
Euler class),

$\displaystyle
I_{\underline{k}}(\underline{a};\underline{K})=\operatorname{ch}\wedge\left[\frac{1}{\mathbf{P}_{2}^{\vee}}\left(\mathbf{P}_{1}\mathbf{K}\mathbf{K}^{\vee}+\widetilde{\mathbf{N}}\mathbf{K}^{\vee}-\mathbf{N}\mathbf{K}^{\vee}\right)\right]_{\emptyset_{\underline{k}}}\,.$
(2.31)

Since we have (see, e.g., [Sha06] for details)

$\displaystyle\operatorname{ch}\left(\mathbf{P}_{1}\mathbf{K}\mathbf{K}^{\vee}+\widetilde{\mathbf{N}}\mathbf{K}^{\vee}-\mathbf{N}\mathbf{K}^{\vee}\right)_{\emptyset_{\underline{k}}}=-\sum_{\begin{subarray}{c}\alpha,\beta=1,\ldots,n\\\
i=1,\ldots,k_{\alpha}\end{subarray}}a_{\beta\alpha}q_{1}^{k_{\beta}-k_{\alpha}+i}\,,$
(2.32)

we obtain the expression above by evaluating the index. ∎

We now write down the vertex function in general.

######  Proposition 2.11.

We denote $(\cdot)_{d}=(\cdot;q_{2}^{-1})_{d}$ and
$\theta(\cdot)=\theta(\cdot;q_{2}^{-1})$. Let
$I=i+\sum_{\alpha^{\prime}>\alpha}\ell_{\alpha^{\prime}}$,
$J=j+\sum_{\alpha^{\prime}>\beta}\ell_{\alpha^{\prime}}$, hence
$c^{\prime}(I)=\alpha$ and $c^{\prime}(J)=\beta$. We also abuse the notation,
$I=(\alpha,i)$ and $J=(\beta,j)$, to denote $\nu_{I}=\nu_{\alpha,i}$ and
$\nu_{J}=\nu_{\beta,j}$ for simplicity. Then, the contribution of the
conformal block $\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$ associated with the poles
(2.29) is given as follows,

$\displaystyle\left.\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})\right|_{\underline{\ell}}=\operatorname{Stab}_{\underline{k};\underline{\ell}}(\underline{a})\mathring{Z}_{\underline{\ell}}\sum_{\underline{\nu}\in\mathsf{P}_{\underline{\ell}}}z^{|\underline{\nu}|}{Z}_{\underline{\nu}}\,.$
(2.33)

We have the so-called perturbative part and the vortex part,

$\displaystyle\mathring{Z}_{\underline{\ell}}$
$\displaystyle=\frac{(q_{1})^{k}}{(q_{2}^{-1})^{k}}\prod_{\begin{subarray}{c}\alpha,\beta=1,\ldots,n\\\
i=1,\ldots,\ell_{\alpha}\end{subarray}}\frac{(K_{\beta}a_{\beta\alpha}q_{1}^{1-i})_{\infty}}{(a_{\beta\alpha}q_{1}^{\ell_{\beta\alpha}+i})_{\infty}}\,,$
(2.34a) $\displaystyle Z_{\underline{\nu}}$
$\displaystyle=\prod_{(\alpha,i)\neq(\beta,j)}\frac{(q_{1}^{j-i+1}a_{\beta\alpha})_{\nu_{\beta,j}-\nu_{\alpha,i}}}{(q_{1}^{j-i}a_{\beta\alpha})_{\nu_{\beta,j}-\nu_{\alpha,i}}}\prod_{\alpha,\beta=1}^{n}\prod_{i=1}^{\ell_{\alpha}}\frac{(q_{1}^{1-i}a_{\beta\alpha})_{-\nu_{\alpha,i}}}{(q_{1}^{1-i}K_{\beta}a_{\beta\alpha})_{-\nu_{\alpha,i}}}\,,$
(2.34b)

which agree with the vertex function of the Grassmannian
$X=T^{\vee}\\!\operatorname{Gr}(k,n)$ (Nakajima quiver variety associated with
$A_{1}$ quiver). The specialization of the elliptic stable envelope (the on-
shell stable envelope) is given by

$\displaystyle\operatorname{Stab}_{\underline{k};\underline{\ell}}(\underline{a})$
$\displaystyle:=\operatorname{Stab}_{2,\underline{k}}(x_{i}=a_{c^{\prime}(i)},\underline{a})$
$\displaystyle=\prod_{1\leq I<J\leq
k}\frac{\theta(q_{1}^{j-i+1}a_{\beta\alpha})}{\theta(q_{1}^{j-i}a_{\beta\alpha})}\prod_{I=1}^{k}\left(\prod_{\beta<\alpha}\frac{\theta(q_{1}^{1-i}a_{\beta\alpha})}{\theta(q_{1}^{1-i}K_{\beta}a_{\beta\alpha})}\right)\frac{\theta(q_{1}^{1-i-2\rho_{I}}zK_{1\ldots\alpha})\theta(K_{\alpha})}{\theta(q_{1}^{1-i}K_{\alpha})\theta(q_{1}^{-2\rho_{I}}zK_{1\ldots\alpha})}\,.$
(2.35)

###### Proof.

The perturbative part can be directly obtained by Lemma 2.10. It is
straightforward to evaluate the residue to obtain the vortex contributions,

$\displaystyle\frac{1}{k!}\oint_{\\{\underline{\nu}\\}}[\differential{\underline{x}}]\,\Phi_{2}(\underline{x},\underline{a})$
$\displaystyle=\mathring{Z}_{\underline{\ell}}Z_{\underline{\nu}}\,.$ (2.36)

We then consider the stable envelope. We have

$\displaystyle\left.\prod_{1\leq I<J\leq
k}\frac{\theta(q_{1}x_{JI})}{\theta(x_{JI})}\right|_{\\{\underline{\nu}\\}}$
$\displaystyle=\prod_{1\leq I<J\leq
k}\frac{\theta(q_{2}^{\nu_{I}-\nu_{J}}q_{1}^{j-i+1}a_{\beta\alpha})}{\theta(q_{2}^{\nu_{I}-\nu_{J}}q_{1}^{j-i}a_{\beta\alpha})}=\prod_{1\leq
I<J\leq
k}q_{1}^{\nu_{I}-\nu_{J}}\frac{\theta(q_{1}^{j-i+1}a_{\beta\alpha})}{\theta(q_{1}^{j-i}a_{\beta\alpha})}$
$\displaystyle=q_{1}^{\sum_{I=1}^{k}2\rho_{I}\nu_{I}}\prod_{1\leq I<J\leq
k}\frac{\theta(q_{1}^{j-i+1}a_{\beta\alpha})}{\theta(q_{1}^{j-i}a_{\beta\alpha})}\,,$
(2.37)

and

$\displaystyle\left.\prod_{I=1}^{k}\left(\prod_{\beta<\alpha}\frac{\theta(a_{\beta}/x_{I})}{\theta(K_{\beta}a_{\beta}/x_{I})}\right)\frac{\theta(zq_{1}^{-2\rho_{I}}K_{1\ldots\alpha}a_{\alpha}/x_{I})\theta(K_{\alpha})}{\theta(K_{\alpha}a_{\alpha}/x_{I})\theta(zq_{1}^{-2\rho_{I}}K_{1\ldots\alpha})}\right|_{\\{\underline{\nu}\\}}$
$\displaystyle=\prod_{I=1}^{k}\left(\prod_{\beta<\alpha}\frac{\theta(q_{2}^{\nu_{I}}q_{1}^{1-i}a_{\beta\alpha})}{\theta(q_{2}^{\nu_{I}}q_{1}^{1-i}K_{\beta}a_{\beta\alpha})}\right)\frac{\theta(q_{2}^{\nu_{I}}q_{1}^{1-i-2\rho_{I}}zK_{1\ldots\alpha})\theta(K_{\alpha})}{\theta(q_{2}^{\nu_{I}}q_{1}^{1-i}K_{\alpha})\theta(q_{1}^{-2\rho_{I}}zK_{1\ldots\alpha})}$
$\displaystyle=\prod_{I=1}^{k}\left(zq_{1}^{-2\rho_{I}}\right)^{\nu_{I}}\left(\prod_{\beta<\alpha}\frac{\theta(q_{1}^{1-i}a_{\beta\alpha})}{\theta(q_{1}^{1-i}K_{\beta}a_{\beta\alpha})}\right)\frac{\theta(q_{1}^{1-i-2\rho_{I}}zK_{1\ldots\alpha})\theta(K_{\alpha})}{\theta(q_{1}^{1-i}K_{\alpha})\theta(q_{1}^{-2\rho_{I}}zK_{1\ldots\alpha})}\,.$
(2.38)

Hence, the specialization of the stable envelope yields

$\displaystyle\operatorname{Stab}_{2,\underline{k}}(\underline{x},\underline{a})\Big{|}_{\\{\underline{\nu}\\}}$
$\displaystyle=z^{\sum_{I=1}^{k}\nu_{I}}\prod_{1\leq I<J\leq
k}\frac{\theta(q_{1}^{j-i+1}a_{\beta\alpha})}{\theta(q_{1}^{j-i}a_{\beta\alpha})}\prod_{I=1}^{k}\left(\prod_{\beta<\alpha}\frac{\theta(q_{1}^{1-i}a_{\beta\alpha})}{\theta(q_{1}^{1-i}K_{\beta}a_{\beta\alpha})}\right)\frac{\theta(q_{1}^{1-i-2\rho_{I}}zK_{1\ldots\alpha})\theta(K_{\alpha})}{\theta(q_{1}^{1-i}K_{\alpha})\theta(q_{1}^{-2\rho_{I}}zK_{1\ldots\alpha})}$
$\displaystyle=z^{|\underline{\nu}|}\operatorname{Stab}_{\underline{k};\underline{\ell}}(\underline{a})\,.$
(2.39)

This completes the proof. ∎

##### 5d gauge theory partition function

We compare the vertex function with 5d gauge theory partition function
(K-theoretic Nekrasov partition function). Let
$\underline{\lambda}=(\lambda_{\alpha})_{\alpha=1,\ldots,n}=(\lambda_{\alpha,1}\geq\lambda_{\alpha,2}\geq\cdots)_{\begin{subarray}{c}\alpha=1,\ldots,n\\\
i=1,\ldots,\infty\end{subarray}}$ be an $n$-tuple partition. The full
partition function of 5d $\mathcal{N}=1$ $\mathrm{U}(n)$ gauge theory on
$\mathbb{C}^{2}\times\mathbb{S}^{1}$ with $n$ fundamental and $n$
antifundamental matters is given by444Compared with [KP15, Kim20], the
$q$-parameters are flipped, $q_{1,2}\leftrightarrow q_{1,2}^{-1}$, and
slightly modify the notation.

$\displaystyle
Z^{\text{5d}}=\sum_{\underline{\lambda}}\mathfrak{q}^{|\underline{\lambda}|}Z^{\text{5d}}_{\underline{\lambda}}\,,$
(2.40)

where

$\displaystyle
Z^{\text{5d}}_{\underline{\lambda}}=Z^{\text{5d}}_{\underline{\lambda}}(\underline{b},\underline{m},\underline{\widetilde{m}};q_{1,2})=\prod_{(\alpha,i)\neq(\beta,j)}\frac{(b_{\beta\alpha}q_{2}^{\lambda_{\alpha,i}-\lambda_{\beta,j}}q_{1}^{i-j};q_{2}^{-1})_{\infty}}{(b_{\beta\alpha}q_{2}^{\lambda_{\alpha,i}-\lambda_{\beta,j}}q_{1}^{i-j+1};q_{2}^{-1})_{\infty}}\prod_{\begin{subarray}{c}\alpha,f=1,\ldots,n\\\
i=1,\ldots,\infty\end{subarray}}\frac{(\frac{\widetilde{m}_{f}}{b_{\alpha}}q_{2}^{\lambda_{\alpha,i}}q_{1}^{i};q_{2}^{-1})_{\infty}}{(\frac{m_{f}}{b_{\alpha}}q_{2}^{\lambda_{\alpha,i}}q_{1}^{i};q_{2}^{-1})_{\infty}}\,.$
(2.41)

We denote the instanton fugacity by $\mathfrak{q}$, the multiplicative Coulomb
moduli by $\underline{b}=\\{b_{\alpha}\\}_{\alpha=1,\ldots,n}$, the mass
parameters by $\underline{m}=\\{m_{f}\\}_{f=1,\ldots,n}$ and
$\underline{\widetilde{m}}=\\{\widetilde{m}_{f}\\}_{f=1,\ldots,n}$.

######  Proposition 2.12.

The 5d gauge theory partition function agrees with the vertex function under
the identification (the root of Higgs branch) as follows,

$\displaystyle\mathfrak{q}=z\,,\quad
b_{\alpha}=\widetilde{m}_{\alpha}=a_{\alpha}q_{1}^{-\ell_{\alpha}}\,,\quad
m_{\alpha}=a_{\alpha}\,,\quad
K_{\alpha}=\frac{m_{\alpha}}{\widetilde{m}_{\alpha}}=q_{1}^{\ell_{\alpha}}\,,\quad\lambda_{\alpha,i}=\nu_{\alpha,\ell_{\alpha}-i+1}\,,\quad(\alpha=1,\ldots,n)\,.$
(2.42)

###### Proof.

Since $K_{\alpha}=q_{1}^{\ell_{\alpha}}$, the length of the partition
$\lambda_{\alpha}$ is restricted by Proposition 2.7, and thus the range of the
index $i$ becomes $\\{1,\ldots,\ell_{\alpha}\\}$ for $\alpha=1,\ldots,n$.
Then, denoting $\ell_{\beta\alpha}=\ell_{\beta}-\ell_{\alpha}$, we have

$\displaystyle Z^{\text{5d}}_{\underline{\lambda}}$
$\displaystyle=\prod_{(\alpha,i)\neq(\beta,j)}\frac{(b_{\beta\alpha}q_{2}^{\lambda_{\alpha,i}-\lambda_{\beta,j}}q_{1}^{i-j};q_{2}^{-1})_{\infty}}{(b_{\beta\alpha}q_{2}^{\lambda_{\alpha,i}-\lambda_{\beta,j}}q_{1}^{i-j+1};q_{2}^{-1})_{\infty}}\prod_{\begin{subarray}{c}\alpha,f=1,\ldots,n\\\
i=1,\ldots,\ell_{\alpha}\end{subarray}}\frac{(\frac{\widetilde{m}_{f}}{b_{\alpha}}q_{2}^{\lambda_{\alpha,i}}q_{1}^{i};q_{2}^{-1})_{\infty}}{(\frac{m_{f}}{b_{\alpha}}q_{2}^{\lambda_{\alpha,i}}q_{1}^{i};q_{2}^{-1})_{\infty}}$
$\displaystyle=\prod_{(\alpha,i)\neq(\beta,j)}\frac{(b_{\beta\alpha}q_{2}^{\lambda_{\alpha,\ell_{\alpha}+1-i}-\lambda_{\beta,\ell_{\beta}+1-j}}q_{1}^{\ell_{\beta\alpha}+j-i};q_{2}^{-1})_{\infty}}{(b_{\beta\alpha}q_{2}^{\lambda_{\alpha,\ell_{\alpha}+1-i}-\lambda_{\beta,\ell_{\beta}+1-j}}q_{1}^{\ell_{\beta\alpha}+j-i+1};q_{2}^{-1})_{\infty}}\prod_{\begin{subarray}{c}\alpha,f=1,\ldots,n\\\
i=1,\ldots,\ell_{\alpha}\end{subarray}}\frac{(\frac{\widetilde{m}_{f}}{b_{\alpha}}q_{2}^{\lambda_{\alpha,\ell_{\alpha}+1-i}}q_{1}^{\ell_{\alpha}+1-i};q_{2}^{-1})_{\infty}}{(\frac{m_{f}}{b_{\alpha}}q_{2}^{\lambda_{\alpha,\ell_{\alpha}+1-i}}q_{1}^{\ell_{\alpha}+1-i};q_{2}^{-1})_{\infty}}\,,$
(2.43)

which agrees with the vertex function under the identification (2.42). ∎

This phenomenon is known to be the triality between the $q$-deformed W-algebra
conformal block and 3d/5d gauge theory partition functions [AHKS13, AHS14,
AH15]. As mentioned in §2.2, the 3d/5d correspondence would be interpreted as
a two-dimensional analog of the one-leg DT/PT correspondence.

####  2.1.3 Elliptic conformal block

We consider an elliptic analog of the conformal block discussed in Proposition
2.4,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$
$\displaystyle=\tr\left[p^{d}\mathsf{Z}^{(k_{1})}_{K_{1}}(a_{1})\mathsf{Z}^{(k_{2})}_{K_{2}}(a_{2})\cdots\mathsf{Z}^{(k_{n})}_{K_{n}}(a_{n})\right]/\tr\left[p^{d}\mathsf{Z}_{K_{1}}(a_{1})\mathsf{Z}_{K_{2}}(a_{2})\cdots\mathsf{Z}_{K_{n}}(a_{n})\right]$
$\displaystyle=\frac{\mathsf{c}_{\underline{k}}(\underline{K})}{k!}\oint_{\gamma}[\differential{\underline{x}}]\,\Phi_{2}(\underline{x},\underline{a})\operatorname{Stab}_{\underline{k}}(\underline{x},\underline{a})\,,$
(2.44)

with the elliptic nome $p\in\mathbb{C}^{\times}$ and the principal part,

$\displaystyle\Phi_{\sigma}(\underline{x},\underline{a})=\prod_{i\neq
j}^{k}\frac{\Gamma(q_{\bar{\sigma}}x_{j}/x_{i};q_{\sigma}^{-1},p)}{\Gamma(x_{j}/x_{i};q_{\sigma}^{-1},p)}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{\Gamma(a_{\alpha}/x_{i};q_{\sigma}^{-1},p)}{\Gamma(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1},p)}\,,$
(2.45)

where we denote by $\Gamma(\cdot)$ the elliptic gamma function (A.4), and the
trace is taken over the Fock space generated by all the free fields and $d$ is
the corresponding degree counting operator. See, e.g., [KP16]. This principal
part is reduced to the previous one presented in (2.12) in the limit $p\to 0$.
This elliptic conformal block is identified with the partition function of 4d
mass-deformed $\mathcal{N}=2$ gauge theory (softly broken to $\mathcal{N}=1$
theory) on $\mathbb{C}_{q_{2}}\times\mathcal{E}_{p}$ [LNP19].

##### Elliptic vertex function

Let us focus on the simplest example, which is an elliptic analog of the
conformal block (2.20) associated with quasimaps to $T^{\vee}\mathbb{P}^{1}$.
We write $\Gamma(\cdot)=\Gamma(\cdot;q_{2}^{-1},p)$ and
$\theta(\cdot)=\theta(\cdot;q_{2}^{-1})$. Then, in the chamber
$\mathscr{C}_{12}:a_{1}\succ a_{2}$, we have555We can similarly obtain the
vertex function for generic $(n,k)$ given as a summation over the inverse
partitions as discussed in §2.1.2.

$\displaystyle\Psi_{\mathscr{C}_{12}}^{(1,0)}(a_{1,2};K_{1,2})$
$\displaystyle=\oint_{\gamma}\frac{\differential{x}}{2\pi\mathrm{i}x}\frac{\Gamma(a_{1,2}/x)}{\Gamma(K_{1,2}a_{1,2}/x)}\frac{\theta(zK_{1}a_{1}/x)\theta(K_{1})}{\theta(K_{1}a_{1}/x)\theta(zK_{1})}\,,$
(2.46a) $\displaystyle\Psi_{\mathscr{C}_{12}}^{(0,1)}(a_{1,2};K_{1,2})$
$\displaystyle=\oint_{\gamma}\frac{\differential{x}}{2\pi\mathrm{i}x}\frac{\Gamma(a_{1,2}/x)}{\Gamma(K_{1,2}a_{1,2}/x)}\frac{\theta(zK_{12}a_{2}/x)\theta(K_{2})}{\theta(K_{2}a_{2}/x)\theta(zK_{12})}\frac{\theta(a_{1}/x)}{\theta(K_{1}a_{1}/x)}\,.$
(2.46b)

Similarly, in the other chamber $\mathscr{C}_{21}:a_{1}\prec a_{2}$, we have

$\displaystyle\Psi_{\mathscr{C}_{21}}^{(1,0)}(a_{1,2};K_{1,2})$
$\displaystyle=\oint_{\gamma}\frac{\differential{x}}{2\pi\mathrm{i}x}\frac{\Gamma(a_{1,2}/x)}{\Gamma(K_{1,2}a_{1,2}/x)}\frac{\theta(zK_{12}a_{1}/x)\theta(K_{1})}{\theta(K_{1}a_{1}/x)\theta(zK_{12})}\frac{\theta(a_{2}/x)}{\theta(K_{2}a_{2}/x)}\,,$
(2.47a) $\displaystyle\Psi_{\mathscr{C}_{21}}^{(0,1)}(a_{1,2};K_{1,2})$
$\displaystyle=\oint_{\gamma}\frac{\differential{x}}{2\pi\mathrm{i}x}\frac{\Gamma(a_{1,2}/x)}{\Gamma(K_{1,2}a_{1,2}/x)}\frac{\theta(zK_{2}a_{2}/x)\theta(K_{2})}{\theta(K_{2}a_{2}/x)\theta(zK_{2})}\,.$
(2.47b)

######  Lemma 2.13.

Taking the contour integral, we have the elliptic vertex function associated
with each vacuum,

$\displaystyle\Psi_{\mathscr{C}_{12}}^{(1,0)}(a_{1,2};K_{1,2})\Big{|}_{\\{a_{1}q_{2}^{-d}\\}}$
$\displaystyle=\frac{\Gamma(q_{2}^{-1},a_{21})}{\Gamma(K_{1},K_{2}a_{21})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{1}&q_{2}^{-1}K_{2}a_{21}\\\
q_{2}^{-1}a_{21}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\,,$ (2.48a)
$\displaystyle\Psi_{\mathscr{C}_{12}}^{(1,0)}(a_{1,2};K_{1,2})\Big{|}_{\\{a_{2}q_{2}^{-d}\\}}$
$\displaystyle=\frac{\theta(zK_{1}a_{12},K_{1})}{\theta(K_{1}a_{12},zK_{1})}\frac{\Gamma(q_{2}^{-1},a_{12})}{\Gamma(K_{2},K_{1}a_{12})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{2}&q_{2}^{-1}K_{1}a_{12}\\\
q_{2}^{-1}a_{12}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\,,$ (2.48b)
$\displaystyle\Psi_{\mathscr{C}_{12}}^{(0,1)}(a_{1,2};K_{1,2})\Big{|}_{\\{a_{1}q_{2}^{-d}\\}}$
$\displaystyle=0\,,$ (2.48c)
$\displaystyle\Psi_{\mathscr{C}_{12}}^{(0,1)}(a_{1,2};K_{1,2})\Big{|}_{\\{a_{2}q_{2}^{-d}\\}}$
$\displaystyle=\frac{\theta(a_{12})}{\theta(K_{1}a_{12})}\frac{\Gamma(q_{2}^{-1},a_{12})}{\Gamma(K_{2},K_{1}a_{12})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{2}&q_{2}^{-1}K_{1}a_{12}\\\
q_{2}^{-1}a_{12}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\,,$ (2.48d)

where ${}_{2}E_{1}$ is the elliptic hypergeometric series (Definition A.4).
These contributions are summarized in a matrix form as follows,

$\displaystyle\Psi_{\mathscr{C}_{12}}(a_{1,2};K_{1,2})$
$\displaystyle=\begin{pmatrix}\Psi_{\mathscr{C}_{12}}^{(1,0)}(a_{1,2};K_{1,2})\\\
\Psi_{\mathscr{C}_{12}}^{(0,1)}(a_{1,2};K_{1,2})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&\displaystyle\frac{\theta(zK_{1}a_{12},K_{1})}{\theta(K_{1}a_{12},zK_{1})}\\\
0&\displaystyle\frac{\theta(a_{12})}{\theta(K_{1}a_{12})}\end{pmatrix}\begin{pmatrix}\displaystyle\frac{\Gamma(q_{2}^{-1},a_{21})}{\Gamma(K_{1},K_{2}a_{21})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{1}&q_{2}^{-1}K_{2}a_{21}\\\
q_{2}^{-1}a_{21}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\\\\[15.00002pt]
\displaystyle\frac{\Gamma(q_{2}^{-1},a_{12})}{\Gamma(K_{2},K_{1}a_{12})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{2}&q_{2}^{-1}K_{1}a_{12}\\\
q_{2}^{-1}a_{12}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\end{pmatrix}\,.$
(2.49)

Similarly, we have

$\displaystyle\Psi_{\mathscr{C}_{21}}(a_{1,2};K_{1,2})$
$\displaystyle=\begin{pmatrix}\Psi_{\mathscr{C}_{21}}^{(1,0)}(a_{1,2};K_{1,2})\\\
\Psi_{\mathscr{C}_{21}}^{(0,1)}(a_{1,2};K_{1,2})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\displaystyle\frac{\theta(a_{21})}{\theta(K_{2}a_{21})}&0\\\
\displaystyle\frac{\theta(zK_{2}a_{21},K_{2})}{\theta(K_{2}a_{21},zK_{2})}&1\end{pmatrix}\begin{pmatrix}\displaystyle\frac{\Gamma(q_{2}^{-1},a_{21})}{\Gamma(K_{1},K_{2}a_{21})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{1}&q_{2}^{-1}K_{2}a_{21}\\\
q_{2}^{-1}a_{21}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\\\\[15.00002pt]
\displaystyle\frac{\Gamma(q_{2}^{-1},a_{12})}{\Gamma(K_{2},K_{1}a_{12})}{{}_{2}E_{1}}\left[\begin{matrix}q_{2}^{-1}K_{2}&q_{2}^{-1}K_{1}a_{12}\\\
q_{2}^{-1}a_{12}\end{matrix};q_{2}^{-1},p;zK_{12}\right]\end{pmatrix}\,,$
(2.50)

from which we obtain the elliptic $R$-matrix as before (Proposition 2.8).

##### Elliptic $q$-KZ equation

Let us consider the difference equation for the two-point elliptic conformal
block.

######  Lemma 2.14.

We have

$\displaystyle\Psi^{(1,0)}_{\mathscr{C}_{12}}(a_{1},pa_{2};K_{1,2})=\Psi^{(1,0)}_{\mathscr{C}_{21}}(a_{1},a_{2};K_{1,2})\Big{|}_{z\to
zK_{2}^{-1}}\,,\quad\Psi^{(0,1)}_{\mathscr{C}_{12}}(a_{1},pa_{2};K_{1,2})=\Psi^{(0,1)}_{\mathscr{C}_{21}}(a_{1},a_{2};K_{1,2})\Big{|}_{z\to
zK_{1}}\,.$ (2.51)

###### Proof.

This immediately follows from the integral formulas (2.46) and (2.47). ∎

Defining the weight factor (see also (2.8)),

$\displaystyle\kappa_{\alpha}=K_{\alpha}q_{1}^{-2k_{\alpha}}\,,$ (2.52)

and imposing the zero weight condition, $K_{12}=q_{1}^{2}$, we may rewrite

$\displaystyle\Psi^{(1,0)}_{\mathscr{C}_{12}}(a_{1},pa_{2};K_{1,2})=\Psi^{(1,0)}_{\mathscr{C}_{21}}(a_{1},a_{2};K_{1,2})\Big{|}_{z\to
z\kappa_{2}^{-1}}\,,\quad\Psi^{(0,1)}_{\mathscr{C}_{12}}(a_{1},pa_{2};K_{1,2})=\Psi^{(0,1)}_{\mathscr{C}_{21}}(a_{1},a_{2};K_{1,2})\Big{|}_{z\to
z\kappa_{2}^{-1}}\,.$ (2.53)

Hence, we have

$\displaystyle\Psi_{\mathscr{C}_{12}}\Big{|}_{a_{2}\to
pa_{2}}=\Psi_{\mathscr{C}_{21}}\Big{|}_{z\to z\kappa_{2}^{-1}}\,.$ (2.54)

We remark that the zero weight condition was first discussed in [FJM+93]. See
also a recent manuscript on this subject [Kon20].

######  Proposition 2.15.

Under the zero weight condition $K_{12}=q_{1}^{2}$, we have the elliptic
$q$-KZ equation,

$\displaystyle\Psi_{\mathscr{C}_{12}}\Big{|}_{\begin{subarray}{c}z\to
z\kappa_{2}\\\ a_{2}\to pa_{2}\end{subarray}}=\Psi_{\mathscr{C}_{21}}=R_{12\to
21}\Psi_{\mathscr{C}_{12}}\,.$ (2.55)

Similarly, we have

$\displaystyle\Psi_{\mathscr{C}_{12}}\Big{|}_{\begin{subarray}{c}z\to
z\kappa_{1}\\\ a_{1}\to pa_{1}\end{subarray}}=R_{21\to
12}\Psi_{\mathscr{C}_{21}}\Big{|}_{\begin{subarray}{c}z\to z\kappa_{1}\\\
a_{1}\to pa_{1}\end{subarray}}=R_{21\to 12}\Big{|}_{\begin{subarray}{c}z\to
z\kappa_{1}\\\ a_{1}\to pa_{1}\end{subarray}}\Psi_{\mathscr{C}_{12}}\,.$
(2.56)

This is the simplest case of the elliptic $q$-KZ equation. Let us discuss a
generic situation as follows.

######  Lemma 2.16.

Let $\Psi_{\mathscr{C}_{1\ldots n}}$ be the elliptic conformal block in the
chamber $\mathscr{C}_{1\ldots n}$. Then, under the zero weight condition
$K_{1}\cdots K_{n}=q_{1}^{2k}$, the following holds,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}^{\underline{k}}\Big{|}_{a_{n}\to
pa_{n}}=\Psi_{\mathscr{C}_{n1\ldots n-1}}^{\underline{k}}\Big{|}_{z\to
z\kappa_{n}^{-1}}\,,$ (2.57)

where $\kappa_{n}$ is the weight factor defined in (2.52).

###### Proof.

Denoting $\Gamma(\cdot)=\Gamma(\cdot;q_{2}^{-1},p)$ and
$\theta(\cdot)=\theta(\cdot,q_{2}^{-1})$, the elliptic conformal block
associated with the configuration $\underline{k}=(k_{1},\ldots,k_{n})$ is
given by

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}^{\underline{k}}$
$\displaystyle=\frac{1}{k!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{i\neq
j}^{k}\frac{\Gamma(q_{1}x_{ij})}{\Gamma(x_{ij})}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{\Gamma(a_{\alpha}/x_{i})}{\Gamma(K_{\alpha}a_{\alpha}/x_{i})}$
$\displaystyle\qquad\times\operatorname*{Sym}_{\underline{x}}\prod_{i<j}^{k}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}\prod_{i=1}^{k}\left[\left(\prod_{\alpha<c(i)}\frac{\theta(a_{\alpha}/x_{i})}{\theta(K_{\alpha}a_{\alpha}/x_{i})}\right)\frac{\theta(zq_{1}^{-2\rho_{i}}K_{i}^{c}a_{c(i)}/x_{i})\theta(K_{c(i)})}{\theta(K_{c(i)}a_{c(i)}/x_{i})\theta(zq_{1}^{-2\rho_{i}}K_{i}^{c})}\right]\,.$
(2.58)

Then, recalling $K_{i}^{c}=K_{1}K_{2}\cdots K_{c(i)}=K_{1\ldots c(i)}$, we
have

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}^{\underline{k}}\Big{|}_{a_{n}\to
pa_{n}}$
$\displaystyle=\frac{1}{k!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{i\neq
j}^{k}\frac{\Gamma(q_{1}x_{ij})}{\Gamma(x_{ij})}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{\Gamma(a_{\alpha}/x_{i})}{\Gamma(K_{\alpha}a_{\alpha}/x_{i})}\operatorname*{Sym}_{\underline{x}}\prod_{i<j}^{k}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}$
$\displaystyle\qquad\times\prod_{i=k_{n}+1}^{k}\left[\frac{\theta(a_{n}/x_{i})}{\theta(K_{n}a_{n}/x_{i})}\left(\prod_{\alpha<c(i)}\frac{\theta(a_{\alpha}/x_{i})}{\theta(K_{\alpha}a_{\alpha}/x_{i})}\right)\frac{\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
c(i)}a_{c(i)}/x_{i})\theta(K_{c(i)})}{\theta(K_{c(i)}a_{c(i)}/x_{i})\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
c(i)})}\right]$
$\displaystyle\qquad\times\prod_{i=1}^{k_{n}}\left[\left(\prod_{\alpha=1}^{n}\frac{\theta(a_{\alpha}/x_{i})}{\theta(K_{\alpha}a_{\alpha}/x_{i})}\right)\frac{\theta(pzq_{1}^{-2\rho_{i}}K_{1\ldots
n}a_{n}/x_{i})\theta(K_{n})}{\theta(pK_{n}a_{n}/x_{i})\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
n})}\right]$ $\displaystyle\stackrel{{\scriptstyle\begin{subarray}{c}x_{i}\to
px_{i}\\\
i=1,\ldots,k_{n}\end{subarray}}}{{=}}\frac{1}{k!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{i\neq
j}^{k}\frac{\Gamma(q_{1}x_{ij})}{\Gamma(x_{ij})}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{\Gamma(a_{\alpha}/x_{i})}{\Gamma(K_{\alpha}a_{\alpha}/x_{i})}\operatorname*{Sym}_{\underline{x}}\prod_{1\leq
i<j\leq k}^{\prime}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}$
$\displaystyle\qquad\times\prod_{i=k_{n}+1}^{k}\left[\frac{\theta(a_{n}/x_{i})}{\theta(K_{n}a_{n}/x_{i})}\left(\prod_{\alpha<c(i)}\frac{\theta(a_{\alpha}/x_{i})}{\theta(K_{\alpha}a_{\alpha}/x_{i})}\right)\frac{\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
c(i)}a_{c(i)}/x_{i})\theta(K_{c(i)})}{\theta(K_{c(i)}a_{c(i)}/x_{i})\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
c(i)})}\right]$
$\displaystyle\qquad\times\prod_{i=1}^{k_{n}}\left[\frac{\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
n}a_{n}/x_{i})\theta(K_{n})}{\theta(K_{n}a_{n}/x_{i})\theta(zq_{1}^{-2\rho_{i}}K_{1\ldots
n})}\right]\,,$ (2.59)

where we denote

$\displaystyle\prod_{1\leq i<j\leq
k}^{\prime}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}=\prod_{1\leq i\leq
k_{n}<j\leq k}\frac{\theta(q_{1}x_{ij})}{\theta(x_{ij})}\prod_{1\leq i<j\leq
k_{n}}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}\prod_{k_{n}+1\leq i<j\leq
k}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}\,.$ (2.60)

Moreover, due to the symmetrization, we may change the ordering of the
$x$-variables,
$(x_{1},\ldots,x_{k})\mapsto(x_{k_{n}+1},\ldots,x_{k},x_{1},\ldots,x_{k_{n}})$,
to compute

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}^{\underline{k}}\Big{|}_{a_{n}\to
pa_{n}}$
$\displaystyle=\frac{1}{k!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{i\neq
j}^{k}\frac{\Gamma(q_{1}x_{ij})}{\Gamma(x_{ij})}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{\Gamma(a_{\alpha}/x_{i})}{\Gamma(K_{\alpha}a_{\alpha}/x_{i})}\operatorname*{Sym}_{\underline{x}}\prod_{1\leq
i<j\leq k}\frac{\theta(q_{1}x_{ji})}{\theta(x_{ji})}$
$\displaystyle\qquad\times\prod_{i=1}^{k-k_{n}}\left[\frac{\theta(a_{n}/x_{i})}{\theta(K_{n}a_{n}/x_{i})}\left(\prod_{\alpha<c(i)}\frac{\theta(a_{\alpha}/x_{i})}{\theta(K_{\alpha}a_{\alpha}/x_{i})}\right)\frac{\theta(zq_{1}^{-2\rho_{i+k_{n}}}K_{1\ldots
c(i)}a_{c(i)}/x_{i})\theta(K_{c(i)})}{\theta(K_{c(i)}a_{c(i)}/x_{i})\theta(zq_{1}^{i+k_{n}-k}K_{1\ldots
c(i)})}\right]$
$\displaystyle\qquad\times\prod_{i=k-k_{n}+1}^{k}\left[\frac{\theta(zq_{1}^{-2\rho_{i+k_{n}-k}}K_{1\ldots
n}a_{n}/x_{i})\theta(K_{n})}{\theta(K_{n}a_{n}/x_{i})\theta(zq_{1}^{i+k_{n}-2k}K_{1\ldots
n})}\right]\,.$ (2.61)

This is identified with $\Psi_{\mathscr{C}_{n1\ldots
n-1}}^{\underline{k}}\Big{|}_{z\to zK_{n}^{-1}q_{1}^{k_{n}}}$ under the zero
weight condition $K_{1\ldots n}=q_{1}^{2k}$. ∎

Let

$\displaystyle\kappa_{\alpha}^{c}=\prod_{\alpha^{\prime}<\alpha}\kappa_{\alpha^{\prime}}\,.$
(2.62)

Then, we have the dynamical elliptic $q$-KZ equation for the elliptic
conformal block.

######  Theorem 2.17.

The elliptic conformal block obeys the dynamical elliptic $q$-KZ equation,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}\Big{|}_{\begin{subarray}{c}z\to
z\kappa_{\alpha}\\\ a_{\alpha}\to
pa_{\alpha}\end{subarray}}=R_{\alpha+1,\alpha}(z\kappa^{c}_{\alpha+1})\cdots
R_{n,\alpha}(z\kappa^{c}_{n})\Big{|}_{a_{\alpha}\to
pa_{\alpha}}R_{1,\alpha}(z)\cdots
R_{\alpha-1,\alpha}(z\kappa^{c}_{\alpha-1})\Psi_{\mathscr{C}_{1\ldots n}}\,.$
(2.63)

The zero weight condition implies the periodicity
$\kappa_{\alpha}^{c}=\kappa_{\alpha+n}^{c}$ for $\alpha=1,\ldots,n$.

This elliptic $q$-KZ solution was first constructed in the context of elliptic
quantum group $U_{q,p}(\widehat{\mathfrak{sl}}_{2})$ [FJM+93, Fel94a, Kon17].
We emphasize that the conformal block of the W-algebra
$\mathrm{W}_{q_{1,2}}(A_{1})$ itself satisfies the $q$-KZ equation, directly
establishing the quantum $q$-Langlands correspondence between the electric
block and the magnetic block. Moreover, we do not need to insert an additional
stable envelope for the elliptic setup compared with the trigonometric
situation [AO16].

####  2.1.4 Trigonometric conformal block

We denote $(\cdot)_{\infty}=(\cdot;q_{2}^{-1})_{\infty}$ and
$[\cdot]=\operatorname{PE}[\cdot]^{-1}$. We consider the trigonometric
conformal block associated with the configuration
$\underline{k}=(k_{1},\ldots,k_{n})$ given by

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}^{\underline{k}}$
$\displaystyle=\frac{1}{k!}\exp\left(\frac{\log z}{\log
q_{2}^{-1}}\sum_{\alpha=1}^{n}\frac{\mu_{\alpha}}{2}\log
a_{\alpha}\right)\oint_{\gamma}[\differential{\underline{x}}]\prod_{i=1}^{k}x_{i}^{-\frac{\log
z}{\log q_{2}^{-1}}}\prod_{i\neq
j}^{k}\frac{(x_{ij})_{\infty}}{(q_{1}x_{ij})_{\infty}}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i})_{\infty}}{(a_{\alpha}/x_{i})_{\infty}}$
$\displaystyle\qquad\times\operatorname*{Sym}_{\underline{x}}\left[\prod_{i<j}^{k}\frac{[q_{1}x_{ji}]}{[x_{ji}]}\prod_{i=1}^{k}\prod_{\alpha<c(i)}\frac{[a_{\alpha}/x_{i}]}{[K_{\alpha}a_{\alpha}/x_{i}]}\right]\,.$
(2.64)

######  Lemma 2.18.

The trigonometric version of Lemma 2.16 holds,

$\displaystyle\Psi^{\underline{k}}_{\mathscr{C}_{1\ldots n}}\Big{|}_{a_{n}\to
q_{2}^{-1}a_{n}}=z^{\frac{\mu_{n}}{2}-k_{n}}\Psi^{\underline{k}}_{\mathscr{C}_{n1\ldots
n-1}}\,,$ (2.65)

where the factor $\frac{\mu_{n}}{2}-k_{n}$ corresponds to the weight
associated with the $n$-th operator (see also (2.52)).

###### Proof.

We have

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}^{\underline{k}}\Big{|}_{a_{n}\to
q_{2}^{-1}a_{n}}$
$\displaystyle=\frac{z^{\frac{\mu_{\alpha}}{2}}}{k!}\exp\left(\frac{\log
z}{\log q_{2}^{-1}}\sum_{\alpha=1}^{n}\frac{\mu_{\alpha}}{2}\log
a_{\alpha}\right)\oint_{\gamma}\prod_{i=1}^{k}\frac{\differential{x}_{i}}{2\pi\mathrm{i}x_{i}}x_{i}^{-\frac{\log
z}{\log q_{2}^{-1}}}\prod_{i\neq
j}^{k}\frac{(x_{ij})_{\infty}}{(q_{1}x_{ij})_{\infty}}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i})_{\infty}}{(a_{\alpha}/x_{i})_{\infty}}$
$\displaystyle\qquad\times\operatorname*{Sym}_{\underline{x}}\prod_{i<j}^{k}\frac{[q_{1}x_{ji}]}{[x_{ji}]}\prod_{i=1}^{k}\left[\frac{[a_{n}/x_{i}]}{[K_{n}a_{n}/x_{i}]}\prod_{\alpha<c(i)}\frac{[a_{\alpha}/x_{i}]}{[K_{\alpha}a_{\alpha}/x_{i}]}\right]$
$\displaystyle\stackrel{{\scriptstyle\begin{subarray}{c}x_{i}\to
q_{2}^{-1}x_{i}\\\
i=1,\ldots,k_{n}\end{subarray}}}{{=}}\frac{z^{\frac{\mu_{\alpha}}{2}-k_{n}}}{k!}\exp\left(\frac{\log
z}{\log q_{2}^{-1}}\sum_{\alpha=1}^{n}\frac{\mu_{\alpha}}{2}\log
a_{\alpha}\right)\oint_{\gamma}\prod_{i=1}^{k}\frac{\differential{x}_{i}}{2\pi\mathrm{i}x_{i}}x_{i}^{-\frac{\log
z}{\log q_{2}^{-1}}}\prod_{i\neq
j}^{k}\frac{(x_{ij})_{\infty}}{(q_{1}x_{ij})_{\infty}}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i})_{\infty}}{(a_{\alpha}/x_{i})_{\infty}}$
$\displaystyle\hskip
100.00015pt\times\operatorname*{Sym}_{\underline{x}}\prod_{1\leq i<j\leq
k}^{\prime}\frac{[q_{1}x_{ji}]}{[x_{ji}]}\prod_{i=k_{n}+1}^{k}\left[\frac{[a_{n}/x_{i}]}{[K_{n}a_{n}/x_{i}]}\prod_{\alpha<c(i)}\frac{[a_{\alpha}/x_{i}]}{[K_{\alpha}a_{\alpha}/x_{i}]}\right]\,,$
(2.66)

where we denote

$\displaystyle\prod_{1\leq i<j\leq
k}^{\prime}\frac{[q_{1}x_{ji}]}{[x_{ji}]}=\prod_{1\leq i\geq k_{n}<j\geq
k}\frac{[q_{1}x_{ij}]}{[x_{ij}]}\prod_{1\leq i<j\leq
k_{n}}\frac{[q_{1}x_{ji}]}{[x_{ji}]}\prod_{k_{n}+1\leq i<j\leq
k}\frac{[q_{1}x_{ji}]}{[x_{ji}]}\,.$ (2.67)

Changing the ordering of the $x$-variables,
$(x_{1},\ldots,x_{k})\to(x_{k_{n}+1},\ldots,x_{k},x_{1},\ldots,x_{k_{n}})$
(the stable envelope is invariant under this change due to the
symmetrization), we obtain the result. ∎

######  Proposition 2.19.

Let $h_{\alpha}$ be an operator that counts the weight of the $\alpha$-th
operator. Then, we have the $q$-KZ equation,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots n}}\Big{|}_{a_{\alpha}\to
q_{2}^{-1}a_{\alpha}}=R_{\alpha+1,\alpha}(q_{2}a_{\alpha+1}/a_{\alpha})\cdots
R_{n,\alpha}(q_{2}a_{n}/a_{\alpha})z^{h_{n}}R_{1,\alpha}(a_{1}/a_{\alpha})\cdots
R_{\alpha-1,\alpha}(a_{\alpha-1}/a_{\alpha})\Psi_{\mathscr{C}_{1\ldots n}}\,.$
(2.68)

Compared with the elliptic case (Theorem 2.17), there is no dependence of the
dynamical parameter $z$ in the trigonometric $q$-KZ equation except for the
factor $z^{h_{n}}$.

###  2.2 Double screening current

We now consider the vertex operator screened by both types of the screening
currents $\mathsf{S}_{1,2}$. We assume $|q_{1,2}|>1$.

######  Definition 2.20.

We define the $(m_{1}|m_{2})$-screened vertex operator,

$\displaystyle\mathsf{Z}_{K}^{(m_{1}|m_{2})}(a)=\oint_{\gamma}[\differential{\underline{x}}]\prod_{\sigma\in\underline{\mathbf{2}}}\left(\prod_{i=1,\ldots,m_{\sigma}}^{\curvearrowleft}\varphi_{\sigma,K}(a/x_{\sigma,i},zK)\mathsf{S}_{\sigma}(x_{\sigma,i})\right)\mathsf{Z}_{K}(a)\,,\qquad[\differential{\underline{x}}]=\prod_{\sigma\in\underline{\mathbf{2}}}\prod_{i=1}^{m_{\sigma}}\frac{\differential{x}_{\sigma,i}}{2\pi\mathrm{i}x_{\sigma,i}}\,.$
(2.69)

We apply the ordered product under the radial ordering configuration,

$\displaystyle\begin{cases}x_{1,m_{1}}\succ x_{1,m_{1}-1}\succ\cdots\succ
x_{1,2}\succ x_{1,1}\succ a\\\ x_{2,m_{2}}\succ x_{2,m_{2}-1}\succ\cdots\succ
x_{2,2}\succ x_{2,1}\succ a\end{cases}$ (2.70)

Since $\mathsf{S}_{1}$ and $\mathsf{S}_{2}$ commute, we do not need to specify
the ordering between the $x_{1}$\- and $x_{2}$-variables.

Let $(\underline{k}_{1}|\underline{k}_{2})=(k_{1,1},k_{1,2},\ldots,k_{1,n}\mid
k_{2,1},k_{2,2},\ldots,k_{2,n})$. We denote
$\bar{\sigma}=\underline{\mathbf{2}}\backslash\sigma$ for
$\sigma\in\underline{\mathbf{2}}$. Then we have the conformal block of the
screened vertex operators in the chamber denoted by $\mathscr{C}_{12\ldots
n}:a_{1}\succ a_{2}\succ\cdots\succ a_{n}$,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{(\underline{k}_{1}|\underline{k}_{2})}(\underline{a};\underline{K})=\bra{0}\mathsf{Z}^{(k_{1,1}|k_{2,1})}_{K_{1}}(a_{1})\mathsf{Z}^{(k_{1,2}|k_{2,2})}_{K_{2}}(a_{2})\cdots\mathsf{Z}^{(k_{1,n}|k_{2,n})}_{K_{n}}(a_{n})\ket{0}/\bra{0}\mathsf{Z}_{K_{1}}(a_{1})\mathsf{Z}_{K_{2}}(a_{2})\cdots\mathsf{Z}_{K_{n}}(a_{n})\ket{0}$
$\displaystyle=\prod_{\sigma\in\underline{\mathbf{2}}}\frac{\mathsf{c}_{\sigma,\underline{k}_{\sigma}}(\underline{K})}{k_{\sigma}!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{\sigma\in\underline{\mathbf{2}}}\Phi_{\sigma}(\underline{x}_{\sigma},\underline{a})\operatorname{Stab}_{\sigma,{\underline{k}}_{\sigma}}(\underline{x}_{\sigma},\underline{a})\prod_{\begin{subarray}{c}i=1,\ldots,k_{1}\\\
j=1,\ldots,k_{2}\end{subarray}}\frac{-q_{12}x_{1,i}x_{2,j}}{(1-q_{2}x_{1,i}/x_{2,j})(1-q_{1}x_{2,j}/x_{1,i})}\,.$
(2.71)

This conformal block computes the intersecting vortex theory supported on
$\mathbb{C}_{q_{1}}$ and $\mathbb{C}_{q_{2}}$ [GLFPP16, PP16, KN21]. From the
geometric point of view, it is constructed from the intersection of (moduli
spaces of) quasimaps, which gives rise to the doubled version of the handsaw
quiver (Fig. 1(b)). It can be seen as $\widehat{A}_{1}$ quiver theory as it
involves two gauge nodes with two bifundamental matters. In fact, this
configuration is closely related to $\mathrm{U}(k_{1}|k_{2})$ supergroup gauge
theory [KP19, Kim23].

$k$$n$$n$ (a) Single screened case

$k_{1}$$k_{2}$$n$$n$ (b) Double screened case

Figure 1: Quivers for $\mathrm{W}_{q_{1,2}}(A_{1})$ block. The red edge
describes the interaction of two nodes (the so-called 0d theory contribution).

##### Pole structure

We reparametrize the $x_{1,2}$-variables,
$(x_{\sigma,i})_{i=1,\ldots,k_{\sigma}}=(x_{\sigma,\alpha,i})_{\alpha=1,\ldots,n,i=1,\ldots,k_{\sigma,\alpha}}$.
There are three possible types of the poles for
$\sigma\in\underline{\mathbf{2}}$,
$\bar{\sigma}=\underline{\mathbf{2}}\backslash\sigma$, (i)
$x_{\sigma,\alpha,i}=a_{\alpha}q_{\sigma}^{\mathbb{Z}_{\leq 0}}$, (ii)
$x_{\sigma,\alpha,i}=x_{\sigma,\alpha,j(<i)}q_{\bar{\sigma}}q_{\sigma}^{\mathbb{Z}_{\leq
0}}$, and (iii) $x_{\sigma,\alpha,i}=x_{\bar{\sigma},\alpha,j}q_{\sigma}$.
Here we take the following poles:

$\displaystyle
x_{\sigma,\alpha,1}=a_{\alpha}q_{\sigma}^{-\nu_{\sigma,\alpha,1}}\,,\qquad$
$\displaystyle
x_{\sigma,\alpha,i}=x_{\sigma,\alpha,i-1}q_{\bar{\sigma}}q_{\sigma}^{-\nu_{\sigma,\alpha,i}+\nu_{\sigma,\alpha,i-1}}=a_{\alpha}q_{\bar{\sigma}}^{i-1}q_{\sigma}^{-\nu_{\sigma,\alpha,i}}\,,$
(2.72)

where $(\nu_{\sigma,\alpha})_{\alpha=1,\ldots,n}$ is a set of inverse
partitions,

$\displaystyle\nu_{1,\alpha}=(0\leq\nu_{1,\alpha,1}\leq\cdots\leq\nu_{1,\alpha,k_{1,\alpha}})\,,\qquad\nu_{2,\alpha}=(-k_{1,\alpha}\leq\nu_{2,\alpha,1}\leq\cdots\leq\nu_{2,\alpha,k_{2,\alpha}})\,.$
(2.73)

The entries of $\nu_{1,\alpha}$ are all non-negative, while those of
$\nu_{2,\alpha}$ can be negative, $\nu_{2,\alpha,i}\geq-k_{1,\alpha}$. We take
the type (iii) poles for the $x_{2}$-variable to have the negative value
through the following path, $x_{1,\alpha,1}=a_{\alpha}$,
$x_{1,\alpha,2}=a_{\alpha}q_{2}$, …, $x_{1,\alpha,d}=a_{\alpha}q_{2}^{d-1}$,
then $x_{2,\alpha,1}=x_{1,\alpha,d}q_{2}=a_{\alpha}q_{2}^{d}$, which
corresponds to $\nu_{2,\alpha,1}=-d$, and also $\nu_{1,\alpha,i}=0$ for
$i=1,\ldots,d$ with
$d=\\#\\{\nu_{1,\alpha,i}=0,i=1,\ldots,k_{1,\alpha}\\}=k_{1,\alpha}-\ell(\nu_{1,\alpha})$
(the length of partition $\lambda$ denoted by
$\ell(\lambda)=\lambda_{1}^{\scalebox{0.5}{$\mathrm{T}$}}$).

######  Example 2.21.

In the case $n=1$ with $k_{1}=k_{1,1}=6$, $k_{2}=k_{2,1}=8$, we may have the
following configuration,

(2.74)

where

$\displaystyle\nu_{1}=(0,0,0,0,1,2)\,,\qquad\nu_{2}=(-4,-4,-3,-3,-1,0,2,3)\,,$
(2.75)

with $\ell(\nu_{1})=2$ and thus $d=4$.

Combining $-\nu_{1,\alpha}$, $\nu_{2,\alpha}^{\scalebox{0.5}{$\mathrm{T}$}}$,
and also the rectangular part of the size $\ell(\nu_{1,\alpha})\times
k_{2,\alpha}$ (the green part in Exemple 2.21), the whole part is interpreted
as (an inverse of) the fat hook partition, which parametrizes the irreducible
representation of supergroup $\mathrm{U}(k_{1,\alpha}|k_{2,\alpha})$. This
counting rule is naturally seen as a two-dimensional analog of the two-leg PT
count: We may have two boundary conditions parametrized by one-dimensional
partitions ($k_{1,\alpha}$ and $k_{2,\alpha}$ for each $\alpha=1,\ldots,n$).

######  Proposition 2.22.

Let $K_{\alpha}=q_{1}^{\mu_{2,\alpha}}q_{2}^{\mu_{1,\alpha}}$,
$\mu_{\sigma,\alpha}\in\mathbb{Z}_{\geq 0}$ for $\sigma=1,2$,
$\alpha=1,\ldots,n$. Then, $\Psi^{\underline{k}}=0$ unless $0\leq
k_{\sigma,\alpha}\leq\mu_{\sigma,\alpha}$.

This is a double version of Proposition 2.7, which is also called the (two-
dimensional version of) pit condition [BFM15]. Moreover, Proposition 2.12 can
be also generalized to the current case: Starting with the 5d $\mathcal{N}=1$
$\mathrm{U}(n)$ gauge theory on
$\mathbb{C}_{q_{1}}\times\mathbb{C}_{q_{2}}\times\mathbb{S}^{1}$ with $n$
fundamental and $n$ antifundamental matters with the Higgsing condition
$K_{\alpha}=m_{\alpha}/\widetilde{m}_{\alpha}=q_{1}^{\ell_{2,\alpha}}q_{2}^{\ell_{1,\alpha}}$,
we obtain the intersecting defect partition function, which realizes the
double screening current situation [DGH10, BTZ11, GLFPP16, PP16, GLFMS17,
NPZ17, KN21]. As mentioned above, the double screening current situation may
be seen as the two-dimensional two-leg PT count. From this point of view, the
3d/5d relation is then interpreted as the two-dimensional version of DT/PT
correspondence, equivalence of D0 brane counting and D2 brane counting. We see
the higher-dimensional uplift of this story in the following section.

##  3 W${}_{q_{1,2,3,4}}(\widehat{A}_{0})$ conformal block

We consider the W-algebra associated with $\widehat{A}_{0}$ quiver (Jordan
quiver) introduced in [KP15], which depends on four parameters
$\\{q_{\sigma}\\}_{\sigma\in\underline{\mathbf{4}}}$ with the condition,

$\displaystyle q_{1}q_{2}q_{3}q_{4}=1\,.$ (3.1)

Identifying $\\{q_{1,2,3,4}\\}$ with the equivariant parameters associated
with $\mathbb{C}^{4}$, this is the Calabi–Yau 4-fold condition.

######  Definition 3.1.

For $A\subseteq\underline{\mathbf{4}}$, we denote
$\bar{A}=\underline{\mathbf{4}}\backslash A$. We define the vertex operators
$\\{\mathsf{S}_{1,2,3,4},\mathsf{Z}_{K}\\}$ associated with $\widehat{A}_{0}$
quiver, obeying the following OPEs for $|q_{\sigma}|>1$,
$\sigma\in\underline{\textbf{4}}$,

$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\sigma}(x^{\prime})}=\frac{(x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}{(q_{\sigma}^{-1}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}\prod_{\sigma^{\prime}\in\bar{\sigma}}\frac{(q_{\overline{\sigma\sigma^{\prime}}}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}{(q_{\sigma^{\prime}}x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}\,,$
(3.2a)
$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\sigma^{\prime}}(x^{\prime})}=\mathscr{S}_{\overline{\sigma\sigma^{\prime}}}(q_{\sigma}x^{\prime}/x)\,,\qquad\wick{\c{\mathsf{S}}_{\sigma^{\prime}}(x^{\prime})\c{\mathsf{S}}_{\sigma}(x)}=\mathscr{S}_{\overline{\sigma\sigma^{\prime}}}(q_{\sigma^{\prime}}x/x^{\prime})\,,\qquad(\sigma\neq\sigma^{\prime})$
(3.2b)
$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{Z}}_{K}(x^{\prime})}=\frac{(Kx^{\prime}/x;q_{\sigma}^{-1})_{\infty}}{(x^{\prime}/x;q_{\sigma}^{-1})_{\infty}}\,,\qquad\wick{\c{\mathsf{Z}}_{K}(x^{\prime})\c{\mathsf{S}}_{\sigma}(x)}=\frac{(q_{\sigma}^{-1}x/x^{\prime};q_{\sigma}^{-1})_{\infty}}{(K^{-1}q_{\sigma}^{-1}x/x^{\prime};q_{\sigma}^{-1})_{\infty}}\,,$
(3.2c) $\displaystyle\mathsf{Z}_{K}(x)z=(Kz)\mathsf{Z}_{K}(x)\,,$ (3.2d)

with the $\mathscr{S}$ function,

$\displaystyle\mathscr{S}_{\sigma\sigma^{\prime}}(x)=\frac{(1-xq_{\sigma})(1-xq_{\sigma^{\prime}})}{(1-x)(1-xq_{\sigma\sigma^{\prime}})}\,.$
(3.3)

For $|q_{\sigma}|<1$, we instead have

$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\sigma}(x^{\prime})}=\frac{(x^{\prime}/x;q_{\sigma})_{\infty}}{(q_{\sigma}x^{\prime}/x;q_{\sigma})_{\infty}}\prod_{\sigma^{\prime}\in\bar{\sigma}}\frac{(q_{\overline{\sigma\sigma^{\prime}}}^{-1}x^{\prime}/x;q_{\sigma})_{\infty}}{(q_{\sigma^{\prime}}^{-1}x^{\prime}/x;q_{\sigma})_{\infty}}\,,$
(3.4a)
$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{Z}}_{K}(x^{\prime})}=\frac{(q_{\sigma}x^{\prime}/x;q_{\sigma})_{\infty}}{(q_{\sigma}Kx^{\prime}/x;q_{\sigma})_{\infty}}\,,\qquad\wick{\c{\mathsf{Z}}_{K}(x^{\prime})\c{\mathsf{S}}_{\sigma}(x)}=\frac{(K^{-1}x/x^{\prime};q_{\sigma})_{\infty}}{(x/x^{\prime};q_{\sigma})_{\infty}}\,.$
(3.4b)

In this case, we do not impose the non-trivial relation between $z$ and
$\mathsf{S}_{\sigma}$ as in (2.2c). We remark that $\mathsf{S}_{\sigma}$ and
$\mathsf{S}_{\sigma^{\prime}}$ commute if $\sigma\neq\sigma^{\prime}$. See
(3.2b).

######  Remark 3.2.

The vertex operator $\mathsf{Z}$ would be identified with (the rank-one case
of) $\phi_{i}$ in [Shi19, §2.3], associated with the framing spaces of the
affine Laumon space. Moreover, (the rank-one case of) their screening current
$S_{i}$ agrees with (one of) ours, $\mathsf{S}$.

######  Lemma 3.3.

For $\sigma\in\underline{\mathbf{4}}$, we write
$\bar{\sigma}=\underline{\mathbf{4}}\backslash\sigma$. For $|q_{\sigma}|>1$,
we may rewrite the screening current OPE as follows,

$\displaystyle\wick{\c{\mathsf{S}}_{\sigma}(x)\c{\mathsf{S}}_{\sigma}(x^{\prime})}$
$\displaystyle=\frac{((x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}(q_{\sigma^{\prime}\sigma^{\prime\prime}}(x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}}{(q_{\sigma^{\prime}}(x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}(q_{\sigma^{\prime\prime}}(x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}}\frac{\theta(q_{\sigma^{\prime}}x^{\prime}/x;q_{\sigma}^{-1})\theta(q_{\sigma^{\prime\prime}}x^{\prime}/x;q_{\sigma}^{-1})}{\theta(x^{\prime}/x;q_{\sigma}^{-1})\theta(q_{\sigma^{\prime}\sigma^{\prime\prime}}x^{\prime}/x;q_{\sigma}^{-1})}\,,\qquad(\sigma^{\prime},\sigma^{\prime\prime}\in\bar{\sigma})$
(3.5a) $\displaystyle=\frac{((x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}}{(q_{\sigma}^{-1}(x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}\prod_{\sigma^{\prime}\in\bar{\sigma}}(q_{\sigma^{\prime}}(x^{\prime}/x)^{\pm
1};q_{\sigma}^{-1})_{\infty}}\frac{\prod_{\sigma^{\prime}\in\bar{\sigma}}\theta(q_{\sigma^{\prime}}x^{\prime}/x;q_{\sigma}^{-1})}{\theta(x^{\prime}/x;q_{\sigma}^{-1})}\,.$
(3.5b)

###  3.1 Single screening current

In this subsection, we set $\mathsf{S}(x)\equiv\mathsf{S}_{4}(x)$ and assume
$|q_{4}|>1$. We apply the same definition for the screened vertex operator as
before (Definition 2.3).

######  Proposition 3.4.

We have the two different integral formulas for the conformal block in the
chamber $\mathscr{C}_{12\ldots n}:a_{1}\succ a_{2}\succ\cdots\succ a_{n}$,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$
$\displaystyle=\bra{0}\mathsf{Z}^{(k_{1})}_{K_{1}}(a_{1})\mathsf{Z}^{(k_{2})}_{K_{2}}(a_{2})\cdots\mathsf{Z}^{(k_{n})}_{K_{n}}(a_{n})\ket{0}/\bra{0}\mathsf{Z}_{K_{1}}(a_{1})\mathsf{Z}_{K_{2}}(a_{2})\cdots\mathsf{Z}_{K_{n}}(a_{n})\ket{0}$
$\displaystyle=\frac{\mathsf{c}_{4,\underline{k}}(\underline{K})}{k!}\oint_{\gamma}[\differential{\underline{x}}]\,\Phi_{4;\sigma^{\prime}}^{\text{F}}(\underline{x},\underline{a})\widetilde{\operatorname{Stab}}_{4;\sigma^{\prime},\underline{k}}^{\text{F}}(\underline{x},\underline{a})\,,$
(3.6a)
$\displaystyle=\frac{\mathsf{c}_{4,\underline{k}}(\underline{K})}{k!}\oint_{\gamma}[\differential{\underline{x}}]\,\Phi_{4}^{\text{M}}(\underline{x},\underline{a})\widetilde{\operatorname{Stab}}_{4,\underline{k}}^{\text{M}}(\underline{x},\underline{a})\,,$
(3.6b)

where the principal parts are defined by

$\displaystyle\Phi_{\sigma;\sigma^{\prime}}^{\text{F}}(\underline{x},\underline{a})$
$\displaystyle=\prod_{i\neq
j}^{k}\frac{(x_{ij};q_{\sigma}^{-1})_{\infty}(q_{\overline{\sigma\sigma^{\prime}}}x_{ij};q_{\sigma}^{-1})_{\infty}}{\prod_{\sigma^{\prime\prime}\in\overline{\sigma\sigma^{\prime}}}(q_{\sigma^{\prime\prime}}x_{ij};q_{\sigma}^{-1})_{\infty}}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})_{\infty}}{(a_{\alpha}/x_{i};q_{\sigma}^{-1})_{\infty}}\,,$
(3.7a) $\displaystyle\Phi_{\sigma}^{\text{M}}(\underline{x},\underline{a})$
$\displaystyle=\prod_{i\neq
j}^{k}\frac{(x_{ij};q_{\sigma}^{-1})_{\infty}}{\prod_{\sigma^{\prime}\in\overline{\sigma}}(q_{\sigma^{\prime}}x_{ij};q_{\sigma}^{-1})_{\infty}}\prod_{\begin{subarray}{c}i=1,\ldots,k\\\
\alpha=1,\ldots,n\end{subarray}}\frac{(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})_{\infty}}{(a_{\alpha}/x_{i};q_{\sigma}^{-1})_{\infty}}\,,$
(3.7b)

with the constant term under a slight modification from (2.13),

$\displaystyle\mathsf{c}_{\sigma,\underline{k}}(\underline{K})=\prod_{i=1}^{k}\frac{\theta(K_{c(i)};q_{\sigma}^{-1})}{\theta(zK_{i}^{c};q_{\sigma}^{-1})}\,.$
(3.8)

We define two different off-shell elliptic (pre-)stable envelopes as follows,

$\displaystyle\widetilde{\operatorname{Stab}}_{\sigma;\sigma^{\prime},\underline{k}}^{\text{F}}(\underline{x},\underline{a})$
$\displaystyle=\operatorname{Sym}\prod_{i<j}^{k}\frac{\prod_{\sigma^{\prime\prime}\in\overline{\sigma\sigma^{\prime}}}\theta(q_{\sigma^{\prime\prime}}x_{ji};q_{\sigma}^{-1})}{\theta(x_{ji};q_{\sigma}^{-1})\theta(q_{\overline{\sigma\sigma^{\prime}}}x_{ji};q_{\sigma}^{-1})}$
$\displaystyle\qquad\times\prod_{i=1}^{k}\frac{\prod_{\alpha<c(i)}\theta(a_{\alpha}/x_{i};q_{\sigma}^{-1})\theta(zK_{i}^{c}a_{c(i)}/x_{i};q_{\sigma}^{-1})\prod_{\alpha>c(i)}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})}{\prod_{\alpha=1}^{k}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})}\,,$
(3.9a)
$\displaystyle\widetilde{\operatorname{Stab}}_{\sigma,\underline{k}}^{\text{M}}(\underline{x},\underline{a})$
$\displaystyle=\operatorname{Sym}\prod_{i<j}^{k}\frac{\theta(q_{1,2,3}x_{ji};q_{\sigma}^{-1})}{\theta(x_{ji};q_{\sigma}^{-1})}$
$\displaystyle\qquad\times\prod_{i=1}^{k}\frac{\prod_{\alpha<c(i)}\theta(a_{\alpha}/x_{i};q_{\sigma}^{-1})\theta(zK_{i}^{c}a_{c(i)}/x_{i};q_{\sigma}^{-1})\prod_{\alpha>c(i)}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})}{\prod_{\alpha=1}^{k}\theta(K_{\alpha}a_{\alpha}/x_{i};q_{\sigma}^{-1})}\,.$
(3.9b)

###### Proof.

It directly follows from Definition 3.1 with Lemma 3.3. ∎

For the moment, the choice of $\sigma^{\prime}\in\bar{4}$ in
$\Phi_{4;\sigma^{\prime}}^{\text{F}}$ and
$\widetilde{\operatorname{Stab}}^{\text{F}}_{4;\sigma^{\prime}}$ is not
unique. It would be determined by the specialization of the parameter
$K_{\alpha}=q_{\sigma^{\prime}}$. The conformal block in the “Fock formula”
(3.6a) is identified as the Coulomb branch localization formula of the 3d ADHM
sigma model [CHK19, CDZ20], where the corresponding matter content is
described by the quiver diagram given in Fig. 2(a). Namely, it turns out to be
the vertex function associated with the quasimaps to the Hilbert scheme of
points on $\mathbb{C}^{2}$ denoted by
$\operatorname{Hilb}_{k}(\mathbb{C}^{2})$ for $n=1$ [DS19], and the
corresponding Quot scheme for $n>1$. In the “MacMahon formula” (3.6b), it
turns out to be similarly the vertex function for the quasimaps to
$\operatorname{Hilb}_{k}(\mathbb{C}^{3})$ [CZ23, Pia23] and also the higher-
rank analog. See the corresponding quiver in Fig. 2(b). One can realize two
different forms based on the same vertex operator as observed before [KN23].

The stable envelopes $\widetilde{\operatorname{Stab}}^{\text{F,M}}$ are
associated with the degree $n$ tensor product of Fock and the MacMahon modules
of quantum toroidal $\mathfrak{gl}_{1}$ as discussed below. We remark that
$\widetilde{\operatorname{Stab}}^{\text{F,M}}$ defined above are not yet
precisely the stable envelopes: The expression of
$\widetilde{\operatorname{Stab}}^{\text{F}}$ does not agree with that given by
Smirnov [Smi18] (in the case $n=1$, hence associated with
$\operatorname{Hilb}_{k}(\mathbb{C}^{2})$) although they have a close form.
The stable envelope for $\mathbb{C}^{3}$ is not yet even constructed. For
generic $(n,k)$, the stable envelopes for $\mathbb{C}^{2}$ and
$\mathbb{C}^{3}$ should be parametrized by $n$-tuple 2d and 3d partitions of
size $k$. For this purpose, it would be plausible to update the definition of
the screened vertex operator (Definition 2.3) in order that it depends on
2d/3d partitions. We may need the tree structure as discussed in [Smi18] for
the case $\mathbb{C}^{2}$. In order to emphasize this point, we would also
call $\widetilde{\operatorname{Stab}}^{\text{F,M}}$ the “pre-stable
envelopes”.

####  3.1.1 Fock formula

##### Pole structure

Let $x_{\alpha,i}=x_{i+\sum_{\beta>\alpha}k_{\beta}}$ for
$i=1,\ldots,k_{\alpha}$ as before. The poles of the contour integral are
parametrized by $n$-tuple partitions
$\underline{\lambda}=(\lambda_{1},\ldots,\lambda_{n})$, hence corresponding to
the $n$-tensor product of the Fock modules of quantum toroidal
$\mathfrak{gl}_{1}$,

$\displaystyle x_{\alpha,1}=a_{\alpha}q_{4}^{-n_{\alpha,1}}\,,\quad
x_{\alpha,i}=x_{\alpha,j(<i)}q_{1}q_{4}^{-n_{\alpha,i}}\quad\text{or}\quad
x_{\alpha,j(<i)}q_{2}q_{4}^{-n_{\alpha,i}}\,.$ (3.10)

Hence, we have

$\displaystyle
x_{\alpha,i}=a_{\alpha}q_{1}^{i_{1}-1}q_{2}^{i_{2}-1}q_{4}^{-\pi_{\alpha,i}}\,,\quad
i=(i_{1},i_{2})\in\lambda_{\alpha}\,,$ (3.11)

where we denote the position of the $i$-th coordinate in the partition
$\lambda_{\alpha}$ by $i=(i_{1},i_{2})$ for simplicity. As pointed out in
[CDZ20], the configuration
$\mathsf{PP}_{\underline{\lambda}}:=\\{\pi_{\alpha,i},i=(i_{1},i_{2})\in\lambda_{\alpha}\\}_{\alpha=1,\ldots,n}$
is interpreted as an $n$-tuple inverse plane partition whose base is fixed by
the partitions $\underline{\lambda}=\\{\lambda_{\alpha}\\}$. Similarly to
Propositions 2.7 and 2.22, we should impose $K_{\alpha}=q_{3}$ to realize this
pole structure. Namely we take $\Phi_{4;3}^{\text{F}}$ for the principal part
of the conformal block. It has been known that this box counting structure
agrees with the one-leg PT3 count.

##### Vertex function

In order to deal with the conformal block, we introduce the combinatorial
notations. Let $\lambda_{\alpha}$ be a partition and
$i=(i_{1},i_{2})\in\lambda_{\alpha}$ for $\alpha=1,\ldots,n$. We denote by
$\lambda_{\alpha}^{\scalebox{0.5}{$\mathrm{T}$}}$ the transposition of
$\lambda_{\alpha}$. We define the arm length and the leg length as follows,

$\displaystyle\textsl{a}_{\alpha}(i)=\lambda_{\alpha,i_{1}}-i_{2}\,,\qquad\textsl{l}_{\alpha}(i)=\lambda_{\alpha,i_{2}}^{\scalebox{0.5}{$\mathrm{T}$}}-i_{1}\,.$
(3.12)

We have the following formula.

######  Lemma 3.5.

Let
$\emptyset_{\underline{\lambda}}=\\{a_{\alpha}q_{1}^{i_{1}-1}q_{2}^{i_{2}-1}\\}_{\alpha=1,\ldots,n,(i_{1},i_{2})\in\lambda_{\alpha}}\in\mathsf{PP}_{\underline{\lambda}}$,
and we denote $(\cdot)_{\infty}=(\cdot;q_{4}^{-1})_{\infty}$. Then, we have
the following combinatorial formula for the contour integral,

$\displaystyle I_{\underline{\lambda}}(\underline{a})$
$\displaystyle=\frac{(q_{4}^{-1})_{\infty}^{k}}{k!}\oint_{\emptyset_{\underline{\lambda}}}[\differential{\underline{x}}]\prod_{i\neq
j}^{k}(x_{ij})_{\infty}\prod_{i,j}^{k}\frac{(q_{12}x_{ij})_{\infty}}{(q_{1,2}x_{ij})_{\infty}}\prod_{\begin{subarray}{c}\alpha=1,\ldots,n\\\
i=1,\ldots,k\end{subarray}}\frac{1}{(a_{\alpha}/x_{i})_{\infty}(q_{12}x_{i}/a_{\alpha})_{\infty}}$
$\displaystyle=\prod_{1\leq\alpha,\beta\leq
n}\left[\prod_{i\in\lambda_{\beta}}(a_{\beta\alpha}q_{1}^{\textsl{l}_{\alpha}(i)+1}q_{2}^{-\textsl{a}_{\beta}(i)})^{-1}_{\infty}\prod_{i\in\lambda_{\alpha}}(a_{\beta\alpha}q_{1}^{-\textsl{l}_{\beta}(i)}q_{2}^{\textsl{a}_{\alpha}(i)+1})^{-1}_{\infty}\right]\,.$
(3.13)

###### Proof.

This can be shown in the same way as Lemma 2.10. We rewrite the contour
integral as the equivariant index,

$\displaystyle
I_{\underline{\lambda}}(\underline{a})=\operatorname{ch}\wedge\left[\frac{1}{\mathbf{P}_{4}^{\vee}}\left(\mathbf{P}_{12}\mathbf{K}\mathbf{K}^{\vee}-\mathbf{N}\mathbf{K}^{\vee}-\mathbf{Q}_{12}\mathbf{K}\mathbf{N}^{\vee}\right)\right]_{\emptyset_{\underline{\lambda}}}\,,$
(3.14)

where $\operatorname{ch}\mathbf{P}_{A}=\prod_{a\in A}(1-q_{a})$ and
$\operatorname{ch}\mathbf{Q}_{A}=\prod_{a\in A}q_{a}$ for
$A\subseteq\underline{\mathbf{4}}$. Noticing (see, e.g., [Nak99, Kim20] for
details)

$\displaystyle\operatorname{ch}\left[\mathbf{P}_{12}\mathbf{K}\mathbf{K}^{\vee}-\mathbf{N}\mathbf{K}^{\vee}-\mathbf{Q}_{12}\mathbf{K}\mathbf{N}^{\vee}\right]_{\emptyset_{\underline{\lambda}}}=-\sum_{1\leq\alpha,\beta\leq
n}a_{\beta\alpha}\left(\sum_{i\in\lambda_{\beta}}q_{1}^{\textsl{l}_{\alpha}(i)+1}q_{2}^{-\textsl{a}_{\beta}(i)}+\sum_{i\in\lambda_{\alpha}}q_{1}^{-\textsl{l}_{\beta}(i)}q_{2}^{\textsl{a}_{\alpha}(i)+1}\right)\,,$
(3.15)

and then evaluating the index, we obtain the formula. ∎

This is the $q$-shifted version of the Nekrasov partition function of
$\mathrm{U}(n)$ gauge theory on
$\mathbb{C}_{q_{1}}\times\mathbb{C}_{q_{2}}\times\mathbb{S}^{1}$ [Nek02,
NO03], which is given by the equivariant integral over the instanton moduli
space isomorphic to the Hilbert scheme of points on $\mathbb{C}^{2}$ and its
higher-rank analog.

We evaluate the contour integral with the poles given in (3.11). The principal
part of the conformal block is given as follows.

######  Theorem 3.6.

For $i=(i_{1},i_{2})$ and $j=(j_{1},j_{2})$, we write
$q_{12}^{i}=q_{12}^{(i_{1},i_{2})}=q_{1}^{i_{1}}q_{2}^{i_{2}}$. We denote
$(\cdot)_{m}=(\cdot;q_{4}^{-1})_{m}$ and
$\theta(\cdot)=\theta(\cdot;q_{4}^{-1})$. We put $K_{\alpha}=q_{3}$ for
$\alpha=1,\ldots,n$. Then, the contribution of the principal part of the
conformal block associated with the poles specified in (3.11) is given as
follows,

$\displaystyle\frac{1}{k!}\oint_{\\{\underline{\pi}\\}}[\differential{\underline{x}}]\,\Phi_{4;3}^{\text{F}}(\underline{x},\underline{a})=\mathring{Z}_{\underline{\lambda}}\sum_{\underline{\pi}\in\mathsf{PP}_{\underline{\lambda}}}Z_{\underline{\pi}}$
(3.16)

where

$\displaystyle\mathring{Z}_{\underline{\lambda}}$
$\displaystyle=\frac{(q_{1,2})_{\infty}^{k}}{(q_{4}^{-1})_{\infty}^{k}(q_{12})_{\infty}^{k}}\prod_{\begin{subarray}{c}1\leq\alpha,\beta\leq
n\\\
i\in\lambda_{\alpha}\end{subarray}}\theta(q_{3}a_{\beta\alpha}q_{i}^{-1})\prod_{1\leq\alpha,\beta\leq
n}\left[\prod_{i\in\lambda_{\beta}}(a_{\beta\alpha}q_{1}^{\textsl{l}_{\alpha}(i)+1}q_{2}^{-\textsl{a}_{\beta}(i)})^{-1}_{\infty}\prod_{i\in\lambda_{\alpha}}(a_{\beta\alpha}q_{1}^{-\textsl{l}_{\beta}(i)}q_{2}^{\textsl{a}_{\alpha}(i)+1})^{-1}_{\infty}\right]\,,$
(3.17a) $\displaystyle Z_{\underline{\pi}}$
$\displaystyle=\prod_{(\alpha,i)\neq(\beta,j)}\frac{(q_{1,2}a_{\beta\alpha}q_{12}^{j}/q_{12}^{i})_{\pi_{\beta,j}-\pi_{\alpha,i}}}{(a_{\beta\alpha}q_{12}^{j}/q_{12}^{i})_{\pi_{\beta,j}-\pi_{\alpha,i}}(q_{12}a_{\beta\alpha}q_{12}^{j}/q_{12}^{i})_{\pi_{\beta,j}-\pi_{\alpha,i}}}\prod_{\begin{subarray}{c}\alpha,\beta=1,\ldots,n\\\
i\in\lambda_{\alpha}\end{subarray}}\frac{(q_{12}a_{\beta\alpha}/q_{12}^{i})_{-\pi_{\alpha,i}}}{(q_{123}a_{\beta\alpha}/q_{12}^{i})_{-\pi_{\alpha,i}}}\,.$
(3.17b)

This is a higher-rank version of the vertex function associated with the
Hilbert scheme of points on $\mathbb{C}^{2}$ given by [DS19, CDZ20]. See also
[CHK19]. In this expression, there is no counting parameter in the summation
over $\underline{\pi}\in\mathsf{PP}_{\underline{\lambda}}$. In fact, the
stable envelope gives rise to the factor $z^{|\underline{\pi}|}$ in the
summation, which is a consequence of the quasiperiodicity of the kernel
function $\varphi_{K}(\cdot,\cdot)$ as seen in Proposition 2.11.

####  3.1.2 MacMahon formula

##### Pole structure

Let $x_{\alpha,i}=x_{i+\sum_{\beta>\alpha}k_{\beta}}$ for
$i=1,\ldots,k_{\alpha}$ as before. The poles of the contour integral are
parametrized by $n$-tuple plane partitions
$\underline{\pi}=(\pi_{1},\ldots,\pi_{n})$,

$\displaystyle x_{\alpha,1}=a_{\alpha}q_{4}^{-n_{\alpha,1}}\,,\quad
x_{\alpha,i}=x_{\alpha,j(<i)}q_{1}q_{4}^{-n_{\alpha,i}}\quad\text{or}\quad
x_{\alpha,j(<i)}q_{2}q_{4}^{-n_{\alpha,i}}\quad\text{or}\quad
x_{\alpha,j(<i)}q_{3}q_{4}^{-n_{\alpha,i}}\,.$ (3.18)

Hence, we have

$\displaystyle
x_{\alpha,i}=a_{\alpha}q_{1}^{i_{1}-1}q_{2}^{i_{2}-1}q_{3}^{i_{3}-1}q_{4}^{-\rho_{\alpha,i}}\,,\quad
i=(i_{1},i_{2},i_{3})\in\pi_{\alpha}\,,$ (3.19)

where
$\mathsf{SP}_{\underline{\pi}}:=\\{\rho_{\alpha,ijk},i=(i_{1},i_{2},i_{3})\in\pi_{\alpha}\\}_{\alpha=1,\ldots,n}$
is interpreted as an $n$-tuple inverse solid partition with the fixed base
given by the plane partition $\underline{\pi}=\\{\pi_{\alpha}\\}$.

######  Proposition 3.7.

Let
$K_{\alpha}=q_{1}^{\mu^{(1)}_{\alpha}}q_{2}^{\mu^{(2)}_{\alpha}}q_{3}^{\mu^{(3)}_{\alpha}}$,
$\mu^{(1,2,3)}_{\alpha}\in\mathbb{Z}_{\geq 0}$ for $\alpha=1,\ldots,n$. Then,
$Z_{\underline{\pi}}=0$ unless
$(\mu^{(1)}_{\alpha}+1,\mu^{(2)}_{\alpha}+1,\mu^{(3)}_{\alpha}+1)\not\in\pi_{\alpha}$.

##### Vertex function

The contour integral evaluated with the poles (3.19) gives rise to the
following vertex function. We denote
$q_{123}^{i}=q_{123}^{(i_{1},i_{2},i_{3})}=q_{1}^{i_{1}}q_{2}^{i_{2}}q_{3}^{i_{3}}$
for $i=(i_{1},i_{2},i_{3})$.

######  Theorem 3.8.

We denote $(\cdot)_{m}=(\cdot;q_{4}^{-1})_{m}$. The contribution of the
principal part of the conformal block associated with the poles specified in
(3.19) is given as follows,

$\displaystyle\frac{1}{k!}\oint_{\\{\underline{\rho}\\}}[\differential{\underline{x}}]\,\Phi_{4}^{\text{M}}(\underline{x},\underline{a})=\mathring{Z}_{\underline{\pi}}\sum_{\underline{\rho}\in\mathsf{SP}_{\underline{\pi}}}Z_{\underline{\rho}}$
(3.20)

where

$\displaystyle\mathring{Z}_{\underline{\pi}}$
$\displaystyle=\frac{(q_{1,2,3})_{\infty}^{k}}{(q_{4}^{-1})_{\infty}^{k}}\operatorname{ch}\wedge\left[\frac{1}{\mathbf{P}_{123}}\left((1-\mathbf{Q}_{1,2,3})\mathbf{K}\mathbf{K}^{\vee}-\mathbf{N}\mathbf{K}^{\vee}+\widetilde{\mathbf{N}}\mathbf{K}^{\vee}\right)\right]_{\emptyset_{\underline{\pi}}}\,,$
(3.21a) $\displaystyle Z_{\underline{\rho}}$
$\displaystyle=\prod_{(\alpha,i)\neq(\beta,j)}\frac{(q_{1,2,3}a_{\beta\alpha}q_{ji})_{\rho_{\beta,j}-\rho_{\alpha,i}}}{(a_{\beta\alpha}q_{ji})_{\rho_{\beta,j}-\rho_{\alpha,i}}}\prod_{\begin{subarray}{c}\alpha,\beta=1,\ldots,n\\\
i\in\pi_{\alpha}\end{subarray}}\frac{(a_{\beta\alpha}q_{i}^{-1})_{-\rho_{\alpha,i}}}{(K_{\beta}a_{\beta\alpha}q_{i}^{-1})_{-\rho_{\alpha,i}}}\,.$
(3.21b)

This is a higher-rank version of the vertex function associated with the
Hilbert scheme of points on $\mathbb{C}^{3}$ given by [CZ23, Pia23]. Again,
there is no counting factor $z^{|\underline{\rho}|}$ in the summation over
$\underline{\rho}\in\mathsf{SP}_{\underline{\pi}}$, which can be imposed by
the stable envelope thanks to the quasiperiodicity of the kernel function
$\varphi_{K}(\cdot,\cdot)$.

###  3.2 Multiple screening current

We consider the conformal block of the multi-screened vertex operators, and
discuss its relation to the enumerative invariants.

######  Definition 3.9.

Let
$\underline{m}=\\{m_{\sigma}\\}_{\sigma\in\underline{\mathbf{4}}}\in\mathbb{Z}_{\geq
0}^{4}$. We define the $\underline{m}$-screened vertex operator,

$\displaystyle\mathsf{Z}_{K}^{(\underline{m})}(a)$
$\displaystyle=\oint_{\gamma}[\differential{\underline{x}}]\prod_{\sigma\in\underline{\mathbf{4}},|q_{\sigma}|>1}\left(\prod_{i=1,\ldots,m_{\sigma}}^{\curvearrowleft}\varphi_{\sigma,K}\left(\frac{a}{x_{\sigma,i}},zK\right)\mathsf{S}_{\sigma}(x_{\sigma,i})\right)\mathsf{Z}_{K}(a)\prod_{\sigma\in\underline{\mathbf{4}},|q_{\sigma}|<1}\left(\prod_{i=1,\ldots,m_{\sigma}}^{\curvearrowright}\varphi_{\sigma,K}\left(\frac{a}{x_{\sigma,i}},zK\right)\mathsf{S}_{\sigma}(x_{\sigma,i})\right)\,.$
(3.22)

When we consider the quadruple screened case ($m_{\sigma}>0$ for all
$\sigma\in\underline{\mathbf{4}}$), one cannot take all the parameters
$|q_{\sigma}|>1$ nor $|q_{\sigma}|<1$ as a consequence of the condition (3.1).
Hence, we should consider both situations in this case. For the other cases,
we can take all the parameters $|q_{\sigma}|>1$ or $|q_{\sigma}|<1$ for
simplicity.

$k$$n$$n$ (a) Single case (F)

$k$$n$$n$ (b) Single case (M)

$k_{1}$$k_{2}$$k_{3}$$n$$n$ (c) Triple case (F)

$k_{1}$$k_{2}$$k_{3}$$k_{4}$$n$$n$ (d) Quadruple case (M)

Figure 2: Quivers for $\mathrm{W}_{q_{1,2,3,4}}(\widehat{A}_{0})$ block. (F)
and (M) stand for the Fock and the MacMahon formula, respectively. The solid
(dashed) lines indicate the chiral (Fermi) multiplets, and the red ones are
multiple 0d theory contributions.

##### Fock formula

We take $K_{\alpha}=q_{4}$ and
$\underline{k}_{\alpha}=(k_{\alpha,1},k_{\alpha,2},k_{\alpha,3},0)$ for all
$\alpha=1,\ldots,n$ to consider the Fock formula. We assume $|q_{1,2,3}|>1$.
Then, we have the contour integral formula of the conformal block as follows,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};q_{4})$
$\displaystyle=\bra{0}\mathsf{Z}^{(\underline{k}_{1})}_{q_{4}}(a_{1})\mathsf{Z}^{(\underline{k}_{2})}_{q_{4}}(a_{2})\cdots\mathsf{Z}^{(\underline{k}_{n})}_{q_{4}}(a_{n})\ket{0}/\bra{0}\mathsf{Z}_{q_{4}}(a_{1})\mathsf{Z}_{q_{4}}(a_{2})\cdots\mathsf{Z}_{q_{4}}(a_{n})\ket{0}$
$\displaystyle=\prod_{\sigma\in\underline{\mathbf{3}}}\frac{\mathsf{c}_{\sigma,\underline{k}}}{k_{\sigma}!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{\sigma\in\underline{\mathbf{3}}}\Phi_{\sigma;4}^{\text{F}}(\underline{x}_{\sigma},\underline{a})\widetilde{\operatorname{Stab}}_{\sigma;4,\underline{k}}^{\text{F}}(\underline{x}_{\sigma},\underline{a})\prod_{\sigma,\sigma^{\prime}\in\underline{\mathbf{3}}}\prod_{\begin{subarray}{c}i=1,\ldots,k_{\sigma}\\\
j=1,\ldots,k_{\sigma^{\prime}}\end{subarray}}\mathscr{S}_{\overline{\sigma\sigma^{\prime}}}(q_{\sigma}x_{\sigma^{\prime},j}/x_{\sigma,i})\,.$
(3.23)

The poles of this contour integral are parametrized by $n$-tuple three-
dimensional partitions with a triple of boundary two-dimensional partitions
$\\{\underline{\lambda}_{\sigma}\\}_{\sigma\in\underline{\mathbf{3}}}$. If
there is no interaction between different directions organized by the
$\mathscr{S}$ functions, it is just a union of three inverse plane partitions
$\\{\mathsf{PP}_{\underline{\lambda}_{\sigma}}\\}_{\sigma\in\underline{\mathbf{3}}}$
similar to $\mathrm{W}_{q_{1,2}}(A_{1})$ algebra as seen in §2.2. Due to the
$\mathscr{S}$ function factors, the pole structure turns out to be compatible
with the PT3 count.

The conformal block (3.23) is identified with the partition function of gauge
origami of D2 branes supported on
$\\{\mathbb{C}_{\sigma}\\}_{\sigma\in\underline{\mathbf{3}}}$, whose matter
content is given by the quiver shown in Fig. 2(c). Each direction gives rise
to the vertex function associated with the quasimaps in the corresponding
geometry, and the total contribution would be given by the equivariant Euler
characteristics of the moduli space of the intersecting sheaves, that we call
the origami vertex function.

##### MacMahon formula

We finally consider the fully generic situation. The conformal block of the
multi-screened vertex operator is given by the following contour integral,

$\displaystyle\Psi_{\mathscr{C}_{1\ldots
n}}^{\underline{k}}(\underline{a};\underline{K})$
$\displaystyle=\bra{0}\mathsf{Z}^{(\underline{k}_{1})}_{K_{1}}(a_{1})\mathsf{Z}^{(\underline{k}_{2})}_{K_{2}}(a_{2})\cdots\mathsf{Z}^{(\underline{k}_{n})}_{K_{n}}(a_{n})\ket{0}/\bra{0}\mathsf{Z}_{K_{1}}(a_{1})\mathsf{Z}_{K_{2}}(a_{2})\cdots\mathsf{Z}_{K_{n}}(a_{n})\ket{0}$
$\displaystyle=\prod_{\sigma\in\underline{\mathbf{4}}}\frac{\mathsf{c}_{\sigma,\underline{k}}}{k_{\sigma}!}\oint_{\gamma}[\differential{\underline{x}}]\prod_{\sigma\in\underline{\mathbf{4}}}\Phi_{\sigma}^{\text{M}}(\underline{x}_{\sigma},\underline{a})\widetilde{\operatorname{Stab}}_{\sigma,\underline{k}}^{\text{M}}(\underline{x}_{\sigma},\underline{a})\prod_{\sigma,\sigma^{\prime}\in\underline{\mathbf{4}}}\prod_{\begin{subarray}{c}i=1,\ldots,k_{\sigma}\\\
j=1,\ldots,k_{\sigma^{\prime}}\end{subarray}}\mathscr{S}_{\overline{\sigma\sigma^{\prime}}}(q_{\sigma}x_{\sigma^{\prime},j}/x_{\sigma,i})\,.$
(3.24)

For $\sigma\in\underline{\mathbf{4}}$ with $|q_{\sigma}|<1$, the principal
part and the stable envelope are understood through the analytic continuation
$\left.\Phi_{\sigma}^{\text{M}}\right|_{q_{1,2,3,4}\to q_{1,2,3,4}^{-1}}$,
$\widetilde{\operatorname{Stab}}_{\sigma,\underline{k}}^{\text{M}}|_{q_{1,2,3,4}\to
q_{1,2,3,4}^{-1}}$. Due to the condition (3.1), we take for example
$|q_{1,2,3}|>1$ and $|q_{4}|<1$. In this case, the fourth-direction should be
differently treated although the final result is expected to be symmetric for
$q_{1,2,3,4}$. Such an asymmetric treatment gives rise to a peculiar sign
factor specific to the 4-fold counting [Nek17d, NP18]. The poles of the
contour integral are parametrized by $n$-tuple four-dimensional partitions
with a quadruple of boundary three-dimensional plane partitions
$\\{\underline{\pi}_{\sigma}\\}_{\sigma\in\underline{\mathbf{4}}}$, and its
structure is expected to be compatible with (the higher-rank version of) the
four-leg PT4 count discussed in the literature [CKM19, Mon22, Liu23, NP23].

The conformal block (3.24) is the partition function of gauge origami of D2
branes on $\\{\mathbb{C}_{\sigma}\\}_{\sigma\in\underline{\mathbf{4}}}$
described by the quiver shown in Fig. 2(d). Again from the geometric point of
view, it would be identified with the fully generic origami vertex function in
$\mathbb{C}^{4}$.

######  Conjecture 3.10.

The conformal blocks (3.23) and (3.24) are the rank-$n$ multi-leg PT vertex of
$\mathbb{C}^{3}$ and $\mathbb{C}^{4}$.

##  Appendix A Special functions

######  Definition A.1 ($q$-hypergeometric series, e.g., [GR04]).

The $q$-hypergeometric series is defined

${}_{r}\phi_{s}\left[\begin{matrix}a_{1}&\cdots&a_{r}\\\
b_{1}&\cdots&b_{s}\end{matrix};q;x\right]=\sum_{d\geq
0}\frac{(a_{1},\ldots,a_{r};q)_{d}}{(b_{1},\ldots,b_{s},q;q)_{d}}\left((-1)^{d}q^{{n\choose
2}}\right)^{1+s-r}x^{d}\,,$ (A.1)

where we denote

$\displaystyle(a_{1},\ldots,a_{r};q)_{d}=\prod_{i=1}^{r}(a_{i};q)_{d}\,.$
(A.2)

######  Proposition A.2 (Heine’s transformation formula).

Let $|x|<1$. We have

$\displaystyle{{}_{2}\phi_{1}}\left[\begin{matrix}a&b\\\
c&-\end{matrix};q;x\right]$
$\displaystyle=\frac{(b,ax;q)_{\infty}}{(c,x;q)_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}c/b&x\\\
ax&-\end{matrix};q;b\right]\quad(|b|<1)$ (A.3a)
$\displaystyle=\frac{(a,bx;q)_{\infty}}{(c,x;q)_{\infty}}{{}_{2}\phi_{1}}\left[\begin{matrix}c/a&x\\\
bx&-\end{matrix};q;a\right]\quad(|a|<1)$ (A.3b)

######  Definition A.3 (Elliptic gamma function).

For $p,q\in\mathbb{C}^{\times}$ with $|p|,|q|<1$, we define

$\displaystyle\Gamma(z;p,q)=\frac{(pqz^{-1};p,q)_{\infty}}{(z;p,q)_{\infty}}\,,$
(A.4)

obeying the functional relation

$\displaystyle\frac{\Gamma(pz;p,q)}{\Gamma(z;p,q)}=\theta(z;q)\,,\qquad\frac{\Gamma(qz;p,q)}{\Gamma(z;p,q)}=\theta(z;p)\,.$
(A.5)

######  Definition A.4 (Elliptic hypergeometric series, e.g., [Ros16]).

We define

$\displaystyle{{}_{r+1}E_{r}}\left[\begin{matrix}a_{1}&\cdots&a_{r+1}\\\
b_{1}&\cdots&b_{r}\end{matrix};q,p;z\right]=\sum_{d\geq
0}z^{d}\frac{(a_{1},\ldots,a_{r+1};q,p)_{d}}{(q,b_{1},\ldots,b_{r};q,p)_{d}}$
(A.6)

where

$\displaystyle(a_{1},\ldots,a_{r};q,p)_{d}=\prod_{i=1}^{r}(a_{i};q,p)_{d}\,,\qquad(a;q,p)_{d}=\prod_{n=0}^{d}\theta(aq^{n};p)\,.$
(A.7)

The parameters should obey the balancing condition,

$\displaystyle a_{1}a_{2}\cdots a_{r+1}=qb_{1}\cdots b_{r}\,.$ (A.8)

## References

  * [AFO17] M. Aganagic, E. Frenkel, and A. Okounkov, _Quantum $q$-Langlands Correspondence_, Trans. Moscow Math. Soc. 79 (2018) 1–83, arXiv:1701.03146 [hep-th].
  * [AH15] M. Aganagic and N. Haouzi, _ADE Little String Theory on a Riemann Surface (and Triality)_ , arXiv:1506.04183 [hep-th].
  * [AHKS13] M. Aganagic, N. Haouzi, C. Kozcaz, and S. Shakirov, _Gauge/Liouville Triality_ , arXiv:1309.1687 [hep-th].
  * [AHS14] M. Aganagic, N. Haouzi, and S. Shakirov, _$A_{n}$ -Triality_, arXiv:1403.3657 [hep-th].
  * [AO16] M. Aganagic and A. Okounkov, _Elliptic stable envelopes_ , J. Am. Math. Soc. 34 (2021) 79–133, arXiv:1604.00423 [math.AG].
  * [BFM15] M. Bershtein, B. Feigin, and G. Merzon, _Plane partitions with a “pit”: generating functions and representation theory_ , Sel. Math. 24 (2018) 21–62, arXiv:1512.08779 [math.CO].
  * [BKP22] Y. Bae, M. Kool, and H. Park, _Counting surfaces on Calabi-Yau 4-folds I: Foundations_ , arXiv:2208.09474 [math.AG].
  * [BKP24] , _Counting surfaces on Calabi-Yau 4-folds II: $\mathrm{DT}$-$\mathrm{PT}_{0}$ correspondence_, arXiv:2402.06526 [math.AG].
  * [BTZ11] G. Bonelli, A. Tanzini, and J. Zhao, _Vertices, Vortices and Interacting Surface Operators_ , JHEP 06 (2012) 178, arXiv:1102.0184 [hep-th].
  * [BZ21] M. Bullimore and D. Zhang, _3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve_, SciPost Phys. 13 (2022) 005, arXiv:2109.10907 [hep-th].
  * [BZJ19] D. Bykov and P. Zinn-Justin, _Higher spin ${{\mathfrak{s}}}{{\mathfrak{l}}}_{2}$R-matrix from equivariant (co)homology_, Lett. Math. Phys. 110 (2020) 2435–2470, arXiv:1904.11107 [math-ph].
  * [CDZ20] S. Crew, N. Dorey, and D. Zhang, _Blocks and Vortices in the 3d ADHM Quiver Gauge Theory_ , JHEP 03 (2021) 234, arXiv:2010.09732 [hep-th].
  * [CHK19] S. Choi, C. Hwang, and S. Kim, _Quantum vortices, M2-branes and black holes_ , arXiv:1908.02470 [hep-th].
  * [CKM19] Y. Cao, M. Kool, and S. Monavari, _K-Theoretic DT/PT Correspondence for Toric Calabi–Yau 4-Folds_ , Commun. Math. Phys. 396 (2022) 225–264, arXiv:1906.07856 [math.AG].
  * [CP91] V. Chari and A. Pressley, _Quantum affine algebras_ , Commun. Math. Phys. 142 (1991) 261–283.
  * [CZ23] Y. Cao and G. Zhao, _Quasimaps to quivers with potentials_ , arXiv:2306.01302 [math.AG].
  * [CZZ23] S. Crew, D. Zhang, and B. Zhao, _Boundaries & localisation with a topological twist_, JHEP 10 (2023) 093, arXiv:2306.16448 [hep-th].
  * [DGH10] T. Dimofte, S. Gukov, and L. Hollands, _Vortex Counting and Lagrangian 3-manifolds_ , Lett. Math. Phys. 98 (2011) 225–287, arXiv:1006.0977 [hep-th].
  * [DN21] M. Dedushenko and N. Nekrasov, _Interfaces and quantum algebras, I: Stable envelopes_ , J. Geom. Phys. 194 (2023) 104991, arXiv:2109.10941 [hep-th].
  * [DN23] , _Interfaces and Quantum Algebras, II: Cigar Partition Function_ , arXiv:2306.16434 [hep-th].
  * [DS19] H. Dinkins and A. Smirnov, _Characters of tangent spaces at torus fixed points and 3d-mirror symmetry_ , Lett. Math. Phys. 110 (2020) 2337–2352, arXiv:1908.01199 [math.AG].
  * [Fel94a] G. Felder, _Conformal Field Theory and Integrable Systems Associated to Elliptic Curves_, Proceedings of the International Congress of Mathematicians, Birkhäuser Basel, 1995, pp. 1247–1255. arXiv:hep-th/9407154.
  * [Fel94b] , _Elliptic quantum groups_ , 11th International Conference on Mathematical Physics (ICMP-11) (Satellite colloquia: New Problems in the General Theory of Fields and Particles, Paris, France, 25-28 Jul 1994), 1994, pp. 211–218. arXiv:hep-th/9412207.
  * [FJM+93] O. Foda, M. Jimbo, T. Miwa, K. Miki, and A. Nakayashiki, _Vertex operators in solvable lattice models_ , J. Math. Phys. 35 (1994) 13–46, arXiv:hep-th/9305100.
  * [FR92] I. B. Frenkel and N. Y. Reshetikhin, _Quantum affine algebras and holonomic difference equations_ , Commun. Math. Phys. 146 (1992) 1–60.
  * [FR97] E. Frenkel and N. Reshetikhin, _Deformations of W-algebras associated to simple Lie algebras_ , Commun. Math. Phys. 197 (1998) 1–32, arXiv:q-alg/9708006.
  * [FRV17] G. Felder, R. Rimányi, and A. Varchenko, _Elliptic dynamical quantum groups and equivariant elliptic cohomology_ , SIGMA 14 (2018) 132, arXiv:1702.08060 [math.RT].
  * [GLFMS17] A. Gorsky, B. Le Floch, A. Milekhin, and N. Sopenko, _Surface defects and instanton–vortex interaction_ , Nucl. Phys. B 920 (2017) 122–156, arXiv:1702.03330 [hep-th].
  * [GLFPP16] J. Gomis, B. Le Floch, Y. Pan, and W. Peelaers, _Intersecting Surface Defects and Two-Dimensional CFT_ , Phys. Rev. D 96 (2017) 045003, arXiv:1610.03501 [hep-th].
  * [GR04] G. Gasper and M. Rahman, _Basic Hypergeometric Series_, Cambridge University Press, 2004.
  * [GRTV12] V. Gorbounov, R. Rimanyi, V. Tarasov, and A. Varchenko, _Cohomology of a flag variety as a Yangian Bethe algebra_ , J. Geom. Phys. 74 (2013) 56, arXiv:1204.5138 [math.AG].
  * [Hao23] N. Haouzi, _A new realization of quantum algebras in gauge theory and Ramification in the Langlands program_ , arXiv:2311.04367 [hep-th].
  * [Her08] D. Hernandez, _Quantum toroidal algebras and their representations_ , Sel. Math. 14 (2009) 701–725, arXiv:0801.2397 [math.QA].
  * [JKOS98] M. Jimbo, H. Konno, S. Odake, and J. Shiraishi, _Elliptic algebra $U_{q,p}(\widehat{\mathfrak{sl}}_{2})$: Drinfeld currents and vertex operators_, Commun. Math. Phys. 199 (1999) 605–647, arXiv:math/9802002.
  * [JLN24] S. Jeong, N. Lee, and N. Nekrasov, _di-Langlands correspondence and extended observables_ , arXiv:2402.13888 [hep-th].
  * [Kim20] T. Kimura, _Instanton Counting, Quantum Geometry and Algebra_, Mathematical Physics Studies, Springer International Publishing, 2021. arXiv:2012.11711 [hep-th].
  * [Kim23] , _Aspects of supergroup gauge theory_ , Int. J. Mod. Phys. A 38 (2023) 2330001, arXiv:2301.05927 [hep-th].
  * [KN21] T. Kimura and F. Nieri, _Intersecting defects and supergroup gauge theory_ , J. Phys. A 54 (2021) 435401, arXiv:2105.02776 [hep-th].
  * [KN23] T. Kimura and G. Noshita, _Gauge origami and quiver W-algebras_ , arXiv:2310.08545 [hep-th].
  * [Kon17] H. Konno, _Elliptic Weight Functions and Elliptic q-KZ Equation_ , J. Int. Sys. 2 (2017) xyx011, arXiv:1706.07630 [math.QA].
  * [Kon18] , _Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group_ , J. Int. Sys. 3 (2018) , arXiv:1803.01540 [math.QA].
  * [Kon20] , _Elliptic quantum groups: Representations and related geometry_, SpringerBriefs in Mathematical Physics, vol. 37, Springer Singapore, 2020.
  * [KP15] T. Kimura and V. Pestun, _Quiver W-algebras_ , Lett. Math. Phys. 108 (2018) 1351–1381, arXiv:1512.08533 [hep-th].
  * [KP16] , _Quiver elliptic W-algebras_ , Lett. Math. Phys. 108 (2018) 1383–1405, arXiv:1608.04651 [hep-th].
  * [KP17] , _Fractional quiver W-algebras_ , Lett. Math. Phys. 108 (2018) 2425–2451, arXiv:1705.04410 [hep-th].
  * [KP19] , _Super instanton counting and localization_ , arXiv:1905.01513 [hep-th].
  * [KP22] , _Fractionalization of quiver variety and $qq$-character_, unpublished (2022) .
  * [Liu23] H. Liu, _The 4-fold Pandharipande–Thomas vertex_ , arXiv:2306.12923 [math.AG].
  * [LNP19] P. Longhi, F. Nieri, and A. Pittelli, _Localization of 4d $\mathcal{N}=1$ theories on $\mathbb{D}^{2}\times\mathbb{T}^{2}$_, JHEP 12 (2019) 147, arXiv:1906.02051 [hep-th].
  * [MNNZ23] Y. Matsuo, S. Nawata, G. Noshita, and R.-D. Zhu, _Quantum toroidal algebras and solvable structures in gauge/string theory_ , Phys. Rept. 1055 (2024) 1–144, arXiv:2309.07596 [hep-th].
  * [MO12] D. Maulik and A. Okounkov, _Quantum Groups and Quantum Cohomology_ , Astérisque 408 (2019) 1–212, arXiv:1211.1287 [math.AG].
  * [Mon22] S. Monavari, _Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds_ , J. Geom. Phys. 174 (2022) 104466, arXiv:2203.11381 [math.AG].
  * [Nak99] H. Nakajima, _Lectures on Hilbert Schemes of Points on Surfaces_, American Mathematical Society, 1999\.
  * [Nek02] N. Nekrasov, _Seiberg-Witten prepotential from instanton counting_ , Adv. Theor. Math. Phys. 7 (2003) 831–864, arXiv:hep-th/0206161.
  * [Nek15] , _BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters_ , JHEP 03 (2016) 181, arXiv:1512.05388 [hep-th].
  * [Nek16] , _BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem_ , Adv. Theor. Math. Phys. 21 (2017) 503–583, arXiv:1608.07272 [hep-th].
  * [Nek17a] , _BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters_ , Commun. Math. Phys. 358 (2018) 863–894, arXiv:1701.00189 [hep-th].
  * [Nek17b] , _BPS/CFT correspondence IV: sigma models and defects in gauge theory_ , Lett. Math. Phys. 109 (2019) 579–622, arXiv:1711.11011 [hep-th].
  * [Nek17c] , _BPS/CFT correspondence V: BPZ and KZ equations from qq-characters_ , arXiv:1711.11582 [hep-th].
  * [Nek17d] , _Magnificent four_ , Adv. Theor. Math. Phys. 24 (2020) 1171–1202, arXiv:1712.08128 [hep-th].
  * [NO03] N. Nekrasov and A. Okounkov, _Seiberg-Witten theory and random partitions_ , Prog. Math. 244 (2006) 525–596, arXiv:hep-th/0306238.
  * [NP18] N. Nekrasov and N. Piazzalunga, _Magnificent Four with Colors_ , Commun. Math. Phys. 372 (2019) 573–597, arXiv:1808.05206 [hep-th].
  * [NP23] , _Global magni $4$icence, or: 4G Networks_, arXiv:2306.12995 [hep-th].
  * [NPZ17] F. Nieri, Y. Pan, and M. Zabzine, _3d Expansions of 5d Instanton Partition Functions_ , JHEP 04 (2018) 092, arXiv:1711.06150 [hep-th].
  * [NS09] N. A. Nekrasov and S. L. Shatashvili, _Supersymmetric vacua and Bethe ansatz_ , Nucl. Phys. Proc. Suppl. 192-193 (2009) 91–112, arXiv:0901.4744 [hep-th].
  * [Oko15] A. Okounkov, _Lectures on K-theoretic computations in enumerative geometry_ , arXiv:1512.07363 [math.AG].
  * [Pia23] N. Piazzalunga, _The one-legged K-theoretic vertex of fourfolds from 3d gauge theory_ , arXiv:2306.12405 [hep-th].
  * [PP16] Y. Pan and W. Peelaers, _Intersecting Surface Defects and Instanton Partition Functions_ , JHEP 07 (2017) 073, arXiv:1612.04839 [hep-th].
  * [Ros16] H. Rosengren, _Elliptic Hypergeometric Functions_ , 2016. arXiv:1608.06161 [math.CA].
  * [RTV14] R. Rimanyi, V. Tarasov, and A. Varchenko, _Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety_ , J. Geom. Phys. 94 (2015) 81–119, arXiv:1411.0478 [math.AG].
  * [Sha06] S. Shadchin, _On F-term contribution to effective action_ , JHEP 08 (2007) 052, arXiv:hep-th/0611278.
  * [Shi19] J. Shiraishi, _Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions_ , J. Int. Sys. 4 (2019) xyz010, arXiv:1903.07495 [math.QA].
  * [SKAO95] 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.
  * [Smi18] A. Smirnov, _Elliptic stable envelope for Hilbert scheme of points in the plane_ , Sel. Math. 26 (2019) 3, arXiv:1804.08779 [math.AG].
  * [Smi21] , _Quantum differential and difference equations for $\mathrm{Hilb}^{n}(\mathbb{C}^{2})$_, arXiv:2102.10726 [math.AG].
  * [Tam23] S. Tamagni, _Stable Envelopes, Vortex Moduli Spaces, and Verma Modules_ , arXiv:2311.03462 [hep-th].
  * [Tsy22] A. Tsymbaliuk, _Shuffle approach towards quantum affine and toroidal algebras_, SpringerBriefs in Mathematical Physics, vol. 49, Springer Singapore, 2023. arXiv:2209.04294 [math.RT].