analytic in chamber $\mathfrak{C}_{C}$, and correspondingly the gauge theory
$T^{3d}_{A_{1}}$ defined on $M_{3}=S^{1}_{\mathcal{C}^{\prime}}\times D^{2}$.
Following Section 3, the integral form of the partition function takes the
form

${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf
N}})=\oint_{C}\frac{dy}{y}\,y^{\frac{\ln(z)}{\ln(q)}}\prod_{d=1}^{L}\frac{\left(\sqrt{q/t}\;y/x_{d,1}\,;q\right)_{\infty}}{\left(\sqrt{t/q}\;y/x_{d,2}\,;q\right)_{\infty}}\;.$
(9.75)

The power of $y$ containing the complexified F.I. parameter $z$ is the
contribution of the mixed topological/gauge anomaly; namely, the boundary
anomaly polynomial is $2\,f\,f_{z}$, where $f$ is the curvature of the
corresponding dynamical $U(1)$ gauge symmetry at the boundary, and $f_{z}$ is
the curvature of the topological symmetry $T_{C}=U(1)$.

The ratio of $q$-Pochhammer symbols in the integrand is the contribution of
the chiral matter, as dictated by the Drinfeld polynomial data. In particular,
the R-charge and $U(1)_{t}$ charge of the chiral multiplets are fixed by the
Drinfeld roots of the polynomials associated to the weights
$\\{\lambda\\}_{d}$.

The denominator stands for the fundamental chiral multiplets with boundary
conditions N, whose scalars and their derivatives have charge $(n,-1,1/2)$
under $U(1)_{J}\times U(1)_{R}\times U(1)_{t}$. The numerator stands for the
anti-fundamental chiral multiplets with boundary conditions D, whose fermions
and their derivatives have charge $(n,1,1/2)$ under $U(1)_{J}\times
U(1)_{R}\times U(1)_{t}$. The ratio of $q$-Pochhammer factors matches the
product of two-points

$\prod_{d=1}^{L}\left\langle
S^{\vee}(y)\,\mathcal{V}_{\\{\lambda\\}_{d}}(\tilde{x}_{d})\right\rangle$
(9.76)

where the $L$ primary operators were defined in (9.40). The notation is as
before, with each mass $x_{d,s}\in T_{H}$ redefined with respect to a center
of mass $\tilde{x}_{d}$ as

$x_{d,s}\equiv\tilde{x}_{d}\,q^{\sigma_{d,s}}\;,\qquad\;\;s=1,2\;.$ (9.77)

The contour is chosen to enclose the infinite number of $q$-shifted poles
between $\infty$ and $x_{k,2}$ for some $k\in\\{1,\ldots,L\\}$. Then, in
center of mass coordinates, and after rescaling,

${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf
N}})=\oint_{C_{k}}\frac{dy}{y}\,y^{\frac{\ln(z)}{\ln(q)}}\prod_{d=1}^{L}\frac{(q^{1-\sigma_{d}}\,y/\tilde{x}_{d}\,;q)_{\infty}}{\left(y/\tilde{x}_{d}\,;q\right)_{\infty}}\;,$
(9.78)

with $\sigma_{d}\equiv\sigma_{d,1}-\sigma_{d,2}$ for all $d$. The half-index
${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf N}})$ and the integral form of the
magnetic block clearly match, since they have the same integrand and contour.

In order to exhibit the analytic properties of the magnetic block in the F.I.
parameter $z$, we can simply evaluate the integral by residue. The result is a
$q$-hypergeometric function which takes the form of a power series expansion
in $z$ with positive radius of convergence around 0, as in the $q$-conformal
block computation.

In the gauge theory $T^{3d}_{A_{1}}$, we can obtain the series form of the
$z$-solution from the onset by imposing Exceptional Dirichlet boundary
conditions ${\bf D_{EX}}$. We start by imposing usual Dirichlet boundary
conditions ${\bf D}$ on the $U(1)$ gauge field. Recall that unlike N, imposing
${\bf D}$ breaks the $U(1)$ bulk gauge symmetry to a global symmetry at the
boundary, which we denote as $U(1)_{\partial}$, with corresponding fugacity
$u$ and generator $\mathfrak{t}_{\partial}$. The index reads

${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf D}})={\rm
Tr}\left[(-1)^{F}\,q^{J+\frac{V}{2}}\,t^{\frac{A-V}{2}}\,u^{\mathfrak{t}_{\partial}}\,z^{\mathfrak{t}_{C}}\,x^{\mathfrak{t}_{H}}\right]\;\;.$
(9.79)

The index takes the form of a sum over all abelian magnetic fluxes $m$ through
the hemisphere $D^{2}$,

${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf
D}})=\frac{1}{(q;q)_{\infty}}\sum_{m\in\mathbb{Z}}\;\prod_{d=1}^{L}\frac{(q^{1+m}\sqrt{t}\,u\,x_{d,2};q)_{\infty}}{(q^{m}\sqrt{t}\,u\,x_{d,1};q)_{\infty}}\;z^{m}\,q^{m}\;.$
(9.80)

$m$ is an integer since it is an element of the cocharacter lattice
$\Lambda_{cochar}=\text{Hom}(U(1),U(1))=\mathbb{Z}$, the space of maps to the
$U(1)$ gauge group.

The exceptional Dirichlet boundary condition is implemented by generating a
flow along the boundary to a new boundary condition. This is done by giving a
vev $c\neq 0$ to, say, the $k$-th fundamental chiral field
$\Phi^{+}_{k,|\partial}=c$. This chiral field has charge $(1,1,1)$ under
$U(1)_{V}\times U(1)_{\partial}\times U(1)_{\mathcal{H}}$, where
$U(1)_{\mathcal{H}}$ is part of the maximal torus of the relevant $U(L)$. In
the index, the weight of the corresponding field is set to 1: following our
choice of vacuum in (9.74), this is $u\,x_{k,1}\,t^{1/2}=1$, meaning we simply
substitute $u=t^{-1/2}\,x_{k,1}^{-1}$ in the entire summand (9.80). We change
to center of mass notation, $x_{d,j}\equiv\tilde{x}_{d}\,q^{\sigma_{d,j}}$
($j=1,2$), and once again set $\sigma_{d}\equiv\sigma_{d,1}-\sigma_{d,2}$ for
all $d$. After rescaling $x_{d,1}$ and $x_{d,2}$ by the appropriate factors of
$q^{\sigma_{d,j}}$, we find in these new center of mass coordinates

${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf
D_{EX,k}}})=\frac{1}{(q;q)_{\infty}}\sum_{m\geq
0}\;\prod_{d=1}^{L}\frac{(q^{1+m}\,\tilde{x}_{d}/\tilde{x}_{k};q)_{\infty}}{(q^{\sigma_{d}+m}\,\tilde{x}_{d}/\tilde{x}_{k};q)_{\infty}}\;z^{m}\,q^{m(1-\sigma_{d})}\;.$
(9.81)

Note the summation index $m$ now runs over the positive integers only, since
the numerator is null otherwise. The factors of $z^{m}$ and
$q^{m(1-\sigma_{d})}$ are interpreted as a mixed $U(1)_{\partial}-U(1)_{C}$
anomaly and a mixed $U(1)_{\partial}-U(1)_{\mathcal{H}}$ anomaly,
respectively. The index is holomorphic in a punctured neighborhood of
$0_{\mathfrak{C}_{C}}$, the origin of the F.I. chamber
$\mathfrak{C}_{C}=\\{|z|<1\\}$.

Up to normalization by the $z$-independent part, we recognize the $k$-th
component of the vector vertex function $\mathbb{V}$ introduced earlier.
Namely,

$\mathbb{V}_{k}=\left[(\tilde{x}_{k})^{\eta}\,\frac{\Theta(q^{\sigma_{k}};q)}{(q;q)_{\infty}}\prod_{d\neq
k}^{L}\frac{\Theta(q^{1-\sigma_{k}}\tilde{x}_{k}/\tilde{x}_{d};q)}{\Theta(\tilde{x}_{k}/\tilde{x}_{d};q)}\right]{\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf
D_{EX,k}}})$ (9.82)

with $\mathbb{V}_{k}$ as in (9.48):

$\mathbb{V}_{k}=(\tilde{x}_{k})^{\eta}\frac{(q^{1-\sigma_{k}};q)_{\infty}}{(q;q)_{\infty}}\prod_{d\neq
k}^{L}\frac{(q^{1-\sigma_{d}}\tilde{x}_{k}/\tilde{x}_{d};q)_{\infty}}{(\tilde{x}_{k}/\tilde{x}_{d};q)_{\infty}}\sum_{m\geq
0}\left(\frac{z}{\prod_{d}q^{\sigma_{d}-1}}\right)^{m}\prod_{d=1}^{L}\frac{(q^{\sigma_{d}}\tilde{x}_{d}/\tilde{x}_{k};q)_{m}}{(q\,\tilde{x}_{d}/\tilde{x}_{k};q)_{m}}\;.$
(9.83)

This is proved by simply invoking the definition of the finite $q$-Pochhammer
symbol

$(q^{m}\,x;q)_{\infty}=\frac{(x;q)_{\infty}}{(x;q)_{m}}\;\;,\qquad
m\in\mathbb{Z}^{+}\;,$ (9.84)

as well as the theta function
$\Theta_{q}(x)=(x\,;q)_{\infty}\,(q/x\,;q)_{\infty}$. Besides the bulk $U(1)$
contribution $(q;q)_{\infty}$, the prefactor in brackets represents the
contributions of various additional degrees of freedom at the boundary: the
$(L-1)$ theta functions on the denominator are the contributions of $(L-1)$ 2d
$\mathcal{N}=(0,2)$ chiral multiplets living on the boundary. We also find $L$
2d $\mathcal{N}=(0,2)$ Fermi multiplets on the boundary, which make up the
numerator theta functions.

The ratio of theta functions in the product is the $\mathcal{N}=(0,2)$ version
of a similar contribution found in the 2d $\mathcal{N}=(2,2)$ context, where
enhanced supersymmetry would require a fixed value $q^{\sigma_{d}}=\hbar$ for
all $d=1,\ldots,L$; there, this ratio represents the contribution of $(L-1)$
boundary 2d $\mathcal{N}=(2,2)$ chiral multiplets, which are used to “flip”
the polarization of hypermultiplets with ${\bf D_{EX,k}}$ boundary conditions.
In our 3d $\mathcal{N}=2$ theory, we initially chose a polarization
$(+,+,\ldots,+)$, as defined in Section 3.4.3, but the vertex function
$\mathbb{V}_{k}$ as we have introduced it is more naturally presented in the
polarization $(-,\ldots,-,+,-,\ldots,-)$, where all “$+$” except for the
$k$-th one are flipped to “$-$”, hence the ratio of theta functions appearing
in (9.82).

We fix the mass parameter chamber as

$\mathfrak{C}_{H}=\\{|\tilde{x}_{1}|<|\tilde{x}_{2}|<\ldots<|\tilde{x}_{L}|\\}\,,$
(9.85)

and further choose the parameters $\sigma_{d,j}$ such that

$1<|q^{\sigma_{d,1}}|<|q^{\sigma_{d,2}}|$ (9.86)

for all $d=1,\ldots,L$, with the understanding that $|q|<1$. The $x$-solution
is by definition holomorphic in a punctured neighborhood of
$0_{\mathfrak{C}_{H}}$, the origin of the chamber $\mathfrak{C}_{H}$. It is
obtained by imposing enriched Neumann boundary conditions
$\bf{N}_{EN,\mathfrak{C}_{H}}$ in the theory $T^{3d}_{A_{1}}$. In what
follows, we simply write ${\bf{N}_{EN}}$ and keep the dependence on the
chamber $\mathfrak{C}_{H}$ implicit.

The support of ${\bf{N}_{EN}}$ consists of attracting submanifolds to the
vacuum $\\{\bf A\\}$, defined via the gradient flow from other vacua $\\{\bf
B\\}$. Crucially, we should include all submanifolds where the image of the
moment map is strictly greater than the image of the moment map for our vacuum
${\bf{A}}$. In this simple example, it is enough to proceed in two steps:
first, we multiply the integrand of the $\bf{N}$ half-index (9.78) by

$\frac{\displaystyle\prod_{d<k}\Theta\left(\tilde{x}_{d}/y\right)\prod_{d>k}\Theta(q^{\sigma_{d}}\tilde{x}_{d}/y)}{\displaystyle\prod_{d=1}^{L}\Theta(q^{\sigma_{d}}\tilde{x}_{d}/y)}\;.$
(9.87)

This insertion has precisely the desired effect of flipping chiral multiplets
from N to D and vice-versa, such that the support is on attracting manifolds
only. The new pole structure is

$\displaystyle
y=q^{-n}\,\tilde{x}_{d}\,,\;\;\;\;\,\qquad\;n=0,1,2,\ldots,\qquad d\geq k\;,$
$\displaystyle
y=q^{n}\,\tilde{x}_{d}\,q^{\sigma_{d}}\,,\qquad\;n=0,1,2,\ldots,\qquad d\leq
k\;,$ (9.88)

and the contour $C$ is deformed so as to enclose only the poles from the first
line, which accumulate at $y=\infty$. Second, the remaining insertions are
determined from anomaly cancellation, for all $k=1,\ldots,L$. In the end, we
recover the expression

${\mathcal{Z}}(T^{3d}_{{A_{1}},{\bf
N_{EN}}_{,\mathfrak{C}_{H}}})=\oint_{C}\frac{dy}{y}\,y^{\frac{\ln(z)}{\ln(q)}}\prod_{d=1}^{L}\frac{(q^{1-\sigma_{d}}\,y/\tilde{x}_{d}\,;q)_{\infty}}{\left(y/\tilde{x}_{d}\,;q\right)_{\infty}}\;\mathfrak{B}_{\mathfrak{C}}$
(9.89)

for the magnetic $x$-solution, where the pole-subtraction matrix
$\mathfrak{B}_{\mathfrak{C}}$ is as before,

$\mathfrak{B}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})=U_{\mathfrak{C}_{H},k}(z,\\{\tilde{x}_{d}\\})\;\frac{\text{Stab}^{Ell}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})}{\displaystyle\prod_{d=1}^{L}\Theta\left(q^{\sigma_{d}}\tilde{x}_{d}/y\right)}\;\text{e}(y,z)^{-1}\,.$
(9.90)

The factor $\text{Stab}^{Ell}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})$
was written in (9.55); it includes the numerator in (9.87) and various other
contributions of 2d $\mathcal{N}=(0,2)$ Fermi and chiral multiplets on the
boundary, which are needed to cancel anomalies, including the bulk mixed
topological/gauge anomaly $\text{e}(y,z)$ in (9.56). There will be leftover
global symmetry mixed anomalies which further need to be canceled: these are
all encoded in the exponential factor
$U_{\mathfrak{C}_{H},k}(z,\\{\tilde{x}_{d}\\})$ as written in (9.57).

As a concrete example, consider the case $L=2$, meaning two surface defect
insertions at center of mass position $\tilde{x}_{1}$ and $\tilde{x}_{2}$ on
$\mathcal{C}$ (with chamber
$\mathfrak{C}_{H}=\\{|\tilde{x}_{1}|<|\tilde{x}_{2}|\\}$), and the reference
vacuum labeled by $k=2$. Our choice of chamber $\mathfrak{C}_{H}$ instructs us
to introduce two boundary chiral multiplets and a Fermi multiplet, which
contribute to the index as

$\frac{\Theta(\tilde{x}_{1}/y)}{\Theta(q^{\sigma_{1}}\tilde{x}_{1}/y)\,\Theta(q^{\sigma_{2}}\tilde{x}_{2}/y)}$
(9.91)

inside the integrand. This will flip the two 3d chiral multiplets from having
D to N boundary conditions, and likewise it will flip one chiral multiplet
from having ${\bf N}$ to ${\bf D}$ boundary conditions, providing the adequate
support on the attracting submanifold under gradient flow. Indeed, note that
evaluation at $y=x_{1}$ makes the numerator vanish, so the $k^{\prime}=1$
vacuum ${\bf{B}}$ is not supported on ${\bf{N}_{EN}}$, in accordance with the
definition of the chamber $\mathfrak{C}_{H}$. The boundary anomalies are
canceled by computing the anomaly polynomial and canceling the quadratic
terms. We find that additional Fermi and chiral multiplets are required, with
contribution

$\frac{\Theta(q^{\sigma_{1}+\sigma_{2}}x_{2}/z\,y)}{\Theta(q^{\sigma_{1}+\sigma_{2}}/z)}\;,$
(9.92)

where $z$ is the F.I. parameter. The leftover anomalies are canceled by (9.56)
and (9.57), for $L=2$ and $k=2$.

The electric solutions are $U_{\hbar}(\widehat{A_{1}})$-algebra $q$-conformal
blocks, which solve the $A_{1}$-type qKZ equations. To engineer them from the
3d gauge theory $T^{3d}_{A_{1}}$, we simply change the support manifold from
$M_{3}=S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}$ to
$M^{\times}_{3}=S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}^{\times}_{q}$,
with the origin removed. The index is as before, but now includes 3d/1d
contributions of 1/2-BPS Wilson loop operators on
$S^{1}_{\mathcal{C}^{\prime}}\times\\{0\\}$ coupled to the 3d gauge theory:

$\text{Stab}^{K}_{i}(y,\vec{x})=\prod_{d=1}^{i-1}(1-y/\tilde{x}_{d})\;\prod_{d=i+1}^{L}(1-q^{-\sigma_{d}}\,y/\tilde{x}_{d})$
(9.93)

Namely, the loops are supported on the K-theoretic stable envelopes.

###  9.4 The 5d gauge theory

Consider a 5d $\mathcal{N}=1$ gauge theory $T^{5d}_{\mathfrak{g}_{\text{o}}}$
with gauge group

$G_{5d}=U(n)\qquad\text{with}\;\;n=L\;,$ (9.94)

and flavor group

$G^{F}_{5d}=U(n^{F})\qquad\text{with}\;\;n^{F}=2\,L\;,$ (9.95)

with the $U(1)$ centers understood as nondynamical. The matter content
consists of $2\,L$ fundamental hypermultiplets in representation
$(n,\overline{n^{F}})$ of $U(n)\times U(n^{F})$. There is a remaining global
symmetry, the topological symmetry

$G^{top}_{5d}=U(1)^{top}\;,$ (9.96)

whose associated charge is the instanton number $k$. The mass for this
topological symmetry is the instanton counting parameter $z$, related to the
5d gauge coupling as $z=e^{-8\pi^{2}\,R_{\mathcal{C}^{\prime}}/g^{2}_{5d}}$.
The bare Chern-Simons level is set to 0. The 5d gauge theory
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ should be understood as a relevant
deformation of a fixed point in the UV, with the deformation scale set by the
gauge coupling116116116This is a particular case of a more general statement
[268]: 5d $SU(n)$ gauge theory with $n^{F}$ fundamental hypermultiplets has a
UV fixed point if $n^{F}+2\,|k_{CS,bare}|\leq 2\,n$, with $k_{CS,bare}$ an
integer whenever $n^{F}$ is even and $k_{CS,bare}$ half an odd integer
whenever $n^{F}$ is odd. In our case, we have $n_{F}=2\,L$, $n=L$, and
$k_{CS,bare}=0$, which saturates the inequality..

__

Figure 18: The 5d $\mathcal{N}=1$ gauge theory and its Higgsing to the
Drinfeld gauge theory $T^{3d}_{A_{1}}$. Here, we focus on the moduli space of
charge $N=1$ vortices. The string theory realization is shown at the bottom,
with branes wrapping a resolved $A_{1}$ singularity. The green and blue colors
stand for D5 branes, while the yellow color stands for D3 branes.

We take $T^{5d}_{\mathfrak{g}_{\text{o}}}$ to be supported on
$M_{5}=S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$,
or on
$M^{\times}_{5}=S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}^{\times}_{q}\times\mathbb{C}_{t}$
with the origin removed, depending on whether we want to make contact with
$\mathcal{W}_{q,t}(A_{1})$ or $U_{\hbar}(\widehat{A_{1}})$ $q$-conformal
blocks, respectively.

First consider the manifold $M_{5}$. The instanton partition function of
$T^{5d}_{A_{1}}$ is computed as the Witten index of a 1d $\mathcal{N}=4$
gauged quantum mechanics on $S^{1}_{\mathcal{C}^{\prime}}$, follwoing (4.11).
We fix the F.I. parameter of the quantum mechanics to lie in the positive
chamber $\zeta>0$. Because $n^{F}=2\,n$ and the 5d bare Chern-Simons level is
0, the Coulomb branch continuum contributes additional spurious UV states to
the index, which will factorize as

${\mathcal{Z}}_{inst}=Z_{inst}\cdot{\mathcal{Z}}_{extra}$ (9.97)

with ${\mathcal{Z}}_{extra}\neq 1$. In the rest of this section, we will only
make statements about ${\mathcal{Z}}_{inst}$, so we won’t need to identify
${\mathcal{Z}}_{extra}$ explicitly. The details of the factorization can be
found in [287, 288, 289, 290].

The quantum mechanical index is a sum over all instanton numbers,

${\mathcal{Z}}_{inst}=\sum_{k=0}^{\infty}{\mathcal{Z}}_{k}\;\frac{z^{k}}{k!}\;.$
(9.98)

where $k!$ is the order of the Weyl group for $U(k)$. The quantum mechanical
index is evaluated in the F.I. chamber $\zeta>0$ with coupling
$g_{QM}\rightarrow 0$, in two steps: first, we integrate over non-zero modes
via Gaussian integrals over massive fluctuations, keeping the zero modes
fixed. Second, we integrate over these zero modes, which are the scalars
$\varphi$ inside the 1d $\mathcal{N}=2$ vector multiplet of the $U(k)$ quantum
mechanics (and the gaugini fermion zero modes). The scalars $\varphi$ have $k$
eigenvalues $I=1,\ldots,k$, and get complexified as
$\phi_{I}=\varphi_{I}+i\,A^{\theta}_{I}$, with $A^{\theta}$ the holonomy
around $S^{1}_{\mathcal{C}^{\prime}}$. Explicitly, we obtain [279, 280, 281,
92, 282]:

$\displaystyle{\mathcal{Z}}_{k}=\oint\prod_{I=1}^{k}\left[\frac{d\phi_{I}}{2\pi
i}\right]{\mathcal{Z}}^{vec}\cdot{\mathcal{Z}}^{fund}\;,$ (9.99)

with

$\displaystyle{\mathcal{Z}}^{vec}=\prod_{\begin{subarray}{c}I\neq J\\\
I,J=1\end{subarray}}^{k}\mbox{sh}(\phi_{I}-\phi_{J})\prod_{I,J=1}^{k}\frac{\mbox{sh}(\phi_{I}-\phi_{J}+{\epsilon}_{q}+{\epsilon}_{t})}{\mbox{sh}(\phi_{I}-\phi_{J}+{\epsilon}_{q})\mbox{sh}(\phi_{I}-\phi_{J}+{\epsilon}_{t})}$
$\displaystyle\qquad\times\prod_{I=1}^{k}\prod_{i=1}^{L}\frac{1}{\mbox{sh}\left(\phi_{I}-a_{i}+({\epsilon}_{q}+{\epsilon}_{t})/2\right)\mbox{sh}\left(\phi_{I}-a_{i}-({\epsilon}_{q}+{\epsilon}_{t})/2\right)}\;,$
(9.100) $\displaystyle
Z^{fund}=\prod_{I=1}^{k}\prod_{s=1}^{2L}\mbox{sh}\left(\phi_{I}-m_{s}\right)\;.$

We used the notation $\mbox{sh}(x)\equiv
2\sinh(R_{\mathcal{C}^{\prime}}\,x/2)$. The factors $Z^{vec}$ and $Z^{fund}$
are the 1-loop determinants for the quantum mechanical modes of the vector
multiplet and fundamental hypermultiplets of the 5d gauge theory,
respectively.

The equivariant parameters appearing in the argument of the $\mbox{sh}(x)$
function are related to the K-theoretic fugacities we have been using in this
paper, as

$\displaystyle q=e^{R_{\mathcal{C}^{\prime}}\,\epsilon_{q}},\qquad
t=e^{-R_{\mathcal{C}^{\prime}}\,\epsilon_{t}},\qquad
e_{i}=e^{-R_{\mathcal{C}^{\prime}}\,a_{i}},\qquad
f_{s}=e^{-R_{\mathcal{C}^{\prime}}\,m_{s}}\;,$ (9.101)

where nonzero holonomies are understood.

In the F.I. chamber $\zeta>0$, the contours are chosen to enclose a subset of
the poles in ${\mathcal{Z}}^{vec}$. The poles that end up contributing to the
residue integral are famously classified by $n$-colored Young tableaux
$\\{\overrightarrow{\boldsymbol{\mu}}\\}=\\{\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2},\ldots,\boldsymbol{\mu}_{n}\\}$
[92, 292]:

$\phi_{I}=a_{i}+\frac{(\epsilon_{q}+\epsilon_{t})}{2}-s_{1}\,\epsilon_{1}-s_{2}\,\epsilon_{2},\text{with}\;(s_{q},s_{t})\in\boldsymbol{\mu}_{i}\;.$
(9.102)

Summing over residues, the instanton partition function is the sum

${\mathcal{Z}}_{inst}(T^{5d}_{A_{1}})=\sum_{\\{\overrightarrow{\boldsymbol{\mu}}\\}}z^{\sum_{i=1}^{n}{\left|\boldsymbol{\mu}_{i}\right|}}\,{\boldsymbol{Z}}^{5d,vec}\cdot{\boldsymbol{Z}}^{5d,fund}\cdot{\boldsymbol{Z}}^{5d,CS}\,,$
(9.103)

with

$\displaystyle{\boldsymbol{Z}}^{5d,vec}=\prod_{i,j=1}^{L}\left[\mathcal{N}_{\boldsymbol{\mu}_{i}\boldsymbol{\mu}_{j}}\left(\frac{e_{i}}{e_{j}}\,;q\right)\right]^{-1}$
(9.104)
$\displaystyle{\boldsymbol{Z}}^{5d,fund}=\prod_{d=1}^{2L}\prod_{i=1}^{L}\mathcal{N}_{\boldsymbol{\emptyset}\,\boldsymbol{\mu}_{i}}\left(\sqrt{\frac{q}{t}}\,\frac{f_{a,s}}{e_{a,i}}\,;q\right)$
(9.105)
$\displaystyle{\boldsymbol{Z}}^{5d,CS}=\prod\limits_{i=1}^{L}\left((-1)^{|\boldsymbol{\mu}_{i}|}q^{\Arrowvert\boldsymbol{\mu}_{i}\Arrowvert^{2}/2}t^{-\Arrowvert\boldsymbol{\mu}_{i}^{t}\Arrowvert^{2}/2}\right)^{L}$
(9.106)

and

$\mathcal{N}_{\boldsymbol{\mu}_{i}\boldsymbol{\mu}_{j}}(Q\,\;q)=\prod\limits_{k,s=1}^{\infty}\frac{\big{(}Q\,q^{\boldsymbol{\mu}_{i,k}-\boldsymbol{\mu}_{j,s}}\,t^{s-k+1}\big{)}_{\infty}}{\big{(}Q\,q^{\boldsymbol{\mu}_{i,k}-\boldsymbol{\mu}_{j,s}}\,t^{s-k}\big{)}_{\infty}}\,\frac{\big{(}Q\,t^{s-k}\big{)}_{\infty}}{\big{(}Q\,t^{s-k+1}\big{)}_{\infty}}\;.$
(9.107)

The notation $\boldsymbol{\mu}_{i,k}$ stands for the length of the $k$-th row
in the partition $\boldsymbol{\mu}_{i}$.

Gauge-vortex duality to 3 dimensions is achieved by freezing each of the $L$
Coulomb moduli $e_{i}$ to a distinct mass among the $2\,L$ ones $f_{s}$. After
turning on vortices of charge $N_{i}$, the Coulomb parameters get shifted as:

$e_{i}=f_{2\,i}\,t^{N_{i}}\,\sqrt{t/q}\;,\qquad\;i=1,\ldots,L\;.$ (9.108)

This truncates each partition $\boldsymbol{\mu}_{i}$: if there are more than
$N_{i}$ rows, the entire summand of the partition function vanishes. In the
gauge-vortex dictionary, the vortex charges $N_{i}$ are identified with the
rank of the 3d gauge group, so let us turn on a single unit of vortex charge
$N=1$ on the Higgs branch of $T^{5d}_{A_{1}}$. At the level of the instanton
partition function, this means we freeze

$e_{k}=x_{k,2}^{-1}\,t\,\sqrt{t/q}\;,\qquad\;\text{for
some}\;k\in\\{1,\ldots,L\\}\;$ (9.109)

and

$e_{i}=x_{i,2}^{-1}\,\sqrt{t/q}\;,\qquad\;\text{for all}\;i\neq k\;.$ (9.110)

(We implicitly relabeled all 5d hypermultiplet masses as 3d chiral multiplet
masses, $f_{a,s}\rightarrow x^{-1}_{d,s}$, see footnote 72). Specializing
(9.103) as such, we recover our previous expression (9.81) for the 3d half-
index with exceptional boundary conditions for a $U(1)$ gauge group and
$U(L)\times U(L)$ flavor group:

$\displaystyle{\mathcal{Z}}(T^{3d}_{A_{1},\bf
D_{EX}})=c_{3d}\cdot{\mathcal{Z}}_{inst}(T^{5d}_{A_{1}})_{e_{i}=x^{-1}_{i,2}\,t^{N_{a,i}}}\,.$
(9.111)

This is the UV version of the IR statement proved in [62], where the Higgsed
5d instanton partition function was identified as a 3d holomorphic block on
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}$.

To recover the electric blocks, one works on $M^{\times}_{5}$ instead, and
simply inserts the Fermi contributions of Wilson loops directly into the
quantum mechanics integrand (9.99). The contour is not modified by this
insertion, and the electric form of the magnetic block follows after the same
Higgsing we performed.

## Acknowledgments

We are grateful to Mina Aganagic, Mykola Dedushenko, Gurbir Dhillon, Ahsan
Khan, Spencer Tamagni, Oleksandr Tsymbaliuk and Edward Witten for valuable
discussions at various stages of this project. NH acknowledges support from
NSF Grant PHY-2207584 and the Sivian Fund at the Institute for Advanced
Studies.

##  Appendix A Basics of Lie theory and Langlands duality

Let us briefly review some Lie theory terminology: Let $G$ be a simple Lie
group117117117The discussion also applies to a compact Lie group with minor
modifications. with Lie algebra $\mathfrak{g}$ of rank $r$. We denote the
maximal torus of $G$ as $T$, with Lie algebra $\mathfrak{h}$, also called the
Cartan subalgebra of $\mathfrak{g}$.

One defines a symmetric bilinear form
$K:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ as118118118For
convenience, We have chosen a normalization with a factor of half the dual
Coxeter number in front of the trace, $1/2\,h^{\vee}$. For reference, the
explicit value of the dual Coxeter number of $\mathfrak{g}$ is
$h^{\vee}_{A_{r}}=r+1$, $h^{\vee}_{B_{r}}=2r-1$, $h^{\vee}_{C_{r}}=r+1$,
$h^{\vee}_{D_{r}}=2r-2$, $h^{\vee}_{E_{6}}=12$, $h^{\vee}_{E_{7}}=18$,
$h^{\vee}_{E_{8}}=30$, $h^{\vee}_{F_{4}}=9$, and $h^{\vee}_{G_{2}}=4$.

$K(X,Y)=\frac{1}{2\,h^{\vee}}\text{Tr}\left(\text{ad}X\,\text{ad}Y\right)$
(A.1)

called the Killing form on $\mathfrak{g}$. The role of this form is to
establish an isomorphism between the Cartan subalgebra $\mathfrak{h}$ and its
dual $\mathfrak{h}^{\vee}$. Namely, for $a=1,\ldots,r$ and
${\mathbb{h}}_{a}\in\mathfrak{h}$, the form $K({\mathbb{h}}_{a},\cdot)$ maps
every element of $\mathfrak{h}$ to a scalar. Then, for every
$\beta\in\mathfrak{h}^{\vee}$, there exists
${\mathbb{h}}_{\beta}\in\mathfrak{h}$ such that
$\beta({\mathbb{h}}_{a})=K({\mathbb{h}}_{a},{\mathbb{h}}_{\beta})$. Equipped
with this isomorphism, it is possible to write a positive-definite scalar
product on the dual space as

$\langle\beta,\gamma\rangle=K(h_{\beta},h_{\gamma})\;.$ (A.2)

For an arbitrary representation of $\mathfrak{g}$, there always exists a basis
$\\{|\lambda\rangle\\}$ such that

${\mathbb{h}}_{a}|\lambda\rangle=\lambda_{a}|\lambda\rangle\;,\;\qquad
a=1,\ldots,r\;.$ (A.3)

The eigenvalues $\lambda_{a}$ make up a vector
$\lambda=(\lambda_{1},\ldots,\lambda_{r})$ called a weight. Weights live in
the dual space $\mathfrak{h}^{\vee}$, meaning
$\lambda({\mathbb{h}}_{a})=\lambda_{a}$. As such, the scalar product between
weights is fixed by the Killing form, and we denote it as (A.2). Weights in
the adjoint representation of $\mathfrak{g}$ are called roots. In particular,
roots are also valued in the space $\mathfrak{h}^{\vee}$, so we can take (A.2)
to define the scalar product on the root space. The dimension of the root
space is the dimension of $\mathfrak{g}$ minus its rank $r$, which is larger
than $r$. This means that roots are linearly dependent, so we fix a basis
$\\{\alpha_{1},\ldots,\alpha_{r}\\}$ in $\mathfrak{h}^{\vee}$. A given root
$\alpha$ can therefore be be written in that basis as
$\alpha=\sum_{a=1}^{r}u_{a}\,\alpha_{a}$, and we define an ordering by
declaring $\alpha$ positive if the first nonzero entry in the vector
$(u_{1},\ldots,u_{r})$ is positive. We denote the set of positive roots as
$\Delta_{+}$. A root $\alpha_{a}$ is called simple root when it cannot be
written as the sum of two elements of $\Delta_{+}$. We denote the set of
simple roots as $\Delta$. Since there are precisely $r$ such simple roots, we
conveniently choose the basis $\\{\alpha_{1},\ldots,\alpha_{r}\\}$ to be
exclusively made out of the simple roots.

The Weyl group $W$ of $\mathfrak{g}$ is generated by the reflections with
respect to the simple roots. Explicitly, a reflection with respect to a simple
root $\alpha_{b}$ is the map

$\alpha_{a}\mapsto\alpha_{a}-\alpha_{b}\frac{2\,\langle\alpha_{a},\alpha_{b}\rangle}{\langle\alpha_{b},\alpha_{b}\rangle}\;.$
(A.4)

In Physics, the Lie group $G$ is called the gauge group, and the choice of a
Cartan subalgebra $\mathfrak{h}$ in $\mathfrak{g}$ specifies a Higgsing, or
breaking, of $G$ to $T\times W$ at a generic point on the moduli space, where
$T\cong U(1)^{r}$ denotes a maximal torus, and $W$ denotes the Weyl group of
$\mathfrak{g}$.

The classification of simple Lie algebras guarantees that when $\mathfrak{g}$
is simply-laced, meaning that when
$\mathfrak{g}=A_{r},D_{r},E_{6},E_{7},E_{8}$, all simple roots have the same
length squared. Meanwhile, when $\mathfrak{g}$ is non simply-laced, meaning
when $\mathfrak{g}=B_{r},C_{r},F_{4},G_{2}$, simple roots are divided into two
sets $\Delta_{l}$ and $\Delta_{s}$, whose elements are respectively called
long roots and short roots. The long roots all have the same length squared,
and the short roots likewise all have the same length squared. The ratio of
the length squared of long roots over short roots is an integer
$\mathfrak{n}_{\mathfrak{g}}$ called the lacing number of $\mathfrak{g}$. When
$\mathfrak{g}$ is simply-laced, long and short roots are one and the same, so
$\mathfrak{n}_{\mathfrak{g}}=1$. In the non simply-laced case, one finds
$\mathfrak{n}_{\mathfrak{g}}=2$ for $\mathfrak{g}=B_{r},C_{r},F_{4}$, while
$\mathfrak{n}_{G_{2}}=3$.

Additionally, one defines a set of $r$ simple coroots of $\mathfrak{g}$,

$\alpha_{a}^{\vee}=2\alpha_{a}/\langle\alpha_{a},\alpha_{a}\rangle\;.$ (A.5)

Note from the denominator that this definition of coroots depends on the
normalization of the scalar product of roots in $\mathfrak{g}$. In order to
give a definition of coroots which is intrinsically independent of such a
normalization, we should understand the coroots of $\mathfrak{g}$ as elements
of the Cartan subalgebra $\mathfrak{h}$, since the roots of $\mathfrak{g}$
were elements of the dual space $\mathfrak{h}^{\vee}$. The existence of
coroots gives a definition of the Langlands dual Lie group ${}^{L}G$, with
corresponding Lie algebra ${}^{L}\mathfrak{g}$, such that the roots of
${}^{L}\mathfrak{g}$ are the coroots of $\mathfrak{g}$. It implies that
${}^{L}\mathfrak{g}=\mathfrak{g}$ whenever $\mathfrak{g}$ is simply-laced.
Moreover, ${}^{L}B_{r}=C_{r}$, ${}^{L}C_{r}=B_{r}$,
${}^{L}F_{4}=\widetilde{F_{4}}$ and ${}^{L}G_{2}=\widetilde{G_{2}}$, by which
we mean that $B_{r}$ and $C_{r}$ get mapped to each other under Langlands
duality, while $F_{4}$ and $G_{2}$ get mapped to themselves, but in a
nontrivial way, since their long and short roots get swapped under duality
(hence the use of a different notation $\widetilde{F_{4}}$ and
$\widetilde{G_{2}}$). Note that ${}^{L}(^{L}\mathfrak{g})=\mathfrak{g}$ for
all simple $\mathfrak{g}$.

The Weyl group ${}^{L}W$ of ${}^{L}\mathfrak{g}$ is generated by the
reflections with respect to its simple roots, which are the simple coroots of
$\mathfrak{g}$. Explicitly, a reflection with respect to a simple coroot
$\alpha^{\vee}_{b}$ is the map

$\alpha_{a}\mapsto\alpha_{a}-\alpha^{\vee}_{b}\frac{2\,\langle\alpha_{a},\alpha^{\vee}_{b}\rangle}{\langle\alpha^{\vee}_{b},\alpha^{\vee}_{b}\rangle}\;.$
(A.6)

It follows from this action and the definition of a coroot that the Weyl
groups of $\mathfrak{g}$ and ${}^{L}\mathfrak{g}$ coincide: ${}^{L}W=W$.

The Cartan matrix of $\mathfrak{g}$ has size $r\times r$, with entries

$C_{ab}=\langle\alpha_{a},\alpha_{b}^{\vee}\rangle\;.$ (A.7)

This definition directly implies that the Cartan matrix of
${}^{L}\mathfrak{g}$ is the transpose of the Cartan matrix of $\mathfrak{g}$.

The fundamental weights of $\mathfrak{g}$ are a distinguished set of $r$
weights, defined as dual to the simple coroots,

$\langle\lambda_{a},\alpha_{b}^{\vee}\rangle=\delta_{ab}\;.$ (A.8)

The expansion coefficients of a weight in the fundamental weight basis are
called Dynkin labels. In particular, the Dynkin labels of a weight in a
finite-dimensional irreducible representation of $\mathfrak{g}$ are always
integers. Finally, one defines $r$ fundamental coweights of $\mathfrak{g}$ as
dual to the simple roots:

$\langle\lambda_{a}^{\vee},\alpha_{b}\rangle=\delta_{ab}\;.$ (A.9)

From the Langlands dual point of view, the fundamental weights of
${}^{L}\mathfrak{g}$ are the fundamental coweights of $\mathfrak{g}$.

The simple roots span a rank $r$ lattice $\Lambda_{rt}$ called the root
lattice of $\mathfrak{g}$. Physically, $\Lambda_{rt}$ is the lattice of
electric charges of massive gauge bosons in 4d Yang-Mills. There exists a dual
rank $r$ lattice $\Lambda_{cowt}$ spanned by the fundamental coweights, called
the coweight lattice of $\mathfrak{g}$. This is the lattice of magnetic
charges of a physical state, as prescribed by the Dirac quantization
condition. Meanwhile, the simple coroots span a rank $r$ lattice
$\Lambda_{cort}$ called the coroot lattice of $\mathfrak{g}$. $\Lambda_{cort}$
is the lattice of magnetic charges of elementary BPS monopoles. The dual rank
$r$ lattice $\Lambda_{wt}$ spanned by the fundamental weights is called the
weight lattice of $\mathfrak{g}$. This is the lattice of electric charges of a
physical state. From the definition of the lattices, it is clear that
$\Lambda_{rt}\subset\Lambda_{wt}$, and that
$\Lambda_{cort}\subset\Lambda_{cowt}$.

We now fix the normalization of the scalar product $\langle,\rangle$ on
$\mathfrak{g}$ for the entire paper. Whenever $\mathfrak{g}$ is simply-laced,
the simple roots $\alpha_{a}$ (which are also the simple coroots) will all
have length squared $2$, meaning $\langle\alpha_{a},\alpha_{a}\rangle=2$ for
all $a=1,\ldots,r$.

Whenever $\mathfrak{g}$ is non simply-laced, we choose to give the long roots
$\alpha_{l,a}$ a length squared $2$, and therefore the short roots
$\alpha_{s,a}$ a length squared $2/\mathfrak{n}_{\mathfrak{g}}$. It follows
that the short coroots $\alpha^{\vee}_{s,a}$ coincide with the long roots,
while the long coroots are $\mathfrak{n}_{\mathfrak{g}}$ times the short
roots:

$\displaystyle\alpha^{\vee}_{s,a}=\alpha_{l,a}\;,\;\;\;\;\,\ \qquad
a=1,\ldots,|\Delta_{l}|$
$\displaystyle\alpha^{\vee}_{l,a}=\mathfrak{n}_{\mathfrak{g}}\cdot\alpha_{s,a}\;,\qquad
a=1,\ldots,|\Delta_{s}|$ (A.10)

For reference, our conventions agree with the ones used in [353], which are
different from the ones used in [31]119119119In [31], Gukov and Witten
normalize their inner products such that short roots have length squared 2,
meaning the long roots will have length $2\,\mathfrak{n}_{\mathfrak{g}}$.
Their short and long coroots are defined as
$\displaystyle\alpha^{\vee}_{s,a}=\alpha_{l,a}/\mathfrak{n}_{\mathfrak{g}}\;,\;\qquad
a=1,\ldots,|\Delta_{l}|$
$\displaystyle\alpha^{\vee}_{l,a}=\alpha_{s,a}\;,\;\;\;\;\;\;\qquad
a=1,\ldots,|\Delta_{s}|\;.$ (A.11) . Our conventions were chosen to match the
existing literature on deformed $\mathcal{W}_{q,t}(\mathfrak{g})$-algebras of
non simply-laced type [23], and their recent string theoretic realizations
[13, 146, 173].

The fundamental weights $w_{a}$ of $\mathfrak{g}$ are defined as dual to the
simple coroots, $\langle w_{a},\alpha_{b}^{\vee}\rangle=\delta_{ab}$. Just
like roots, arbitrary weights live in the dual space $\mathfrak{h}^{\vee}$, so
the Killing form also fixes the scalar product between weights. The expansion
coefficients of a weight in the fundamental weight basis are called Dynkin
labels. In particular, the Dynkin labels of a weight in a finite-dimensional
irreducible representation of $\mathfrak{g}$ are always integers. The
fundamental coweights $w^{\vee}_{a}$ of $\mathfrak{g}$ are likewise defined as
dual to the simple roots,
$\langle\lambda_{a}^{\vee},\alpha_{b}\rangle=\delta_{ab}$.

Langlands duality exchanges the Lie algebras $\mathfrak{g}$ and
${}^{L}\mathfrak{g}$ by taking the transpose of the Cartan matrix $C_{ab}$. It
follows that the simple roots and fundamental weights of $\mathfrak{g}$ are
mapped to the simple coroots and fundamental coweights of
${}^{L}\mathfrak{g}$, respectively. The only simple Lie algebras which are
exchanged under Langlands duality are $B_{r}$ and $C_{r}$. The other simple
Lie algebras are mapped to themselves, although nontrivially for $G_{2}$ and
$F_{4}$, since long and short roots are swapped in ${}^{L}G_{2}$ and
${}^{L}F_{4}$.

For completeness, we mention here the existence of intermediate lattices: let
$\Lambda_{char}$ be a character lattice of $\mathfrak{g}$, which is any
lattice lying between the root and weight lattices,

$\Lambda_{rt}\subset\Lambda_{char}\subset\Lambda_{wt}\;.$ (A.12)

The lattice $\Lambda_{char}$ is also called the group of characters, which is
the space of maps

$\Lambda_{char}=\text{Hom}(T,U(1))$ (A.13)

from the maximal torus $T$ of $G$ to $U(1)$. The choice of such a lattice is
an important ingredient in the classification of compact simple Lie groups, as
$G$ is uniquely determined from $\mathfrak{g}$ and the choice of a lattice
$\Lambda_{char}$.

The lattice $\Lambda_{char}$ admits a dual $\Lambda_{cochar}$, called the
cocharacter lattice of $\mathfrak{g}$, which lies between the coroot and
coweight lattices,

$\Lambda_{cort}\subset\Lambda_{cochar}\subset\Lambda_{cowt}\;.$ (A.14)

The lattice $\Lambda_{cochar}$ is the space of maps

$\Lambda_{cochar}=\text{Hom}(^{L}T,U(1))$ (A.15)

from the maximal torus ${}^{L}T$ of ${}^{L}G$ to $U(1)$. Equivalently, it is
the space of maps $\text{Hom}(U(1),T)$.

The lattices $\Lambda_{char}$ and $\Lambda_{cochar}$ also play a role in
probing the physics of extended operators in 4d, most notably using Wilson or
’t Hooft line operators [370, 30], or the surface operators we study here [31,
32]. A sharper statement, then, is that the lattice of electric charges of $G$
is really $\Lambda_{char}$, while the lattice of magnetic charges is
$\Lambda_{cochar}$. From the point of view of the gauge groups, this amounts
to specifying the global form of $G$ and ${}^{L}G$, respectively. Because the
quantum $q$-Langlands program is formulated at the level of the Lie algebras
$\mathfrak{g}$ and ${}^{L}\mathfrak{g}$, the global form of the gauge groups
is ignored in the correspondence.

When we discuss 4d $\mathcal{N}=4$ Super Yang-Mills of type $G$ and its
surface defects, the group $G$ will be simply-connected in this paper. This is
because we will want $\text{T}:\tau\mapsto\tau+1$ to be a symmetry of the
gauge coupling $\tau=\frac{\theta}{2\pi}+i\,\frac{4\pi}{g^{2}_{4d}}$, and is
argued as follows: the symmetry reflects the invariance of the theory under a
shift of the theta angle $\theta$ by $2\pi$, where $\theta$ appears in the
Yang-Mills action through an instanton term:

$-i\,\theta\,k\;,\;\;\;\text{where}\;\;k=\frac{-1}{8\pi^{2}}\int_{M_{4}}\text{Tr}F\wedge
F\;.$ (A.16)

The trace Tr is a negative-definite quadratic form on $\mathfrak{g}$,
normalized so that if $\alpha^{\vee}_{s}$ is a short coroot, then
$\text{Tr}(\alpha^{\vee}_{s}\cdot\alpha^{\vee}_{s})=-2$. With this
normalization120120120A different normalization of the trace Tr would rescale
the definition of the instanton number by a factor of a root length, which we
decide against for clarity of presentation. We therefore adopt the same
normalization on the short coroot length., and assuming $k$ is an integer (the
instanton number), the action $\text{T}:\theta\mapsto\theta+2\pi$, or
equivalently $\text{T}:\tau\mapsto\tau+1$, is a symmetry only if $G$ is
simply-connected121121121The 4-manifold $M_{4}$ should also be closed. For us,
the manifold $M_{4}$ we typically encounter is $\mathbb{C}^{2}$, and therefore
not closed; this causes a non-compactness of the instanton moduli space in the
IR, which we cure via an $\Omega$-background
$\mathbb{C}_{q}\times\mathbb{C}_{t}$ [92, 93, 292].. If $G$ is not simply-
connected, the symmetry is no longer T, but instead
$\text{T}^{p}:\tau\mapsto\tau+p$, for some integer $p$. The quantum
$q$-Langlands correspondence (with or without ramification) we study is well-
defined at least for $p=1$, hence the requirement that $G$ be simply-
connected. In that case, the character lattice will coincide with the weight
lattice:

$\Lambda_{char}=\Lambda_{wt}\;.$ (A.17)

The simple-connectedness of $G$ further implies that the Langlands dual group
${}^{L}G$ will be of adjoint type, meaning that its root and weight lattices
coincide. Equivalently, this sets the cocharacter lattice of $G$ to coincide
with its coroot lattice:

$\Lambda_{cochar}=\Lambda_{cort}\;.$ (A.18)

##  Appendix B A quick guide to the 3d $\mathcal{N}=2$ half-index

In an effort to keep our presentation self-contained, we summarize here the
various building blocks that make up the 3d $\mathcal{N}=2$ half-index (3.19):

${\mathcal{Z}}(T^{3d}_{\mathfrak{g}_{\text{o}}})={\rm
Tr}\left[(-1)^{F}\,q^{J+\frac{R}{2}}\;{\bf{x}}^{\Pi}\right]\;\;.$ (B.1)

We use the following notations:

$(x\,;q)_{\infty}=\prod_{k=0}^{\infty}\left(1-q^{k}x\right)\;,$ (B.2)

denotes the $q$-Pochhammer symbol, and

$\Theta(x\,;q)=(x\,;q)_{\infty}\,(q/x\,;q)_{\infty}\;.$ (B.3)

the genus 1 odd theta function.

A 3d chiral multiplet $\Phi$ satisfies chirality constraints with respect to
differential operators in superspace:
$\overline{D}_{+}\Phi=0=\overline{D}_{-}\Phi$, with off-shell superfield
content

$\Phi=\phi+\theta^{+}\psi_{+}+\theta^{-}\psi_{-}+\theta^{+}\theta^{-}\,F+\ldots\;.$
(B.4)

The fields $\phi$ and $\psi_{\alpha}$ are respectively a complex boson and a
complex fermion, the field $F$ is an auxiliary complex boson set to 0 by the
equations of motion. We denoted by “$\ldots$” additional terms which will not
contribute to the index. The 3d chiral multiplet decomposes on the boundary
into a (0,2) chiral multiplet,

$\Phi_{|\partial}=\phi+\theta^{+}\psi_{+}-i\theta^{+}\overline{\theta}^{+}\partial_{+}\phi\;,$
(B.5)

which satisfies $\overline{D}_{+}\Phi_{|\partial}=0$, and a (0,2) Fermi
multiplet,

$\Psi_{|\partial}={\psi}_{-}+\theta^{+}\,G-i\theta^{+}\overline{\theta}^{+}\partial_{+}{\psi}_{-}\;,$
(B.6)

which satisfies $\overline{D}_{+}\Psi_{|\partial}=0$. On-shell, the auxiliary
field $G$ is equal to $\partial_{\bot}\phi$, and
$\Psi_{|\partial}=\overline{D}_{+}\Phi|_{\theta^{-}=\overline{\theta}^{-}}=0$.

The bulk action for the 3d chiral multiplet couples to a boundary Fermi
multiplet via a J-term, as
$\Psi_{|\partial}\,\partial_{\bot}\Phi_{|\partial}$. In components, this means
the action contains the term ${\psi}_{-}\,\partial_{\bot}\psi_{+}$. In order
to avoid boundary terms in the equations of motion, one should therefore
either set the fermion $\psi_{-}=0$ or the fermion $\psi_{+}=0$ at the
boundary. Following the terminology of [72], we call the supersymmetric
completion of these boundary conditions Neumann (N) and Dirichlet (D),
respectively. Then, a boundary condition N sets to zero the entire Fermi
multiplet $\Psi_{|\partial}$; in particular, $\psi_{-}=0$ and
$\partial_{\bot}\phi=0$ at the boundary. Meanwhile, a boundary condition D
sets to zero the entire chiral multiplet $\Phi_{|\partial}=0$; in particular,
$\psi_{+}=0$ and $\phi=0$ at the boundary.

It is also possible to consider a “deformed” Dirichlet boundary condition
$\text{D}_{c}$, by setting $\Phi_{|\partial}=c$, with $c\in\mathbb{C}$ a
constant chiral multiplet at the boundary; in particular, $\psi_{+}=0$ and
$\phi=c$ at the boundary. This boundary condition will break $U(1)_{R}$ unless
$\phi$ has R-charge 0. In isolation, the R-charge of such a chiral field is
too low for $\text{D}_{c}$ to flow to a superconformal boundary condition in
the IR; however, such a flow is possible if the field is part of a full
fledged gauge theory (with the gauge group partially broken at the boundary
because of $\text{D}_{c}$). This is the typical situation we encounter in this
paper: some of our chiral multiplets have $\text{D}_{c}$ boundary conditions,
but they are always part of a gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$
whose IR boundary conditions are superconformal.

We denote by $U(1)_{J}$ the action rotating the boundary plane
$\mathbb{R}^{1,1}$. Suppose then that the 3d chiral has charge $+1$ under a
flavor symmetry $U(1)_{x}$, with associated fugacity $x$, and charge $\rho$
under the R-symmetry $U(1)_{R}$. In the decomposition into (0,2) chiral and
Fermi multiplets, the operators in $\overline{Q}_{+}$-cohomology counted by
the index are $\phi$ (the top component of the chiral multiplet) and its
derivatives $\partial^{n}_{z}\phi$, as well as ${\psi}_{-}$ (the top component
of the Fermi multiplet) and its derivatives $\partial^{n}_{z}{\psi}_{-}$, and
products thereof. Under $U(1)_{x}\times U(1)_{R}\times U(1)_{J}$, the
operators $\partial^{n}_{z}\phi$ carry charge $(1,\rho,n)$, while
$\partial^{n}_{z}{\psi}_{-}$ carry charge $(-1,1-\rho,n+\frac{1}{2})$.

When the 3d chiral multiplet is given Neumann boundary conditions N, it
follows that the operators $\partial^{n}_{z}{\psi}_{-}$ are set to zero, so
only the bosons $\partial^{n}_{z}\phi$ contribute to the index:

$\prod_{n=0}^{\infty}\left(1-q^{n+\frac{\rho}{2}}\,x\right)^{-1}=\left(x\,q^{\frac{\rho}{2}};q\right)^{-1}_{\infty}\;.$
(B.7)

Meanwhile, when the 3d chiral multiplet is given Dirichlet boundary conditions
D, the operators $\partial^{n}_{z}\phi$ are set to zero, so only the fermions
$\partial^{n}_{z}{\psi}_{-}$ contribute to the index:

$\prod_{n=0}^{\infty}\left(1-q^{n+(1-\frac{\rho}{2})}/x\right)=\left(q^{1-\frac{\rho}{2}}/x;q\right)_{\infty}\;.$
(B.8)

For the 3d chiral multiplet $\Phi$ with deformed Dirichlet boundary conditions
$\text{D}_{c}$, the fermions $\partial^{n}_{z}{\psi}_{-}$ likewise contribute
to the index, but now carry charge $(0,1,n+\frac{1}{2})$ under $U(1)_{x}\times
U(1)_{R}\times U(1)_{J}$. Indeed, $U(1)_{x}$ is broken at the boundary, and
$U(1)_{R}$ is unbroken only when $\Phi$ has R-charge 0. At the level of the
index, one can simply specialize the fugacity $x\rightarrow 1$ in the D
boundary condition. For a single chiral multiplet, this gives a contribution

$\left(q\;;q\right)_{\infty}\;.$ (B.9)

When the chiral multiplet is part of a gauge theory
$T^{3d}_{\mathfrak{g}_{\text{o}}}$, one should first impose D and then
specialize the corresponding fugacity $x\rightarrow 1$ inside the entire
index.

A 3d $\mathcal{N}=2$ gauge multiplet can be written as a gauge invariant
linear multiplet [260]:

$\Sigma=\mathfrak{a}-\theta\overline{\lambda}+\overline{\theta}\lambda+i\overline{\theta}\theta\mathfrak{D}+\overline{\theta}\sigma^{ij}\theta
F_{ij}+\ldots\;,$ (B.10)

which satisfies
${D}_{\alpha}{D}^{\alpha}\Sigma_{3d}=0=\overline{D}^{\alpha}\overline{D}_{\alpha}\Sigma_{3d}$.
In the multiplet, one finds the field strength $F_{ij}$ for a connection
$A_{i}$, a real scalar $\mathfrak{a}$ and off-shell auxiliary field
$\mathfrak{D}$, and a complex fermion $\lambda_{\alpha}$. The 3d linear
multiplet $\Sigma$ decomposes on the boundary into a (0,2) chiral multiplet,

$S_{|\partial}=(\mathfrak{a}+iA_{\bot})-2\,\theta^{+}\overline{\lambda}_{+}\ldots\;,$
(B.11)

which satisfies $\overline{D}_{+}S_{|\partial}=0$, and a (0,2) gauge
multiplet, or alternatively a chiral field-strength superfield

$\Upsilon_{|\partial}=\lambda_{-}+\theta^{+}(F_{01}+i\,\mathfrak{D}^{\prime})+\ldots\;,$
(B.12)

which satisfies $\overline{D}_{+}\Upsilon_{|\partial}=0$, with
$\mathfrak{D}^{\prime}$ an auxiliary field equal to
$\mathfrak{D}-\partial_{\bot}\mathfrak{a}$ on-shell.

The Neumann boundary condition on $\Sigma$ sets $F_{\bot\mu|\partial}=0$, and
in particular leaves an unbroken gauge symmetry on the 2d boundary. In
contrast, the Dirichlet boundary condition on $\Sigma$ sets
$A_{\mu|\partial}=0$, which implies the local gauge symmetry on the 2d
boundary is broken to a global symmetry. The supersymmetric completion of
these boundary conditions will also be denoted as Neumann (N) and Dirichlet
(${\bf D}$).

The boundary condition ${\bf N}$ preserves a subset of fields present in the
$\Upsilon_{|\partial}$ multiplet only. Let us first ignore the surviving gauge
invariance at the boundary, and treat the symmetry there as global. The
operators in $\overline{Q}_{+}$-cohomology counted by the index are the
gaugino $\lambda_{-}$, its covariant derivatives $D^{n}_{z}\lambda_{-}$, and
products thereof. If we denote the gauge group by $G_{3d}$, then under
$G_{3d}\times U(1)_{R}\times U(1)_{J}$, the operators $D^{n}_{z}\lambda_{-}$
carry charge $(\text{adj},1,n+\frac{1}{2})$. Factorizing the adjoint
contribution as the contribution of $\text{rank}(G_{3d})$ null weights and the
roots, the index is

$\left(q\;;q\right)^{\text{rank}(G_{3d})}_{\infty}\prod_{\alpha_{a}\in\text{roots}\,[G_{3d}]}\left(q\,y_{\alpha_{a}}\;;q\right)_{\infty}\;,$
(B.13)

with $y_{\alpha_{a}}$ valued in the maximal torus of $G_{3d}$. Imposing gauge
invariance is done by integrating over these eigenvalues and inserting the
Vandermonde factor $\frac{1}{W_{G_{3d}}}(1-y_{\alpha_{a}})$:

$\frac{\left(q\;;q\right)^{\text{rank}(G_{3d})}_{\infty}}{W_{G_{3d}}}\oint\frac{dy}{y}\prod_{\alpha_{a}\in\text{roots}\,[G_{3d}]}\left(y_{\alpha_{a}}\;;q\right)_{\infty}\;,$
(B.14)

with $W(G_{3d})$ the Weyl group of $G_{3d}$. In our work, $G_{3d}$ is always a
product of unitary gauge groups. For instance, for $G_{3d}=U(N)$, the above
reads

$\left(q\;;q\right)^{N}_{\infty}\oint\prod_{i=1}^{N}\frac{dy_{i}}{y_{i}}\prod_{j\neq
i}^{N}\left(\,y_{i}/y_{j}\;;q\right)_{\infty}\;.$ (B.15)

The boundary condition ${\bf D}$ preserves a subset of fields in the
$S_{|\partial}$ multiplet instead. The operators in
$\overline{Q}_{+}$-cohomology counted by the index are
$\mathfrak{a}+iA_{\bot}$ and its derivatives, or more precisely
$D_{z}\mathfrak{a}+iF_{z\bot}$ and its derivatives; the reasoning here is that
even though the gauge symmetry is broken at the boundary to a global symmetry
$G_{|\partial}$, there is a residual gauge symmetry along the $x_{\bot}$
coordinate, under which $\mathfrak{a}+iA_{\bot}$ is not invariant, but the
field strength $D_{z}\mathfrak{a}+iF_{z\bot}$ is. The operators
$D^{n+1}_{z}\mathfrak{a}+iD^{n}_{z}F_{z\bot}$ carry charge
$(\text{adj},0,n+\frac{1}{2})$ under $G_{|\partial}\times U(1)_{R}\times
U(1)_{J}$. Factorizing the adjoint contribution as the contribution of
$\text{rank}(G_{3d})$ null weights and the roots, the index is

$\left(q\;;q\right)^{-\text{rank}(G_{|\partial})}_{\infty}\prod_{\alpha_{a}\in\text{roots}\,[G_{|\partial}]}\left(q\,y_{\alpha_{a}}\;;q\right)^{-1}_{\infty}\;,$
(B.16)

with $y_{\alpha_{a}}$ valued in the maximal torus of the boundary global
symmetry $G_{|\partial}$.

Crucially, there are non-perturbative contributions to this ${\bf D}$ half-
index coming from BPS monopole operators, which should be counted as they are
in $\overline{Q}_{+}$-cohomology. A monopole operator in the 3d bulk of a
$\mathcal{N}=2$ theory is a disorder operator given by a singular solution to

$F=\star D\mathfrak{a}\,,\qquad\;D\star\mathfrak{a}=0\;.$ (B.17)

In the abelian case $G_{3d}=U(1)$, the basic solution is a Dirac monopole
$\mathfrak{a}=m/2\,r$, with $r$ is the radial distance from the singularity
and $m$ is an integer, due to the quantization of flux through a 2-sphere
centered at the origin:

$\oint_{S^{2}}\frac{F}{2\pi}=m\in\mathbb{Z}\;.$ (B.18)

Likewise, boundary monopoles contribute to the index. This time, a monopole
operator at the origin should be surrounded by a hemisphere $D^{2}$, with
boundary $S^{1}_{D^{2}}$. Crucially, $F_{z\bot}$ and $\mathfrak{a}$ are left
unconstrained at the boundary, so the presence of such a monopole is
compatible with the Dirichlet boundary condition ${\bf D}$, which we impose to
trivialize the $U(1)$-bundle at the boundary $S^{1}_{D^{2}}$. The curvature
integral now takes the form

$\frac{1}{2\pi}\oint_{D^{2}}F+\frac{1}{2\pi}\oint_{S^{1}_{D^{2}}}A=m\in\mathbb{Z}\;.$
(B.19)

But $\oint_{S^{1}_{D^{2}}}A=0$ from the boundary condition
$A_{\mu|\partial}=0$, so the monopole charge $m$ is nothing but the quantized
flux through the hemisphere:

$\frac{1}{2\pi}\oint_{D^{2}}F=m\in\mathbb{Z}\;.$ (B.20)

More generally, when $G_{3d}$ is nonabelian, we embed the Dirac monopole into
$G_{3d}$, so by definition $m$ is an element of the cocharacter lattice
$m\in\Lambda_{cochar}=\text{Hom}(U(1),T_{G_{3d}})$, the space of maps to the
maximal torus $T_{G_{3d}}$. The index takes the form of a sum over all abelian
magnetic fluxes $m$ on the hemisphere $D^{2}$:

$\left(q\;;q\right)^{-\text{rank}(G_{|\partial})}_{\infty}\sum_{m\in\Lambda_{cochar}}\;\prod_{\alpha_{a}\in\text{roots}\,[G_{|\partial}]}\left(q^{1+m\cdot\alpha_{a}}\,y_{\alpha_{a}}\;;q\right)^{-1}_{\infty}\;\ldots\;,$
(B.21)

where the boundary fugacity $y_{\alpha_{a}}$ is now shifted by $q^{m}$,
reflecting the fact that the corresponding state acquires spin $m$ due to the
monopole. The dots “$\ldots$” stand for the contribution of the remaining
matter content of the theory, as well as boundary mixed ’t Hooft anomalies.

We now write the 2d degrees of freedom living exclusively on the boundary.
Namely, consider a 2d $(2,0)$ chiral multiplet
$\Phi_{|\partial}={\phi}+\ldots$ with charge $+1$ under a flavor symmetry
$U(1)_{x}$ and charge $\rho$ under the R-symmetry $U(1)_{R}$. The operators in
$\overline{Q}_{+}$-cohomology counted by the index are $\phi$,
$\partial_{z}\overline{\phi}$, their derivatives, and products thereof. Under
$U(1)_{x}\times U(1)_{R}\times U(1)_{J}$, the operators $\partial^{n}_{z}\phi$
carry charge $(1,\rho,n)$, while $\partial^{n+1}_{z}\overline{\phi}$ carry
charge $(-1,-\rho,n+1)$. Their contribution to the index is

$\prod_{n=0}^{\infty}\left(1-q^{n+\frac{\rho}{2}}\,x\;;q\right)_{\infty}^{-1}\left(1-q^{n+1-\frac{\rho}{2}}\,x^{-1}\;;q\right)_{\infty}^{-1}=\frac{1}{\Theta\left(x\,q^{\frac{\rho}{2}};q\right)}\;.$
(B.22)

Note that the index counts more operators compared to the 3d chiral multiplet
with Neumann boundary conditions (B.7). Indeed, by the supersymmetry
transformation $\overline{Q}_{+}\overline{\phi}\sim\overline{\psi}_{+}$, the
operators $\partial^{n+1}_{z}\overline{\phi}$ in the 2d chiral multiplet are
$\overline{Q}_{+}$-closed after imposing the Dirac equation
$\partial_{z}\overline{\psi}_{+}=0$, so they are counted by the index. This is
no longer true for the 3d chiral, as $\partial_{z}\overline{\psi}_{+}\neq 0$.

Similarly, consider a 2d $(2,0)$ Fermi multiplet
$\Psi_{|\partial}={\psi}_{-}+\ldots$ with charge $-1$ under a flavor symmetry
$U(1)_{x}$ and charge $1-\rho$ under the R-symmetry $U(1)_{R}$. The operators
in $\overline{Q}_{+}$-cohomology counted by the index are
$\partial^{n}_{z}{\psi}_{-}$, $\partial^{n}_{z}\overline{\psi}_{-}$, their
derivatives, and products thereof. Under $U(1)_{x}\times U(1)_{R}\times
U(1)_{J}$, the operators $\partial^{n}_{z}{\psi}_{-}$ carry charge
$(-1,1-\rho,n+\frac{1}{2})$, while $\partial^{n}_{z}\overline{\psi}_{-}$ carry
charge $(1,\rho-1,n+\frac{1}{2})$. Their contribution to the index is

$\prod_{n=0}^{\infty}\left(1-q^{n+1-\frac{\rho}{2}}\,x^{-1}\;;q\right)_{\infty}^{-1}\left(1-q^{n+\frac{\rho}{2}}\,x\;;q\right)_{\infty}^{-1}=\frac{1}{\Theta\left(x\,q^{\frac{\rho}{2}};q\right)}\;.$
(B.23)

Finally, consider a 2d $(2,0)$ gauge multiplet with gauge group $G$, which we
wrote before in terms of a chiral field-strength superfield $\Upsilon$ (B.12).
The R-charge of the multiplet is uniquely fixed to be 1, since the gauge
fields must have R-charge 0. The operators in $\overline{Q}_{+}$-cohomology
counted by the index are the gauginos $\lambda_{-}$,
$D^{n}_{z}\overline{\lambda}_{-}$, their covariant derivatives, and products
thereof. Note that when we discussed the 3d gauge multiplet with ${\bf N}$
boundary conditions (B.13), $D^{n}_{z}\overline{\lambda}_{-}$ was not
$\overline{Q}_{+}$-closed, by the 3d equations of motion. Under $G\times
U(1)_{R}\times U(1)_{J}$, the operators $D^{n}_{z}{\lambda}_{-}$ carry charge
$(\text{adj},1,n+\frac{1}{2})$, while $D^{n+1}_{z}\overline{\lambda}_{-}$
carry charge $(\text{adj},-1,n+\frac{3}{2})$.

If $G$ were a global symmetry then the index would be

$\left(q\;;q\right)^{2\,\text{rank}(G)}_{\infty}\prod_{\alpha_{a}\in\text{roots}\,[G]}\left(q\,y_{\alpha_{a}}\;;q\right)_{\infty}\left(q\,y^{-1}_{\alpha_{a}}\;;q\right)_{\infty}\;,$
(B.24)

with $y_{\alpha_{a}}$ valued in the maximal torus of $G$. In this expression,
we factorized (twice) the adjoint contribution as that of $\text{rank}(G)$
null weights and the roots. Imposing gauge invariance is done by integrating
over the $y_{\alpha_{a}}$ eigenvalues and inserting the Vandermonde factor
$\frac{1}{W_{G}}(1-y_{\alpha_{a}})$:

$\frac{\left(q\;;q\right)^{2\,\text{rank}(G)}_{\infty}}{W_{G}}\oint\frac{dy}{y}\prod_{\alpha_{a}\in\text{roots}\,[G]}\Theta\left(y_{\alpha_{a}};q\right)\;.$
(B.25)

## References

  * Beilinson and Drinfeld [1991] A. Beilinson and V. Drinfeld, “Quantization of hitchin’s integrable system and hecke eigensheaves”, 1991.
  * Feigin and Frenkel [1992] B. Feigin and E. Frenkel, “Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras”, Int. J. Mod. Phys. A 7S1A (1992) 197–215.
  * Teschner [2011] J. Teschner, “Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I”, Adv. Theor. Math. Phys. 15 (2011), no. 2, 471–564, arXiv:1005.2846.
  * Furlan et al. [1993] P. Furlan, A. C. Ganchev, R. Paunov, and V. B. Petkova, “Solutions of the Knizhnik-Zamolodchikov equation with rational isospins and the reduction to the minimal models”, Nucl. Phys. B 394 (1993) 665–706, hep-th/9201080.
  * Frenkel et al. [2016] E. Frenkel, S. Gukov, and J. Teschner, “Surface Operators and Separation of Variables”, JHEP 01 (2016) 179, arXiv:1506.07508.
  * Feigin and Frenkel [1990] B. Feigin and E. Frenkel, “Quantization of the drinfeld-sokolov reduction”, Physics Letters B 246 (1990), no. 1, 75–81.
  * de Boer and Tjin [1994] J. de Boer and T. Tjin, “The Relation between quantum W algebras and Lie algebras”, Commun. Math. Phys. 160 (1994) 317–332, hep-th/9302006.
  * Drinfeld and Sokolov [1984] V. G. Drinfeld and V. V. Sokolov, “Lie algebras and equations of Korteweg-de Vries type”, J. Sov. Math. 30 (1984) 1975–2036.
  * De Sole et al. [2013] A. De Sole, V. G. Kac, and D. Valeri, “Classical -Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras”, Communications in Mathematical Physics 323 (2013), no. 2, 663–711, arXiv:1207.6286.
  * Dotsenko and Fateev [1984] V. S. Dotsenko and V. A. Fateev, “Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models”, Nucl. Phys. B240 (1984) 312\.
  * Arakawa et al. [2019] T. Arakawa, T. Creutzig, and A. R. Linshaw, “W-algebras as coset vertex algebras”, Invent. Math. 218 (2019) 145–195, arXiv:1801.03822.
  * Frenkel [1995] E. Frenkel, “Affine algebras, Langlands duality and Bethe ansatz”, in “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)”.  6 1995.  q-alg/9506003.
  * Aganagic et al. [2017] M. Aganagic, E. Frenkel, and A. Okounkov, “Quantum q-Langlands Correspondence”, arXiv:1701.03146.
  * Jimbo [1985] M. Jimbo, “A q difference analog of U(g) and the Yang-Baxter equation”, Lett. Math. Phys. 10 (1985) 63–69.
  * Jimbo [1986] M. Jimbo, “A q Analog of u (Gl (n+1)), Hecke Algebra and the Yang-Baxter Equation”, Lett. Math. Phys. 11 (1986) 247.
  * Drinfeld [1985] V. Drinfeld, “Hopf algebras and the quantum yang-baxter equation”, Proceedings of the USSR Academy of Sciences 32 (1985) 254–258.
  * Drinfeld [1986] V. G. Drinfeld, “Quantum groups”, Zap. Nauchn. Semin. 155 (1986) 18–49.
  * Pasquier and Saleur [1990] V. Pasquier and H. Saleur, “Common Structures Between Finite Systems and Conformal Field Theories Through Quantum Groups”, Nucl. Phys. B 330 (1990) 523–556.
  * Sevostyanov [2001] A. Sevostyanov, “Drinfeld-Sokolov reduction for quantum groups and deformations of W-algebras”, arXiv Mathematics e-prints, 2001 math/0107215, arXiv:math/0107215.
  * Shiraishi et al. [1996] 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.
  * Feigin and Frenkel [1996] B. Feigin and E. Frenkel, “Quantum W algebras and elliptic algebras”, Commun. Math. Phys. 178 (1996) 653–678, arXiv:q-alg/9508009.
  * Awata et al. [1996] H. Awata, H. Kubo, S. Odake, and J. Shiraishi, “Quantum W(N) algebras and Macdonald polynomials”, Commun. Math. Phys. 179 (1996) 401–416, arXiv:q-alg/9508011.
  * Frenkel and Reshetikhin [1998] E. Frenkel and N. Reshetikhin, “Deformations of $\mathcal{W}$-algebras associated to simple Lie algebras”, Comm. Math. Phys. 197 (1998), no. 1, 1–32, arXiv:q-alg/9708006.
  * Kojima [2019] T. Kojima, “Quadratic relations of the deformed w-superalgebra wq,t(sl(2—1))”, arXiv: Quantum Algebra, 2019.
  * Kojima [2021] T. Kojima, “Quadratic relations of the deformed w-superalgebra wq,ta(m,n)”, Journal of Physics A: Mathematical and Theoretical 54 (2021).
  * Harada et al. [2021] K. Harada, Y. Matsuo, G. Noshita, and A. Watanabe, “$q$-deformation of corner vertex operator algebras by Miura transformation”, JHEP 04 (2021) 202, arXiv:2101.03953.
  * Kimura and Pestun [2015] T. Kimura and V. Pestun, “Quiver W-algebras”, arXiv:1512.08533.
  * Kimura and Pestun [2018] T. Kimura and V. Pestun, “Quiver elliptic W-algebras”, Lett. Math. Phys. 108 (2018), no. 6, 1383–1405, arXiv:1608.04651.
  * Kimura and Pestun [2017] T. Kimura and V. Pestun, “Fractional quiver W-algebras”, arXiv:1705.04410.
  * Kapustin and Witten [2007] A. Kapustin and E. Witten, “Electric-Magnetic Duality And The Geometric Langlands Program”, Commun. Num. Theor. Phys. 1 (2007) 1–236, hep-th/0604151.
  * Gukov and Witten [2006] S. Gukov and E. Witten, “Gauge Theory, Ramification, And The Geometric Langlands Program”, hep-th/0612073.
  * Gukov and Witten [2010] S. Gukov and E. Witten, “Rigid Surface Operators”, Adv. Theor. Math. Phys. 14 (2010), no. 1, 87–178, arXiv:0804.1561.
  * Gaiotto and Witten [2009a] D. Gaiotto and E. Witten, “Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory”, J. Statist. Phys. 135 (2009)a 789–855, arXiv:0804.2902.
  * Gaiotto and Witten [2009b] D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory”, Adv. Theor. Math. Phys. 13 (2009)b, no. 3, 721–896, arXiv:0807.3720.
  * Frenkel and Gaiotto [2020] E. Frenkel and D. Gaiotto, “Quantum Langlands dualities of boundary conditions, D-modules, and conformal blocks”, Commun. Num. Theor. Phys. 14 (2020), no. 2, 199–313, arXiv:1805.00203.
  * Gaiotto and Witten [2021] D. Gaiotto and E. Witten, “Gauge Theory and the Analytic Form of the Geometric Langlands Program”, arXiv:2107.01732.
  * Jeong and Nekrasov [2020] S. Jeong and N. Nekrasov, “Opers, surface defects, and Yang-Yang functional”, Adv. Theor. Math. Phys. 24 (2020), no. 7, 1789–1916, arXiv:1806.08270.
  * Jeong et al. [2023] S. Jeong, N. Lee, and N. Nekrasov, “Parallel surface defects, Hecke operators, and quantum Hitchin system”, arXiv:2304.04656.
  * Nahm [1978] W. Nahm, “Supersymmetries and their Representations”, Nucl. Phys. B 135 (1978) 149.
  * Witten [2009] E. Witten, “Geometric Langlands From Six Dimensions”, arXiv:0905.2720.
  * Seiberg [1997] N. Seiberg, “New theories in six-dimensions and Matrix description of M theory on $T^{5}$ and $T^{5}/\mathbb{Z}_{2}$”, Phys. Lett. B408 (1997) 98–104, arXiv:hep-th/9705221.
  * Vafa [1998] C. Vafa, “Geometric origin of Montonen-Olive duality”, Adv. Theor. Math. Phys. 1 (1998) 158–166, hep-th/9707131.
  * Harvey et al. [1995] J. A. Harvey, G. W. Moore, and A. Strominger, “Reducing S duality to T duality”, Phys. Rev. D 52 (1995) 7161–7167, hep-th/9501022.
  * Bershadsky et al. [1995] M. Bershadsky, A. Johansen, V. Sadov, and C. Vafa, “Topological reduction of 4-d SYM to 2-d sigma models”, Nucl. Phys. B 448 (1995) 166–186, hep-th/9501096.
  * Frenkel and Reshetikhin [1992] I. B. Frenkel and N. Y. Reshetikhin, “Quantum affine algebras and holonomic difference equations”, Commun. Math. Phys. 146 (1992) 1–60.
  * Knizhnik and Zamolodchikov [1984] V. G. Knizhnik and A. B. Zamolodchikov, “Current Algebra and Wess-Zumino Model in Two-Dimensions”, Nucl. Phys. B 247 (1984) 83–103.
  * Reshetikhin [1992] N. Reshetikhin, “Jackson-type integrals, bethe vectors, and solutions to a difference analog of the knizhnik-zamolodchikov system”, Letters in Mathematical Physics 26 (1992) 153–165.
  * Matsuo [1993a] A. Matsuo, “Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations”, Communications in Mathematical Physics 151 (1993)a, no. 2, 263 – 273.
  * Matsuo [1993b] A. Matsuo, “Quantum algebra structure of certain Jackson integrals”, Communications in Mathematical Physics 157 (1993)b, no. 3, 479 – 498.
  * Tarasov and Varchenko [1993] V. Tarasov and A. Varchenko, “Jackson integral representations for solutions of the quantized Knizhnik-Zamolodchikov equation”, hep-th/9311040.
  * Varchenko [1994] A. Varchenko, “Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equation, and difference equations for q-hypergeometric functions”, Communications in Mathematical Physics 162 (1994), no. 3, 499–528.
  * Tarasov and Varchenko [1997] V. Tarasov and A. Varchenko, “Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups”, eprint arXiv:q-alg/9703044, 1997 q–alg/9703044, arXiv:q-alg/9703044.
  * Aganagic and Okounkov [2017] M. Aganagic and A. Okounkov, “Quasimap counts and Bethe eigenfunctions”, Moscow Math. J. 17 (2017), no. 4, 565–600, arXiv:1704.08746.
  * Deligne [1970] P. Deligne, “Equations différentielles à points singuliers réguliers”, Springer-Verlag, 1970.
  * Aganagic and Okounkov [2016] M. Aganagic and A. Okounkov, “Elliptic stable envelope”, arXiv:1604.00423.
  * Alday et al. [2010] L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories”, Lett. Math. Phys. 91 (2010) 167–197, arXiv:0906.3219.
  * Kanno et al. [2010] S. Kanno, Y. Matsuo, S. Shiba, and Y. Tachikawa, “N=2 gauge theories and degenerate fields of Toda theory”, Phys. Rev. D 81 (2010) 046004, arXiv:0911.4787.
  * Bouwknegt and Pilch [1998] P. Bouwknegt and K. Pilch, “On deformed W algebras and quantum affine algebras”, Adv. Theor. Math. Phys. 2 (1998) 357–397, math/9801112.
  * Chari and Pressley [1991] V. Chari and A. Pressley, “Quantum affine algebras”, Communications in Mathematical Physics 142 (1991), no. 2, 261 – 283.
  * Drinfeld [1988] V. G. Drinfeld, “A New realization of Yangians and quantized affine algebras”, Sov. Math. Dokl. 36 (1988) 212–216.
  * Frenkel and Reshetikhin [2000] E. Frenkel and N. Reshetikhin, “The q-characters of representations of quantum affine algebras and deformations of w-algebras”, in Contemporary Math 248 (2000) arXiv:math/9810055.
  * Aganagic et al. [2013] M. Aganagic, N. Haouzi, C. Kozcaz, and S. Shakirov, “Gauge/Liouville Triality”, arXiv:1309.1687.
  * Aganagic et al. [2014] M. Aganagic, N. Haouzi, and S. Shakirov, “$A_{n}$-Triality”, arXiv:1403.3657.
  * Aganagic and Haouzi [2015] M. Aganagic and N. Haouzi, “ADE Little String Theory on a Riemann Surface (and Triality)”, arXiv:1506.04183.
  * Beem et al. [2014] C. Beem, T. Dimofte, and S. Pasquetti, “Holomorphic Blocks in Three Dimensions”, JHEP 12 (2014) 177, arXiv:1211.1986.
  * Cecotti and Vafa [1991] S. Cecotti and C. Vafa, “Topological antitopological fusion”, Nucl. Phys. B 367 (1991) 359–461.
  * Cecotti and Vafa [1993] S. Cecotti and C. Vafa, “On classification of N=2 supersymmetric theories”, Commun. Math. Phys. 158 (1993) 569–644, hep-th/9211097.
  * Hori et al. [2000] K. Hori, A. Iqbal, and C. Vafa, “D-branes and mirror symmetry”, hep-th/0005247.
  * Gaiotto et al. [2015a] D. Gaiotto, G. W. Moore, and E. Witten, “An Introduction To The Web-Based Formalism”, arXiv:1506.04086.
  * Gaiotto et al. [2015b] D. Gaiotto, G. W. Moore, and E. Witten, “Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions”, arXiv:1506.04087.
  * Yoshida and Sugiyama [2014] Y. Yoshida and K. Sugiyama, “Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^{1}\times D^{2}$”, arXiv:1409.6713.
  * Dimofte et al. [2018] T. Dimofte, D. Gaiotto, and N. M. Paquette, “Dual boundary conditions in 3d SCFT’s”, JHEP 05 (2018) 060, arXiv:1712.07654.
  * Bullimore et al. [2016] M. Bullimore, T. Dimofte, D. Gaiotto, and J. Hilburn, “Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory”, JHEP 10 (2016) 108, arXiv:1603.08382.
  * Bullimore et al. [2021] M. Bullimore, S. Crew, and D. Zhang, “Boundaries, Vermas, and Factorisation”, JHEP 04 (2021) 263, arXiv:2010.09741.
  * Okazaki [2021] T. Okazaki, “Abelian mirror symmetry of $\mathcal{N}$ = (2, 2) boundary conditions”, JHEP 03 (2021) 163, arXiv:2010.13177.
  * Crew et al. [2021] S. Crew, N. Dorey, and D. Zhang, “Blocks and Vortices in the 3d ADHM Quiver Gauge Theory”, JHEP 03 (2021) 234, arXiv:2010.09732.
  * Dedushenko and Nekrasov [2021] M. Dedushenko and N. Nekrasov, “Interfaces and Quantum Algebras, I: Stable Envelopes”, arXiv:2109.10941.
  * Drummond et al. [2009] J. M. Drummond, J. M. Henn, and J. Plefka, “Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory”, JHEP 05 (2009) 046, arXiv:0902.2987.
  * Costello [2013] K. Costello, “Supersymmetric gauge theory and the Yangian”, arXiv:1303.2632.
  * Costello et al. [2018] K. Costello, E. Witten, and M. Yamazaki, “Gauge Theory and Integrability, I”, ICCM Not. 06 (2018), no. 1, 46–119, arXiv:1709.09993.
  * Nekrasov and Shatashvili [2009] N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and Four Dimensional Gauge Theories”, in “Proceedings, 16th International Congress on Mathematical Physics (ICMP09): Prague, Czech Republic, August 3-8, 2009”, pp. 265–289.  2009\.  arXiv:0908.4052.
  * Maulik and Okounkov [2012] D. Maulik and A. Okounkov, “Quantum Groups and Quantum Cohomology”, arXiv e-prints, 2012 arXiv:1211.1287, arXiv:1211.1287.
  * Nekrasov and Pestun [2012] N. Nekrasov and V. Pestun, “Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories”, arXiv:1211.2240.
  * Nekrasov et al. [2018] N. Nekrasov, V. Pestun, and S. Shatashvili, “Quantum geometry and quiver gauge theories”, Commun. Math. Phys. 357 (2018), no. 2, 519–567, arXiv:1312.6689.
  * Dorey [1998] N. Dorey, “The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms”, JHEP 11 (1998) 005, hep-th/9806056.
  * Dorey et al. [1999] N. Dorey, T. J. Hollowood, and D. Tong, “The BPS spectra of gauge theories in two-dimensions and four-dimensions”, JHEP 05 (1999) 006, hep-th/9902134.
  * Abrikosov [1957] A. A. Abrikosov, “On the Magnetic properties of superconductors of the second group”, Sov. Phys. JETP 5 (1957) 1174–1182.
  * Nielsen and Olesen [1973] H. B. Nielsen and P. Olesen, “Vortex Line Models for Dual Strings”, Nucl. Phys. B 61 (1973) 45–61.
  * Shifman and Yung [2004] M. Shifman and A. Yung, “NonAbelian string junctions as confined monopoles”, Phys. Rev. D 70 (2004) 045004, hep-th/0403149.
  * Hanany and Tong [2004] A. Hanany and D. Tong, “Vortex strings and four-dimensional gauge dynamics”, JHEP 04 (2004) 066, hep-th/0403158.
  * Nekrasov and Shatashvili [2009] N. A. Nekrasov and S. L. Shatashvili, “Quantum integrability and supersymmetric vacua”, Prog. Theor. Phys. Suppl. 177 (2009) 105–119, arXiv:0901.4748.
  * Nekrasov [2003] N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting”, Adv. Theor. Math. Phys. 7 (2003), no. 5, 831–864, arXiv:hep-th/0206161.
  * Losev et al. [2003] A. S. Losev, A. Marshakov, and N. A. Nekrasov, “Small instantons, little strings and free fermions”, in “From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan”, pp. 581–621.  2 2003.  hep-th/0302191.
  * Nekrasov and Witten [2010] N. Nekrasov and E. Witten, “The Omega Deformation, Branes, Integrability, and Liouville Theory”, JHEP 09 (2010) 092, arXiv:1002.0888.
  * Witten [1993] E. Witten, “Phases of N=2 theories in two-dimensions”, Nucl. Phys. B 403 (1993) 159–222, hep-th/9301042.
  * Dorey et al. [2011] N. Dorey, S. Lee, and T. J. Hollowood, “Quantization of Integrable Systems and a 2d/4d Duality”, JHEP 10 (2011) 077, arXiv:1103.5726.
  * Chen et al. [2011] H.-Y. Chen, N. Dorey, T. J. Hollowood, and S. Lee, “A New 2d/4d Duality via Integrability”, JHEP 09 (2011) 040, arXiv:1104.3021.
  * Aganagic and Shakirov [2016] M. Aganagic and S. Shakirov, Gauge/Vortex duality and AGT, pp. 419–448.  2016\.  arXiv:1412.7132.
  * Frenkel and Mukhin [2001] E. Frenkel and E. Mukhin, “Combinatorics of q characters of finite dimensional representations of quantum affine algebras”, Commun. Math. Phys. 216 (2001) 23–57, math/9911112.
  * Nekrasov [2016] N. Nekrasov, “BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters”, JHEP 03 (2016) 181, arXiv:1512.05388.
  * Nekrasov [2008] N. Nekrasov, “On the bps/cft correspondence, lecture at the university of amsterdam string theory group seminar”, 2008, unpublished.
  * Nakajima [1999] H. Nakajima, “Quiver varieties and finite dimensional representations of quantum affine algebras”, arXiv Mathematics e-prints, 1999 math/9912158, arXiv:math/9912158.
  * Okounkov [2015] A. Okounkov, “Lectures on K-theoretic computations in enumerative geometry”, arXiv:1512.07363.
  * Ciocan-Fontanine et al. [2011] I. Ciocan-Fontanine, B. Kim, and D. Maulik, “Stable quasimaps to GIT quotients”, arXiv e-prints, 2011 arXiv:1106.3724, arXiv:1106.3724.
  * Bullimore et al. [2019] M. Bullimore, A. Ferrari, and H. Kim, “Twisted indices of 3d $\mathcal{N}$ = 4 gauge theories and enumerative geometry of quasi-maps”, JHEP 07 (2019) 014, arXiv:1812.05567.
  * Dedushenko and Nekrasov [2023] M. Dedushenko and N. Nekrasov, “Interfaces and Quantum Algebras, II: Cigar Partition Function”, arXiv:2306.16434.
  * Crew et al. [2023] S. Crew, D. Zhang, and B. Zhao, “Boundaries & Localisation with a Topological Twist”, arXiv:2306.16448.
  * Ishtiaque et al. [2023] N. Ishtiaque, S. F. Moosavian, and Y. Zhou, “Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical R-Matrices for Superspin Chains from The Bethe/Gauge Correspondence”, arXiv:2308.12333.
  * Dinkins [2020a] H. Dinkins, “Symplectic Duality of $T^{*}Gr(k,n)$”, arXiv e-prints, 2020a arXiv:2008.05516, arXiv:2008.05516.
  * Dinkins [2020b] H. Dinkins, “3d mirror symmetry of the cotangent bundle of the full flag variety”, arXiv e-prints, 2020b arXiv:2011.08603, arXiv:2011.08603.
  * Rimányi et al. [2022] R. Rimányi, A. Smirnov, Z. Zhou, and A. Varchenko, “Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes”, Int. Math. Res. Not. 2022 (2022), no. 13, 10016–10094, arXiv:1902.03677.
  * Botta and Rimanyi [2023] T. M. Botta and R. Rimanyi, “Bow varieties: Stable envelopes and their 3d mirror symmetry”, arXiv e-prints, 2023 arXiv:2308.07300, arXiv:2308.07300.
  * Pushkar et al. [2020] P. P. Pushkar, A. Smirnov, and A. M. Zeitlin, “Baxter Q-operator from quantum K-theory”, Adv. Math. 360 (2020) 106919, arXiv:1612.08723.
  * Koroteev et al. [2021] P. Koroteev, P. P. Pushkar, A. V. Smirnov, and A. M. Zeitlin, “Quantum K-theory of quiver varieties and many-body systems”, Selecta Math. 27 (2021), no. 5, 87, arXiv:1705.10419.
  * Okounkov [2021] A. Okounkov, “Inductive construction of stable envelopes”, Letters in Mathematical Physics 111 (2021) 1–56.
  * Constable et al. [2003] N. R. Constable, J. Erdmenger, Z. Guralnik, and I. Kirsch, “Intersecting D-3 branes and holography”, Phys. Rev. D 68 (2003) 106007, hep-th/0211222.
  * Lin et al. [2004] H. Lin, O. Lunin, and J. M. Maldacena, “Bubbling AdS space and 1/2 BPS geometries”, JHEP 10 (2004) 025, hep-th/0409174.
  * Lin and Maldacena [2006] H. Lin and J. M. Maldacena, “Fivebranes from gauge theory”, Phys. Rev. D 74 (2006) 084014, hep-th/0509235.
  * Gomis and Matsuura [2007] J. Gomis and S. Matsuura, “Bubbling surface operators and S-duality”, JHEP 06 (2007) 025, arXiv:0704.1657.
  * Chen et al. [2023] Y. Chen, M. Heydeman, Y. Wang, and M. Zhang, “Probing Supersymmetric Black Holes with Surface Defects”, arXiv:2306.05463.
  * Alday and Tachikawa [2010] L. F. Alday and Y. Tachikawa, “Affine SL(2) conformal blocks from 4d gauge theories”, Lett. Math. Phys. 94 (2010) 87–114, arXiv:1005.4469.
  * Kanno and Tachikawa [2011] H. Kanno and Y. Tachikawa, “Instanton counting with a surface operator and the chain-saw quiver”, JHEP 06 (2011) 119, arXiv:1105.0357.
  * Nekrasov [2019] N. Nekrasov, “BPS/CFT correspondence IV: sigma models and defects in gauge theory”, Lett. Math. Phys. 109 (2019), no. 3, 579–622, arXiv:1711.11011.
  * Ashok et al. [2020a] S. K. Ashok, M. Billo, M. Frau, A. Lerda, and S. Mahato, “Surface defects from fractional branes. Part I”, JHEP 07 (2020)a 051, arXiv:2005.02050.
  * Ashok et al. [2020b] S. K. Ashok, M. Billò, M. Frau, A. Lerda, and S. Mahato, “Surface defects from fractional branes. Part II”, JHEP 08 (2020)b 058, arXiv:2005.03701.
  * Witten [1995] E. Witten, “Some comments on string dynamics”, in “STRINGS 95: Future Perspectives in String Theory”, pp. 501–523.  7 1995.  hep-th/9507121.
  * Strominger [1996] A. Strominger, “Open p-branes”, Phys. Lett. B 383 (1996) 44–47, hep-th/9512059.
  * Douglas and Moore [1996] M. R. Douglas and G. W. Moore, “D-branes, quivers, and ALE instantons”, arXiv:hep-th/9603167.
  * Hanany and Tong [2003] A. Hanany and D. Tong, “Vortices, instantons and branes”, JHEP 07 (2003) 037, hep-th/0306150.
  * Haouzi [To appear] N. Haouzi, “Massive deformations of scfts from quantum algebras”,  To appear.
  * Martinec and Warner [1996] E. J. Martinec and N. P. Warner, “Integrable systems and supersymmetric gauge theory”, Nucl. Phys. B 459 (1996) 97–112, hep-th/9509161.
  * Donagi and Witten [1996] R. Donagi and E. Witten, “Supersymmetric Yang-Mills theory and integrable systems”, Nucl. Phys. B 460 (1996) 299–334, hep-th/9510101.
  * Gaiotto [2012] D. Gaiotto, “N=2 dualities”, JHEP 08 (2012) 034, arXiv:0904.2715.
  * Gaiotto et al. [2009] D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin Systems, and the WKB Approximation”, arXiv:0907.3987.
  * Spaltenstein [1982] N. Spaltenstein, “Classes unipotentes et sous-groupes de borel”,  1982\.
  * Chacaltana and Distler [2010] O. Chacaltana and J. Distler, “Tinkertoys for Gaiotto Duality”, JHEP 11 (2010) 099, arXiv:1008.5203.
  * Chacaltana et al. [2013] O. Chacaltana, J. Distler, and Y. Tachikawa, “Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories”, Int. J. Mod. Phys. A 28 (2013) 1340006, arXiv:1203.2930.
  * Balasubramanian [2014] A. K. Balasubramanian, “Describing codimension two defects”, JHEP 07 (2014) 095, arXiv:1404.3737.
  * Balasubramanian and Distler [2018] A. Balasubramanian and J. Distler, “Masses, Sheets and Rigid SCFTs”, arXiv:1810.10652.
  * Haouzi and Schmid [2017] N. Haouzi and C. Schmid, “Little String Origin of Surface Defects”, JHEP 05 (2017) 082, arXiv:1608.07279.
  * Haouzi and Schmid [2016] N. Haouzi and C. Schmid, “Little String Defects and Bala-Carter Theory”, arXiv:1612.02008.
  * Logares and Martens [2008] M. Logares and J. Martens, “Moduli of Parabolic Higgs Bundles and Atiyah Algebroids”, arXiv e-prints, 2008 arXiv:0811.0817, arXiv:0811.0817.
  * Wyllard [2009] N. Wyllard, “A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories”, JHEP 11 (2009) 002, arXiv:0907.2189.
  * Schiffmann and Vasserot [2012] O. Schiffmann and E. Vasserot, “Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2̂”, arXiv e-prints, 2012 arXiv:1202.2756, arXiv:1202.2756.
  * Braverman et al. [2014] A. Braverman, M. Finkelberg, and H. Nakajima, “Instanton moduli spaces and $\mathscr{W}$-algebras”, arXiv:1406.2381.
  * Haouzi and Kozçaz [2017] N. Haouzi and C. Kozçaz, “The abcdefg of little strings”, arXiv:1711.11065.
  * Katz et al. [1998] S. Katz, P. Mayr, and C. Vafa, “Mirror symmetry and exact solution of 4-D N=2 gauge theories: 1.”, Adv. Theor. Math. Phys. 1 (1998) 53–114, hep-th/9706110.
  * Mironov et al. [2013a] A. Mironov, A. Morozov, Y. Zenkevich, and A. Zotov, “Spectral Duality in Integrable Systems from AGT Conjecture”, JETP Lett. 97 (2013)a 45–51, arXiv:1204.0913.
  * Mironov et al. [2013b] A. Mironov, A. Morozov, B. Runov, Y. Zenkevich, and A. Zotov, “Spectral dualities in XXZ spin chains and five dimensional gauge theories”, JHEP 12 (2013)b 034, arXiv:1307.1502.
  * Mitev et al. [2015] V. Mitev, E. Pomoni, M. Taki, and F. Yagi, “Fiber-Base Duality and Global Symmetry Enhancement”, JHEP 04 (2015) 052, arXiv:1411.2450.
  * Finkelberg et al. [2002] M. Finkelberg, D. Gaitsgory, and A. Kuznetsov, “Uhlenbeck spaces for $A^{2}$ and affine Lie algebra $\hat{sl}_{n}$”, arXiv Mathematics e-prints, 2002 math/0202208, arXiv:math/0202208.
  * Braverman [2004] A. Braverman, “Instanton counting via affine Lie algebras. 1. Equivariant J functions of (affine) flag manifolds and Whittaker vectors”, in “CRM Workshop on Algebraic Structures and Moduli Spaces”.  1 2004.  math/0401409.
  * Braverman and Etingof [2004] A. Braverman and P. Etingof, “Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential”, math/0409441.
  * Feigin et al. [2008] B. Feigin, M. Finkelberg, A. Negut, and L. Rybnikov, “Yangians and cohomology rings of Laumon spaces”, arXiv e-prints, 2008 arXiv:0812.4656, arXiv:0812.4656.
  * Negut [2011] A. Negut, “Affine Laumon spaces and integrable systems”, arXiv:1112.1756.
  * Kozcaz et al. [2011] C. Kozcaz, S. Pasquetti, F. Passerini, and N. Wyllard, “Affine sl(N) conformal blocks from N=2 SU(N) gauge theories”, JHEP 01 (2011) 045, arXiv:1008.1412.
  * Wyllard [2011a] N. Wyllard, “W-algebras and surface operators in N=2 gauge theories”, J. Phys. A 44 (2011)a 155401, arXiv:1011.0289.
  * Wyllard [2011b] N. Wyllard, “Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals”, JHEP 02 (2011)b 114, arXiv:1012.1355.
  * Nekrasov and Tsymbaliuk [2021] N. Nekrasov and A. Tsymbaliuk, “Surface defects in gauge theory and KZ equation”, arXiv:2103.12611.
  * Nekrasov [2017] N. Nekrasov, “BPS/CFT correspondence V: BPZ and KZ equations from qq-characters”, arXiv:1711.11582.
  * Jeong et al. [2021] S. Jeong, N. Lee, and N. Nekrasov, “Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations”, JHEP 10 (2021) 120, arXiv:2103.17186.
  * Bouwknegt and Schoutens [1993] P. Bouwknegt and K. Schoutens, “W symmetry in conformal field theory”, Phys. Rept. 223 (1993) 183–276, arXiv:hep-th/9210010.
  * Gaitsgory [2007] D. Gaitsgory, “Twisted Whittaker model and factorizable sheaves”, arXiv e-prints, 2007 arXiv:0705.4571, arXiv:0705.4571.
  * Gaitsgory [2016] D. Gaitsgory, “Quantum Langlands Correspondence”, arXiv e-prints, 2016 arXiv:1601.05279, arXiv:1601.05279.
  * Kazhdan and Lusztig [1993] D. Kazhdan and G. Lusztig, “Tensor structures arising from affine lie algebras. i”, Journal of the American Mathematical Society 6 (1993), no. 4, 905–947.
  * Frenkel [2007] E. Frenkel, “Lectures on the Langlands program and conformal field theory”, in “Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry”, pp. 387–533.  2007\.  hep-th/0512172.
  * Frenkel and Reshetikhin [1996] E. Frenkel and N. Reshetikhin, “Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1”, Communications in Mathematical Physics 178 (1996), no. 1, 237–264, arXiv:q-alg/9505025.
  * Frenkel et al. [2022] E. Frenkel, D. Hernandez, and N. Reshetikhin, “Folded quantum integrable models and deformed W-algebras”, Lett. Math. Phys. 112 (2022), no. 4, 80, arXiv:2110.14600.
  * Kashiwara and Oh [2022] M. Kashiwara and S.-j. Oh, “The $(q,t)$-Cartan matrix specialized at $q=1$”, arXiv e-prints, 2022 arXiv:2201.11918, arXiv:2201.11918.
  * Chen and Kimura [2018] H.-Y. Chen and T. Kimura, “Quantum integrability from non-simply laced quiver gauge theory”, JHEP 06 (2018) 165, arXiv:1805.01308.
  * Lukyanov and Pugai [1996a] S. L. Lukyanov and Y. Pugai, “Bosonization of ZF algebras: Direction toward deformed Virasoro algebra”, J. Exp. Theor. Phys. 82 (1996)a 1021–1045, hep-th/9412128.
  * Lukyanov and Pugai [1996b] S. L. Lukyanov and Y. Pugai, “Multipoint local height probabilities in the integrable RSOS model”, Nucl. Phys. B 473 (1996)b 631–658, hep-th/9602074.
  * Haouzi and Kozçaz [2019] N. Haouzi and C. Kozçaz, “Supersymmetric Wilson Loops, Instantons, and Deformed ${\cal W}$-Algebras”, arXiv:1907.03838.
  * Zenkevich [2021] Y. Zenkevich, “Mixed network calculus”, JHEP 12 (2021) 027, arXiv:2012.15563.
  * Kimura and Noshita [2023] T. Kimura and G. Noshita, “Gauge origami and quiver W-algebras”, arXiv:2310.08545.
  * Chari and Pressley [1994] V. Chari and A. Pressley, “Quantum affine algebras and their representations”, arXiv:hep-th/9411145.
  * Chari [1994a] V. Chari, “Minimal affinizations of representations of quantum groups: The Rank 2 case”, hep-th/9410022.
  * Chari [1994b] V. Chari, “Minimal affinizations of representations of quantum groups: The U-q(g) module structure”, arXiv:hep-th/9411144.
  * Chari [1991] P. A. Chari, Vyjayanthi, “Fundamental representations of yangians and singularities of r-matrices.”, Journal für die reine und angewandte Mathematik 417 (1991) 87–128.
  * Knight [1995] H. Knight, “Spectra of tensor products of finite dimensional representations of yangians”, Journal of Algebra 174 (1995), no. 1, 187–196.
  * Nakai and Nakanishi [2008] W. Nakai and T. Nakanishi, “On Frenkel-Mukhin algorithm for q-character of quantum affine algebras”, arXiv e-prints, 2008 arXiv:0801.2239, arXiv:0801.2239.
  * Leclerc and Hernandez [2013] B. Leclerc and D. Hernandez, “A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules”, arXiv e-prints, 2013 arXiv:1303.0744, arXiv:1303.0744.
  * Reshetikhin and Semenov-Tian-Shansky [1990] N. Y. Reshetikhin and M. A. Semenov-Tian-Shansky, “Central extensions of quantum current groups”, Lett. Math. Phys. 19 (1990) 133–142.
  * Ding and Etingof [1994] J.-t. Ding and P. Etingof, “Center of a quantum affine algebra at the critical level”, hep-th/9403064.
  * Kim [2016] H.-C. Kim, “Line defects and 5d instanton partition functions”, JHEP 03 (2016) 199, arXiv:1601.06841.
  * Mironov et al. [2016] A. Mironov, A. Morozov, and Y. Zenkevich, “Ding–Iohara–Miki symmetry of network matrix models”, Phys. Lett. B762 (2016) 196–208, arXiv:1603.05467.
  * Bourgine et al. [2016] J.-E. Bourgine, Y. Matsuo, and H. Zhang, “Holomorphic field realization of SHc and quantum geometry of quiver gauge theories”, JHEP 04 (2016) 167, arXiv:1512.02492.
  * Jeong and Zhang [2020] S. Jeong and X. Zhang, “A note on chiral trace relations from qq-characters”, JHEP 04 (2020) 026, arXiv:1910.10864.
  * Haouzi and Oh [2021] N. Haouzi and J. Oh, “On the Quantization of Seiberg-Witten Geometry”, JHEP 01 (2021) 184, arXiv:2004.00654.
  * Haouzi [2020] N. Haouzi, “Quantum geometry and $\theta$-angle in five-dimensional super Yang-Mills”, JHEP 09 (2020) 035, arXiv:2005.13565.
  * Assel and Sciarappa [2018] B. Assel and A. Sciarappa, “Wilson loops in 5d $\mathcal{N}=1$ theories and S-duality”, JHEP 10 (2018) 082, arXiv:1806.09636.
  * Chang et al. [2017] C.-M. Chang, O. Ganor, and J. Oh, “An index for ray operators in 5d $E_{n}$ SCFTs”, JHEP 02 (2017) 018, arXiv:1608.06284.
  * Frenkel et al. [2020] E. Frenkel, P. Koroteev, D. S. Sage, and A. M. Zeitlin, “q-Opers, QQ-Systems, and Bethe Ansatz”, arXiv:2002.07344.
  * Nekrasov [2018] N. Nekrasov, “BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters”, Commun. Math. Phys. 358 (2018), no. 3, 863–894, arXiv:1701.00189.
  * Haouzi [2020] N. Haouzi, “Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N}=4$ Gauge Theories”, arXiv:2009.08989.
  * Kimura et al. [2018] T. Kimura, H. Mori, and Y. Sugimoto, “Refined geometric transition and $qq$-characters”, JHEP 01 (2018) 025, arXiv:1705.03467.
  * Ding and Iohara [1996] J. Ding and K. Iohara, “Generalization and Deformation of Drinfeld quantum affine algebras”, arXiv:q-alg/9608002.
  * Miki [2007] K. Miki, “A $(q,\gamma)$ analog of the $w_{1+\infty}$ algebra”, J. Math. Phys. 48 (2007).
  * Neguţ [2013] A. Neguţ, “Quantum toroidal and shuffle algebras”, arXiv e-prints, 2013 arXiv:1302.6202, arXiv:1302.6202.
  * Burban and Schiffmann [2005] I. Burban and O. Schiffmann, “On the Hall algebra of an elliptic curve, I”, arXiv Mathematics e-prints, 2005 math/0505148, arXiv:math/0505148.
  * Schiffmann [2010] O. Schiffmann, “Drinfeld realization of the elliptic Hall algebra”, arXiv e-prints, 2010 arXiv:1004.2575, arXiv:1004.2575.
  * Feigin et al. [2010] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Quantum continuous $\mathfrak{gl}_{\infty}$: Semi-infinite construction of representations”, arXiv e-prints, 2010 arXiv:1002.3100, arXiv:1002.3100.
  * Feigin et al. [2011] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Quantum continuous $\mathfrak{gl}_{\infty}$: Tensor products of fock modules and wn-characters”, Kyoto Journal of Mathematics 51 06 (2011).
  * Bershtein et al. [2015] M. Bershtein, B. Feigin, and G. Merzon, “Plane partitions with a “pit”: generating functions and representation theory”, arXiv e-prints, 2015 arXiv:1512.08779, arXiv:1512.08779.
  * Bourgine et al. [2016] J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang, and R.-D. Zhu, “Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d Super Yang-Mills”, PTEP 2016 (2016), no. 12, 123B05, arXiv:1606.08020.
  * Feigin et al. [2020] B. Feigin, M. Jimbo, E. Mukhin, and I. Vilkoviskiy, “Deformations of $\mathcal{W}$ algebras via quantum toroidal algebras”, arXiv e-prints, 2020 arXiv:2003.04234, arXiv:2003.04234.
  * Feigin and Tsymbaliuk [2009] B. Feigin and A. Tsymbaliuk, “Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra”, arXiv e-prints, 2009 arXiv:0904.1679, arXiv:0904.1679.
  * Schiffmann and Vasserot [2009] O. Schiffmann and E. Vasserot, “The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^{2}$”, arXiv e-prints, 2009 arXiv:0905.2555, arXiv:0905.2555.
  * Feigin et al. [2009] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida, “A commutative algebra on degenerate CP1 and Macdonald polynomials”, Journal of Mathematical Physics 50 (2009), no. 9, 095215–095215, arXiv:0904.2291.
  * Bourgine et al. [2017] J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo, and R.-D. Zhu, “(p, q)-webs of DIM representations, 5d $\mathcal{N}=1$ instanton partition functions and qq-characters”, JHEP 11 (2017) 034, arXiv:1703.10759.
  * Bourgine and Jeong [2020] J.-E. Bourgine and S. Jeong, “New quantum toroidal algebras from 5D $\mathcal{N}$ = 1 instantons on orbifolds”, JHEP 05 (2020) 127, arXiv:1906.01625.
  * Bourgine [2021] J.-E. Bourgine, “Intertwining operator and integrable hierarchies from topological strings”, JHEP 05 (2021) 216, arXiv:2101.09925.
  * Bourgine [2022] J.-E. Bourgine, “Shifted quantum groups and matter multiplets in supersymmetric gauge theories”, arXiv:2205.01309.
  * Awata et al. [2012] H. Awata, B. Feigin, and J. Shiraishi, “Quantum Algebraic Approach to Refined Topological Vertex”, JHEP 03 (2012) 041, arXiv:1112.6074.
  * Zenkevich [2022] Y. Zenkevich, “Hanany-Witten brane crossing and Ding-Iohara-Miki algebra”, arXiv e-prints, 2022 arXiv:2212.14808, arXiv:2212.14808.
  * Gaiotto and Rapčák [2019] D. Gaiotto and M. Rapčák, “Vertex Algebras at the Corner”, JHEP 01 (2019) 160, arXiv:1703.00982.
  * Liu [2022] H. Liu, “A representation-theoretic approach to $qq$-characters”,  2022\.
  * Neguţ [2020] A. Neguţ, “The R-matrix of the quantum toroidal algebra”, arXiv:2005.14182.
  * Awata et al. [2019] H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake, and Y. Zenkevich, “The MacMahon $R$-matrix”, JHEP 04 (2019) 097, arXiv:1810.07676.
  * Bayındırlı et al. [2023] M. B. Bayındırlı, D. N. Demirtaş, C. Kozçaz, and Y. Zenkevich, “On R-matrix formulation of qq-characters”, arXiv:2310.02587.
  * Nakajima [2000] H. Nakajima, “t-analogue of the q-characters of finite dimensional representations of quantum affine algebras”, Physics and Combinatorics, 2000 196–219, arXiv:math/0009231.
  * Nakajima [2001] H. Nakajima, “Quiver varieties and $t$-analogs of $q$-characters of quantum affine algebras”, Annals of Mathematics 160 (2001) 1057–1097.
  * Nakajima [2002] H. Nakajima, “t-analogs of q-characters of quantum affine algebras of type $a_n$, $d_n$”, arXiv: Quantum Algebra, 2002.
  * Nakajima [2010] H. Nakajima, “t-analogs of q-characters of quantum affine algebras of type e6, e7, e8”, arXiv: Quantum Algebra, 2010 257–272.
  * Frenkel and Hernandez [2011] E. Frenkel and D. Hernandez, “Langlands duality for finite-dimensional representations of quantum affine algebras”, Lett. Math. Phys. 96 (2011) 217–261, arXiv:0902.0447.
  * Dijkgraaf and Vafa [2009] R. Dijkgraaf and C. Vafa, “Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems”, arXiv:0909.2453.
  * Itoyama et al. [2010] H. Itoyama, K. Maruyoshi, and T. Oota, “The Quiver Matrix Model and 2d-4d Conformal Connection”, Prog. Theor. Phys. 123 (2010) 957–987, arXiv:0911.4244.
  * Mironov et al. [2010] A. Mironov, A. Morozov, and S. Shakirov, “Conformal blocks as Dotsenko-Fateev Integral Discriminants”, Int. J. Mod. Phys. A25 (2010) 3173–3207, arXiv:1001.0563.
  * Morozov and Shakirov [2010] A. Morozov and S. Shakirov, “The matrix model version of AGT conjecture and CIV-DV prepotential”, JHEP 08 (2010) 066, arXiv:1004.2917.
  * Maruyoshi [2016] K. Maruyoshi, “$\beta$-Deformed Matrix Models and 2d/4d Correspondence”, arXiv:1412.7124.
  * Finkelberg and Rybnikov [2010] M. Finkelberg and L. Rybnikov, “Quantization of Drinfeld Zastava in type A”, arXiv e-prints, 2010 arXiv:1009.0676, arXiv:1009.0676.
  * Nakajima [2012] H. Nakajima, “Handsaw quiver varieties and finite W-algebras”, Moscow Math. J. 12 (2012) 633, arXiv:1107.5073.
  * Finkelberg and Tsymbaliuk [2017] M. Finkelberg and A. Tsymbaliuk, “Multiplicative slices, relativistic Toda and shifted quantum affine algebras”, arXiv e-prints, 2017 arXiv:1708.01795, arXiv:1708.01795.
  * Neguţ [2022] A. Neguţ, “Toward AGT for Parabolic Sheaves”, Int. Math. Res. Not. 2022 (2022), no. 9, 6512–6539, arXiv:1911.02963.
  * Creutzig et al. [2022] T. Creutzig, D.-E. Diaconescu, and M. Ma, “Affine Laumon spaces and iterated W-algebras”, arXiv:2206.01600.
  * Kim [2009] S. Kim, “The Complete superconformal index for N=6 Chern-Simons theory”, Nucl. Phys. B 821 (2009) 241–284, arXiv:0903.4172, [Erratum: Nucl.Phys.B 864, 884 (2012)].
  * Imamura and Yokoyama [2011] Y. Imamura and S. Yokoyama, “Index for three dimensional superconformal field theories with general R-charge assignments”, JHEP 04 (2011) 007, arXiv:1101.0557.
  * Kapustin and Willett [2011] A. Kapustin and B. Willett, “Generalized Superconformal Index for Three Dimensional Field Theories”, arXiv:1106.2484.
  * Alday et al. [2013] L. F. Alday, D. Martelli, P. Richmond, and J. Sparks, “Localization on Three-Manifolds”, JHEP 10 (2013) 095, arXiv:1307.6848.
  * Cecotti et al. [2014] S. Cecotti, D. Gaiotto, and C. Vafa, “$tt^{*}$ geometry in 3 and 4 dimensions”, JHEP 05 (2014) 055, arXiv:1312.1008.
  * Fujitsuka et al. [2014] M. Fujitsuka, M. Honda, and Y. Yoshida, “Higgs branch localization of 3d $\mathscr{N}$ = 2 theories”, PTEP 2014 (2014), no. 12, 123B02, arXiv:1312.3627.
  * Benini and Peelaers [2014] F. Benini and W. Peelaers, “Higgs branch localization in three dimensions”, JHEP 05 (2014) 030, arXiv:1312.6078.
  * Tanaka et al. [2015] A. Tanaka, H. Mori, and T. Morita, “Superconformal index on $\mathbb{RP}^{2}\times\mathbb{S}^{1}$ and mirror symmetry”, Phys. Rev. D 91 (2015) 105023, arXiv:1408.3371.
  * Schellekens and Warner [1986] A. N. Schellekens and N. P. Warner, “Anomalies and Modular Invariance in String Theory”, Phys. Lett. B 177 (1986) 317–323.
  * Schellekens and Warner [1987] A. N. Schellekens and N. P. Warner, “Anomalies, Characters and Strings”, Nucl. Phys. B 287 (1987) 317.
  * Witten [1987] E. Witten, “Elliptic Genera and Quantum Field Theory”, Commun. Math. Phys. 109 (1987) 525.
  * Gadde and Gukov [2014] A. Gadde and S. Gukov, “2d Index and Surface operators”, JHEP 03 (2014) 080, arXiv:1305.0266.
  * Benini et al. [2015] F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic Genera of 2d ${\mathcal{N}}$ = 2 Gauge Theories”, Commun. Math. Phys. 333 (2015), no. 3, 1241–1286, arXiv:1308.4896.
  * Benini et al. [2014] F. Benini, R. Eager, K. Hori, and Y. Tachikawa, “Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups”, Lett. Math. Phys. 104 (2014) 465–493, arXiv:1305.0533.
  * Gadde et al. [2014] A. Gadde, S. Gukov, and P. Putrov, “Walls, Lines, and Spectral Dualities in 3d Gauge Theories”, JHEP 05 (2014) 047, arXiv:1302.0015.
  * Gadde et al. [2016] A. Gadde, S. Gukov, and P. Putrov, “Fivebranes and 4-manifolds”, Prog. Math. 319 (2016) 155–245, arXiv:1306.4320.
  * Aganagic et al. [2017] M. Aganagic, K. Costello, J. McNamara, and C. Vafa, “Topological Chern-Simons/Matter Theories”, arXiv:1706.09977.
  * Closset et al. [2013] C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. Komargodski, “Supersymmetric Field Theories on Three-Manifolds”, JHEP 05 (2013) 017, arXiv:1212.3388.
  * Costello et al. [2020] K. Costello, T. Dimofte, and D. Gaiotto, “Boundary Chiral Algebras and Holomorphic Twists”, arXiv:2005.00083.
  * Okazaki [2019] T. Okazaki, “Abelian dualities of $\mathcal{N}=(0,4)$ boundary conditions”, JHEP 08 (2019) 170, arXiv:1905.07425.
  * Gaiotto and Okazaki [2019] D. Gaiotto and T. Okazaki, “Dualities of Corner Configurations and Supersymmetric Indices”, JHEP 11 (2019) 056, arXiv:1902.05175.
  * Benini and Zaffaroni [2015] F. Benini and A. Zaffaroni, “A topologically twisted index for three-dimensional supersymmetric theories”, JHEP 07 (2015) 127, arXiv:1504.03698.
  * Closset and Kim [2016] C. Closset and H. Kim, “Comments on twisted indices in 3d supersymmetric gauge theories”, JHEP 08 (2016) 059, arXiv:1605.06531.
  * Intriligator and Seiberg [1996] K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories”, Phys. Lett. B 387 (1996) 513–519, hep-th/9607207.
  * Aharony et al. [1997] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. J. Strassler, “Aspects of N=2 supersymmetric gauge theories in three-dimensions”, Nucl. Phys. B 499 (1997) 67–99, hep-th/9703110.
  * Jafferis [2012] D. L. Jafferis, “The Exact Superconformal R-Symmetry Extremizes Z”, JHEP 05 (2012) 159, arXiv:1012.3210.
  * Bullimore and Zhang [2021] M. Bullimore and D. Zhang, “3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve”, arXiv:2109.10907.
  * Sugiyama and Yoshida [2020] K. Sugiyama and Y. Yoshida, “Supersymmetric indices on $I\times T^{2}$, elliptic genera and dualities with boundaries”, Nucl. Phys. B 960 (2020) 115168, arXiv:2007.07664.
  * de Boer et al. [1997] J. de Boer, K. Hori, Y. Oz, and Z. Yin, “Branes and mirror symmetry in N=2 supersymmetric gauge theories in three-dimensions”, Nucl. Phys. B 502 (1997) 107–124, hep-th/9702154.
  * Gomis and Passerini [2006] J. Gomis and F. Passerini, “Holographic Wilson Loops”, JHEP 08 (2006) 074, arXiv:hep-th/0604007.
  * Assel and Gomis [2015] B. Assel and J. Gomis, “Mirror Symmetry And Loop Operators”, JHEP 11 (2015) 055, arXiv:1506.01718.
  * Gomis and Passerini [2007] J. Gomis and F. Passerini, “Wilson Loops as D3-Branes”, JHEP 01 (2007) 097, hep-th/0612022.
  * Intriligator et al. [1997] K. A. Intriligator, D. R. Morrison, and N. Seiberg, “Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces”, Nucl. Phys. B 497 (1997) 56–100, hep-th/9702198.
  * Achucarro and Vachaspati [2000] A. Achucarro and T. Vachaspati, “Semilocal and electroweak strings”, Phys. Rept. 327 (2000) 347–426, hep-ph/9904229.
  * Argyres et al. [1996] P. C. Argyres, M. R. Plesser, and N. Seiberg, “The Moduli space of vacua of N=2 SUSY QCD and duality in N=1 SUSY QCD”, Nucl. Phys. B 471 (1996) 159–194, hep-th/9603042.
  * Hanany and Witten [1997] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics”, Nucl. Phys. B 492 (1997) 152–190, hep-th/9611230.
  * Braverman et al. [2019] A. Braverman, M. Finkelberg, and H. Nakajima, “Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian”, Adv. Theor. Math. Phys. 23 (2019) 75–166, arXiv:1604.03625.
  * Finkelberg [2017] M. Finkelberg, “Double affine Grassmannians and Coulomb branches of 3d N=4 quiver gauge theories”, arXiv e-prints, 2017 arXiv:1712.03039, arXiv:1712.03039.
  * Bullimore et al. [2017] M. Bullimore, T. Dimofte, and D. Gaiotto, “The Coulomb Branch of 3d ${\mathcal{N}=4}$ Theories”, Commun. Math. Phys. 354 (2017), no. 2, 671–751, arXiv:1503.04817.
  * Bourget et al. [2021] A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, “Branes, Quivers, and the Affine Grassmannian”, arXiv:2102.06190.
  * Atiyah et al. [1978] M. Atiyah, N. J. Hitchin, V. Drinfeld, and Y. Manin, “Construction of Instantons”, Phys. Lett. A 65 (1978) 185–187.
  * Witten [1982] E. Witten, “Constraints on supersymmetry breaking”, Nuclear Physics B 202 (1982), no. 2, 253–316.
  * Alvarez-Gaume [1986] L. Alvarez-Gaume, “SUPERSYMMETRY AND INDEX THEORY”, in “1984 NATO ASI on Supersymmetry”.  1986\.
  * Moore et al. [2000] G. W. Moore, N. Nekrasov, and S. Shatashvili, “Integrating over Higgs branches”, Commun. Math. Phys. 209 (2000) 97–121, hep-th/9712241.
  * Losev et al. [1998] A. Losev, G. W. Moore, and S. L. Shatashvili, “M & m’s”, Nucl. Phys. B522 (1998) 105–124, arXiv:hep-th/9707250.
  * Moore et al. [2000] G. W. Moore, N. Nekrasov, and S. Shatashvili, “D particle bound states and generalized instantons”, Commun. Math. Phys. 209 (2000) 77–95, hep-th/9803265.
  * Hwang et al. [2015] C. Hwang, J. Kim, S. Kim, and J. Park, “General instanton counting and 5d SCFT”, JHEP 07 (2015) 063, arXiv:1406.6793, [Addendum: JHEP04,094(2016)].
  * Nekrasov and Shadchin [2004] N. Nekrasov and S. Shadchin, “ABCD of instantons”, Commun. Math. Phys. 252 (2004) 359–391, hep-th/0404225.
  * Shadchin [2005] S. Shadchin, “On certain aspects of string theory/gauge theory correspondence”, other thesis, 2 2005.
  * Witten [1995] E. Witten, “Sigma models and the ADHM construction of instantons”, J. Geom. Phys. 15 (1995) 215–226, hep-th/9410052.
  * Douglas [1999] M. R. Douglas, “Branes within branes”, NATO Sci. Ser. C 520 (1999) 267–275, arXiv:hep-th/9512077.
  * Bergman et al. [2014] O. Bergman, D. Rodríguez-Gómez, and G. Zafrir, “Discrete $\theta$ and the 5d superconformal index”, JHEP 01 (2014) 079, arXiv:1310.2150.
  * Bao et al. [2014] L. Bao, V. Mitev, E. Pomoni, M. Taki, and F. Yagi, “Non-Lagrangian Theories from Brane Junctions”, JHEP 01 (2014) 175, arXiv:1310.3841.
  * Hayashi et al. [2014] H. Hayashi, H.-C. Kim, and T. Nishinaka, “Topological strings and 5d $T_{N}$ partition functions”, JHEP 06 (2014) 014, arXiv:1310.3854.
  * Taki [2013] M. Taki, “Notes on Enhancement of Flavor Symmetry and 5d Superconformal Index”, arXiv:1310.7509.
  * Alba et al. [2011] V. A. Alba, V. A. Fateev, A. V. Litvinov, and G. M. Tarnopolskiy, “On combinatorial expansion of the conformal blocks arising from AGT conjecture”, Lett. Math. Phys. 98 (2011) 33–64, arXiv:1012.1312.
  * Nekrasov and Okounkov [2006] N. Nekrasov and A. Okounkov, “Seiberg-Witten theory and random partitions”, Prog. Math. 244 (2006) 525–596, arXiv:hep-th/0306238.
  * Jeffrey and Kirwan [1993] L. Jeffrey and F. Kirwan, “Localization for nonabelian group actions”, arXiv:9307001.
  * Cordova and Shao [2014] C. Cordova and S.-H. Shao, “An Index Formula for Supersymmetric Quantum Mechanics”, arXiv:1406.7853.
  * Hori et al. [2015] K. Hori, H. Kim, and P. Yi, “Witten Index and Wall Crossing”, JHEP 01 (2015) 124, arXiv:1407.2567.
  * Pedder et al. [2008] C. Pedder, J. Sonner, and D. Tong, “The Geometric Phase in Supersymmetric Quantum Mechanics”, Phys. Rev. D 77 (2008) 025009, arXiv:0709.0731.
  * Sonner and Tong [2009] J. Sonner and D. Tong, “Berry Phase and Supersymmetry”, JHEP 01 (2009) 063, arXiv:0810.1280.
  * de Boer et al. [2009a] J. de Boer, J. Manschot, K. Papadodimas, and E. Verlinde, “The Chiral ring of AdS(3)/CFT(2) and the attractor mechanism”, JHEP 03 (2009)a 030, arXiv:0809.0507.
  * de Boer et al. [2009b] J. de Boer, K. Papadodimas, and E. Verlinde, “Black Hole Berry Phase”, Phys. Rev. Lett. 103 (2009)b 131301, arXiv:0809.5062.
  * Wong [2015] K. Wong, “Berry’s connection, Kähler geometry and the Nahm construction of monopoles”, JHEP 12 (2015) 147, arXiv:1511.01052.
  * Dedushenko [2022] M. Dedushenko, “Remarks on Berry Connection in QFT, Anomalies, and Applications”, arXiv:2211.15680.
  * Galakhov [2023] D. Galakhov, “BPS States Meet Generalized Cohomology”, arXiv:2303.05538.
  * Hwang et al. [2017] C. Hwang, P. Yi, and Y. Yoshida, “Fundamental Vortices, Wall-Crossing, and Particle-Vortex Duality”, JHEP 05 (2017) 099, arXiv:1703.00213.
  * Crew et al. [2020] S. Crew, N. Dorey, and D. Zhang, “Factorisation of 3d $\mathcal{N}$ = 4 twisted indices and the geometry of vortex moduli space”, JHEP 08 (2020), no. 08, 015, arXiv:2002.04573.
  * Koroteev and Sciarappa [2018] P. Koroteev and A. Sciarappa, “Quantum Hydrodynamics from Large-n Supersymmetric Gauge Theories”, Lett. Math. Phys. 108 (2018), no. 1, 45–95, arXiv:1510.00972.
  * Koroteev [2021] P. Koroteev, “A-type Quiver Varieties and ADHM Moduli Spaces”, Commun. Math. Phys. 381 (2021), no. 1, 175–207, arXiv:1805.00986.
  * Nieri et al. [2018] F. Nieri, Y. Pan, and M. Zabzine, “3d Expansions of 5d Instanton Partition Functions”, JHEP 04 (2018) 092, arXiv:1711.06150.
  * Smirnov [2018] A. Smirnov, “Elliptic stable envelope for Hilbert scheme of points in the plane”, arXiv:1804.08779.
  * Dinkins and Smirnov [2021] H. Dinkins and A. Smirnov, “Euler characteristic of stable envelopes”, arXiv e-prints, 2021 arXiv:2108.07202, arXiv:2108.07202.
  * KING [1994] A. D. KING, “MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS”, The Quarterly Journal of Mathematics 45 12 (1994) 515–530, https://academic.oup.com/qjmath/article-pdf/45/4/515/4391718/45-4-515.pdf.
  * Ciocan-Fontanine and Kim [2009] I. Ciocan-Fontanine and B. Kim, “Moduli stacks of stable toric quasimaps”, arXiv e-prints, 2009 arXiv:0908.4446, arXiv:0908.4446.
  * Kim [2010] B. Kim, “Stable quasimaps to holomorphic symplectic quotients”, arXiv e-prints, 2010 arXiv:1005.4125, arXiv:1005.4125.
  * Ciocan-Fontanine and Kim [2013] I. Ciocan-Fontanine and B. Kim, “Wall-crossing in genus zero quasimap theory and mirror maps”, arXiv e-prints, 2013 arXiv:1304.7056, arXiv:1304.7056.
  * Tseng and You [2016] H.-H. Tseng and F. You, “K-theoretic quasimap invariants and their wall-crossing”, arXiv e-prints, 2016 arXiv:1602.06494, arXiv:1602.06494.
  * Aharony [2000] O. Aharony, “A Brief review of ’little string theories”’, Class. Quant. Grav. 17 (2000) 929–938, arXiv:hep-th/9911147.
  * Kutasov [2002] D. Kutasov, “Introduction to little string theory”, ICTP Lect. Notes Ser. 7 (2002) 165–209.
  * Kapustin [2001] A. Kapustin, “On the universality class of little string theories”, Phys. Rev. D 63 (2001) 086005, hep-th/9912044.
  * Johnson and Myers [1997] C. V. Johnson and R. C. Myers, “Aspects of type IIB theory on ALE spaces”, Phys. Rev. D 55 (1997) 6382–6393, hep-th/9610140.
  * Cachazo et al. [2001] F. Cachazo, S. Katz, and C. Vafa, “Geometric transitions and N=1 quiver theories”, hep-th/0108120.
  * Cachazo et al. [2002] F. Cachazo, B. Fiol, K. A. Intriligator, S. Katz, and C. Vafa, “A Geometric unification of dualities”, Nucl. Phys. B 628 (2002) 3–78, hep-th/0110028.
  * Strominger [1995] A. Strominger, “Massless black holes and conifolds in string theory”, Nucl. Phys. B 451 (1995) 96–108, hep-th/9504090.
  * Greene et al. [1996] B. R. Greene, D. R. Morrison, and C. Vafa, “A Geometric realization of confinement”, Nucl. Phys. B 481 (1996) 513–538, hep-th/9608039.
  * Procházka and Rapčák [2019] T. Procházka and M. Rapčák, “$\mathcal{W}$ -algebra modules, free fields, and Gukov-Witten defects”, JHEP 05 (2019) 159, arXiv:1808.08837.
  * Gaiotto and Rapcak [2022] D. Gaiotto and M. Rapcak, “Miura operators, degenerate fields and the M2-M5 intersection”, JHEP 01 (2022) 086, arXiv:2012.04118.
  * Haouzi and Jeong [To appear] N. Haouzi and S. Jeong, “Brane intersections for the quantum toroidal algebra”,  To appear.
  * Gaiotto and Witten [2012] D. Gaiotto and E. Witten, “Knot Invariants from Four-Dimensional Gauge Theory”, Adv. Theor. Math. Phys. 16 (2012), no. 3, 935–1086, arXiv:1106.4789.
  * Aspinwall and Gross [1996] P. S. Aspinwall and M. Gross, “The SO(32) heterotic string on a K3 surface”, Phys. Lett. B387 (1996) 735–742, arXiv:hep-th/9605131.
  * Bershadsky et al. [1996] M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and C. Vafa, “Geometric singularities and enhanced gauge symmetries”, Nucl. Phys. B481 (1996) 215–252, arXiv:hep-th/9605200.
  * Slodowy [1980] P. Slodowy, “Simple singularities and simple algebraic groups”,  1980\.
  * Happel et al. [1980] D. Happel, U. Preiser, and C. M. Ringel, “Binary polyhedral groups and euclidean diagrams”, manuscripta mathematica 31 (1980), no. 1-3, 317–329.
  * Drukker et al. [2011] N. Drukker, D. Gaiotto, and J. Gomis, “The Virtue of Defects in 4D Gauge Theories and 2D CFTs”, JHEP 06 (2011) 025, arXiv:1003.1112.
  * Tachikawa [2011] Y. Tachikawa, “N=2 S-duality via Outer-automorphism Twists”, J. Phys. A 44 (2011) 182001, arXiv:1009.0339.
  * Witten [2011] E. Witten, “Fivebranes and Knots”, arXiv:1101.3216.
  * Hanany et al. [2013] A. Hanany, N. Mekareeya, and S. S. Razamat, “Hilbert Series for Moduli Spaces of Two Instantons”, JHEP 01 (2013) 070, arXiv:1205.4741.
  * Cremonesi et al. [2014] S. Cremonesi, G. Ferlito, A. Hanany, and N. Mekareeya, “Coulomb Branch and The Moduli Space of Instantons”, JHEP 12 (2014) 103, arXiv:1408.6835.
  * Bourget et al. [2021] A. Bourget, A. Hanany, and D. Miketa, “Quiver origami: discrete gauging and folding”, JHEP 01 (2021) 086, arXiv:2005.05273.
  * Geiss et al. [2014] C. Geiss, B. Leclerc, and J. Schröer, “Quivers with relations for symmetrizable Cartan matrices I : Foundations”, arXiv e-prints, 2014 arXiv:1410.1403, arXiv:1410.1403.
  * Braverman et al. [2018] A. Braverman, M. Finkelberg, and H. Nakajima, “Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II”, Adv. Theor. Math. Phys. 22 (2018) 1071–1147, arXiv:1601.03586.
  * Nakajima and Weekes [2019] H. Nakajima and A. Weekes, “Coulomb branches of quiver gauge theories with symmetrizers”, arXiv:1907.06552.
  * Cecotti and Del Zotto [2012] S. Cecotti and M. Del Zotto, “4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case”, JHEP 10 (2012) 190, arXiv:1207.7205.
  * Dey et al. [2017] A. Dey, A. Hanany, P. Koroteev, and N. Mekareeya, “On Three-Dimensional Quiver Gauge Theories of Type B”, JHEP 09 (2017) 067, arXiv:1612.00810.
  * Nakajima [2016] H. Nakajima, “Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, I”, Adv. Theor. Math. Phys. 20 (2016) 595–669, arXiv:1503.03676.
  * Okounkov and Smirnov [2022] A. Okounkov and A. Smirnov, “Quantum difference equation for Nakajima varieties”, Invent. Math. 229 (2022), no. 3, 1203–1299, arXiv:1602.09007.
  * Hitchin [1987] N. J. Hitchin, “The Selfduality equations on a Riemann surface”, Proc. Lond. Math. Soc. 55 (1987) 59–131.
  * Simpson [1991] C. T. Simpson, “Nonabelian Hodge theory”, in “Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990)”, pp. 747–756.  Math. Soc. Japan, Tokyo, 1991.
  * Simpson [1992] C. T. Simpson, “Higgs bundles and local systems”, Publications Mathématiques de l’IHÉS 75 (1992) 5–95.
  * KONNO [1993] H. KONNO, “Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface”, Journal of the Mathematical Society of Japan 45 (1993), no. 2, 253 – 276.
  * Nakajima [1996] H. Nakajima, “Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces”, in “Moduli of vector bundles (Sanda, 1994; Kyoto, 1994)”, vol. 179 of Lecture Notes in Pure and Appl. Math., pp. 199–208.  Dekker, New York, 1996.
  * Donagi and Pantev [2008] R. Y. Donagi and T. Pantev, “Geometric langlands and non-abelian hodge theory”, Surveys in differential geometry 13 (2008) 85–116.
  * Frenkel [2006] E. Frenkel, “Ramifications of the geometric Langlands Program”, math/0611294.
  * Witten [2007] E. Witten, “Gauge theory and wild ramification”, arXiv:0710.0631.
  * Dorey et al. [1996] N. Dorey, C. Fraser, T. J. Hollowood, and M. A. C. Kneipp, “S duality in N=4 supersymmetric gauge theories with arbitrary gauge group”, Phys. Lett. B 383 (1996) 422–428, hep-th/9605069.
  * Argyres et al. [2006] P. C. Argyres, A. Kapustin, and N. Seiberg, “On S-duality for non-simply-laced gauge groups”, JHEP 06 (2006) 043, hep-th/0603048.
  * Witten [1979] E. Witten, “Dyons of Charge e theta/2 pi”, Phys. Lett. B 86 (1979) 283–287.
  * Beardon [1995] A. F. Beardon, “The geometry of discrete groups”,  1995\.
  * Chang et al. [2014] C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Little String Amplitudes (and the Unreasonable Effectiveness of 6D SYM)”, JHEP 12 (2014) 176, arXiv:1407.7511.
  * Elliott and Pestun [2019] C. Elliott and V. Pestun, “Multiplicative Hitchin Systems and Supersymmetric Gauge Theory”, Selecta Math. 25 (2019) 64, arXiv:1812.05516.
  * Buscher [1987] T. H. Buscher, “A Symmetry of the String Background Field Equations”, Phys. Lett. B 194 (1987) 59–62.
  * Amariti and Formigoni [2020] A. Amariti and G. Formigoni, “A note on 4d $N=3$ from little string theory”, Nucl. Phys. B 958 (2020) 115108, arXiv:2003.05983.
  * Ooguri and Vafa [2000] H. Ooguri and C. Vafa, “Knot invariants and topological strings”, Nucl. Phys. B 577 (2000) 419–438, hep-th/9912123.
  * Noumi and Shiraishi [2012] M. Noumi and J. Shiraishi, “A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators”, arXiv e-prints, 2012 arXiv:1206.5364, arXiv:1206.5364.
  * Mironov et al. [2021] A. Mironov, A. Morozov, and Y. Zenkevich, “Duality in elliptic Ruijsenaars system and elliptic symmetric functions”, Eur. Phys. J. C 81 (2021), no. 5, 461, arXiv:2103.02508.
  * Koroteev and Shakirov [2020] P. Koroteev and S. Shakirov, “The quantum DELL system”, Lett. Math. Phys. 110 (2020), no. 5, 969–999, arXiv:1906.10354.
  * Etingof [1995] P. I. Etingof, “Quantum integrable systems and representations of Lie algebras”, J. Math. Phys. 36 (1995) 2636–2651, hep-th/9311132.
  * Atiyah [1984] M. F. Atiyah, “INSTANTONS IN TWO-DIMENSIONS AND FOUR-DIMENSIONS”, Commun. Math. Phys. 93 (1984) 437–451.
  * Ashwinkumar et al. [2018] M. Ashwinkumar, J. Cao, Y. Luo, M.-C. Tan, and Q. Zhao, “Little Strings, Quasi-topological Sigma Model on Loop Group, and Toroidal Lie Algebras”, Nucl. Phys. B 928 (2018) 469–498, arXiv:1512.07629.
  * Seiberg [1997] N. Seiberg, “Why is the matrix model correct?”, Phys. Rev. Lett. 79 (1997) 3577–3580, hep-th/9710009.
  * Aharony et al. [1998] O. Aharony, M. Berkooz, S. Kachru, N. Seiberg, and E. Silverstein, “Matrix description of interacting theories in six-dimensions”, Adv. Theor. Math. Phys. 1 (1998) 148–157, hep-th/9707079.
  * Braverman and Finkelberg [2010] A. Braverman and M. Finkelberg, “Pursuing the Double Affine Grassmannian I: Transversal Slices via Instantons on $A_{k}$-Singularities”, Duke Math. J. 152 (2010) 175–206, arXiv:0711.2083.
  * Kapustin [2006] A. Kapustin, “Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality”, Phys. Rev. D 74 (2006) 025005, hep-th/0501015.