since ultimately we are interested in the 1d quantum mechanics obtained by
shrinking the boundary back to a point at the origin; this identifies
$\text{Stab}^{K}_{\\{{\bf A^{\prime}}\\}}$ as a K-theoretic stable envelope
[103, 13]. This shrinking procedure is achieved by sending $q\rightarrow 0$;
as a quick check, this is consistent with the index, since $\lim_{q\to
0}\Theta(x\,;q)=(1-x)$, so the 2d boundary theory on $T^{2}$ indeed reduces to
a 1d quantum mechanics on $S^{1}_{\mathcal{C}^{\prime}}$.

From the above presentation, it is clear that magnetic solutions can be
recovered from the electric ones via pairing with a specific covector $W$. For
instance, in the case of $z$-solutions, consider a magnetic block with
exceptional Dirichlet boundary conditions in vacuum ${\\{{\bf A}\\}}$, and
denote it as $\mathbb{V}_{\\{{\bf A}\\}}$; then there exists a covector $W$
such that

$\sum_{\\{{\bf A^{\prime}}\\}}W_{\\{{\bf A^{\prime}}\\}}\,{\bf F}^{\\{{\bf
A^{\prime}}\\}}_{\\{{\bf A}\\}}=\mathbb{V}_{\\{{\bf A}\\}}\;,$ (3.55)

with ${\bf F}^{\\{{\bf A^{\prime}}\\}}_{\\{{\bf A}\\}}$ the electric $z$
solution from (3.53). For details on the existence of $W$, we refer the reader
to the geometric arguments invoked in the unramified context [13, 103]. The
same arguments apply here, without any major modification; see also Section 5.

##  4 Tame ramification from 5d gauge theory

The 3-dimensional Drinfeld quiver gauge theory
$T^{3d}_{\mathfrak{g}_{\text{o}}}$ has a 5-dimensional origin, as we now
explain.

Let $T^{5d}_{\mathfrak{g}_{\text{o}}}$ be a 5d quiver gauge theory with
$\mathcal{N}=1$ supersymmetry, supported on the manifold
$M_{5}=S^{1}_{\mathcal{C}^{\prime}}(R_{\mathcal{C}^{\prime}})\times\mathbb{C}^{2}$.
It admits a $SU(2)_{R}$ R-symmetry. The quiver shape is once again the Dynkin
diagram of the simply-laced Lie algebra $\mathfrak{g}_{\text{o}}=ADE$ of rank
$r=\text{rk}(\mathfrak{g})$, and the gauge group is a product of unitary
groups

$G_{5d}=\prod_{a=1}^{r}U(n_{a})\;,$ (4.1)

where the $n_{a}$ are all positive integers585858For each $a=1,\ldots,r$ the
center $U(1)\subset U(n_{a})$ is actually nondynamical, so the gauge group
should really be written as $\prod_{a=1}^{r}SU(n_{a})$. Nevertheless, we will
later place the theory on the $\Omega$-background, where the $U(1)$ centers
will become physically sensible. For instance, the instanton partition
function of the theory in that background depends on $n_{a}$ equivariant
Coulomb parameters, not $n_{a}-1$, so we will usually keep the gauge group as
$G_{5d}$..

The bifundamental matter between gauge nodes consists of hypermultiplets in
the representation $\oplus_{b>a}\,\Delta_{ab}\,(n_{a},\overline{n_{b}})$ of
the group $\prod_{a,b}U(n_{a})\times U(n_{b})$. The operator $\Delta_{ab}$ is
the incidence matrix of $\mathfrak{g}_{\text{o}}$, with entry $1$ if there is
a link connecting nodes $a$ and $b$ in the Dynkin diagram, and $0$ otherwise.
The quiver contains a total of $r-1$ such bifundamental hypermultiplets.

The remaining matter consists of fundamental hypermultiplets, taken in
representation $(n_{a},\overline{n^{F}_{a}})$ of $U(n_{a})\times
U(n^{F}_{a})$, with $U(n^{F}_{a})$ the corresponding flavor symmetry group on
node $a$:

$G^{F}_{5d}=\prod_{a=1}^{r}U(n^{F}_{a})\;,$ (4.2)

where $n^{F}_{a}$ are non-negative integers.

There is an additional global symmetry in 5d,

$G^{top}_{5d}=\prod_{a=1}^{r}U(1)^{top}_{a}=U(1)^{r}\;,$ (4.3)

the so-called topological symmetry. For all $a=1,\ldots,r$, the symmetry
arises from the conserved current ${\cal
J}_{5d,a}=\frac{1}{8\pi^{2}}\mbox{Tr}(F_{a}\wedge F_{a})$, and the associated
charge is called the instanton number $k_{a}$. The mass parameters associated
to this topological symmetry are instanton counting parameters $z_{a}$, which
in 5d are related to the gauge couplings as

$z_{a}=e^{-8\pi^{2}\,R_{\mathcal{C}^{\prime}}/(g^{5d}_{a})^{2}}\;,\qquad\;\;a=1,\ldots,\text{rk}(\mathfrak{g})\;.$
(4.4)

There are various physical constraints to take into account.

First, as a supersymmetric gauge theory in 5 dimensions,
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ is non-renormalizable, and should be
understood as an IR relevant deformation of some UV fixed point. The $r$ gauge
coupling $1/(g^{5d}_{a})^{2}$ have mass dimension 1, and set the scale of the
deformation. The requirement of having a UV fixed point imposes constraints on
the ranks of the various groups [268]. For instance, in the rank 1 case,
meaning a single $SU(n)$ gauge group, $T^{5d}_{A_{1}}$ has a UV fixed point if
$n^{F}+2\,|\kappa_{CS,bare}|\leq 2\,n$, where $\kappa_{CS,bare}$ is a bare
Chern-Simons level. Such inequalities exist for all
$\mathfrak{g}_{\text{o}}$-type quivers.

Furthermore, we will see that the 3d theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$
always arises as a codimension-2 vortex solution on the Higgs branch of
$T^{5d}_{\mathfrak{g}_{\text{o}}}$595959The vortices in question can be
described as (non-abelian versions of) semi-local vortices [269, 129]. In
particular, our 5d theories should be “fully Higgsable” to begin with. Their
existence implies that the ranks of the flavor groups $n^{F}_{a}$ have to be
“sufficiently large” integers. In our rank 1 example, assuming $n^{F}$ is even
so we can choose the bare Chern-Simons level to be $\kappa_{CS,bare}=0$, the
existence of vortex solutions imposes the constraint $n^{F}\geq 2\,n$ on the
number of fundamental hypermultiplets606060Whenever $n^{F}<2\,n$, full
Higgsing is also possible, but only on a baryonic Higgs branch [270]. Here,
our gauge group is taken to be $U(n)$ instead of $SU(n)$, and baryons are not
invariant under the additional gauged $U(1)$. It follows that the baryonic
Higgs branch is lifted, and full Higgsing is only possible for $n^{F}\geq
2\,n$. An identical argument was invoked by Hanany and Witten, but in 3
dimensions instead of 5 [271]..

Lastly, we will eventually make contact with various quantum field theories in
lower dimensions after compactification. Perhaps the most natural reduction is
on $S^{1}_{\mathcal{C}^{\prime}}$, down to a 4d $\mathcal{N}=2$ theory, but
even then, this limit is subtle: there are a priori many ways to scale the 5d
variables as $R_{\mathcal{C}^{\prime}}\rightarrow 0$ which can result in
sensible 4d theories. Among them, the most naive one gives a 4d
$\mathcal{N}=2$ $\mathfrak{g}_{\text{o}}$-type quiver gauge theory with the
“same” matter content as the 5d one, and gauge couplings which remain finite.
We would like these theories to be asymptotically free or conformal in 4d, so
the non-positivity of the 1-loop beta function is imposed as a constraint. In
our rank 1 $U(n)$ example, this translates to a constraint on the number of
flavors616161Gauge theories in 4 dimensions where the number of fundamental
hypermultiplets saturates the inequality, such as $n^{F}=2\,n$ in the rank 1
case, are sometimes called _asymptotically conformal_ [83]. The expression
stands for the theory becoming superconformal as the hypermultiplets become
massless.: $n^{F}\leq 2\,n$. Such inequalities exist for all
$\mathfrak{g}_{\text{o}}$-type quiver gauge theories. This compactification is
useful to make contact with the representation theory of quantum groups,
namely the character of finite-dimensional representations of Yangians [83,
84].

Scaling the 5d variables differently, another possible reduction on
$S^{1}_{\mathcal{C}^{\prime}}$ will result in a 4d SCFT of class
${\mathcal{S}}$ [134, 133]. In our rank 1 $SU(n)$ example, the SCFT constraint
in the limit would saturate the previous inequality to $n^{F}=2\,n$.
Generically, and unlike the previous 4d limit, the Lagrangian description is
lost in this reduction, since the gauge couplings become infinite as
$R_{\mathcal{C}^{\prime}}\rightarrow 0$, meaning there is no energy scale at
which we can provide a weakly coupled gauge theory description626262Some 5d
theories are known to have more than one inequivalent Lagrangian descriptions
to begin with, for instance theories which are related by spectral duality
[147]. When this is the case, it is typically possible for one of these
Lagrangian descriptions to survive the 4d limit, a point we will revisit in
Section 8.3.1.. This compactification is useful to make contact with the AGT
correspondence [56].

If we imagine compactifying $\mathbb{C}^{2}\subset M_{5}$ on $T^{2}$ instead,
the reduction on this torus yields a 3d $\mathcal{N}=4$ quiver gauge theory
whose Coulomb branch is a slice of the affine Grassmanian, see [272, 273, 274]
and recently [275]; there, constraints similar to the above determine how
“balanced” the theory is, which in turn fixes the behavior in the IR. In our
rank 1 example, a theory satisfying the constraint $n^{F}\leq 2\,n$ would be
called “good” in the terminology of [34]. Further reducing on
$S^{1}_{\mathcal{C}^{\prime}}$ and at low energies, this compactification is
useful to make contact with Gukov and Witten’s description of surface defects
in 4d Super Yang-Mills via 2d sigma models [31, 140].

In the rest of this work, we decide to impose constraints which are physically
sensible in most (and often all) of the above scenarios. First, the flavor
content of our theories $T^{5d}_{\mathfrak{g}_{\text{o}}}$ requires the bare
Chern-Simons levels to be integers, which we further specialize to be zero out
of convenience, $\kappa_{CS,bare}=0$. There are however nonzero effective
Chern-Simons levels $\kappa_{CS}\neq 0$ generated at 1-loop; assuming all
masses are positive, these levels are uniquely determined from the ranks of
the gauge and flavor groups. Second, we require a sufficiently large number of
fundamental hypermultiplets to guarantee a fully Higgsable 5d theory, and the
existence of vortex solutions on that Higgs branch. Lastly, in the language of
the 4d reductions discussed above, we only consider theories with vanishing
1-loop beta function. In our rank 1 example, this means $n^{F}=2\,n$ already
in 5d, so we are considering the gauge group $G_{5d}=U(n)$ and flavor group
$G^{F}_{5d}=U(2\,n)$. This case will be treated in detail in Section 9.

For a generic quiver made up of unitary gauge groups, the asymptotic
conformality constraint takes the form of $\text{rk}(\mathfrak{g})$ equations:

$\sum_{b=1}^{\text{rk}(\mathfrak{g})}C_{ab}\;n_{b}=n^{F}_{a}\;,\qquad\;a=1,\ldots,\text{rk}(\mathfrak{g})\;,$
(4.5)

where $C_{ab}=\langle\alpha_{a},\alpha_{b}^{\vee}\rangle$ is the Cartan matrix
of $\mathfrak{g}_{\text{o}}$. We will define the quiver theory
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ by specifying its fundamental
hypermultiplet content: a vector of $r=\text{rk}(\mathfrak{g})$ non-negative
integers $(n^{F}_{1},n^{F}_{2},\ldots,n^{F}_{r})$, where $n^{F}_{a}$ denotes
the rank of the $a$-th flavor group $U(n^{F}_{a})$ (and the integers cannot
all be zero). This fixes the right-hand side of the linear system (4.5), which
either has no solution or a unique integer-valued solution
$(n_{1},n_{2},\ldots,n_{r})\in\left(\mathbb{Z}^{+}\right)^{r}$. The solution,
whenever it exists, spells out the gauge content of
$T^{5d}_{\mathfrak{g}_{\text{o}}}$, meaning the ranks of the gauge groups are
uniquely fixed by the ranks of the flavor groups and the constraint.

We have chosen to set the Chern-Simons levels to 0 on all nodes,
$\kappa_{CS,bare,a}=0$. Given the fundamental and bifundamental matter content
of the quiver $T^{5d}_{\mathfrak{g}_{\text{o}}}$, and fixing the sign of the
corresponding masses to be positive, the bare Chern-Simons level on node $a$
is corrected quantum mechanically at 1-loop to give an effective Chern-Simons
coupling

$\kappa_{CS,a}=\kappa_{CS,bare,a}+\frac{1}{2}\left(n^{F}_{a}+\sum_{b<a}\Delta^{ba}\,n_{b}-\sum_{b>a}\Delta_{ab}\,n_{b}\right)\;,$
(4.6)

with $\Delta_{ab}=1$ or $0$ depending on whether the gauge nodes $a$ and $b$
are linked or not in the Dynkin diagram of $\mathfrak{g}_{\text{o}}$. After
imposing the constraint (4.5) and setting $\kappa_{CS,bare,a}=0$, this
simplifies to

$\kappa_{CS,a}=n_{a}-\sum_{b>a}\Delta_{ab}\,n_{b}\;.$ (4.7)

for all nodes $a=1,\ldots,\text{rk}(\mathfrak{g})$.

###  4.1 The instanton partition function

We are interested in counting self-dual instantons on $\mathbb{C}^{2}$ inside
$M_{5}=S^{1}_{\mathcal{C}^{\prime}}(R_{\mathcal{C}^{\prime}})\times\mathbb{C}^{2}$.
In 5 dimensions and in the moduli space approximation, the dynamics of
instantons is captured by a supersymmetric sigma model whose target space is
the instanton moduli space. The partition function of the 5d theory
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ on $S^{1}_{\mathcal{C}^{\prime}}$
factorizes into the contribution of a 1-loop exact perturbative part, counting
W-bosons along with their fermionic supersymmetric partners, and an instanton
part:

$Z(T^{5d}_{\mathfrak{g}_{\text{o}}})=Z_{pert}(T^{5d}_{\mathfrak{g}_{\text{o}}})\cdot
Z_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})\;,\qquad\;\;\;\;Z_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})=\sum_{\vec{k}=0}^{\infty}\;\prod_{a=1}^{r}{Z}_{k_{a}}\;z_{a}^{k_{a}}\;,$
(4.8)

where the $a$-th entry of the vector $\vec{k}=(k_{1},\ldots,k_{r})$ labels an
instanton number $k_{a}$ on node $a$ of the quiver. We will neglect the
perturbative part in what follows, since it does not enter the conformal block
physics. The instanton sector is a sum over all instanton numbers, with the
counting parameters $z_{a}$ encoding the 5d gauge couplings.

As is well-known, the instanton moduli space suffers both from IR
singularities at infinity of $\mathbb{C}^{2}$, and from UV singularities due
to coincident zero-size instantons, which implies the quantum mechanics
description is incomplete. A powerful way to resolve such singularities is to
make use of the ADHM construction [276], which provides a UV completion as a
1d $\mathcal{N}=4$ gauged quantum mechanics on
$S^{1}_{\mathcal{C}^{\prime}}$636363The supersymmetry of the gauged quantum
mechanics in question is sometimes called $\mathcal{N}=4$B. It arises as the
1-dimensional reduction of 2d $\mathcal{N}=(0,4)$ supersymmetry.. The
instanton dynamics are captured by the Higgs branch of this quantum mechanics,
whose gauge group is $\prod_{a=1}^{r}U(k_{a})$. Note that the instanton number
$k_{a}$ now also denotes the rank of the $a$-th gauge group in this UV quantum
mechanics. The instanton partition function takes the form of a Witten index
[277, 278], which can be evaluated via equivariant localization [279, 280,
281, 92, 282]. In the weak coupling regime of the UV quantum mechanics, where
$g_{QM,a}\rightarrow 0$ for all $a=1,\ldots,r$, the index reduces to Gaussian
integrals around saddle points, meaning it can be evaluated exactly.
Meanwhile, the UV incomplete quantum mechanics description is recovered in the
low energy regime, after taking the couplings of the gauged quantum mechanics
to infinity, $g_{QM,a}\rightarrow\infty$. More precisely, if $E$ is the energy
scale at which we wish to study the 5d BPS states, the decoupling of the extra
UV degrees of freedom will occur when $g_{QM,a}^{-2/3}\,E\ll
1$646464Technically, we are not varying the couplings $g_{QM,a}$ in those
arguments, but rather the radius $R_{\mathcal{C}^{\prime}}$ of the circle on
which the quantum mechanics is defined. Namely, the BPS index is properly
defined for $R^{3}_{\mathcal{C}^{\prime}}\,g_{QM,a}^{2}\gg 1$, and the weak
coupling regime where we perform the localization computation is instead
defined for $R^{3}_{\mathcal{C}^{\prime}}\,g_{QM,a}^{2}\ll 1$. This should be
interpreted as varying the radius $R_{\mathcal{C}^{\prime}}$ continuously from
$\infty$ to 0 in the quantity $R^{3}_{\mathcal{C}^{\prime}}\,g_{QM,a}^{2}$,
with the couplings $g_{QM,a}^{2}$ kept fixed and large enough
($g_{QM,a}^{2}\gg E^{3}$) to decouple the UV degrees of freedom. In the main
text, the dependence on $R_{\mathcal{C}^{\prime}}$ was kept implicit not to
overburden the expressions; it is easily reinstated by dimensional analysis..
The instanton partition of this quantum mechanics once again takes the form of
a sum over all instanton numbers,

${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})=\sum_{\vec{k}=0}^{\infty}\;\prod_{a=1}^{r}{\mathcal{Z}}_{k_{a}}\;\frac{z_{a}^{k_{a}}}{k_{a}!}\;.$
(4.9)

A subtle but important point is that the UV index ${\mathcal{Z}}_{inst}$ will
sometimes count more states than the “true” instanton generating function
$Z_{inst}$ of the low energy QFT in the decoupling regime, so the two
quantities are _not_ equal in general. The discrepancy between the two
partition functions can be factored out, as it represents the contribution of
extra spurious states in the UV. We denote the contribution of these unwanted
5d decoupled bulk states as $Z_{extra}$, and so

${\mathcal{Z}}_{inst}\equiv Z_{inst}\cdot{\mathcal{Z}}_{extra}$ (4.10)

The quantity ${\mathcal{Z}}_{extra}$ has been determined for all classical
gauge groups and various types of matter [283, 284]. For some specific gauge
groups and matter contents, it can happen that the UV ADHM quantum mechanics
admits a realization in string theory [285, 286]; when this is the case, the
contribution ${\mathcal{Z}}_{extra}$ can most easily be understood from moduli
associated to light instantonic branes “escaping to infinity,” away from the
heavier branes on which the 5d gauge theory lives. These non-compact flat
directions are moduli coming from the Coulomb branch of the quantum mechanics.
Before we say more about these Coulomb moduli, let us discuss the index
${\mathcal{Z}}_{inst}$.

In $\mathcal{N}=2$ notation, the index is written with respect to two
supercharges $Q_{+}$ and $\overline{Q}_{+}$. These are the same supercharges
we previously used when we discussed the 3d half-index (3.19). Explicitly,

${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})={\rm
Tr}\left[(-1)^{F}\,q^{J_{1}+\frac{R}{2}}\;t^{J_{2}-\frac{R}{2}}\;{\mathfrak{m}}^{\Sigma}\right]\;\;.$
(4.11)

$F$ is the fermion number operator. $J_{1}$ and $J_{2}$ are valued in the
Cartan subalgebra of $U(1)\times U(1)$, which acts by rotating
$\mathbb{C}\times\mathbb{C}$; they have half-integer eigenvalues.

The variable $\log(\mathfrak{m})$ collectively stands for all chemical
potentials conjugate to the Cartan generator $\Sigma$ of the 5d gauge group
$G_{5d}$, 5d flavor group $G^{F}_{5d}$, and 5d topological symmetry group
$G^{top}_{5d}$. These groups are all understood as global symmetries from the
point of view of the quantum mechanics. Explicitly,

${\mathfrak{m}}^{\Sigma}=\prod_{a=1}^{r}z_{a}^{k_{a}}\prod_{i=1}^{n_{a}}e_{a,i}^{\mathfrak{t}_{a,i}}\prod_{s=1}^{n^{F}_{a}}f_{a,s}^{\mathfrak{t}^{F}_{a,s}}\;,$
(4.12)

where each $e_{a,i}$ is a 5d Coulomb parameter with associated Cartan
generator $\mathfrak{t}_{a,i}$, and $f_{a,s}$ is a 5d hypermultiplet mass,
with associated Cartan generator $\mathfrak{t}^{F}_{a,s}$. As before, $z_{a}$
encode the 5d gauge couplings, and $k_{a}$ are the instanton numbers.

The R-symmetry of the $\mathcal{N}=4$ quantum mechanics is $SU(2)_{r}\times
SU(2)_{R}$, and the operator $R$ in the index stands for the Cartan generator
of $U(1)_{R}\subset SU(2)_{R}$. As usual, the presence of the R-symmetry twist
is required to preserve supersymmetry. Note that $J_{1}+\frac{R}{2}$,
$J_{2}-\frac{R}{2}$, and $\Sigma$ all commute with the supercharges.

From the 5-dimensional perspective, we are describing an $\Omega$-background
[92, 93, 94]: just as in 3 dimensions, we view $M_{5}$ as a $\mathbb{C}^{2}$
bundle over $S^{1}_{\mathcal{C}^{\prime}}(R_{\mathcal{C}^{\prime}})$, and let
$(X_{\mathcal{C}^{\prime}},X_{1},X_{2})$ be coordinates on
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}^{2}$, so that the identification

$\displaystyle(0,X_{1},X_{2})\sim(2\pi
R_{\mathcal{C}^{\prime}},\,q\,X_{1},\,t^{-1}\,X_{2})$ (4.13)

holds. That is, the chemical potentials $q$ and $t$ are equivariant parameters
for the $U(1)_{q}\times U(1)_{t}$ action on $\mathbb{C}\times\mathbb{C}\subset
M_{5}$. We denote this $\Omega$-background as
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$.

The index is expected to be well-defined only if the theory is fully gapped.
In our quiver quantum mechanics, we will turn on all possible chemical
potentials (equivariant parameters) for the global symmetries, but even then,
a continuum may remain due to the vector multiplet scalars $\varphi_{a}$, also
called Coulomb branch moduli. Whenever that is the case, we should not expect
the index to have a sensible expansion in its parameters with integer
coefficients. The scalars $\varphi_{a}$ are neutral under all global
symmetries, meaning such a continuum cannot be lifted by the chemical
potentials we turn on. Instead, because our quiver quantum mechanics is
exclusively made up of $r$ unitary gauge groups, we can lift the continuum by
turning on $r$ Fayet-Iliopoulos (F.I.) parameters $\zeta_{a}$, one for each
gauge group in the quiver quantum mechanics.

Just like in the 3-dimensional context, the F.I.-parameter space of the
quantum mechanics is divided into chambers. In the interior of a chamber, the
gauge group is broken to a finite subgroup, while on a boundary between
chambers, a continuous unbroken subgroup will emerge, and with it the
existence of a Coulomb branch.

For generic values of $\zeta_{a}$, deep inside a chamber, the F.I. terms
generate a mass gap, because the vector multiplet scalars $\varphi_{a}$
acquire a mass proportional to $g^{2}_{QM,a}\,\zeta_{a}$; there, the Witten
index is expected to have a sensible integer expansion in its fugacities, and
not depend on the parameters $\zeta_{a}$656565When we evaluate the index by
localization, we are taking the limit $g_{QM,a}\rightarrow 0$, which means we
are simultaneously taking the limit $g_{QM,a}\,\zeta_{a}\rightarrow\pm\infty$
in order to produce a mass gap and guarantee a sensible index.. As the
$\zeta_{a}$ are varied continuously from one chamber to another, the index is
expected to jump, a phenomenon known as wall-crossing in quantum mechanics.
For instance, in the rank 1 case of a $U(1)$ gauged quantum mechanics, there
are two chambers $\zeta>0$ and $\zeta<0$, and a Coulomb branch continuum
arises at the boundary $\zeta=0$, at the root of the Higgs branch. The
continuum contributes a factor ${\mathcal{Z}}_{extra}$ to the index formula
(4.11), which factorizes as (4.10).

In the rest of this work, we fix all F.I. parameters to lie in a generic
chamber, away from the walls, say $\zeta_{a}$ all chosen generic and positive.
In the class of quivers we consider, ${\mathcal{Z}}_{extra}\neq 1$ for our
choice of flavor symmetry and bare Chern-Simons levels. For instance, for a
single $SU(n)$ gauge group in 5 dimensions and zero bare Chern-Simons level,
the Coulomb branch continuum contributes additional states to the index if the
number of fundamental hypermultiplets is twice the rank of the gauge group,
$n^{F}=2\,n$ [287, 288, 289, 290]. But this is precisely the matter content we
consider in this paper, as our constraint (4.5) takes the form $n^{F}=2\,n$ in
the rank 1 case, see Section 9. It follows that our index computation of
${\mathcal{Z}}_{inst}$ will differ from the QFT quantity $Z_{inst}$. Beyond
the rank 1 case, it will always be the case that ${\mathcal{Z}}_{extra}\neq 1$
for our class of quiver gauge theories.

Since we are ultimately interested in comparing gauge theory partition
functions to $q$-deformed conformal blocks, it is natural to ask if
${\mathcal{Z}}_{extra}$ also has a natural interpretation in the chiral
algebra context. For $\mathfrak{g}=A_{r}$ quivers, the ${\mathcal{Z}}_{extra}$
contributions have been attributed to the presence of additional free bosons
in the $\mathcal{W}_{q,t}(A_{r})$ $q$-conformal block [291]. In effect, one
considers not just the algebra $\mathcal{W}_{q,t}(A_{r})$, but rather the
direct sum $\mathcal{W}_{q,t}(A_{r})\oplus u(1)_{q,t}$ with an additional
$q$-deformed Heisenberg algebra. The modern perspective on the presence of
this additional Heisenberg algebra is that the blocks really are defined via
the Fock representation of a quantum toroidal algebra
$U_{q,t}(\hat{\hat{\mathfrak{gl}_{1}}})$ [186]. In our work, we will not worry
about singling out ${\mathcal{Z}}_{extra}$ explicitly. We will simply compute
the UV index ${\mathcal{Z}}_{inst}$ as a whole and the equality to
$\mathcal{W}_{q,t}(\mathfrak{g})$ $q$-conformal blocks will hold only up to
these additional $u(1)_{q,t}$ contributions. We leave the detailed
investigation of ${\mathcal{Z}}_{extra}$ for general quivers to future work,
and in particular the interpretation of such factors whenever
$\mathfrak{g}\neq A_{r}$.

Going back to the evaluation of the index deep inside a given F.I.-parameter
chamber and in the $g_{QM,a}\rightarrow 0$ limit, one first integrates over
the non-zero modes by performing Gaussian integrals over massive fluctuations,
keeping the zero modes fixed. Then, one integrates over these zero modes,
which are the 1d $\mathcal{N}=2$ vector multiplet scalars $\varphi_{a}$, with
$a=1,\ldots,r$ labeling the quiver node. As usual, these scalars get
complexified by the holonomy around the circle
$S^{1}_{\mathcal{C}^{\prime}}(R_{\mathcal{C}^{\prime}})$: we denote the
complex eigenvalues as $\phi_{a,I}=\varphi_{a,I}+i\,A^{\theta}_{a,I}$,
666666There are also fermion zero modes coming from the gaugini in the
$\mathcal{N}=2$ vector multiplet. These contribute Fermi multiplets to the
index integrand, whose form is that of the Vandermonde determinant for the
gauge group of the quantum mechanics, $\prod_{a=1}^{r}U(k_{a})$. where
$I=1,\ldots,k_{a}$. For a given instanton configuration $\vec{k}$, there are a
total of $|\vec{k}|=\sum_{a=1}^{r}k_{a}$ such complex eigenvalues valued on an
infinite cylinder. Explicitly, one finds676767We have absorbed a phase
$(-1)^{k_{a}\,\kappa_{CS,a}}$ in the definition of the instanton counting
parameter $z^{k_{a}}_{a}$; this phase is due to the nonzero effective Chern-
Simons levels.

$\displaystyle{\mathcal{Z}}_{k_{a}}=\oint\prod_{I=1}^{k_{a}}\left[\frac{d\phi_{a,I}}{2\pi
i}\right]{\mathcal{Z}}^{vec}_{a}\cdot{\mathcal{Z}}^{fund}_{a}\cdot\prod_{b>a}^{r}{\mathcal{Z}}^{bif}_{a,b}\;.$
(4.14)

The factors $Z^{vec}_{a}$, $Z^{fund}_{a}$, and $Z^{bif}_{a,b}$ are the 1-loop
determinants for the quantum mechanical modes of the vector multiplets, the
fundamental hypermultiplets and the bifundamental hypermultiplets of the 5d
gauge theory, respectively. The determinant ${\mathcal{Z}}^{fund}_{a}$ is the
contribution of 1d $\mathcal{N}=2$ Fermi multiplets, while
${\mathcal{Z}}^{vec}_{a}$ and ${\mathcal{Z}}^{bif}_{a,b}$ are contributions of
Fermi _and_ chiral multiplets.

The chiral multiplets are responsible for providing poles to the integrand.
The contours can be specified using an “$i{\epsilon}$” prescription [92, 292],
or using the Jeffrey-Kirwan residue [293]. The latter method was recently used
to compute partition functions of 2-dimensional supersymmetric theories [249,
248], and was applied to our quantum mechanical context soon after [282, 294,
295]. In particular, the integral should be treated carefully near the zeroes
of the chiral multiplets, where the integrand blows up and the assumption that
the non-zero modes have a large mass breaks down. Moreover, our integration
variables $\phi_{a,I}$ are $\mathbb{C}^{\times}$-valued, so require
regularization with an IR cutoff at the ends of the cylinders686868This last
point is intimately related to the identification of the factors
${\mathcal{Z}}_{extra}$. Namely one should look for $\phi_{a,I}$-poles at
$\infty$ in the integrand of the index, signaling the appearance of a Coulomb
branch continuum.; for details, see [282].

Regardless of the residue prescription one uses, in the end one finds that the
instanton partition function is evaluated exactly as

${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})=\sum_{\\{\overrightarrow{\boldsymbol{\mu}}\\}}\;\prod_{a=1}^{r}z_{a}^{\sum_{i=1}^{n_{a}}{\left|\boldsymbol{\mu}_{a,i}\right|}}\,{\boldsymbol{Z}}^{5d,vec}_{a}\cdot{\boldsymbol{Z}}^{5d,fund}_{a}\cdot{\boldsymbol{Z}}^{5d,CS}_{a}\cdot\prod^{r}_{b>a}{\boldsymbol{Z}}^{5d,bif}_{a,b}\,.$
(4.15)

The sum is over a collection of 2d partitions (Young tableaux), one for each
$U(1)$ Coulomb parameter in the gauge group $G_{5d}$:

$\\{\overrightarrow{\boldsymbol{\mu}}\\}=\\{\boldsymbol{\mu}_{a,i}\\}_{a=1,\ldots,r\,;\;\;i=1,\ldots,n_{a}}\;.$
(4.16)

Every such partition describes a configuration of instantons at the fixed
point of the $U(1)_{q}\times U(1)_{t}\times U(1)^{\sum_{a}n_{a}}$ equivariant
action. In particular, the instantons sit at the origin of
$\mathbb{C}_{q}\times\mathbb{C}_{t}$.

Each of the factors in the summand is best expressed in terms of the function

${\mathcal{N}}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}(Q\,;q)\equiv\prod\limits_{k,k^{\prime}=1}^{\infty}\frac{\big{(}Q\,q^{\boldsymbol{\mu}_{a,i,k}-\boldsymbol{\mu}_{b,j,k^{\prime}}}\,t^{k^{\prime}-k+1}\,;q\big{)}_{\infty}}{\big{(}Q\,q^{\boldsymbol{\mu}_{a,i,k}-\boldsymbol{\mu}_{b,j,k^{\prime}}}\,t^{k^{\prime}-k}\,;q\big{)}_{\infty}}\,\frac{\big{(}Q\,t^{k^{\prime}-k}\,;q\big{)}_{\infty}}{\big{(}Q\,t^{k^{\prime}-k+1}\,;q\big{)}_{\infty}}\,.$
(4.17)

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

Then, for a given quiver node $a$, the factor

$\displaystyle{\boldsymbol{Z}}^{5d,vec}_{a}=\prod_{i,j=1}^{n_{a}}\left[\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{a,j}}\left(\frac{e_{a,i}}{e_{a,j}}\,;q\right)\right]^{-1}$
(4.18)

represents the contribution of a 5d vector multiplet, where each $e_{a,i}$
denotes a corresponding Coulomb parameter.

The factor

$\displaystyle{\boldsymbol{Z}}^{5d,fund}_{a}=\prod_{s=1}^{n^{F}_{a}}\prod_{i=1}^{n_{a}}\mathcal{N}_{\boldsymbol{\emptyset}\,\boldsymbol{\mu}_{a,i}}\left(\sqrt{\frac{q}{t}}\,\frac{f_{a,s}}{e_{a,i}}\,;q\right)\;.$
(4.19)

represents the contribution of a 5d fundamental hypermultiplet, in
representation $(n_{a},\overline{n^{F}_{a}})$ of $U(n_{a})\times
U(n^{F}_{a})$, where each $f_{a,s}$ denotes a corresponding mass parameter.

The factor

$\displaystyle{\boldsymbol{Z}}^{5d,CS}_{a}=\prod\limits_{i=1}^{n_{a}}\left(T_{\boldsymbol{\mu}_{a,i}}\right)^{\kappa_{CS,a}}\;.$
(4.20)

represents the contribution of the effective Chern-Simons term of level
$\kappa_{CS,a}$ on node $a$ (4.7), where
$T_{\boldsymbol{\mu}_{a,i}}=(-1)^{|\boldsymbol{\mu}_{a,i}|}\;q^{\Arrowvert\boldsymbol{\mu}_{a,i}\Arrowvert^{2}/2}\;t^{-\Arrowvert\boldsymbol{\mu}^{t}_{a,i}\Arrowvert^{2}/2}$.

Finally, the factor

$\displaystyle{\boldsymbol{Z}}^{5d,bif}_{a,b}=\prod_{i=1}^{n_{a}}\prod_{j=1}^{n_{b}}\left[\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}\left(f^{bif}_{a,b}\,\frac{e_{a,i}}{e_{b,j}}\,;q\right)\right]^{\Delta_{ab}}\;.$
(4.21)

represents the contribution of a 5d bifundamental hypermultiplet in
representation $\oplus_{b>a}\,\Delta_{ab}\,(n_{a},\overline{n_{b}})$ of
$\prod_{a,b}U(n_{a})\times U(n_{b})$, where $\Delta_{ab}$ is the incidence
matrix. We turn on corresponding masses $f^{bif}_{a,b}$, one for each link in
the quiver. In order to match to the conventions of the existing
$\mathcal{W}_{q,t}(\mathfrak{g}_{\text{o}})$-algebra literature696969Our
definition of the deformed
$\mathcal{W}_{q,t}(\mathfrak{g}_{\text{o}})$-algebra relied on the choice of a
deformed Cartan matrix of type $\mathfrak{g}_{\text{o}}$; here, we choose to
follow the conventions of [23, 58] and deform the Cartan matrices as (2.7),
which fixes all bifundamental masses $f^{bif}_{a,b}$. For a slightly different
definition of the deformed
$\mathcal{W}_{q,t}(\mathfrak{g}_{\text{o}})$-algebra and its associated Cartan
matrix, it is also possible to keep all masses arbitrary, see [27, 29]
instead. Ultimately, this is not a crucial distinction, as there is freedom to
rescale $f^{bif}_{a,b}$ away by redefining the other parameters. In the
quantum affine algebra, this is the freedom to rescale the evaluation
parameters., we set all bifundamental masses $f^{bif}_{a,b}$ to

$f^{bif}_{a,b}=\sqrt{q/t}\;.$ (4.22)

######  Remark 4.1.

In Section 3, we computed the half-index of $T^{3d}_{\mathfrak{g}_{\text{o}}}$
on $\mathbb{R}^{1,1}\times\mathbb{R}_{-}$ by enumerating all 3d fields in the
UV and specifying 1/2-BPS boundary conditions. We could have instead
considered the adiabatic approximation where the 3d supersymmetric path
integral localizes to the supersymmetric quantum mechanics on the moduli space
of vortices, analogously to the ADHM computation we did in 5d where the target
space of the quantum mechanics was the moduli space of instantons707070In
particular, the dependence of the supersymmetric ground states on the various
masses is controlled by a Berry connection [66, 296, 297, 298, 299, 240, 300].
For recent developments, see [77, 262, 301, 302]..

This approach has some advantages; for example, the equality between 5d
instanton and 3d vortex partition functions is manifest when comparing the
Witten indices of the two quantum mechanics. Moreover, the electric blocks are
more naturally computed in this language: instead of defining a 3d/1d index as
we did, one simply introduces the additional chiral and Fermi multiplets
directly to the 1d vortex quantum mechanics. For explicit illustrations, see
for instance [303, 173, 304]. However, this presentation also has drawbacks,
which is why we ultimately avoided it: deformed conformal blocks are only
known in free field formalism, which _are_ the 3d half-indices we computed,
and the natural language to best describe their analytic properties is in
terms of exceptional Dirichlet and enriched Neumann boundary conditions, which
is obscured in the quantum mechanics. Furthermore, the dictionary we found to
the representation theory of the quantum affine algebra
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ is only manifest in the 3d picture.

###  4.2 Gauge-vortex duality on the Higgs branch, and the 5d origin of the
blocks

In this section, we analyze the gauge-vortex duality between the 5d gauge
theory $T^{5d}_{\mathfrak{g}_{\text{o}}}$ on its Higgs branch and the 3d
Drinfeld gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$.

Consider first the 5d manifold
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}^{2}$ in the absence of
$\Omega$-deformation. Correspondingly, we set $q,t\rightarrow 1$, which
further turns off all bifundamental masses
$f^{bif}_{a,b}=\sqrt{q/t}\rightarrow 1$. At the root of the Higgs branch, all
Coulomb moduli are frozen to (a subset of) the hypermultiplet masses; this
process will be described in detail in the next Section.

We move onto the Higgs branch proper by turning on the real F.I. terms. We are
specifically interested in the existence of 1/2-BPS vortices carrying
$N_{a,i}$ units of charge:

$N_{a,i}=\frac{1}{2\pi}\int_{\mathbb{C}}F_{a,i}\;,$ (4.23)

where $F_{a,i}$ is the field strength associated to the $i$-th $U(1)$ gauge
field, lying in the maximal torus of $U(n_{a})$ on quiver node $a$, and the
integral is over the complex line transverse to the vortices.

Typically, such a nonzero vortex charge results in additional singular surface
operator insertions in spacetime, but in the presence of an
$\Omega$-background, the effect is much milder. To see this, let us restore
_half_ of the $\Omega$-background, meaning we keep $q\rightarrow 1$, but
consider an $\Omega$-background on $\mathbb{C}_{t}$, with $t$ arbitrary. In
this background, the theory has an effective description as a 3d
$\mathcal{N}=2$ theory on $S^{1}_{\mathcal{C}^{\prime}}$ with an infinite
number of massive modes set by the scale $t$ [81]. The 5d Coulomb moduli
always appear in the combination

$e_{a,i}+t^{X_{2}\,D_{a,i,X_{2}}}\;,$ (4.24)

where the operator $D_{a,i,X_{2}}=\partial_{X_{2}}+A_{a,i,X_{2}}$ is the
covariant derivative on $\mathbb{C}_{t}$, with complex coordinate $X_{2}$
[94].

Because of the $\Omega$-background, turning on $N_{a,i}$ units of vortex
charge (4.23) will no longer introduce singular surface operators: the only
effect will be to shift the values of the 4d Coulomb moduli as

$e_{a,i}\longrightarrow e_{a,i}\,t^{N_{a,i}}\;.$ (4.25)

This follows from setting the gauge field to $A_{a,i,X_{2}}=N_{a,i}/X_{2}$ in
(4.24). In the end, even though we started on the Higgs branch where vortex
solutions are well-defined, the $\Omega$-background effectively “puts us back”
onto the Coulomb branch, but only on a lattice of spacing set by positive
powers of $t$717171In the large $N$ limit, with $t\rightarrow 1$ and
$N_{a,i}\rightarrow\infty$, we can in fact probe the entire continuous Coulomb
branch instead of just a lattice. For related perspectives, see [27, 305, 306,
307]..

As we have seen, at the root of the Higgs branch, all Coulomb moduli in
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ are frozen to masses according to the
coweight decomposition (4.42); this remains true in the $\Omega$-background,
and we write

$e_{a,i}=f_{b,s}\,\sqrt{t/q}^{1+\\#_{a,i}}\;,$ (4.26)

for some $b\in\\{1,\ldots,r\\}$ and $s\in\\{1,\ldots,n^{F}_{b}\\}$. The factor
$\sqrt{t/q}^{1+\\#_{a,i}}$ is a refinement due to the $\Omega$-background. In
particular, because $T^{5d}_{\mathfrak{g}_{\text{o}}}$ is a quiver theory, the
bifundamental masses $\sqrt{q/t}$ between adjacent nodes now appear explicitly
in the above Higgsing formula: in our conventions, $\\#_{a,i}$ is a non-
negative integer which records the length of the “string” from node $a$ where
the Coulomb modulus $e_{a,i}$ is located, to node $b$ where the hypermultiplet
mass $f_{b,s}$ is located.

After turning on vortices of charge $N_{a,i}$, the Coulomb parameters get
further shifted as (4.25):

$e_{a,i}=f_{b,s}\,\sqrt{t/q}^{\;1+\\#_{a,i}}\,t^{N_{a,i}}\;.$ (4.27)

As we explained, this should be interpreted as the theory being taken away
from the root of its Higgs branch and pushed back onto its Coulomb branch, but
only on an integer-valued lattice in powers of $t$, due to the transverse
$\Omega$-background.

The equality of the 5d instanton partition function and 3d half-index is
manifest in the presence of the $\Omega$-background. To see this, one relies
on the following property of the function (4.17): if $N$ is a positive
integer, then

$\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}\left(\sqrt{q^{2}/t^{2}}\;t^{-N}\,;q\right)=0\;\;\text{unless}\;\;l\left(\boldsymbol{\mu}_{b,j}\right)\leq
l\left(\boldsymbol{\mu}_{a,i}\right)+N\;,$ (4.28)

where $l\left(\boldsymbol{\mu}_{b,j}\right)$ is the number of rows in the
partition $\boldsymbol{\mu}_{b,j}$, labeling the equivariant Coulomb modulus
$e_{b,j}$. After specializing all the Coulomb moduli as (4.27), the
fundamental matter factor ${\boldsymbol{Z}}^{5d,fund}_{b}$ (4.19) in the
summand ${\mathcal{Z}}_{inst}$ will necessarily contain a factor

$N_{\boldsymbol{\emptyset}\,\boldsymbol{\mu}_{b,j}}\left(\sqrt{q^{2}/t^{2}}\;t^{-N_{b,j}};q\right)\;,$
(4.29)

for some positive integer $N_{b,j}$. It follows that the entire summand
vanishes unless the partition $\boldsymbol{\mu}_{b,j}$ has no more than
$N_{b,j}$ rows. Note there is no restriction on the number of columns. All the
remaining partitions $\boldsymbol{\mu}_{b^{\prime},j^{\prime}}$ are likewise
truncated to have a finite number of rows, either directly from
${\boldsymbol{Z}}^{5d,fund}$ as above, or from previously truncated partitions
via the bifundamental factors ${\boldsymbol{Z}}^{5d,bif}$.

Whenever two partitions $\boldsymbol{\mu}_{a,i}$ and $\boldsymbol{\mu}_{b,j}$
have no more than $N_{a,i}$ and $N_{b,j}$ rows respectively, a further
factorization occurs,

$\displaystyle\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}(Q\,;q)$
$\displaystyle=\prod_{k=1}^{N_{a,i}}\prod_{k^{\prime}=1}^{N_{b,j}}\frac{\big{(}Q\,q^{\boldsymbol{\mu}_{a,i,k}-\boldsymbol{\mu}_{b,j,k^{\prime}}}\,t^{k^{\prime}-k+1}\,;q\big{)}_{\infty}}{\big{(}Q\,q^{\boldsymbol{\mu}_{a,i,k}-\boldsymbol{\mu}_{b,j,k^{\prime}}}\,t^{k^{\prime}-k}\,;q\big{)}_{\infty}}\,\frac{\big{(}Q\,t^{k^{\prime}-k}\,;q\big{)}_{\infty}}{\big{(}Q\,t^{k^{\prime}-k+1}\,;q\big{)}_{\infty}}$
$\displaystyle\;\;\times\mathcal{N}_{\boldsymbol{\mu}_{a,i}\,\boldsymbol{\emptyset}}(Q\,t^{N_{b,j}};q)\;\mathcal{N}_{\boldsymbol{\emptyset}\boldsymbol{\mu}_{b,j}}(Q\,t^{-{N_{a,i}}};q)\;.$
(4.30)

Since all partitions are truncated, this factorization happens for all
functions $\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}$ in the
summand ${\mathcal{Z}}_{inst}$, and it simplifies drastically, to find:

######  Proposition 4.1.

The truncated 5d partition function is a 3d half-index with exceptional
Dirichlet boundary conditions, for a choice of polarization
$\\{+,\ldots,+\\}$, in a vacuum labeled by truncated Young tableaux.

We can reverse this argument to learn about the combinatorial nature of the
contours for the 3d half-index with Neumann boundary conditions. To see how
that happens, consider again

$\displaystyle{\mathcal{Z}}(T^{3d}_{{\mathfrak{g}_{\text{o}}},{\bf
N}})=\frac{1}{|W_{G^{3d}}|}\prod_{a=1}^{r}\prod_{i=1}^{N_{a}}\oint_{C}\frac{dy_{a,i}}{y_{a,i}}\,{z_{a}}^{\frac{\ln(y_{a,i})}{\ln(q)}}\cdot
I_{a}^{3d,vec}\cdot I_{a}^{3d,flavor}\cdot\prod_{b>a}I_{a,b}^{3d,bif}\;.$
(4.31)

In view of the above analysis, we know that the vacua are all labeled by Young
diagrams $\\{{\boldsymbol{\mu}}_{a,i}\\}$ with no more than $N_{a,i}$ rows.
Then, let the contours enclose poles that are precisely of this form. Assuming
$|q|,|t|<1$, these poles are at

$y_{a,i,k}=x_{d,s}\,q^{-\boldsymbol{\mu}_{a,i,k}}\,t^{k-N_{a,i}}\;,\qquad\;\;k=1,\ldots,N_{a,i}\;,$
(4.32)

with $a=1,\ldots,\text{rk}(\mathfrak{g})$. The notation $y_{a,i,k}$ for the 3d
Coulomb moduli reflects that original integration variables $y_{a,i}$ split
after choosing the contours:

$\prod_{a=1}^{r}\prod_{i=1}^{N_{a}}y_{a,i}\longrightarrow\prod_{a=1}^{r}\prod_{i=1}^{n_{a}}\prod_{k=1}^{N_{a,i}}y_{a,i,k}\;.$
(4.33)

Correspondingly, the 3d gauge symmetry group breaks as

$G^{3d}=\prod_{a=1}^{r}U(N_{a})\qquad\longrightarrow\qquad\prod_{a=1}^{r}\prod_{i=1}^{n_{a}}U(N_{a,i})\;,$
(4.34)

where $N_{a}=\sum_{i=1}^{n_{a}}N_{a,i}$.

By gauge-vortex duality, the right-hand side can be reinterpreted purely in
5-dimensional terms: the integers $n_{a}$ are the ranks of the 5d gauge groups
$U(n_{a})$, and the ranks of 3d gauge groups $N_{a,i}$ are the vortex charges
in the 5d theory.

The 3d half-index evaluated via residues at (4.32) yields

${\mathcal{Z}}(T^{3d}_{{\mathfrak{g}_{\text{o}}},{\bf
N}})=\sum_{\\{\overrightarrow{\boldsymbol{\mu}}\\}}\text{res}_{\overrightarrow{\boldsymbol{\mu}}}\left[I^{3d}(y)\right]\;.$
(4.35)

Namely, both the 3d and 5d partition functions (4.15) are sums over the same
set of truncated partitions. Both sums are expansions in the parameters
$z_{a}$, which encode the gauge couplings in 5d and the F.I. parameters in 3d.
The expressions are analytic in chamber $\mathfrak{C}_{C}$, by construction.

In order to further compare the summands, the above residue as we have written
it here actually yields an infinite answer, so it proves useful to first
normalize it by the trivial residue contribution, to obtain a finite answer:

${\mathcal{Z}}(T^{3d}_{{\mathfrak{g}_{\text{o}}},{\bf
N}})=c_{3d}\sum_{\\{\overrightarrow{\boldsymbol{\mu}}\\}}\left[\frac{I^{3d}(y_{\\{\overrightarrow{\boldsymbol{\mu}}\\}})}{I^{3d}(y_{\\{\boldsymbol{\boldsymbol{\emptyset}}\\}})}\right].$
(4.36)

Here, the “constant” $c_{3d}$ stands for the residue at an empty partition,

$c_{3d}\equiv\text{res}_{\boldsymbol{\emptyset}}I^{3d}(y)\;.$ (4.37)

Next, the 5d vector multiplet contributions (4.18) become

$\displaystyle{\boldsymbol{Z}}^{5d,vec}_{a}=\frac{I_{a}^{3d,vec}(y_{\\{\boldsymbol{\mu}_{a}\\}})}{I_{a}^{3d,vec}(y_{\\{\boldsymbol{\emptyset}\\}})}\cdot
V^{vec}\;,$ (4.38)

where the right-hand side is the contribution of the 3d vector multiplets
(3.27), and leftover factors we denote as $V^{vec}$.

The 5d bifundamental hypermultiplet factors (4.21) become

$\displaystyle{\boldsymbol{Z}}^{5d,bif}_{a,b}=\frac{I^{3d,bif}_{a,b}(y_{\\{\boldsymbol{\mu}_{a}\\}},y_{\\{\boldsymbol{\mu}_{b}\\}})}{I^{3d,bif}_{a,b}(y_{\\{\boldsymbol{\emptyset}\\}},y_{\\{\boldsymbol{\emptyset}\\}})}\cdot
V^{bif}\;,$ (4.39)

where the right-hand side is the contribution of the 3d bifundamental
hypermuliplets (3.31), and leftover factors we denote as $V^{bif}$.

Collecting the above leftover factors, along with the fundamental
hypermultiplets and the effective Chern-Simons contributions, we find after
some algebra:

$\displaystyle{\boldsymbol{Z}}^{5d,fund}_{a}\cdot{\boldsymbol{Z}}^{5d,CS}_{a}\cdot
V^{vec}\cdot
V^{bif}=\frac{I^{3d,flavor}_{a}(y_{\\{\boldsymbol{\mu}_{a}\\}},\\{x_{d,s}\\})}{I^{3d,flavor}_{a}(y_{\\{\boldsymbol{\emptyset}\\}},\\{x_{d,s}\\})}\;.$
(4.40)

The right-hand side is the contribution of the chiral matter content of the 3d
Drinfeld quiver, (3.40). By definition, it contains an equal number of
fundamental and anti-fundamental chiral multiplets contributions. In
performing this last identification, we implicitly relabeled all 5d
hypermultiplet masses as 3d chiral multiplet masses, $f_{a,s}\rightarrow
x^{-1}_{d,s}$727272It may not be immediately obvious that such a relabeling is
a priori possible, but the decomposition we write in (4.42) ensures that we
have the same number of masses in 5d and in 3d, so the map $f_{a,s}\rightarrow
x^{-1}_{d,s}$ is well-defined..

After specializing the 5d Coulomb parameters, the sums (4.15) and (4.36) are
indexed by the same set of truncated partitions, and the summands match term-
by-term, up to the proportionality constant $c_{3d}$; we conclude once again
that the indices agree, up to trivial residue contribution737373A slightly
more complete treatment would further take into account the perturbative
contributions to the partition functions, which we have ignored throughout.:

$\displaystyle{\mathcal{Z}}(T^{3d}_{\mathfrak{g}_{\text{o}},\bf
D_{EX}})=c_{3d}\cdot{\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})_{e_{a,i}\propto\;x^{-1}_{d,s}\,t^{N_{a,i}}}\;.$
(4.41)

In the $q$-conformal block language, this gives a 5-dimensional description of
the magnetic $z$-solutions for the $\mathcal{W}_{q,t}(\mathfrak{g})$-algebra
blocks.

###  4.3 From classical to quantum groups via the Higgs mechanism

Nothing in our treatment of the 5-dimensional gauge theory
$T^{5d}_{\mathfrak{g}}$ so far hints at the relevance of a quantum affine
symmetry algebra $U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$. We will now argue
that this symmetry arises on the Higgs branch, in the presence of the
$\Omega$-background.

First, consider a trivial spacetime without $\Omega$-background, and define a
set containing all hypermultiplet masses $f_{a,s}$ and Coulomb moduli
$e_{b,i}$ in $T^{5d}_{\mathfrak{g}}$; at a generic point on the Coulomb
branch, there is no natural notion of ordering on this set, but in a Higgs
vacuum, it is possible to define a partial order $``\geq"$ on it. Concretely,
each hypermultiplet mass will be a maximal element of this ordered set, and
the Coulomb moduli are ordered according to how we “freeze” them to the
masses. Namely, given a mass $f_{a,s}$, we freeze a Coulomb modulus by
declaring $f_{a,s}\geq e_{a,i}$ as set elements, where the index
$a\in\\{1,\ldots,\text{rk}(\mathfrak{g})\\}$ is the _same_ on both sides. The
remaining Coulomb moduli are ordered by further freezing. All in all, this
procedure results in $n^{F}$ partially ordered sets
${\mathcal{K}}^{\geq}_{a,s}$, each starting with a mass $f_{a,s}$ as a maximal
element (note that the singleton set $\\{f_{a^{\prime},s^{\prime}}\\}^{\geq}$
is allowed). The superscript notation $\\{\ldots\\}^{\geq}$ is here to
emphasize that the elements are ordered with respect to $``\geq"$. Different
choices of freezing patterns result in different orderings and therefore
different sets ${\mathcal{K}}^{\geq}_{a,s}$, which describe distinct Higgs
vacua.

A key observation is that the partial ordering we defined on the set of masses
and frozen Coulomb moduli is well-known in the theory of simple Lie algebras:
it coincides with the partial ordering on the weight lattice of
${}^{L}\mathfrak{g}$. Equivalently, working with the coweight lattice of
$\mathfrak{g}$, two coweights $\lambda^{\vee}$ and $\lambda^{\prime\vee}$ are
ordered as $\lambda^{\vee}\geq\lambda^{\prime\vee}$ if
$\lambda^{\vee}-\lambda^{\prime\vee}$ is a linear combination of positive
coroots with non-negative coefficients. This suggests a map from 5d quiver
data to Lie algebras, by identifying each mass $f_{a,s}$ as a dominant weight
in the $a$-th fundamental representation, and each Coulomb modulus $e_{a,i}$
as the $a$-th positive simple root of $\mathfrak{g}$.

So far, all freezing patterns as we have described them are realized on the
root of the Higgs branch, where the F.I. terms are zero, but the existence of
finite tension BPS vortices imposes further constraints: these solutions
appear after the real F.I. terms are turned on, and this typically breaks
supersymmetry when the gauge groups are nonabelian. The Coulomb freezing
patterns which will preserve supersymmetry are found by solving the F-term and
D-term moment map equations for the quiver gauge theory. For example, if
$\mathfrak{g}=A_{1}$, a necessary condition for unbroken supersymmetry with
nonzero F.I. terms is that two Coulomb moduli $e_{1,i}$ and
$e_{1,i^{\prime}\neq i}$ _cannot_ be frozen to one and the same mass
$f_{1,s}$747474In string theory, the freezing of $e_{1,i}$ and
$e_{1,i^{\prime}\neq i}$ to the same mass $f_{1,s}$ via branes is an example
of undesirable “s-configuration,” as defined by Hanany-Witten [271]. Similar
constraints exist for the higher rank Lie algebras, though they take more
complicated forms. In particular, we are not aware of a field theory
characterization of supersymmetry-preserving patterns for general simple Lie
algebras $\mathfrak{g}$.

We will not aim to classify all supersymmetry-preserving Coulomb freezing
patterns in this work. Instead, we will content ourselves with sufficient
conditions for their existence, and leave the important question of whether
they are also necessary conditions to future work. One such sufficient
condition is that the patterns should be labeled by coweights sitting inside
fundamental representations757575This condition will also be justified from
the string theory engineering of the 5d gauge theory in Section 6, via the
McKay correspondence..

Then, we will henceforth label the masses $f_{a,s}$ not just by any dominant
coweight, but by the fundamental coweight $w^{\vee}_{a}$ (with multiplicity
$s$), and all $n_{a}$ Coulomb parameters in $U(n_{a})$ by one and the same
positive simple coroot $\alpha^{\vee}_{a}$. A frozen Coulomb modulus
$f_{a,s}\geq e_{a,i}$ is then labeled by the coweight
$w^{\vee}_{a}-\alpha^{\vee}_{a}$, consistent with the definition of the
partial ordering on $\mathfrak{g}$ and the fact that the coweight
$w^{\vee}_{a}$ is indeed higher than the coweight
$w^{\vee}_{a}-\alpha^{\vee}_{a}$. A Coulomb parameter $e_{a\pm 1,i^{\prime}}$
on a neighboring node can in turn be frozen to the same mass $f_{a,s}$, after
first freezing it to $e_{a,i}$; the operation will now be labeled by the
coweight $w^{\vee}_{a}-\alpha^{\vee}_{a}-\alpha^{\vee}_{a\pm 1}$. Proceeding
in this way for all Coulomb moduli, we end up with a partially ordered set of
$\mathfrak{g}$-coweights $\\{\vec{\underline{\lambda^{\vee}}}\\}$, all written
as a fundamental coweight minus a finite sum of positive simple coroots:

$\underline{\lambda^{\vee}_{s}}=w^{\vee}_{a}-\sum_{b=1}^{\text{rk}(\mathfrak{g})}h_{b,s}\,\alpha^{\vee}_{b}\;,\qquad
s=1,\ldots,n^{F}\;,$ (4.42)

for some $a\in\\{1,\ldots,\text{rk}(\mathfrak{g})\\}$, and $h_{b,s}$ non-
negative integers. Such a decomposition always exists, but is not unique in
general. The physical requirement of keeping supersymmetry unbroken after
turning on the F.I. terms will put additional constraints on the allowed
integers $h_{b}$. Note that consistency of our notations implies

$\sum_{s=1}^{n^{F}}h_{b,s}=n_{b}\;,$ (4.43)

with $n_{b}$ the rank of the gauge group $U(n_{b})$. In term of
$\mathfrak{g}$-coweights, the asymptotic conformality constraint (4.5) takes
an elegant form,

$\sum_{s=1}^{n^{F}}\underline{\lambda^{\vee}_{s}}=0\;,$ (4.44)

which follows at once from (4.42).

So far, the relevant quiver algebra has been $\mathfrak{g}$ throughout. We
will now see that if the above analysis is repeated in an $\Omega$-background,
the relevant representation theory is that of the quantum affine algebra of
$\mathfrak{g}$ instead. The reasoning is as before, meaning we can still
define a partial ordering on the set of 5d hypermultiplet masses $f_{a,s}$ and
Coulomb moduli $e_{a,i}$, where the masses $f_{a,s}$ are once again the
maximal elements. However, the freezing of Coulomb moduli now depends
explicitly on the $\Omega$-background spacetime equivariant parameters $q$ and
$t$, following (4.27). In order for our partial ordering to be compatible with
the Higgsing prescription (4.27), it is not hard to show that the following
must hold for two consecutive elements in the partially ordered set:

$\displaystyle f_{a,s}\geq e_{a,i}$ whenever $\displaystyle
e_{a,i}=f_{a,s}\;\hbar^{-1/2}\,t^{N}\;,$ $\displaystyle e_{a,i}\geq
e_{a+1,i^{\prime}}$ whenever $\displaystyle
e_{a+1,i^{\prime}}=e_{a,i}\;\hbar^{-1/2}\,t^{N^{\prime}}\;,$ $\displaystyle
e_{a,i}\geq e_{a-1,i^{\prime}}$ whenever $\displaystyle
e_{a-1,i^{\prime}}=e_{a,i}\;\hbar^{+1/2}\,t^{-N^{\prime\prime}}\;,$ (4.45)

with $N,N^{\prime},N^{\prime\prime}>0$, and $\hbar\equiv q/t$. In particular,
the dependence on $\hbar$ between neighboring quiver nodes encodes the
contribution of the bifundamental masses $f^{bif}_{a,b}=\hbar^{1/2}$. We are
once again only interested in freezing configurations which do not break
supersymmetry with nonzero F.I. terms. The frozen configurations do not depend
on whether or not the $\Omega$-background is turned on, so there are still
$n^{F}$ sequences ${\mathcal{K}}^{\geq}_{a,s}$ of ordered elements satisfying
(4.3); each sequence ${\mathcal{K}}^{\geq}_{a,s}$ starts with a mass $f_{a,s}$
as a maximal element. What the $\Omega$-background does, however, is give a
dual description of the fully Higgsed 5d theory as the vortex theory
$T^{3d}_{\mathfrak{g}}$. Applying the freezing algorithm to each 5d Coulomb
modulus in the 5d instanton partition function results in (4.40), where the
right-hand side is the chiral matter contribution to the 3d half-index,
encoding the Drinfeld polynomial data of the quantum affine weights
$\\{{\lambda}\\}$. This immediately implies:

__

Figure 9: The construction from 5 dimensions of the weight
$\lambda^{\prime}=Y_{3}Y^{-1}_{3}$ found in the second fundamental
representation $\widehat{V_{2}}$ of $U_{\hbar}(\widehat{D_{4}})$. The maximal
elements are the masses $f_{a,s}$, and the freezing of Coulomb moduli
$e_{a,i}$ is featured explicitly, following the Higgsing rules (4.3). See
(2.4) for the realization of the weight as a $q$-primary insertion, and (3.2)
for the realization as the chiral matter of a 3d Drinfeld quiver on the Higgs
branch of the 5d one.

######  Proposition 4.2.

Let $V_{a,\lambda_{s}}$ be a weight space appearing in the eigenspace
decomposition of a $U_{\hbar}(\widehat{\mathfrak{g}_{\text{o}}})$ fundamental
representation $\widehat{V_{a}}=\bigoplus_{\lambda_{s}}\,V_{a,\lambda_{s}}$ as
$U_{\hbar}(\mathfrak{g}_{\text{o}})$-modules. Let
$\\{{\mathcal{K}}^{\geq}_{a,s}\\}$ be the set of all sequences of ordered
elements with a 5d mass $f_{a,s}$ as a maximal element. Then the map
$\\{{\mathcal{K}}^{\geq}_{a,s}\\}\rightarrow V_{a,\lambda_{s}}$ is surjective.

We checked that this Proposition holds for all weight spaces in all
fundamental representations of $U_{\hbar}(\widehat{\mathfrak{g}_{\text{o}}})$,
for $\mathfrak{g}_{\text{o}}=A_{r\leq 4}$, $\;\mathfrak{g}_{\text{o}}=D_{r\leq
6}$, and $\mathfrak{g}_{\text{o}}=E_{6}$, meaning we identified the Drinfeld
polynomials of all weights in those cases. For $\mathfrak{g}_{\text{o}}=E_{7}$
and $\mathfrak{g}_{\text{o}}=E_{8}$, we only performed a partial check for
some of the weights. As we mentioned, it would be important to clarify whether
the map is also injective. The non simply-laced version of this Proposition is
written in 7.1.

###  4.4 Comparison with the Frenkel-Mukhin algorithm

We would like to compare our proposal to build weight spaces of quantum affine
fundamental representations with the one used in the construction of the
Frenkel-Reshetikhin $\hbar$-characters [61, 99]. By that, we mean a Frenkel-
Mukhin type of algorithm alluded to in (2.37) and (2.34), where we start with
a “fundamental weight” operator

$\Lambda_{a}(x)\equiv\;:\exp\left(\,\sum_{k\neq 0}{w_{a}[k]\over(q^{k\over
2}-q^{-\,{k\over 2}})(t^{k\over 2}-t^{-\,{k\over 2}})}\,t^{-\,{k\over
2}}\,x^{k}\right):\;,\qquad\;\;\;\;a\in\\{1,\ldots,\text{rk}(\mathfrak{g})\\}\;,$
(4.46)

and then proceed to build all the weight spaces in the fundamental
representation by successive applications of positive “simple root
operators”767676In the $\hbar$-character literature, the role of the
$\Lambda_{a}$ operators is played by the $\mathcal{Y}_{a}$-operators (2.23),
and the role of the $E_{a}$ operators is played by operators commonly called
$A_{a}$ [23, 99, 58].

$E_{a}(x)\equiv\;:\exp\left(\sum_{k\neq 0}{\alpha_{a}[k]\over(q^{k\over
2}-q^{-k\over 2})(t^{k\over 2}-t^{-\,{k\over 2}})}\,t^{-\,{k\over
2}}\,x^{k}\right):\;,\qquad\;\;\;\;a\in\\{1,\ldots,\text{rk}(\mathfrak{g})\\}\;.$
(4.47)

__

Figure 10: Example of sequences in a 5d quiver gauge theory of type $A_{2}$.
The maximal elements are the masses $f_{a,s}$, and the freezing of Coulomb
moduli $e_{a,i}$ is featured explicitly. After Higgsing, the resulting quantum
affine weights are grouped by color to denote $q$-primary operator insertions
in the $\mathcal{W}_{q,t}(A_{2})$ conformal block. Here, we have a first
$q$-primary labeled by a set of 3 weights
$\\{\lambda\\}_{1}=\\{Y_{1},\,Y_{2}Y^{-1}_{1},\,Y^{-1}_{2}\\}$, and a second
$q$-primary labeled by a set of 2 weights
$\\{\lambda\\}_{1}=\\{Y_{2},\,Y^{-1}_{2}\\}$. In this example, all the weights
happen to be realized as in the Frenkel-Mukhin algorithm, as suggested by the
picture.

In order to compare to our 5d gauge theory Higgsing construction, consider a
sequence ${\mathcal{K}}^{\geq}_{a,s}=\\{f_{a,s},\ldots\\}_{\geq}$, and
identify each mass $f_{a,s}$ as a highest weight operator $\Lambda_{a}$, and
each frozen Coulomb modulus $e_{a,i}$ in the sequence as the operator $E_{a}$,
with some $\hbar$-shifts of the argument. In the Frenkel-Mukhin algorithm, the
$a$-th fundamental representation would then be built as

$:\Lambda_{a}(x):\;\;\;\rightarrow\;\;\;:\Lambda_{a}(x)\,E_{a}(x\,\hbar^{-1/2}):\;\;\;\rightarrow\;\;\;\ldots$
(4.48)

where the “$\ldots$” stands for all the remaining weights. We claim that this
procedure disagrees in general with the Higgsing one from 5d gauge theory. For
the simplest counterexample, consider the weight
$:\Lambda^{-1}_{2}(\hbar^{-3}x):$ in $\mathcal{W}_{q,t}(A_{3})$, constructed
via a sequence

$\displaystyle:\Lambda_{2}(x):\;\;\;$
$\displaystyle\rightarrow\;\;\;:\Lambda_{2}(x)\,E_{2}(x\,\hbar^{-1/2}):$
$\displaystyle\rightarrow\;\;\;:\Lambda_{2}(x)\,E_{2}(x\,\hbar^{-1/2})\,E_{1}(x\,\hbar^{-1/2}):$
$\displaystyle\rightarrow\;\;\;:\Lambda_{2}(x)\,E_{2}(x\,\hbar^{-1/2})\,E_{1}(x\,\hbar^{-1/2})\,E_{3}(x\,\hbar^{-1/2}):$
$\displaystyle\rightarrow\;\;\;:\Lambda_{2}(x)\,E_{2}(x\,\hbar^{-1/2})\,E_{1}(x\,\hbar^{-1/2})\,E_{3}(x\,\hbar^{-1/2})\,E_{2}(x\,\hbar^{-1/2}):\;,$
(4.49)

in the Frenkel-Mukhin algorithm. The last line is the weight
$:\Lambda^{-1}_{2}(\hbar^{-3}x):$. By inspection of Figure 11, the “tree” to
get to that lowest weight is not the same in gauge theory. This is because the
truncation algorithm (4.3) only acts on nearest neighbors $a\pm 1$ for a
Coulomb modulus on node $a$. It is remarkable that both procedures are
distinct and yet yield the correct weight.

__

Figure 11: This Higgsing pattern engineers the quantum affine weights $Y_{2}$
and $Y^{-1}_{2}$ (and the associated $q$-primary operator of
$\mathcal{W}_{q,t}(A_{3})$ is labeled by
$\\{\lambda\\}=\\{Y_{2},\,Y^{-1}_{2}\\}$). In this example, the construction
of the weight $Y^{-1}_{2}$ does _not_ follow the Frenkel-Mukhin algorithm, as
is clear from the truncation path.

##  5 On the enumerative geometry of tame ramification

The aim of the following Section is to recast our results in the language of
enumerative geometry and the counting of quasimaps. The standard reference is
the set of notes [103] by Okounkov. For recent applications in our context,
see [105, 77, 106, 107, 108] in Physics and [308, 111, 109, 110, 309] in
Mathematics. Throughout this section, the underlying Lie algebra will be
simply-laced $\mathfrak{g}=\mathfrak{g}_{\text{o}}$, of rank $r$. Before we
introduce ramification, it will prove worthwhile to briefly recall the
analysis in the unramified Langlands correspondence [53].

###  5.1 Review: Vertex functions for the unramified Langlands correspondence

In the unramified setup, the 3d quiver gauge theory of interest has unbroken
$\mathcal{N}=4$ supersymmetry. This implies that its Higgs branch $X$ is a
Nakajima quiver variety, meaning a hyper-Kähler quotient

$X=T^{*}\text{Rep}({\cal Q})/\\!\\!/\\!\\!/\\!\\!/G_{V}\;,$ (5.1)

with

$\text{Rep}({\cal Q})=\oplus_{a\rightarrow
b}\text{Hom}(V_{a},V_{b})\oplus_{a}\text{Hom}(V_{a},W_{a})\;,$ (5.2)

and “$\rightarrow$” denotes an arrow in the Dynkin diagram of
$\mathfrak{g}_{\text{o}}$777777In physics terminology, the Hom’s as written
represent multiplets for the 3d $\mathcal{N}=4$ quiver, _not_ their
decomposition as a $\mathcal{N}=2$ quiver.

$G_{V}=\prod_{a=1}^{r}GL(V_{a})\;,\qquad G_{H}=\prod_{a=1}^{r}GL(W_{a})\;.$
(5.3)

The vector space $V_{a}$ has dimension $\text{dim}(V_{a})=N_{a}$, while the
vector space $W_{a}$, also called framing, has dimension
$\text{dim}(V_{a})=N^{F}_{a}$.

We fix maximal tori $T_{V}$ and $T_{H}$ of $G_{V}$ and $G_{H}$, respectively.
The mass parameters $x_{a,d}$ for $a=1,\ldots,\text{rk}(\mathfrak{g})$ and
$i=1,\ldots,N^{F}_{a}$ are $T_{H}$-equivariant parameters valued in
$K_{T_{H}}(\text{pt})$. We choose a generic chamber $\mathfrak{C}_{H}$ by
choosing generic $x_{a,d}$’s.

The quotient (5.1) is in particular a geometric invariant theory (GIT)
quotient, which depends on a choice of stability conditions [310]. We fulfill
these stability conditions by choosing a generic chamber $\mathfrak{C}_{C}$
(generic positive F.I. terms). Crucially, the quotient is the holomorphic
symplectic reduction of a cotangent bundle, so there is a natural action of a
group

$\mathbb{C}^{\times}_{\hbar}\subset\text{Aut}(X)\;,$ (5.4)

which acts by rescaling the cotangent directions with weight $\hbar^{-1}\in
K_{\mathbb{C}^{\times}_{\hbar}}(\text{pt})$. We write

$T=\mathbb{C}^{\times}_{\hbar}\times T_{H}\;.$ (5.5)

The set of massive isolated vacua we have been calling $\\{{\bf{A}}\\}$ is the
$T$-fixed point locus of $X$, also denoted as $X^{T}$.

Mathematically, our moduli spaces of vortices are known as moduli spaces of
quasimaps QM$(X)$787878The maps we consider are all stable, meaning they are
allowed unstable values in the GIT quotient, but only at finitely many points
of the domain $\mathbb{P}^{1}$. [104, 311, 312], of the form

$f:\mathbb{P}^{1}\rightarrow X$ (5.6)

In practice, the map $f$ induces pull-back bundles $f^{*}V_{a}$ and
$f^{*}W_{a}$ on $\mathbb{P}^{1}$ (with $V_{a}$ and $W_{a}$ as in (5.2)) and
one works with this moduli space of bundles.

On the domain, $\mathbb{P}^{1}$, we fix two marked points $\\{0\\}$ and
$\\{\infty\\}$, and consider the subset of quasimaps
$QM_{\text{nonsing},\infty}(X)$ for which $f(\\{\infty\\})$ is stable.
Physically, we are asking for the gauge field to be flat at $\\{\infty\\}$.
This space is not compact, because point-like vortices have nontrivial first
Chern class, by definition, and so can be placed as close to $\\{\infty\\}$ as
desired without ever being allowed there. This IR divergence was cured in
Section 3 by introducing an $\Omega$-background. In the quasimap terminology,
we let

$\mathbb{C}^{\times}_{q}=\text{Aut}(\mathbb{P}^{1},\\{0\\},\\{\infty\\})\;$
(5.7)

rescale $\mathbb{P}^{1}$ with weight $q\in
K_{\mathbb{C}^{\times}_{q}}(\text{pt})$. Quasimaps
$QM_{\text{nonsing},\infty}(X)$ fixed under $\mathbb{C}^{\times}_{q}$ can only
have singularities at the fixed point $\\{0\\}$ (since $\\{\infty\\}$ is by
definition non-singular), making the moduli space compact. Additionally, all
matter fields should a approach an isolated vacuum $\\{{\bf{A}}\\}$ at
infinity. Then, we are restricting ourselves to a subset of
$QM_{\text{nonsing},\infty}(X)$ which sends $\\{\infty\\}$ to a fixed point of
$X^{T}$.

The vertex function ${\bf{V}}$ is a generating function of equivariant Euler
characteristics of the symmetrized virtual structure sheaf over the moduli
space of quasimaps, graded by the degree (the vortex number). The counting
parameter $z$ is also called Kähler parameter (the F.I. parameter). For
$\text{rk}(\mathfrak{g})=r$, a degree $k$ quasimap will have counting
parameter

$z^{deg}=z^{k_{1}}_{1}\,z^{k_{2}}_{2}\,\ldots\,z^{k_{r}}_{r}\;,$ (5.8)

such that $\sum_{a=1}^{r}k_{r}=k$.

In practice, vertex functions are computed via Atiyah-Bott localization using
$\mathbb{C}^{\times}_{q}$-equivariant K-theory on
$QM_{\text{nonsing},\infty}(X)$. Then, given the quiver data (5.2) and the
choice of a chamber $\mathfrak{C}_{C}$ (stability condition), the vertex
function can be expressed in some normalization797979These are perturbative
contributions, or degree 0 quasimaps, which we have ignored. Back in Section
3, we also ignored such contributions in the computation of the 3d half-index;
as we noted, they can be recovered by computing instead the 3d hemisphere
partition function on $S^{1}\times D^{2}$ via localization. For details on the
subtleties of the background and the normalization of the vertex function, see
for instance [74, 107, 106]. as the following integral:

${\bf{V}}\;=\frac{1}{|W_{G_{V}}|}\prod_{a=1}^{r}\prod_{i=1}^{N_{a}}\oint_{\Gamma}\frac{dy_{a,i}}{y_{a,i}}\,e^{\frac{\ln(z^{\\#}_{a})\ln(y_{a,i})}{\ln(q)}}\,{\mathcal{F}}^{\prime}(y_{a,i})\;\left((q-\hbar)\,T^{1/2};q\right)_{\infty}\;.$
(5.9)

Here, $\left(x;q\right)_{\infty}$ is the usual $q$-Pochhammer symbol, and
$z^{\\#}_{a}$ is a rescaling of the quasimap counting parameter $z_{a}$ in
(5.8) by factors of $\hbar$ and $q$. $T^{1/2}X\in K_{T}(X)$ stands for a
choice of polarization, which picks out half of the tangent bundle $TX$. It is
a solution in $K_{T}(X)$ of the equation

$T^{1/2}X+\hbar^{-1}\otimes\left(T^{1/2}X\right)^{\vee}=TX\;,$ (5.10)

where $(\ldots)^{\vee}$ denotes the dual space of $(\ldots)$.

The insertion ${\mathcal{F}}^{\prime}(y)$ in the integrand is chosen to be
$q$-periodic and nonsingular at the $T_{H}$-equivariant points $x_{a,d}$.
Equivalently, we write the insertion as

${\mathcal{F}}^{\prime}(y)=\frac{{\mathcal{F}}(y)}{\Theta(T^{1/2}X)}\;,$
(5.11)

with the requirement that this ratio be invariant under $y\rightarrow q\,y$.
This implies ${\mathcal{F}}(y)$ is a section of a line bundle valued in the
$T$-equivariant elliptic cohomology of $X$. The line bundle $\Theta(T^{1/2}X)$
is also called an elliptic Thom line bundle. The contour $\Gamma$ is a sum of
several cycles which enclose the $T_{H}$-equivariant parameters $x_{a,d}$ and
their $q$-shifts in the denominator of the integrand. The insertion
${\mathcal{F}}(y)$, when restricted to a class on a particular component in
$X^{T}$, reduces the contour to enclose only those equivariant parameters
$x_{a,d}$ which pick out the vacuum $\\{{\bf{A}}\\}$.

For the Nakajima variety $X$, a canonical choice of decomposition of the
tangent bundle is

$T^{1/2}X=\sum_{a}V_{a}\otimes
W^{*}_{a}+\sum_{a,b}(\Delta_{ab}-\delta_{ab})\,V_{a}\otimes V^{*}_{b}\;,$
(5.12)

where we are abusing notations by identifying the vector space $T^{1/2}X$ with
its character under the $T_{V}\times T_{H}\times\mathbb{C}^{\times}_{\hbar}$
torus action.

By inspection of the components of the integral ${\bf{V}}$, we recognize the
magnetic blocks, the free field chiral correlators of the
$\mathcal{W}_{q,t}(\mathfrak{g})$ algebra. Since many aspects of the proof are
identical in the unramified and the ramified cases, we will review it here and
highlight the differences. First, the Kähler variables $z_{a}$ are related to
the highest weight $\mu_{0}$ of the deformed Heisenberg algebra Fock space
representation as

$z_{a}=q^{\langle\mu_{0},\alpha_{a}\rangle}\;.$ (5.13)

If we choose coordinates

$V_{a}=\sum_{i=1}^{N_{a}}y_{a,i}\,\hbar^{a/2}\;,\qquad\;\;W_{a}=\sum_{d=1}^{N^{F}_{a}}x_{a,d}\,\hbar^{(a-1)/2}\;,$
(5.14)

then the contribution from $\text{Hom}(V_{a},W_{a})$ in (5.2) to
$\left((q-\hbar)\,T^{1/2};q\right)_{\infty}$ is

$\prod_{i=1}^{N_{a}}\prod_{d=1}^{N^{F}_{a}}\frac{\left(q\,y_{a,i}/\hbar^{1/2}\,x_{a,d};q\right)_{\infty}}{\left(\hbar\,y_{a,i}/\hbar^{1/2}\,x_{a,d};q\right)_{\infty}}\;.$
(5.15)

Since $\hbar=q/t$, one recognizes the product of 2-point functions

$\prod_{i=1}^{N_{a}}\prod_{d=1}^{N^{F}_{a}}\left\langle
S_{a}^{\vee}(y)\,{\overline{V}}_{b}^{\vee}(x)\right\rangle$ (5.16)

of a screening current with the $a$-th fundamental vertex operator of Frenkel
and Reshetikhin, see (2.89). Equivalently, this factor is readily identified
as the hypermultiplet contribution to the half-index of the corresponding 3d
$\mathcal{N}=4$ gauge theory, with the appropriate $\mathcal{N}=(2,2)$
boundary conditions.

Likewise, the contribution from $\text{Hom}(V_{a},V_{a})$ to
$\left((q-\hbar)\,T^{1/2};q\right)_{\infty}$ is

$\prod_{1\leq i\neq j\leq
N_{a}}\frac{\left(\hbar\,y_{a,i}/y_{a,j};q\right)_{\infty}}{\left(q\,y_{a,i}/y_{a,j};q\right)_{\infty}}\;,$
(5.17)

which reproduces the contribution (2.83),

$\prod_{1\leq i<j\leq N_{a}}\left\langle
S_{a}^{\vee}(y_{a,i})\,S_{a}^{\vee}(y_{a,j})\right\rangle\;,$ (5.18)

up to ratio of Theta functions. From the point of view of the $q$-difference
equations solved by ${\bf{V}}$, the ratio of Theta functions is an
inconsequential factor of $\hbar^{\\#}$ which can be reabsorbed in the
definition of $z_{a}$ in (5.13), hence the notation $z_{a}\rightarrow
z^{\\#}_{a}$.

Finally, the contribution from $\text{Hom}(V_{a},V_{b})$ ($b>a$) to
$\left((q-\hbar)\,T^{1/2};q\right)_{\infty}$ is

$\prod_{i=1}^{N_{a}}\prod_{j=1}^{N_{b}}\frac{\left(q\,\hbar^{a/2}\,y_{a,i}/\hbar^{b/2}\,y_{b,j};q\right)_{\infty}}{\left(\hbar\,\hbar^{a/2}\,y_{a,i}/\hbar^{b/2}\,y_{b,j};q\right)_{\infty}}\;,$
(5.19)

which reproduces the contribution (2.84),

$\prod_{i=1}^{N_{a}}\prod_{j=1}^{N_{b}}\left\langle
S_{a}^{\vee}(y_{a,i})\,S_{b}^{\vee}(y_{b,j})\right\rangle\;.$ (5.20)

Therefore, the integrands agree. As far as the integration contours are
concerned, one notes that a choice of insertion
${\mathcal{F}}^{\prime}(y_{a,i})$ in ${\bf{V}}$, and in particular of a
component supported in $X^{T}$, is in one-to-one correspondence with a choice
of screening charges for the magnetic
$\mathcal{W}_{q,t}(\mathfrak{g})$-algebra block. This completes the
identification.

Electric blocks, which are solutions to the
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ qKZ equations, are likewise vertex
functions. More precisely, the qKZ equations being matrix equations, it is
more accurate to call their solutions vector vertex functions: these are
generating functions of quasimaps with descendant insertions at
$\\{0\\}\in\mathbb{P}^{1}$. These are K-theoretic stable envelope insertions,
which are certain classes in $K_{T}(X)$ which provide the stable basis of $X$
[103].

The content of the unramified quantum $q$-Langlands isomorphism is that there
exists a certain Whittaker covector whose pairing with the vector electric
block equals the scalar magnetic $\mathcal{W}_{q,t}(\mathfrak{g})$-algebra
block. Physically, this is why electric blocks arise as 3d partition functions
on $\mathbb{C}^{\times}_{q}\times S^{1}_{\mathcal{C}^{\prime}}$ with the
origin removed, while magnetic blocks are partition functions on
$\mathbb{C}_{q}\times S^{1}_{\mathcal{C}^{\prime}}$, with the puncture at
$\\{0\\}$ closed up.

Note that both the electric and magnetic blocks so far are $z$-solutions,
since they are by construction analytic in the Kähler variables $z_{a}$ in
chamber $\mathfrak{C}_{C}$. To construct the $x$-solutions instead, analytic
in the equivariant parameters in chamber $\mathfrak{C}_{H}$, one further
inserts the corresponding elliptic stable envelope at
$\\{\infty\\}\in\mathbb{P}^{1}$ via evaluation map, that is to say an
assignment of an elliptic cohomology class for each $T$-fixed point. The
existence and uniqueness of the envelopes for this problem was proved in [55].

###  5.2 Enter tame ramification

The 3d Drinfeld quiver gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$ was
defined in Section 3.1. In particular, let

$G_{V}=\prod_{a=1}^{r}GL(V_{a})\;,$ (5.21)

with $\text{dim}(V_{a})=N_{a}$ as in the unramified case. The framing is

$\displaystyle\mathfrak{g}=A_{r}\;:\;\ \,\,\,\qquad
G_{H}=GL(W^{+}_{1})\times\left[\prod_{a=2}^{r-1}GL(W^{\pm}_{a})\right]\times
GL(W^{-}_{r})\;,$
$\displaystyle\mathfrak{g}=D_{r}\;,E_{r}\;:\;\;\;\;G_{H}=GL(W^{+}_{1})\times\left[\prod_{a=2}^{r-2}GL(W^{\pm}_{a})\right]\times
GL(W^{-}_{r-1})\times GL(W^{-}_{r})\;,$ (5.22)

where the dimensions $\text{dim}(W^{\pm}_{a})=L^{\pm}_{a}$ are defined in
terms of Drinfeld polynomial data, see Proposition 3.1. The vector spaces
$W^{\pm}_{a}$ are graded by an integer $d$ labeling subsets of quantum affine
weights, as $L^{\pm}_{a}=\sum_{d=1}^{L}L^{\pm}_{d,a}$ with
$L^{\pm}_{d,a}=\sum_{s=1}^{J_{d}}\deg(\mathcal{A}^{\pm}_{\lambda_{d,s},a})$.

We fix maximal tori $T_{V}\subset G_{V}$ and $T_{H}\subset G_{H}$. The
(complexified) mass parameters $x^{\pm}_{d,a}$ for $d=1,\ldots,L$ and
$a\in\\{1,\ldots,\text{rk}(^{L}\mathfrak{g})\\}$ are the $T_{H}$-equivariant
parameters. Equivalently, we will also use the notation $x_{d,s}$ for
$d=1,\ldots,L$ and $s=1,\ldots,J_{d}$, to emphasize that each equivariant
parameter is in one-to-one correspondence with a weight $\lambda_{d,s}$ taken
in some fundamental representation of
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$.

The Higgs branch $X$ is now only a GIT quotient

$X=\text{Rep}({\cal Q})/\\!\\!/G_{V}\;,$ (5.23)

As far as the stability condition on $X$ is concerned, we work in the same
generic chamber $\mathfrak{C}_{C}$ as before.

Because the GIT quotient $X$ is no longer a holomorphic symplectic reduction
of a cotangent bundle, at first sight, $\mathbb{C}^{\times}_{\hbar}$ (which
used to rescale the cotangent directions of such a bundle) appears to have
lost its geometric meaning in the ramified setting. However, $X$ has another
interpretation: it is also the moduli space of degree
$(N_{1},N_{2},\ldots,N_{r})$ quasimaps into

$X^{\prime}=T^{*}\text{Rep}({\cal
Q}^{\prime})/\\!\\!/\\!\\!/\\!\\!/G^{\prime}_{V}\;,$ (5.24)

with $\text{Rep}({\cal Q}^{\prime})$ defined by the 5d $\mathcal{N}=1$ quiver
diagram of Section 4:

$\text{Rep}({\cal Q}^{\prime})=\oplus_{a\rightarrow
b}\text{Hom}(V^{\prime}_{a},V^{\prime}_{b})\oplus_{a}\text{Hom}(V^{\prime}_{a},W^{\prime}_{a})\;,$
(5.25)

and

$G^{\prime}_{V}=\prod_{a=1}^{r}GL(V^{\prime}_{a})\;,\qquad
G^{\prime}_{H}=\prod_{a=1}^{r}GL(W^{\prime}_{a})\;,$ (5.26)

with dimensions $\text{dim}(V^{\prime}_{a})=n_{a}$ and
$\text{dim}(W^{\prime}_{a})=n^{F}_{a}$. Crucially, $X^{\prime}$ _is_ a
Nakajima quiver variety, so the $\mathbb{C}^{\times}_{\hbar}$ still retains a
geometric meaning in the ramified setting, as

$\mathbb{C}^{\times}_{\hbar}\subset\text{Aut}(X^{\prime})\;.$ (5.27)

This interpretation of $X$ in terms of quasimaps to $X^{\prime}$ is the
mathematical formulation of gauge-vortex duality from Section 4.2.

Therefore, we will denote as before

$T=\mathbb{C}^{\times}_{\hbar}\times T_{H}\;,$ (5.28)

and $X^{T}$ will be the $T$-fixed point locus of $X$.

In the definition of $T_{H}$, there is an equivariant parameter $x_{d,s}$ for
every quantum affine weight $\lambda_{d,s}$, with a constraint

$\sum_{s=1}^{J_{d}}\left[\deg(\mathcal{A}^{+}_{\lambda_{d,s},a})-\deg(\mathcal{A}^{-}_{\lambda_{d,s},a})\right]=0\;,\qquad
a=1,\ldots,r\;,\;\;\;d=1,\ldots,L$ (5.29)

on the degrees of their Drinfeld polynomials. This constraint translates to
$\text{dim}(W^{-}_{d,a})-\text{dim}(W^{+}_{d,a})=0$ for the dimensions of the
vector spaces. Physically, this means all fundamental chiral multiplets
labeled by “$-$” and anti-fundamental chiral multiplets labeled by “$+$” will
be grouped in pairs, so by analogy with the unramified case, it is still
possible to “erase” half the arrows in the quiver diagram of
$T^{3d}_{\mathfrak{g}_{\text{o}}}$. This intuitive argument was made precise
by Okounkov [115], see also the related concept of partial polarization
recently introduced in [108]. Explicitly, we define

$T^{\pm}X=T^{\text{vec}}X+\sum_{d}T^{\pm}_{\\{\lambda\\}_{d}}X\;,$ (5.30)

where

$T^{\text{vec}}X=\sum_{a,b}(\Delta_{ab}-\delta_{ab})\,V_{a}\otimes V^{*}_{b}$
(5.31)

is the same as in the unramified setting, and the spaces

$T^{\pm}_{\\{\lambda\\}_{d}}X=\sum_{a}V_{a}\otimes W^{\pm*}_{d,a}$ (5.32)

are unique to tame ramification. In $K_{T}(X)$, we will decompose the tangent
bundle as

$\bigg{[}T^{\text{vec}}X+\hbar^{-1}\otimes\left(T^{\text{vec}}X\right)^{\vee}\bigg{]}+\sum_{d}\bigg{[}T^{+}_{\\{\lambda\\}_{d}}X+\left(T^{-}_{\\{\lambda\\}_{d}}X\right)^{\vee}\bigg{]}=TX\;$
(5.33)

As before, we consider the moduli space of quasimaps
$QM_{\text{nonsing},\infty}(X)$ which sends $\\{\infty\\}$ to a fixed point of
$X^{T}$, and its associated vertex function ${\bf{V}}$808080For 3d
$\mathcal{N}=2$ theories and their associated GIT quotients (5.23), it is
perhaps more accurate to call ${\bf{V}}$ a “big K-theoretic $I$ function”
[313, 104, 314].. The degree $k=\sum_{a=1}^{r}k_{r}$ of the quasimaps is
called vortex number from the point of view of $X$, or instanton number from
the point of view of $X^{\prime}$. We keep the notation
$\mathbb{C}^{\times}_{q}$ for the automorphism of the $\mathbb{P}^{1}$ domain
with $\\{0\\}$ and $\\{\infty\\}$ as marked points. The vertex function takes
the form:

$\displaystyle{\bf{V}}\;=\frac{1}{|W_{G_{V}}|}\prod_{a=1}^{r}\prod_{i=1}^{N_{a}}\oint_{\Gamma}$
$\displaystyle\frac{dy_{a,i}}{y_{a,i}}\,e^{\frac{\ln(z^{\\#}_{a})\ln(y_{a,i})}{\ln(q)}}\,{\mathcal{F}}^{\prime}(y_{a,i})\cdot$
$\displaystyle\cdot\left((q-\hbar)\,T^{\text{vec}};q\right)_{\infty}\prod_{d=1}^{L}\left(T^{+}_{\\{\lambda\\}_{d}}-T^{-}_{\\{\lambda\\}_{d}};q\right)_{\infty}\;.$
(5.34)

The insertion ${\mathcal{F}}^{\prime}(y)$ in the integrand is chosen to be
$q$-periodic and nonsingular at the $T_{H}$-equivariant points $x_{d,s}$:

${\mathcal{F}}^{\prime}(y)=\frac{{\mathcal{F}}(y)}{\Theta(T^{+}X)}\;,$ (5.35)

with ${\mathcal{F}}(y)$ the section of a line bundle valued in the
$T$-equivariant elliptic cohomology of $X$. Here, $\Theta(T^{+}X)$ is the
analog of the square root of the line bundle $\Theta(TX)$ one had in the
unramified context. More precisely, it is called an attractive line bundle for
$X$ in the terminology of Okounkov [115]. The existence of this bundle in our
context has important consequences: it guarantees the existence of elliptic
stable envelopes, following the same proof as in the unramified case [55].
This is why it sensible to construct $x$-solutions to the qKZ-equations
starting from the $z$-solutions V. Having defined the Vertex function, We want
to show:

######  Proposition 5.1.

The integral ${\bf{V}}$ is a magnetic block of the
$\mathcal{W}_{q,t}(\mathfrak{g})$ algebra with $L$ $q$-primary vertex
operators insertions
$:\prod_{d=1}^{L}\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d}):$.

To prove this proposition, choose coordinates

$V_{a}=\sum_{i=1}^{N_{a}}y_{a,i}\,\hbar^{a/2}\;,\qquad\;\;\;W^{\pm}_{d,a}=\sum_{s=1}^{J_{d}}\sum_{k=1}^{\deg(\mathcal{A}^{\pm}_{\lambda_{d,s},a})}\tilde{x}_{d,s}\,\hbar^{(a-1)/2}\,\hbar^{\mathfrak{a}^{\pm}_{\lambda_{d,s},a,k}}\;,$
(5.36)

where $\hbar^{-\mathfrak{a}^{\pm}_{\lambda_{d,s},a,k}}$ are the Drinfeld roots
of the polynomials $\mathcal{A}^{\pm}_{\lambda_{d,s},a}$. The framing-
independent contribution to
$\left((q-\hbar)\,T^{\text{vec}};q\right)_{\infty}$ is the same as in the
unramified case, so we do not repeat here. The contribution to
$\prod_{d=1}^{L}\left(T^{+}_{\\{\lambda\\}_{d}}-T^{-}_{\\{\lambda\\}_{d}};q\right)_{\infty}$
is specific to tame ramification, and evaluates to

$\prod_{d=1}^{L}\prod_{i=1}^{N_{a}}\prod_{b=1}^{\text{rk}(\mathfrak{g})}\prod_{s=1}^{J_{d}}\frac{\prod_{k^{\prime}=1}^{\deg(\mathcal{A}^{+}_{\lambda_{d,s^{\prime}},b})}\left(y_{a,I}/(\hbar^{-1/2}\,\hbar^{\mathfrak{a}^{+}_{\lambda_{d,s^{\prime}},b,k^{\prime}}}\,x_{d,s});q\right)_{\infty}}{\prod_{k=1}^{\deg(\mathcal{A}^{-}_{\lambda_{d,s},b})}\left(y_{a,I}/(\hbar^{-1/2}\,\hbar^{\mathfrak{a}^{-}_{\lambda_{d,s},b,i}}\,x_{d,s});q\right)_{\infty}}\;.$
(5.37)

Since $\hbar=q/t$, one recognizes the product of two-point functions

$\prod_{d=1}^{L}\prod_{i=1}^{N_{a}}\left\langle
S_{a}^{\vee}(y_{a,i})\,\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d})\right\rangle$
(5.38)

with the screening currents (2.87).

It remains to justify the appearance of the line bundle $\Theta(T^{+}X)$ (with
$T^{+}$ defined in (5.30)) in the denominator of the
${\mathcal{F}}^{\prime}(y)$ insertion. For this purpose, the parametrization
(2.76) of the $T_{H}$-equivariant parameters $x_{d,s}$ is not particularly
illuminating. Instead, we relabel $x_{d,s}$ as $x^{\pm}_{d,a}$, with
$d=1,\ldots,L$ and $a=1,\ldots,r$, with a constraint: whenever $a$ and $b>a$
label adjacent nodes in the Dynkin diagram of $\mathfrak{g}$, the
$T_{H}$-equivariant parameters $x^{-}_{d,a}$ and $x^{+}_{d,b}$ will stand for
one and the same parameter $x^{\pm}_{d,a}$. It is not hard to show that there
is a one-to-one mapping between the labeling of parameters as $x^{\pm}_{d,a}$
and our previous labeling as $x_{d,s}$ with $s$ running over all weights in
$\\{{\lambda}\\}_{d}$. Then, define center of mass and relative
$T_{H}$-parameters as

$x^{\pm}_{d,a}=\tilde{x}_{d}\,\hbar^{{\sigma}^{\pm}_{d,a}}\;,\qquad\;\;a=1,\ldots,r\;.$
(5.39)

We proceed to choose new coordinates

$V_{a}=\sum_{i=1}^{N_{a}}y_{a,i}\,\hbar^{a/2}\;,\qquad\;\;\;W^{\pm}_{d,a}=\tilde{x}_{d}\;\hbar^{(a-1)/2}\;\hbar^{\overline{\sigma}^{\pm}_{d,a}}\;\sum_{s=1}^{J_{d}}\sum_{p=1}^{\deg(\mathcal{A}^{\pm}_{\lambda_{d,s},a})}\hbar^{\mathfrak{a}^{\pm}_{\lambda_{d,s},a,p}}\;.$
(5.40)

This makes it clear that for a given $\tilde{x}_{d}$, the poles in the
integrand _only_ depend on parameters labeled by “$-$”, while the numerators
are $q$-Pochhammer symbols which depend on “$+$” parameters instead. It
follows that the line bundle $\Theta(T^{+}X)$ will remove the numerators, and
introduce instead new “$+$”-dependent poles, in complete analogy with the
unramified case. By construction, since the vector spaces $W^{+}_{d,a}$ and
$W^{-}_{d,a}$ always appear in pairs for all $d$ and $a$, the number of new
“$+$” poles will equal the number of “$-$” poles already present; the contour
$\Gamma$ is defined as a sum of cycles enclosing the parameters
$\tilde{x}_{d}$ along with all their $q$-shifts. Ultimately, we are only
interested in specific components of $X^{T}$ supported on the vacuum
$\\{{\bf{A}}\\}$, which will reduce the contour $\Gamma$ to enclose only the
relevant poles for that vacuum. In [62, 63, 64], it was shown that for handsaw
quivers, poles are combinatorially labeled by truncated Young diagrams, with
no more than $N_{a,i}$ rows. This remains true of general Drinfeld quivers,
and the proof is the same as in those papers (see Section 4). From the point
of view of the GIT quotient $X$, this labeling of fixed points is an
unexpected result, since the integer $N_{a,i}$ only labels the number of
moduli $y_{a,i}$ in the maximal torus $T_{V_{a}}\subset GL(V_{a})$. Instead,
from the point of view of the Nakajima variety $X^{\prime}$, the integer
$N_{a,i}$ has another meaning as the first Chern class of the $i$-th $U(1)$
gauge bundle in the maximal torus $T_{V^{\prime}_{a}}\subset
GL(V^{\prime}_{a})$, where the combinatorial interpretation in terms of Young
diagrams becomes explicit [62]. This completes the proof of the Proposition.

In order to obtain the vector electric blocks, which are solutions to the
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ qKZ equations, one proceeds as in
the unramified case and considers quasimaps with descendant insertions at
$\\{0\\}\in\mathbb{P}^{1}$. Equivalently, a scalar magnetic
$\mathcal{W}_{q,t}(\mathfrak{g})$-algebra block is recovered from a vector
electric block by pairing the latter with the appropriate Whittaker covector
functional.

Both the electric and magnetic blocks are $z$-solutions, analytic in the
Kähler variables $z_{a}$ for a choice of chamber $\mathfrak{C}_{C}$.
Meanwhile, the existence of $x$-solutions, analytic in the $T_{H}$-equivariant
parameters for a choice of chamber $\mathfrak{C}_{H}$, is guaranteed from the
existence of elliptic stable envelopes in the ramified setting. As we have
seen, this follows from the decomposition of the tangent bundle $TX$ as
(5.33), and viewing $\Theta(T^{+}X)$ as an attractive line bundle. The
elliptic stable envelopes are inserted at $\\{\infty\\}\in\mathbb{P}^{1}$.

##  6 Ramification and little string theory

###  6.1 String Theory Construction

Our starting point is type IIB string theory on the background

${\mathbb{C}}^{2}_{\Gamma}\times\mathbb{C}^{3}\;.$ (6.1)

The space ${\mathbb{C}}^{2}_{\Gamma}$ is the minimal resolution by blowup of a
Du Val (or Kleinian) singularity ${\mathbb{C}}^{2}/\Gamma$, with $\Gamma$ a
discrete subgroup of $SU(2)$, labeled by an $\mathfrak{g}_{\text{o}}=ADE$ Lie
algebra of rank $r$. As is well-known, such a resolution consists of $r$
2-spheres, which intersect following the Dynkin diagram of
$\mathfrak{g}_{\text{o}}$.

We decouple gravity by taking the string coupling limit $g_{s}\rightarrow 0$,
which does not entirely trivialize the physics [41, 280]. In particular, one
obtains a 6-dimensional chiral string theory on $M_{6}$, the $(2,0)$ little
string theory of type $\mathfrak{g}_{\text{o}}$, with the same
$\mathfrak{g}_{\text{o}}=ADE$ that labeled the resolved singularity
${\mathbb{C}}^{2}_{\Gamma}$ in the full 10-dimensional type IIB. The little
string theory is not a local quantum field theory, it contains strings of
finite tension equal to the square of the string mass, $m_{s}^{2}$, again
inherited from the type IIB fundamental strings.

For a gentle introduction to the elementary properties of little strings, the
standard reviews are [315, 316]. For a precise account of how the Wightman
axioms of quantum field theory are modified in little strings, see [317].

In the regime where the string length is taken to be very small, or
equivalently $m_{s}\rightarrow\infty$, we lose the one scale of the theory and
recover a point particle theory, a $(2,0)$ superconformal field theory (SCFT),
again labeled by the Lie algebra $\mathfrak{g}_{\text{o}}$. For the time
being, we will keep $m_{s}$ finite, and will only take the limit to infinity
when we make contact with the (undeformed) quantum geometric Langlands
program.

The moduli of the $(2,0)$ little string all have their origin in the parent
10-dimensional type IIB superstring, as periods of various 2-forms along the
2-cycles $S^{2}_{a}$ of the resolved singularity ${\mathbb{C}}^{2}_{\Gamma}$.
There are $r$ compact moduli, which are periods associated with the R-R 2-form
$C^{(2)}$:

$m_{s}^{2}\int_{S^{2}_{a}}C^{(2)}\,,\;\qquad a=1,\ldots,r\,.$ (6.2)

The corresponding compact directions all have radius $m_{s}^{2}$. One also
finds $4r$ non-compact moduli, which are the periods associated with the NS-NS
B-field $B^{(2)}$ and a triplet of self-dual 2-forms $\omega_{I,J,K}$:

$\frac{m_{s}^{4}}{g_{s}}\int_{S^{2}_{a}}\omega_{I,J,K}\,,\qquad\frac{m_{s}^{2}}{g_{s}}\int_{S^{2}_{a}}B^{(2)}\,,\;\qquad
a=1,\ldots,r\,.$ (6.3)

The 2-forms $\omega_{I,J,K}$ arise here since ${\mathbb{C}}^{2}_{\Gamma}$ is a
hyperkähler manifold. We have chosen a canonical normalization of the moduli
such that they all have mass dimension 2, ensuring no dimensional factors in
their kinetic terms.

In the little string limit $g_{s}\rightarrow 0$, one requires that all these
periods remain fixed. Therefore, the moduli space of the $(2,0)$ little string
is simply

$\left(\mathbb{R}^{4}\times S^{1}\right)^{r}/W(\mathfrak{g}_{\text{o}})\;,$
(6.4)

where the quotient by Weyl group $W(\mathfrak{g}_{\text{o}})$ is performed not
to overcount moduli.

The R-symmetry of the little string theory is $SU(2)_{R}$, which acts by
rotating the $3\,r$ scalars that are the periods of $\omega_{J}$,
$\omega_{K}$, and the NS-NS 2-form $B^{(2)}$. The operator $R$ in the index
stands for the Cartan generator of $U(1)_{R}\subset SU(2)_{R}$, which only
rotates the periods of $\omega_{J}$ and $\omega_{K}$.

Next, we perform a compactification of the type IIB background on a Riemann
surface:

${\mathbb{C}}^{2}_{\Gamma}\times
M_{6}\;,\;\qquad\text{where}\;\;M_{6}=\mathcal{C}\times\mathbb{C}\times\mathbb{C}\;.$
(6.5)

As in the rest of this work, the Riemann surface $\mathcal{C}$ is an infinite
cylinder818181One immediate reason for this restriction is that the Riemann
surface better be flat for the background to be a solution of type IIB string
theory. A more technical reason is that the deformed $\mathcal{W}$-algebras
were naturally related to gauge theories _compactified on a circle_ , see
Sections 3 and 4. Without much additional work, the framework we developed in
this paper can be used to further study the complex line
$\mathcal{C}=\mathbb{C}$ by considering an appropriately scaled
$R_{\mathcal{C}}\rightarrow\infty$ limit, or the torus $\mathcal{C}=T^{2}$ by
identifying the ends of the cylinder. In those cases, the relevant quantum
algebra would be a Yangian and a quantum elliptic algebra, respectively. The
finite-dimensional representation theory of these algebras is essentially the
same as that of the quantum affine algebra. of radius $R_{\mathcal{C}}$,

$\mathcal{C}=\mathbb{R}\times S^{1}(R_{\mathcal{C}})\;.$ (6.6)

The $(2,0)$ little string compactified on the circle of radius
$R_{\mathcal{C}}$ enjoys T-duality, where it becomes equivalent to a non-
chiral little string theory compactified on a circle of radius

$R_{\mathcal{C}^{\prime}}=\frac{1}{m_{s}^{2}\,R_{\mathcal{C}}}\;,$ (6.7)

the (1,1) little string theory. Equivalently, this (1,1) little string is the
$g_{s}\rightarrow 0$ limit of the 10-dimensional type IIA superstring on our
background (6.5)828282Note that the existence of T-duality in compactified
little string is a strong indication that it is not a local quantum field
theory..

From the IIB perspective, the moduli space of vacua of the $(2,0)$ little
string compactified on $S^{1}(R_{\mathcal{C}})$ is again (6.4), where the
periods (6.2) and (6.3) are simply rescaled by the radius $R_{\mathcal{C}}$.
From the IIA perspective, this is also the moduli space of the $(1,1)$ little
string compactified on $S^{1}(R_{\mathcal{C}^{\prime}})$, where $4\,r$ moduli
are as in the uncompactified theory (again up to radius), and $r$ new compact
moduli appear as the holonomy $A^{\theta}_{a}$ of gauge fields around
$S^{1}(R_{\mathcal{C}^{\prime}})$, related to the R-R 2-form period in type
IIB by

$A^{\theta}_{a}=R_{\mathcal{C}}\,m^{2}_{s}\int_{S^{2}_{a}}C^{(2)}\;,$ (6.8)

with periodicity $2\,\pi/R_{\mathcal{C}^{\prime}}$.

###  6.2 D5 branes and 5d gauge theory

In order to make contact with the Langlands program and its (tame)
ramification, we need to introduce punctures on the Riemann surface
$\mathcal{C}$. In string theory, a natural way to achieve this is to introduce
D-branes at points on $\mathcal{C}$. In our type IIB background, there are a
priori many such branes which fit this requirement. The correct branes to
consider here are D5 branes wrapping 2-cycles of the resolved $ADE$
singularity ${\mathbb{C}}^{2}_{\Gamma}$, and wrapping $\mathbb{C}^{2}\subset
M_{6}=\mathcal{C}\times\mathbb{C}\times\mathbb{C}$. The relevance of these
branes in our context was first noted in [64, 140].

In order to be more quantitative, we introduce some notations: by the McKay
correspondence, the second homology group
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\mathbb{Z})$ is identified with the root
lattice $\Lambda_{rt}$ of $\mathfrak{g}$. Indeed, the group
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\mathbb{Z})$ is spanned by $r$ vanishing
2-cycles $\\{S^{2}_{a}\\}$. Correspondingly, as is well-known, $\Lambda$ is
spanned by $r$ positive simple roots $\alpha_{a}$. Furthermore, the
intersection pairing in homology is identified with the Cartan Killing form of
$\mathfrak{g}_{\text{o}}$, up to an overall sign:

$\\#(S^{2}_{a}\cap S^{2}_{b})=-C_{ab}\;,\qquad\;\;\;\;a,b=1,\ldots,r\;,$ (6.9)

where $C_{ab}$ is the Cartan matrix of $\mathfrak{g}_{\text{o}}$.

Then, we first introduce $n_{a}$ D5 branes wrapping the compact 2-cycle
$S^{2}_{a}$, for all $a=1,\ldots,r$, and $\mathbb{C}^{2}\subset M_{6}$. This
results in a total of $n=\sum_{a=1}^{r}n_{a}$ D5 branes, and a net non-zero D5
brane R-R (magnetic) flux, measured by a nontrivial homology class $[S]\in
H_{2}({\mathbb{C}}^{2}_{\Gamma},\mathbb{Z})$. By abuse of notation, we write
$[S]$ in a basis of positive simple roots as

$[S]=-\sum_{a=1}^{r}\,n_{a}\,\alpha_{a}\;\;\in\,\Lambda_{rt}\;,$ (6.10)

with $n_{a}$ non-negative integers.

We also introduce the second relative homology group
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\partial\left({\mathbb{C}}^{2}_{\Gamma}\right),\mathbb{Z})$.
Its elements are 2-cycles of ${\mathbb{C}}^{2}_{\Gamma}$ which are allowed a
nontrivial boundary at infinity
$\partial\left({\mathbb{C}}^{2}_{\Gamma}\right)$. The group is spanned by $r$
non-compact 2-cycles $\\{{S^{2}_{a}}^{*}\\}$, each of which is constructed as
the fiber of the cotangent bundle ${T^{*}}{S^{2}_{a}}$ over a generic point on
the $a$-th 2-sphere $S^{2}_{a}$. The McKay correspondence further identifies
this relative homology group with the weight lattice $\Lambda_{wt}$ of
$\mathfrak{g}_{\text{o}}$. Equivalently, the $r$ non-compact 2-cycle
$\\{{S^{2}_{a}}^{*}\\}$ are identified with the $r$ fundamental weights
$w_{a}$ of $\mathfrak{g}_{\text{o}}$. It follows that the pairing between
positive simple roots and fundamental weights in $\mathfrak{g}_{\text{o}}$
translates in geometry to

$\\#(S^{2}_{a}\cap{S^{2}_{b}}^{*})=\delta_{ab}\;,\qquad\;\;\;\;a,b=1,\ldots,r\;.$
(6.11)

Note that the homology group $H_{2}({\mathbb{C}}^{2}_{\Gamma},\mathbb{Z})$ is
a subgroup of the relative homology group
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\partial\left({\mathbb{C}}^{2}_{\Gamma}\right),\mathbb{Z})$,
since compact 2-cycles are elements of
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\partial\left({\mathbb{C}}^{2}_{\Gamma}\right),\mathbb{Z})$
with trivial boundary at infinity. This is the homological version of the more
familiar statement that the root lattice $\Lambda_{rt}$ is a sublattice of the
weight lattice $\Lambda_{wt}$ in $\mathfrak{g}_{\text{o}}$.

Correspondingly, we introduce $n^{F}_{a}$ D5 branes wrapping the non-compact
2-cycle ${S^{2}_{a}}^{*}$, for all $a=1,\ldots,r$, and $\mathbb{C}^{2}\subset
M_{6}$. This results in a total of $n^{F}=\sum_{a=1}^{r}n^{F}_{a}$ D5 branes,
and an associated R-R flux measured by the class $[S^{*}]\in
H_{2}({\mathbb{C}}^{2}_{\Gamma},\partial\left({\mathbb{C}}^{2}_{\Gamma}\right),\mathbb{Z})$.
Again by abuse of notation, we write $[S^{*}]$ in a basis of fundamental
weights as

$[S^{*}]=\sum_{a=1}^{r}\,n_{a}^{F}\,w_{a}\;\;\in\,\Lambda_{wt}\;,$ (6.12)

where $n_{a}^{F}$ are non-negative integers sometimes called Dynkin labels. In
this basis, the fundamental weight $w_{a}$ is the vector
$w_{a}=[0,\ldots,0,1,0,\ldots,0]$ of size $r$ with a 1 in the $a$-th entry.
The homology class (6.12) is written as
$[S^{*}]=[n_{1}^{F},n_{2}^{F},n_{3}^{F},\ldots,n_{r}^{F}]$.

All in all, the net D5 brane R-R flux is encoded in the homology class
$[S+S^{*}]$, understood as an element of the lattice $\Lambda_{wt}$.

In the absence of branes, the IIB background has 16 supercharges, solely due
to the background ${\mathbb{C}}^{2}_{\Gamma}$. In order to preserve
supersymmetry after introducing the wrapped D5 branes, one needs to properly
align the triplet of self-dual 2-forms
$\vec{\omega}=(\omega_{I},\omega_{J},\omega_{K})$: as far as the $n^{F}$ D5
branes wrapping non-compact 2-cycles are concerned, supersymmetry is preserved
only if the vectors $\int_{{S^{2}_{a}}^{*}}\vec{\omega}$ point in the same
direction, meaning the central charges are aligned for all $a=1,\ldots,r$.
Then, let us explicitly choose $\vec{\omega}$ to satisfy

$\int_{{S^{2}_{a}}^{*}}\omega_{I}>0\,,\qquad\int_{{S^{2}_{a}}^{*}}\omega_{J}=0\,,\qquad\int_{{S^{2}_{a}}^{*}}\omega_{K}=0\,.$
(6.13)

With this choice of periods, the $n^{F}$ D5 branes are 1/2-BPS, meaning 8
supercharges remain.

In order for the $n$ D5 branes to preserve the same supersymmetry, one
considers the periods of the 2-forms through the compact 2-cycles $S^{2}_{a}$,
corresponding to the choice of a metric on ${\mathbb{C}}^{2}_{\Gamma}$. For
all $a=1,\ldots,r$, we will explicitly choose

$\tau^{6d}_{a}\equiv\int_{S^{2}_{a}}\left(\frac{m^{2}_{s}}{g_{s}}\,\omega_{I}+i\,C^{(2)}\right)\,,\qquad\int_{S^{2}_{a}}\omega_{J}=0\,,\qquad\int_{S^{2}_{a}}\omega_{K}=0\;,$
(6.14)

with $\text{Re}(\tau^{6d}_{a})>0$838383The choice of sign in
$\text{Re}(\tau^{6d}_{a})>0$ will also ensure the convergence of the various
$z$-solution blocks we computed in this work., along with

$\int_{S^{2}_{a}}B^{(2)}=0\;.$ (6.15)

With this choice of periods, the D5 branes wrapping the compact and non-
compact 2-cycles break the same half supersymmetry, so the background still
has 8 supercharges.

Our discussion so far was phrased in the full type IIB background. In the
$(2,0)$ little string, one simply considers the D5 branes as above, but in the
$g_{s}\rightarrow 0$ limit. Inspired by the nomenclature of [280], one option
would be to call the branes in that limit “d branes.” However, we choose to
keep the terminology “D branes” throughout, and the context will hopefully
make it clear whether the string coupling is null or not.

We consider an intermediate energy scale $E$ well below the string scale,
$E\ll m_{s}$. At that scale, and in the limit $g_{s}\rightarrow 0$, the theory
on the D5 branes is a 5-dimensional gauge theory with $\mathcal{N}=1$
supersymmetry, on the manifold
$S^{1}(R_{\mathcal{C}^{\prime}})\times\mathbb{C}^{2}$ [64].

The theory is not simply 4-dimensional on $\mathbb{C}^{2}$, due to the
presence of the cylinder $\mathcal{C}=S^{1}(R_{\mathcal{C}})\times\mathbb{R}$
in the transverse directions. Indeed, strings will wind around the cylinder
circle, so one should expect a Kaluza-Klein tower of states on the T-dual
circle of radius

$R_{\mathcal{C}^{\prime}}=\frac{1}{m_{s}^{2}\,R_{\mathcal{C}}}\;,$ (6.16)

and one should therefore expect that the low-energy physics depends on the
scale $R_{\mathcal{C}^{\prime}}$. Phrased differently, the D5 branes are
points on the circle $S^{1}(R_{\mathcal{C}})$, but they have an equivalent
description as D6 branes wrapping $S^{1}(R_{\mathcal{C}^{\prime}})$ in the
(1,1) little string, by T-duality. From this T-dual picture, it is clear that
the low energy theory on the D6 branes really is a 5d theory wrapping the
circle $S^{1}(R_{\mathcal{C}^{\prime}})$.

The characterization of the 5d gauge theory comes from geometry. Since the D5
branes are wrapping 2-cycles in ${\mathbb{C}}^{2}_{\Gamma}$, they are
described at energy scale $E$ by a quiver gauge theory [128, 318], of shape
the Dynkin diagram of the Lie algebra $\mathfrak{g}_{\text{o}}$848484In the
analysis of Douglas and Moore [128], our wrapped D5 branes would have been
treated as (fractional) D3 branes probing the orbifold singularity. The
authors show that in the full type IIB background, the effective gauge theory
on the branes is described by an _affine_ quiver, of rank $r+1$. The affine
nature of the quiver is due to $n_{0}$ (full) D3 branes transverse to the
$ADE$ surface, resulting in an additional diagonally embedded $U(n_{0})$ gauge
group. Its inverse gauge coupling is the sum of inverse gauge couplings for
all quiver nodes, but this sum equals $1/g_{s}$, the string coupling of type
IIB string theory. It follows that in the little string limit
$g_{s}\rightarrow 0$, the $U(n_{0})$ gauge fields are effectively frozen,
explaining why our quiver gauge theories are only of _finite_ Dynkin type
[319, 320].. In 5 dimensions, the inverse gauge couplings $1/(g^{5d}_{a})^{2}$
have mass dimension 1. Since we would like the gauge theory to be weakly
coupled at energy $E$, the inequality $E\,(g^{5d}_{a})^{2}\ll 1$ should hold
for all $a=1,\ldots,r$. In terms of the dimensionless couplings
$\tau^{6d}_{a}$, which scale as $1/(m_{s}\,(g^{5d}_{a})^{2})$, this is
equivalent to requiring $E/m_{s}\ll\tau^{6d}_{a}$. The gauge group of the
quiver is

$G_{5d}=\prod_{a=1}^{r}U(n_{a})\;,$ (6.17)

where the ranks $n_{a}$ are the number of D5 branes wrapping the compact
2-cycle $S^{2}_{a}$ (6.10). Each $U(n_{a})$ gauge group contains a massive
$U(1)$ center by the Green-Schwarz mechanism, so the gauge groups should
really be $SU(n_{a})$. Nevertheless, because the 5d theory is eventually
placed on an $\Omega$-background
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$, it
convenient to simply drop the special unitary condition and treat all $n_{a}$
equivariant parameters on an equal footing. The positions $e_{a,i}$ of the
$n=\sum_{a=1}^{r}\,n_{a}$ D5 branes on $\mathcal{C}$ are the 5d Coulomb
moduli.

The flavor symmetry of the quiver is

$G^{F}_{5d}=\prod_{a=1}^{r}U(n_{a}^{F})\;,$ (6.18)

where the ranks $n_{a}^{F}$ are the number of D5 branes wrapping the non-
compact 2-cycle ${S^{2}_{a}}^{*}$ (6.12). These fundamental hypermultiplets
are in representation $(n_{a},\overline{n^{F}_{a}})$ of $U(n_{a})\times
U(n^{F}_{a})$, they arise from the zero modes of open strings at the
intersection of D5 branes wrapping the compact 2-cycle $S^{2}_{a}$ and the
non-compact 2-cycle ${S^{2}_{a}}^{*}$ on the fiber. The positions $f_{a,s}$ of
the $n^{F}=\sum_{a=1}^{r}\,n_{a}^{F}$ D5 branes on $\mathcal{C}$ are masses
for these hypermultiplets.

Finally858585We do not consider the open strings between D5 branes wrapping
${S^{2}_{a}}^{*}$ and ${S^{2}_{b}}^{*}$, the intersection pairing of which is
dictated by the quadratic form matrix of $\mathfrak{g}_{\text{o}}$ (the
inverse Cartan matrix): the resulting bifundamental hypermultiplets represent
nondynamical degrees of freedom, which we have not kept track of throughout
this work., there are hypermultiplets in the bifundamental representation
$\oplus_{b>a}\,\Delta_{ab}\,(n_{a},\overline{n_{b}})$ of the group
$\prod_{a,b}U(n_{a})\times U(n_{b})$, where $\Delta_{ab}=1$ if there is a link
connecting nodes $a$ and $b$, and $\Delta_{ab}=0$ otherwise. They arise from
the zero modes of open strings with one end on the D5 branes wrapping
$S^{2}_{a}$, and the other end on D5 branes wrapping $S^{2}_{b}$.

The $\tau^{6d}_{a}$ variables (6.14) are not moduli of
$T^{5d}_{\mathfrak{g}_{\text{o}}}$. Indeed, the 2-forms in the definition of
$\tau^{6d}_{a}$ are valued in all six dimensions of
$M_{6}=\mathcal{C}\times\mathbb{C}^{2}$, so they are not dynamical. Instead,
they are parameters: they determine the $r$ gauge couplings of the quiver
theory.

The periods of the 2-forms $\omega_{J}$, $\omega_{K}$, and of the NS-NS
B-field $B^{(2)}$ through the 2-cycles $S^{2}_{a}$ are the F.I. parameters of
$T^{5d}_{\mathfrak{g}_{\text{o}}}$, set to zero for now as in (6.14).

The Coulomb moduli, hypermultiplet masses and F.I. parameters all start out as
real variables, but get complexified by the holonomy around the (T-dual)
cylinder circle $S^{1}(R_{\mathcal{C}^{\prime}})$.

The 5-dimensional quiver gauge theory is the theory
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ from Section 4. There, we also imposed $r$
additional constraint equations:

$\sum_{b=1}^{r}C_{ab}\;n_{b}=n^{F}_{a}\;,\qquad\;a=1,\ldots,r\;,$ (6.19)

with $C_{ab}$ the Cartan matrix of $\mathfrak{g}_{\text{o}}$; if we were to
reduce on $S^{1}(R_{\mathcal{C}^{\prime}})$, these constraints would imply the
vanishing of the beta function. In the string language, the constraints
translate to requiring the net D5 brane R-R flux to be zero. From the
intersection pairing (6.9), our constraint equations are equivalent to
requiring the class $[S+S^{*}]$ to vanish in homology. In practice, given a
fixed class $[S^{*}]$, the class $[S]$ is uniquely determined (if it exists)
from solving the constraint

$[S+S^{*}]=0\;.$ (6.20)

For instance, for $\mathfrak{g}_{\text{o}}=A_{1}$, only the even classes
$[S^{*}]=[2\,n]$ yield solutions $[S]$ for $n$ a non-negative integer. This is
simply the homological version of the more familiar gauge theory statement
that the beta function vanishes in 4 dimensions if and only if the number of
fundamental hypermultiplets is twice the rank, $n^{F}=2\,n$.

We realize the ADHM instanton quantum mechanics of
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ in the little string picture by introducing
$k_{a}$ 1/2-BPS D1 branes which wrapping the compact 2-cycle $S^{2}_{a}$, for
all $a=1,\ldots,r$ [286], and then taking the $g_{s}\rightarrow 0$ limit.
There are a total of $k=\sum_{a=1}^{r}k_{a}$ such branes, each located at a
point $\phi_{a,I}$ on $\mathcal{C}$. The effective theory on the D1 branes is
a 1-dimensional $\mathcal{N}=4$ quantum mechanics on
$S^{1}(R_{\mathcal{C}^{\prime}})$, for the same reason that the theory on the
D5 branes is 5-dimensional on $S^{1}(R_{\mathcal{C}^{\prime}})$.

Equivalently, in the T-dual frame, we are studying the $(1,1)$ little string
on $S^{1}(R_{\mathcal{C}^{\prime}})$ with $n+n^{F}$ D6 branes and $k$ D2
branes wrapping the circle. The Witten index of the ADHM quantum mechanics
(4.11) is the partition function of the $(1,1)$ little string on
$M^{\prime}_{6}=\mathcal{C}^{\prime}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$
in the presence of the branes868686Qualitatively, a localization argument will
show that the partition function of the bulk little string with D6 brane
defects is the same as the partition function of the brane defects
_themselves_ : away from the defect branes, the supersymmetry is restored to
the 16 supercharges of the bulk little string, where the index trivializes due
to the pairwise cancellation of bosons and fermions in the enhanced
supersymmetry. The only nonzero contributions to the index come from the
defect branes.:

${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}_{\text{o}}})={\rm
Tr}\left[(-1)^{F}\,q^{J_{1}+\frac{R}{2}}\;t^{J_{2}-\frac{R}{2}}\;{\mathfrak{m}}^{\Sigma}\right]\;\;.$
(6.21)

The integral representation of the index is now understood as the contribution
of strings stretching between the various branes: the integrand is made up of
1-loop determinants which are the contributions of D2/D2 and D2/D6 strings,
while the integration variables are the D2 brane positions on
$\mathcal{C}^{\prime}$. In the IIA language, these variables are parameterized
by $\phi_{a,I}=\varphi_{a,I}+i\,A^{\theta}_{a,I}$, with $A^{\theta}_{a,I}$ the
holonomy around $S^{1}(R_{\mathcal{C}^{\prime}})$. The integration over the D2
brane positions enforces the gauge invariance of the group
$\prod_{a=1}^{r}U(k_{a})$.

###  6.3 D3 branes and 3d gauge theory

The 3d gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$ also has a realization
in the $(2,0)$ little string. To see this, we force the D5 brane theory on its
Higgs branch by turning on some of the periods:

$\int_{S^{2}_{a}}\omega_{J}=0\,,\qquad\int_{S^{2}_{a}}\omega_{K}=0\,,\qquad\int_{S^{2}_{a}}B^{(2)}>0\,.$
(6.22)

Correspondingly, this turns on $r$ real F.I. parameters (complexified as
usual) for $T^{5d}_{\mathfrak{g}_{\text{o}}}$, while the complex F.I.
parameters remain null. With the F.I. parameters aligned as above, a common
supersymmetry is preserved, and the remaining R-symmetry is $U(1)_{R}\subset
SU(2)_{R}$. In the little string, we bind the $n$ D5 branes wrapping the
cycles $\\{S^{2}_{a}\\}$ to (a subset of) the $n^{F}$ D5 branes wrapping the
cycles $\\{{S^{2}_{a}}^{*}\\}$, and move their positions $e_{a,i}$ to coincide
with the positions $f_{a,s}$ on the cylinder $\mathcal{C}$. This results in a
new configuration of $n^{F}$ D5 branes wrapping non-compact 2-cycles only. We
denote these 2-cycles as $\\{{S^{2}_{s}}^{*}\\}$, with corresponding homology
classes in $H_{2}(X,\partial X,\mathbb{Z})$ given by

$[S_{s}^{*}]=\underline{\lambda_{s}}\;\;\in\,\Lambda_{wt}\;,$ (6.23)

The class $[S_{s}^{*}]$ can be expanded as

$[S_{s}^{*}]=[S_{a}^{*}]-\sum_{b=1}^{r}h_{s,b}\,[S_{b}]\;,$ (6.24)

where $h_{s,b}$ are non-negative integers and $[S_{a}^{*}]=w_{a}$ and
$[S_{a}]=\alpha_{a}$ are the classes associated to the $a$-th fundamental
weight and $a$-th simple root, respectively. This decomposition is the same as
(4.42) in the gauge theory $T^{5d}_{\mathfrak{g}_{\text{o}}}$, where we had
specified which Coulomb moduli get tuned to which mass. Once all Coulomb
moduli are frozen in this way, the hypermultiplets become massless and can
acquire a vev, defining the Higgs branch. The $n^{F}$ classes $[S_{s}^{*}]$
are not independent, since we required the net D5 brane R-R flux to vanish
(6.20). This constraint now reads

$\sum_{s=1}^{n^{F}}[S_{s}^{*}]=0\;,$ (6.25)

which is the homological version of (4.44).

In type IIB, the 1/2-BPS semi-local vortex solutions on the Higgs branch of
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ are D3 branes at points on the cylinder
$\mathcal{C}$, wrapping the compact 2-cycles of $\mathbb{C}^{2}_{\Gamma}$ and
one of the two complex lines, say $\mathbb{C}_{q}$ [129]. Overall, we
introduce a total of $N=\sum_{a=1}^{r}N_{a}$ D3 branes, where $N_{a}$ of them
wrap $S^{2}_{a}$, with corresponding $H_{2}(X,\mathbb{Z})$ homology class

$[N]=\sum_{a=1}^{r}\,N_{a}\,\alpha_{a}\;\;\in\,\Lambda\;.$ (6.26)

From the point of view of the D5 brane theory, the D3 branes are codimension-2
and end on them, which this turns on magnetic flux, consistent with the vortex
interpretation [271]. Recall that on $\mathbb{C}^{2}_{\Gamma}$, the D5 branes
wrap holomorphic cycles; meanwhile, the D3 branes are Lagrangian, intersecting
the D5 branes transversely in $\mathbb{C}^{2}_{\Gamma}$. The low energy
effective theory on the D3 branes is the 3d $\mathcal{N}=2$ Drinfeld quiver
gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$: the quiver shape is the
Dynkin diragram of $\mathfrak{g}_{\text{o}}$, and the theory is supported on
the manifold $S^{1}(R_{\mathcal{C}^{\prime}})\times\mathbb{C}_{q}$878787Once
again, the D3 brane theory is 3-dimensional and not simply 2-dimensional with
$\mathcal{N}=(2,2)$ supersymmetry on $\mathbb{C}_{q}$, because of the
fundamental strings wrapping the cylinder $\mathcal{C}$, meaning there is a
Kaluza Klein tower of states on the T-dual cylinder $\mathcal{C}^{\prime}$.
Alternatively, $T^{3d}_{\mathfrak{g}_{\text{o}}}$ is the effective theory on
D4 branes wrapping $S^{1}(R_{\mathcal{C}^{\prime}})$ in the $(1,1)$ little
string.. The little string couplings $\tau^{6d}_{a}$ in (6.14) are the F.I.
parameters. The zero modes of D3/D3 strings on a node $a$ produce the 3d
$\mathcal{N}=4$ supersymmetric vector multiplets, while the D3/D3 strings
between neighboring nodes are responsible for the $\mathcal{N}=4$
bifundamental hypermultiplets. The $n^{F}$ D5 branes further break half the
supersymmetry, so D3/D5 strings provide the 3d $\mathcal{N}=2$ fundamental and
anti-fundamental chiral matter888888Gauge-vortex duality in gauge theory is
now realized as a large $N$-duality in the little string [62, 98]. The
argument was originally given in the unramified limit $q=t$ by Dijkgraaf and
Vafa [226], see also the early works [321, 322]. In fact, in the
$\Omega$-background, $N$ does not have to be large at all for the duality to
hold, and can even be equal to 1, as we will do in the Examples Section..

We found from our gauge theory analysis that this chiral matter is naturally
labeled by weight spaces of a quantum affine algebra
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$, and furthermore that these weight
spaces are recorded in the half-index of $T^{3d}_{\mathfrak{g}_{\text{o}}}$.
This is the partition function of the theory on the D3
branes898989Alternatively, we argued that the ADHM instanton partition
function of $T^{5d}_{\mathfrak{g}_{\text{o}}}$ was realized as the Witten
index of the gauged quantum mechanics on $k$ D1 branes wrapping the 2-cycles
$\\{S^{2}_{a}\\}$. We could follow the same route here, and consider the
gauged quantum mechanics on the $k$ D1 branes in the presence of D3 and D5
branes; the half-index of $T^{3d}_{\mathfrak{g}_{\text{o}}}$ coincides with
the Witten index of this quantum mechanics.. Therefore, these weight spaces
can now be reinterpreted as the zero modes of D3/D5 strings in $(2,0)$ little
string theory on $M_{6}=\mathcal{C}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$.
Equivalently, they the zero modes of D4/D6 strings in $(1,1)$ little string
theory on
$M^{\prime}_{6}=\mathcal{C}^{\prime}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$
are labeled by quantum affine weights.

Following our dictionary to $q$-conformal blocks, we conclude that
$\mathcal{W}_{t,q}(\mathfrak{g})$ magnetic blocks are $(2,0)$ little string
partition functions on $M_{6}$ in the presence of D3 and D5 branes. Each such
block has a presentation as the half-index on
$S^{1}_{\mathcal{C}^{\prime}}\times D^{2}$ for the effective theory
$T^{3d}_{\mathfrak{g}_{\text{o}}}$ on the D3 branes. The $z$-solutions will be
analytic in a given $\tau^{6d}_{a}$-parameter chamber, and are constructed by
imposing exceptional Dirichlet boundary conditions on
$S^{1}_{\mathcal{C}^{\prime}}\times S^{1}_{D^{2}}$. The $x$-solutions will be
analytic in a given mass-parameter chamber, the positions of the D5 branes on
$\mathcal{C}$, and are constructed by imposing enriched Neumann boundary
conditions on $S^{1}_{\mathcal{C}^{\prime}}\times S^{1}_{D^{2}}$.

The $U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ electric blocks are constructed
in the same way, except one should consider the $(2,0)$ little string on the
manifold
$M^{\times}_{6}=\mathcal{C}\times\mathbb{C}^{\times}_{q}\times\mathbb{C}_{t}$
in the presence of D3 and D5 branes. $M^{\times}_{6}$ differs from $M_{6}$
only by removing the origin of $\mathbb{C}_{q}$, so the D3 and D5 branes are
now supported on $\mathbb{C}^{\times}_{q}\times\\{S^{2}_{a}\\}$ and
$\mathbb{C}^{\times}_{q}\times\mathbb{C}_{t}\times\\{{S^{2}_{s}}^{*}\\}$
respectively, with appropriate boundary conditions at the excised origin.

###  6.4 The conformal limit

The $(2,0)$ little string is supported on $M_{6}=\mathcal{C}\times{\bf
M}_{4}$, with ${\bf M}_{4}=\mathbb{C}_{q}\times\mathbb{C}_{t}$. We now take
the limit to the $(2,0)$ SCFT:

$m_{s}\rightarrow\infty\;,$ (6.27)

where we require $M_{6}$, and in particular the cylinder
$\mathcal{C}=S^{1}_{\mathcal{C}}\times\mathbb{R}$, to remain a fixed
background throughout. The gauge theory $T^{5d}_{\mathfrak{g}_{\text{o}}}$ on
the D5 branes was supported on $S^{1}_{\mathcal{C}^{\prime}}\times{\bf
M}_{4}$, where $S^{1}_{\mathcal{C}^{\prime}}$ is the T-dual circle of radius

$R_{\mathcal{C}^{\prime}}=\frac{1}{m^{2}_{s}\,R_{\mathcal{C}}}\;.$ (6.28)

It follows that the radius $R_{\mathcal{C}^{\prime}}\rightarrow 0$ in the
conformal limit, since $R_{\mathcal{C}}$ is kept fixed. We further keep the
$\Omega$-background parameters $\epsilon_{q}$ and $\epsilon_{t}$ fixed on
${\bf M}_{4}$, and so all the quantization parameters

$q=e^{R_{\mathcal{C}^{\prime}}\,\epsilon_{q}}\,,\qquad
t=e^{-R_{\mathcal{C}^{\prime}}\,\epsilon_{t}}\,,\qquad\hbar=q/t\,,$ (6.29)

trivialize to 1. This is why we lose the quantum algebra deformations in the
limit, and recover the usual chiral algebras on $\mathcal{C}$.

$T^{5d}_{\mathfrak{g}_{\text{o}}}$ becomes a 4-dimensional theory with
$\mathcal{N}=2$ supersymmetry supported on
$\mathbb{C}_{\epsilon_{q}}\times\mathbb{C}_{\epsilon_{t}}$. The 6d little
string moduli (6.2) and (6.3) are kept fixed in the limit and become moduli of
the 6d SCFT; $r$ of these moduli are $m_{s}^{2}\tau^{6d}_{a}$, so we must have
$\tau^{6d}_{a}\rightarrow 0$. But these are the inverse gauge couplings of the
theory on the D5 branes, so the 4-dimensional theory no longer has a
description as a weakly coupled gauge theory909090Except in rare “accidental”
instances, see Section 8.3.1.. The constraint (6.19) becomes a conformal
condition in 4d, namely the vanishing of the beta function. Since
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ originated as a description of a
codimension-2 defect in the little string, we deduce that it becomes a 4d
theory of class $S$ in the limit [133, 134].

Likewise, the Drinfeld gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$ on the
D3 branes is initially supported on
$M_{3}=S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}$. After taking the
conformal limit, it becomes a 2-dimensional theory with $\mathcal{N}=(2,2)$
supersymmetry supported on $\mathbb{C}_{\epsilon_{q}}$ instead. The gauge
couplings of $T^{3d}_{\mathfrak{g}_{\text{o}}}$ are $m_{s}^{-4}$ times fixed
moduli of the (2,0) SCFT, so the Lagrangian description of the theory is again
lost in the limit.

In the following Sections, it will prove useful to reinterpret the manifold
$M_{6}$ as a torus fibration $T^{2}=S^{1}_{q}(R_{q})\times S^{1}_{t}(R_{t})$:

$T^{2}\hookrightarrow M_{6}\hookrightarrow
M_{4}=\mathcal{C}\times\mathbb{R}_{+}\times\mathbb{R}_{+}\;.$ (6.30)

This is the perspective that makes the relation to the geometric Langlands
program manifest. Indeed, in the conformal limit, and reducing on $T^{2}$, we
obtain 4d Super Yang-Mills on
$M_{4}=\mathcal{C}\times\mathbb{R}_{+}\times\mathbb{R}_{+}$. The D5 branes
wrapping $T^{2}$ become D3 branes wrapping
$\mathbb{R}_{+}\times\mathbb{R}_{+}\subset M_{4}$, so they flow to
codimension-2 defects at points on $\mathcal{C}$ in the limit. These are
Gukov-Witten monodromy defects [31, 32], as we will analyze in detail in
Section 8.

It would be important to make precise contact with other recent descriptions
of this background. For instance, it is known that there exist $Y_{L,M,N}$
chiral algebras found “at the corner” of $\mathbb{R}_{+}\times\mathbb{R}_{+}$
in 4d Super Yang-Mills [216, 323]. In particular, for $\mathfrak{g}=A_{n}$,
the fact that our D5 branes realize certain Fock modules in the
$\mathcal{W}_{q,t}(A_{n})$-algebra is consistent with this picture: in the
conformal limit, we have $Y_{n,0,0}=u(1)\oplus\mathcal{W}_{\beta}(A_{n})$, and
Gukov-Witten defects should describe Verma modules of the algebra, via a free
field realization just as in our construction. This also fits into a larger
program where such modules are known to be realized as certain representations
of the affine Yangian of $\mathfrak{gl}(1)$ [324], while in the $q$-deformed
context they should naturally appear as representation of the quantum toroidal
algebra $U_{q,t}(\hat{\hat{\mathfrak{gl}_{1}}})$ [174, 26, 325].

If we consider the starting manifold to be $M^{\times}_{6}$, with the origin
removed, then $M_{4}=\mathcal{C}\times\mathbb{R}\times\mathbb{R}_{+}$, and the
D5 branes flow in the conformal limit to monodromy defects at points on
$\mathcal{C}$ and wrapping $\mathbb{R}\times\mathbb{R}_{+}$. Gaiotto and
Witten explained that imposing Nahm-type boundary conditions at the origin of
$\mathbb{R}_{+}$ naturally leads to a description of the system via
$\mathfrak{g}$-opers, see Section 8.4 in [326]. It would be important to
understand how the classical limit of our $\mathcal{W}(\mathfrak{g})$-algebra
construction reduces to this description.

##  7 The non simply-laced case in gauge and string theory

So far, our realization of the $q$-conformal blocks in gauge and string theory
was discussed in the simply-laced case $\mathfrak{g}_{\text{o}}$. However, the
$\mathcal{W}_{q,t}(\mathfrak{g})$ and
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ blocks of Section 2 were defined for
an arbitrary simple Lie algebra $\mathfrak{g}$. This begs the question of
whether or not the cases $\mathfrak{g}=B_{r},C_{r},G_{2},F_{4}$ could also be
understood in gauge and string theory. The purpose of this Section is to
explain how this is done, and point out some open questions.

__

Figure 12: Construction of the non simply-laced Lie algebras starting from the
simply-laced ones via folding.

The construction of the non simply-laced Lie algebras from the simply-laced
ones is well-known: one observes that a non simply-laced Lie algebra
$\mathfrak{g}$ is always a subalgebra of a simply-laced Lie algebra
$\mathfrak{g}_{\text{o}}$ invariant under an $A$-action, where $A$ is a non-
trivial outer automorphism group of $\mathfrak{g}_{\text{o}}$. The group $A$
is abelian919191The outer automorphism group of $D_{4}$ is in fact nonabelian,
as it is the symmetric group $S_{3}$. For us, only the abelian subgroup
$\mathbb{Z}_{3}$ will be relevant., and it is a well-known fact that the outer
automorphisms of $\mathfrak{g}_{\text{o}}$ are in one-to-one correspondence
with the automorphisms of its Dynkin diagram. It follows at once that the only
possible outer automorphism groups are either $A=\mathbb{Z}_{2}$ or
$A=\mathbb{Z}_{3}$, see figure 12. The lacing number
$\mathfrak{n}_{\mathfrak{g}}$ is the order of the group $A$.

Let $\mathfrak{a}\in A$, and let $\Delta$ be the set of simple roots of
$\mathfrak{g}_{\text{o}}$. As reviewed in Appendix A, the simple roots of
$\mathfrak{g}$ split into two sets: First, we take the long roots of
$\mathfrak{g}$ to be the simple roots of $\mathfrak{g}_{\text{o}}$ invariant
under $A$-action,

$\displaystyle\Delta_{l}=\left\\{\alpha\;|\;\alpha\in\Delta,\;\alpha=\mathfrak{a}(\alpha)\right\\}\;.$
(7.1)

In our convention, long roots have length squared $2$, the same as the simple
roots of $\mathfrak{g}_{\text{o}}$. Second, the short roots of $\mathfrak{g}$
are the simple roots of $\mathfrak{g}_{\text{o}}$ which belong in orbits of
$A$ of length $\mathfrak{n}_{\mathfrak{g}}$,

$\displaystyle\Delta_{s}$
$\displaystyle=\left\\{\frac{1}{\mathfrak{n}_{\mathfrak{g}}}\left(\alpha+\mathfrak{a}(\alpha)+\ldots+\mathfrak{a}^{\mathfrak{n}_{\mathfrak{g}}-1}(\alpha)\right)\;|\;\alpha\in\Delta,\;\alpha\neq\mathfrak{a}(\alpha)\right\\}\;.$
(7.2)

These short roots of $\mathfrak{g}$ have length squared
$2/\mathfrak{n}_{\mathfrak{g}}$, per our convention.

###  7.1 String theory realization

The above construction has a natural realization in string theory [327, 328]
(see also the extension of the usual simply-laced McKay correspondence can to
the non simply-laced case [329, 330]). In the simply-laced case, our starting
point was the type IIB string theory on $\mathbb{C}^{2}_{\Gamma}\times M_{6}$,
with $\mathbb{C}^{2}_{\Gamma}$ a resolved $\mathfrak{g}_{\text{o}}$
singularity, and the $(2,0)$ little string living on
$M_{6}=\mathcal{C}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$. If we view
$M_{6}$ as the fibration (6.30), then it is natural to consider the following
twisted background:

$\frac{\mathbb{C}^{2}_{\Gamma}\times
S^{1}_{q}(R_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t}(R_{t})\times M_{4}\;.$ (7.3)

Namely, as we go around the circle $S^{1}_{q}$, the resolved $ADE$ singularity
$\mathbb{C}^{2}_{\Gamma}$ comes back to itself, but only up to the action of a
generator $\mathfrak{a}\in\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$ of the
outer automorphism group.

In particular, this is a non-trivial action on the root lattice of
$\mathfrak{g}_{\text{o}}$, and therefore a non-trivial $\mathfrak{a}$-action
on the homology group $H_{2}(\mathbb{C}^{2}_{\Gamma},\mathbb{Z})$. Once we
introduce $n$ D5 branes (and later $N$ D3 branes as vortices) wrapping the
compact 2-cycles $\\{S^{2}_{a}\\}$ in
$H_{2}(\mathbb{C}^{2}_{\Gamma},\mathbb{Z})$, this action will permute the
branes, and the configuration of D5 branes (and D3 branes) is henceforth
required to be invariant under this action.

The same considerations apply to the weight lattice of
$\mathfrak{g}_{\text{o}}$, and therefore to the relative homology group
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\partial\left({\mathbb{C}}^{2}_{\Gamma}\right),\mathbb{Z})$:
the configuration of $n^{F}$ flavor D5 branes wrapping the non-compact
2-cycles $\\{{S^{2}_{a}}^{*}\\}$ also has to be invariant under
$\mathfrak{a}$-action.

A simple coroot of $\mathfrak{g}$ is a sum of simple roots of
$\mathfrak{g}_{\text{o}}$, where all the roots are in a single $A$-orbit.
Correspondingly, the $n$ D5 branes wrapping $\\{S^{2}_{a}\\}$ in the fibered
geometry have a net magnetic flux measured by an element of $\Lambda_{cort}$,
the coroot lattice of $\mathfrak{g}$; the corresponding homology class is
denoted as

$[S^{\vee}]=\sum_{a=1}^{r}\,n_{a}\,\alpha^{\vee}_{a}\;\;\in\,\Lambda_{cort}\;.$

Likewise, a fundamental coweight $w_{a}^{\vee}$ of $\mathfrak{g}$ is a sum of
fundamental weights of $\mathfrak{g}_{\text{o}}$ where all the weights are in
a single $A$-orbit. We conclude that the $n^{F}$ D5 branes wrapping
$\\{{S^{2}_{a}}^{*}\\}$ in the fibered geometry have a net magnetic flux
measured by an element of $\Lambda_{cowt}$, the coweight lattice of
$\mathfrak{g}$; the corresponding relative homology class is denoted as
$[{S^{*}}^{\vee}]$.

$[{S^{*}}^{\vee}]=\sum_{a=1}^{r}\,n^{F}_{a}\,w^{\vee}_{a}\;\;\in\,\Lambda_{cowt}\;.$

In order to make sense of the class
$[{S+S^{*}}^{\vee}]=[S^{\vee}]+[{S^{*}}^{\vee}]$, we simply consider
$[S^{\vee}]$ as a class in relative homology with trivial boundary at
infinity. Equivalently, the coroot lattice $\Lambda_{cort}$ being a sublattice
of the coweight lattice $\Lambda_{cowt}$, the net flux $[{S+S^{*}}^{\vee}]$
can always be written as a coweight of $\mathfrak{g}$.

Just as in the simply-laced case, we impose the asymptotic conformality
constraint $[{S+S^{*}}^{\vee}]=0$, which once again takes the form of $r$
equations:

$\sum_{b=1}^{r}C_{ab}\;n_{b}=n^{F}_{a}\;,\qquad\;a=1,\ldots,r\;,$ (7.4)

with $C_{ab}=\langle\alpha_{a},\alpha_{b}^{\vee}\rangle$ the Cartan matrix of
$\mathfrak{g}$. In the limit $g_{s}\rightarrow 0$, the background (7.3)
describes the $ADE$ $(2,0)$ little string on

$\frac{S^{1}_{q}(R_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t}(R_{t})\times M_{4}\;,$ (7.5)

subjected to the $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist. The D5 and
D3 branes likewise survive the little string limit and are subjected to the
twist929292Though we refrain from doing so in this Section, we comment in
passing that if one further takes the limit $m_{s}\rightarrow\infty$, the
compactified $(2,0)$ $ADE$ little string reduces to a compactified $(2,0)$
$ADE$ SCFT, again subjected to the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist. This SCFT background has
been invoked in various contexts to engineer non simply-laced gauge groups and
defects; notable examples can be found in the works [331, 332, 333]..

###  7.2 On the notion of moduli space of vacua for a non simply-laced quiver

We start with a basic question: are there non simply-laced gauge theories?

In the simply-laced case, both $T^{5d}_{\mathfrak{g}_{\text{o}}}$ and
$T^{3d}_{\mathfrak{g}_{\text{o}}}$ arose as the low energy effective gauge
theory on the $n$ D5 and $N$ D3 branes in the $(2,0)$ little string,
respectively. After the $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist, is
it still sensible to define non simply-laced quiver gauge theories
$T^{5d}_{\mathfrak{g}}$ and $T^{3d}_{\mathfrak{g}}$ as the effective theories
on the D5 and D3 branes in the background (7.5)?

Let us first consider the 3d quiver gauge theory
$T^{3d}_{\mathfrak{g}_{\text{o}}}$ on the $N$ D3 branes, where the $r$ gauge
nodes $U(N_{a})$ are linked by bifundamental hypermultiplets following the
shape of the Dynkin diagram of a simply-laced Lie algebra