$\mathfrak{g}_{\text{o}}$. An immediate obstacle to considering
$T^{3d}_{\mathfrak{g}}$ with $\mathfrak{g}$ non simply-laced is that there is
no well-defined notion of bifundamental matter in that case, and in particular
no associated Lagrangian: there are always at least two adjacent nodes $a$ and
$b$ in the Dynkin diagram of $\mathfrak{g}$ for which the roots $\alpha_{a}$
and $\alpha_{b}$ have different lengths, and the representation theory of the
supersymmetry algebra does not allow for a hypermultiplet to exist for these
nodes.

Not all is lost, however. Most notably, it was conjectured in [334, 335] that
for any simple Lie group $G$, the moduli space of $G$-instantons on
$\mathbb{C}^{2}$ could be studied via the Coulomb branch of an auxiliary
3-dimensional $\widehat{\mathfrak{g}}$-shaped quiver (Note the fundamental
matter has $\mathcal{N}=4$ supersymmetry in that setup instead of our
$\mathcal{N}=2$ Drinfeld flavors). One implication is that even though a non
simply-laced quiver does not make sense as a gauge theory, the notion of a
Coulomb branch as a variety could potentially still be well-defined. The
conjecture was rigorously established in [272]: even though $\mathfrak{g}$ is
non simply-laced, its Cartan matrix is symmetrizable, and a meaningful
$\mathfrak{g}$-type Coulomb branch is defined via a folding action on the
Coulomb branch of a $\mathfrak{g}_{\text{o}}$-theory, whose Dynkin graph is
simply-laced. Namely, folding is realized via the $\mathfrak{a}$-action on the
$\mathfrak{g}_{\text{o}}$-type Coulomb branch, whose fixed point gives a
definition of a $\mathfrak{g}$-type Coulomb branch variety. For physical
considerations on folding and other subtleties in discrete gauging, see the
recent work [336].

Physically, let $\psi_{a}(X_{1})$ be a vector multiplet field of the simply-
laced theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$, where the index
$a\in\\{1,\ldots,r\\}$ denotes the node of the $\mathfrak{g}_{\text{o}}$-type
quiver, and $X_{1}$ is a complex coordinate on $\mathbb{C}_{q}$, where the D3
branes are supported. We require $\psi_{a}$ to obey

$\psi_{a}(e^{2\pi i}X_{1})=\mathfrak{a}\cdot\psi_{a}(X_{1})\;,\qquad
a\in\\{1,\ldots,r\\}\;,$ (7.6)

where $\mathfrak{a}\in A$ is acting on the field $\psi_{a}$ on the right-hand
side.

In order to define the “vector multiplet” of a tentative non simply-laced
theory $T^{3d}_{\mathfrak{g}}$, we consider a subset of the fields $\psi_{a}$
in $T^{3d}_{\mathfrak{g}_{\text{o}}}$ which split into two sets: first, there
exist fields $\psi_{a}$ which are fixed points of the $\mathfrak{a}$-action.
We call these fields $\psi_{a,long}(X_{1})$, where $a$ now labels a long root
of $\mathfrak{g}$. In other words, for such fields,

$\displaystyle\psi_{a,long}(e^{2\pi i}X_{1})=\psi_{a,long}(X_{1})\;.$ (7.7)

Second, we organize the fields of $T^{3d}_{\mathfrak{g}_{\text{o}}}$ which
belong in $A$-orbits of length $\mathfrak{n}_{\mathfrak{g}}>1$ into a sum
$\psi_{a}+\mathfrak{a}\cdot\psi_{a}+\ldots+\mathfrak{a}^{\mathfrak{n}_{\mathfrak{g}}-1}\cdot\psi_{a}$.
The sum gives a well-defined single-valued field on the
$\mathfrak{n}_{\mathfrak{g}}$-cover of
$S^{1}_{q}/\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}\times\mathbb{R}$, with
covering map $X_{1}\mapsto X_{1}^{\mathfrak{n}_{\mathfrak{g}}}$; we call such
a field $\psi_{a,short}$, with $a$ labeling a short root of $\mathfrak{g}$.
Note in particular that the ranks of the gauge groups $U(N_{a})$ belonging in
the same $A$-orbit must be equal in the original
$T^{3d}_{\mathfrak{g}_{\text{o}}}$ theory.

The same considerations extend to chiral fields in adjoint and fundamental
representations on a given node $a$. For instance, the chiral matter of
$T^{3d}_{\mathfrak{g}}$ originates from D3/D5 strings with the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist.

The above considerations suggest a definition of what we mean by a
$T^{3d}_{\mathfrak{g}}$ theory: when $a$ labels a long root in $\mathfrak{g}$,
the gauge and matter field content on node $a$ is identical to the one in the
original theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$. When $a$ labels a short
root in $\mathfrak{g}$, the gauge and matter field content on node $a$ is
derived from $T^{3d}_{\mathfrak{g}_{\text{o}}}$ by restricting to $A$-orbits
of length $\mathfrak{n}_{\mathfrak{g}}$ as above. In particular, all gauge and
flavor groups which belong in the same
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-orbit will have identical ranks.
Between two long nodes or two short nodes, the bifundamental matter is well-
defined with a Lagrangian origin found in $T^{3d}_{\mathfrak{g}_{\text{o}}}$.
Between a long node and a short node, the “bifundamental matter” is at best
exotic and non-Lagrangian, and at worst physically ill-defined. At any rate,
the construction makes it clear that the notion of $T^{3d}_{\mathfrak{g}}$
Coulomb branch is sensible as a variety, at least classically.

It turns out that given a general symmetrizable Cartan matrix [337], there
exist different possible definitions of a Coulomb branch, each with distinct
advantages; for instance, a construction939393Think of a simply-laced theory
$T^{3d}_{\mathfrak{g}_{\text{o}}}$ supported on
$S^{1}_{\mathcal{C}^{\prime}}(R_{\mathcal{C}}^{\prime})\times\mathbb{C}_{q}$
as an assignment of various vector bundles over the ring
$R=\mathbb{C}[X_{1}]$, with $X_{1}$ a coordinate on $\mathbb{C}_{q}$ [338].
Then, define a “fractional quiver” $T^{3d}_{\mathfrak{g}}$ by assigning a
different ring $R_{a}=\mathbb{C}[X_{1,a}]$ to each node $a=1,\ldots,r$ in the
Dynkin graph of $\mathfrak{g}$ [29, 339]. Our D3 branes in the string theory
background (7.3) realize a particular instance of this construction.  proposed
by Nakajima-Weekes [339] lends itself to various geometric interpretations,
such as the identification of $\mathfrak{g}$-type Coulomb branches as
generalized slices of the affine Grassmanian. For the non simply-laced Lie
algebras $\mathfrak{g}=B_{r},C_{r},G_{2},F_{4}$ of interest to us, the
different definitions of the Coulomb branch all happen to yield isomorphic
varieties, though this need not be true for more general symmetrizable Cartan
matrices.

In the Physics literature, $\mathfrak{g}$-type Coulomb branches have been
defined for supersymmetric quiver gauge theories from 1 to 5 dimensions, in a
similar fashion to what we presented here in 3 dimensions [340, 341, 13, 146,
29, 173].

For instance, the quiver theory $T^{5d}_{\mathfrak{g}}$ is similarly defined
as the effective theory on the $n$ D5 branes in the little string background
(7.5). We let $\widetilde{\psi}_{a}(X_{1},X_{2})$ denote a field of the
simply-laced theory $T^{5d}_{\mathfrak{g}_{\text{o}}}$ on node $a$ of the
quiver, with $(X_{1},X_{2})$ coordinates on
$\mathbb{C}_{q}\times\mathbb{C}_{t}$, where our D5 branes are supported. We
submit the background to the $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist
as before, meaning there is a monodromy around $X_{1}=0$ (but not around
$X_{2}=0$), and we obtain

$\widetilde{\psi}_{a}(e^{2\pi
i}X_{1},X_{2})=\mathfrak{a}\cdot\widetilde{\psi}_{a}(X_{1},X_{2})\;,\qquad
a\in\\{1,\ldots,r\\}\;,$ (7.8)

as in (7.6). The fields $\widetilde{\psi}_{a,long}$ and
$\widetilde{\psi}_{a,short}$ are defined in the same way as their 3d
counterparts $\psi_{a,long}$ and $\psi_{a,short}$.

In practice, the gauge content of a theory $T^{5d}_{\mathfrak{g}}$ can be
defined by first specifying the fundamental hypermultiplet content: a vector
of $r$ 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 (7.4), which either has no solution or a unique solution
$(n_{1},n_{2},\ldots,n_{r})\in\left(\mathbb{Z}^{+}\right)^{r}$. The solution,
whenever it exists, defines the gauge content of $T^{5d}_{\mathfrak{g}}$,
meaning the ranks of the gauge groups are uniquely fixed by the ranks of the
flavor groups, together with the constraint.

###  7.3 Supersymmetric indices

It is clear that a Higgs branch $X_{\mathfrak{g}}$ cannot be sensibly defined,
at least not in the usual sense. For instance, the dimension of this tentative
branch would necessarily come out a non-integer rational number, due to the
problematic bifundamental hypermultiplet. Nevertheless, it follows from the
work of Nakajima [342] that a K-theoretic count of quasimaps
$\mathbb{P}^{1}\rightarrow X_{\mathfrak{g}}$ _is_ well-defined949494Reversing
the argument, the existence of those quasimaps suggests that there should
exist a more general definition of a Higgs branch, which would reduce to the
usual one in the simply-laced case.. For us, $X_{\mathfrak{g}}$ would have
been the moduli space of “vortices” of the non simply-laced quiver theory
$T^{5d}_{\mathfrak{g}}$, which is ill-defined, but the half-index
${\mathcal{Z}}(T^{3d}_{\mathfrak{g}})$ has combinatorial meaning [13, 146,
343, 173]959595Physically, we would say that Coulomb branch localization is
well-defined on $T^{3d}_{\mathfrak{g}}$ and yields the integral form
${\mathcal{Z}}(T^{3d}_{\mathfrak{g}},{\bf N})$ of the half-index.. By abuse of
notation, we will sometimes refer to $T^{3d}_{\mathfrak{g}}$ as the theory on
the vortices of $T^{5d}_{\mathfrak{g}}$, keeping in mind that by “vortex” on a
“Higgs branch,” we formally mean the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist of the physically sound
simply-laced version.

Various enumerative counts of BPS solitons in non-simply laced quiver theories
have been computed in this fashion: instanton partition functions on
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$ from
index theorems [146, 29] a generalized ADHM quantum mechanics [173]969696It
was argued in that work that a bifundamental hypermultiplet between a short
node and a long node should be reproduced by the contribution of
$\mathfrak{n}_{\mathfrak{g}}$ $(0,2)$ usual Fermi multiplets in the ADHM
quantum mechanics, as many as the number of arrows between the two nodes of
the non simply-laced quiver. It would be important to derive this conjecture
from first principles., the Hilbert series of $G$-type instantons on
$\mathbb{C}^{2}$ [335], and various partition functions on $S^{3}$ or
superconformal indices on $S^{2}\times S^{1}$, $S^{3}\times S^{1}$ and Lens
space [341].

The 3d half-index is once again defined as a trace, but only counting BPS
states in orbits of the automorphism group
$A=\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$, whether they are bulk or
boundary degrees of freedom. Our twisted background (7.3) implies that the
index will now be a Taylor series in the fugacity
$q^{1/(2\,\mathfrak{n}_{\mathfrak{g}})}$ in our previous conventions (3.20)
(with $\mathfrak{n}_{\mathfrak{g}}=1,2,$ or $3$). In the simply-laced case,
where $\mathfrak{n}_{\mathfrak{g}}=1$, the index was a Taylor series in
$q^{1/2}$. We decide to keep this feature universal across all quivers,
including the non simply-laced ones, by simply rescaling $q\rightarrow
q^{\mathfrak{n}_{\mathfrak{g}}}$:

${\mathcal{Z}}(T^{3d}_{\mathfrak{g}})={\rm
Tr}\left[(-1)^{F}\,q^{\mathfrak{n}_{\mathfrak{g}}(J+\frac{V}{2})}\;t^{\frac{A-V}{2}}\;z^{\mathfrak{t}_{C}}\;x^{\mathfrak{t}_{H}}\right]\;\;.$
(7.9)

The only effect of this rescaling is to give a different $q$-weight to the BPS
states according to whether they are associated to long or short roots in the
$\mathfrak{g}$-quiver.

The index is evaluated as before in terms of $q$-Pochhammer symbols and Theta-
functions, with the following modifications

$\left(x_{a}\,;q\right)_{\infty}\;\longrightarrow\;\left(x_{a}\,;q^{r_{a}}\right)_{\infty}\;,\qquad\;\;\;\Theta\left(x_{a}\,;q\right)\;\longrightarrow\;\Theta\left(x_{a}\,;q^{r_{a}}\right)\;.$
(7.10)

where

$r_{a}=\begin{cases}\mathfrak{n}_{\mathfrak{g}}&\text{when node $a$ labels a
long root in the $\mathfrak{g}$-quiver\ ,}\\\ 1&\text{when node $a$ labels a
short root in the $\mathfrak{g}$-quiver\ .}\end{cases}$ (7.11)

Note that by definition of the Theta function, the “flips” of boundary
conditions from Neumann to Dirichlet (and vice-versa) in the index can be
extended without trouble to the non simply-laced case. For instance, a 3d
chiral field on node $a$ with D boundary conditions is dual to a 3d chiral
multiplet with N boundary conditions after introducing a boundary
$\mathcal{N}=(0,2)$ Fermi multiplet:

$\left(q^{r_{a}}/x_{a}\,;q^{r_{a}}\right)_{\infty}=\frac{\Theta\left(x_{a}\,;q^{r_{a}}\right)}{\left(x_{a}\,;q^{r_{a}}\right)_{\infty}}\;.$
(7.12)

For the bifundamental matter contributions between two gauge nodes $a$ and
$b$, we replace

$\left(x_{a}/x_{b}\,;q\right)_{\infty}\;\longrightarrow\;\left(x_{a}/x_{b}\,;q^{r_{ab}}\right)_{\infty}\;,\qquad\;\;\;\Theta\left(x_{a}/x_{b}\,;q\right)\;\longrightarrow\;\Theta\left(x_{a}/x_{b}\,;q^{r_{ab}}\right)$
(7.13)

where $r_{ab}=\text{gcd}(r_{a},r_{b})$ is the greatest common divisor of
$r_{a}$ and $r_{b}$. Explicitly, the bifundamental matter contribution to the
index takes the form

$\displaystyle I_{a,b}^{bif}(y_{a,i},y_{b,j})=\prod_{1\leq i\leq
N_{a}}\prod_{1\leq j\leq
N_{b}}\left[\frac{(\sqrt{q^{r_{ab}}\,t}\,y_{a,i}/y_{b,j};q^{r_{ab}})_{\infty}}{(\sqrt{q^{r_{ab}}/t}\,y_{a,i}/y_{b,j};q^{r_{ab}})_{\infty}}\right]^{\Delta_{ab}}$
(7.14)

between two gauge nodes $a$ and $b$ in the Dynkin diagram of
$T^{3d}_{\mathfrak{g}}$. Note in particular that if the roots $\alpha_{a}$ and
$\alpha_{b}$ have different lengths, the above contribution is non-Lagrangian
by construction.

Repeating the derivation of Section 3, we find that after folding, the 3d
half-index with exceptional Dirichlet boundary conditions
${\mathcal{Z}}(T^{3d}_{\mathfrak{g},{\bf D_{EX}}})$ is well-defined, and
matches the expression for the $\mathcal{W}_{q,t}(\mathfrak{g})$-algebra
magnetic block analytic in the chamber $\mathfrak{C}_{C}$ [146, 173]. It is
likewise expected that
${\mathcal{Z}}(T^{3d}\mathfrak{g}_{\text{o}},\,\bf{N}_{EN})$ should be well-
defined, but the geometric interpretation of elliptic stable envelopes is not
well-established in the non simply-laced case, so we will not comment on it
further. For analogous results in the unramified case, see [13].

The various blocks and 3d half-indices can be derived from the instanton
quantum mechanics of the 5-dimensional theory $T^{5d}_{\mathfrak{g}}$, just as
in the simply-laced case. The non simply-laced instanton quantum mechanics is
again defined via folding, by simply imposing the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist on our previous simply-laced
ADHM quantum mechanics.

At the level of the Witten index, this translates to rescaling the fugacity
$q\rightarrow q^{\mathfrak{n}_{\mathfrak{g}}}$, as we have done in the 3d
index:

${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}})={\rm
Tr}\left[(-1)^{F}\,q^{\mathfrak{n}_{\mathfrak{g}}(J_{1}+\frac{R}{2})}\;t^{J_{2}-\frac{R}{2}}\;{\mathfrak{m}}^{\Sigma}\right]\;\;.$
(7.15)

The index is evaluated as a sum over $\sum_{a=1}^{r}n_{a}$ 2-dimensional
partitions, one for each $U(1)$ equivariant Coulomb parameter:

$\\{\overrightarrow{\boldsymbol{\mu}}\\}=\\{\boldsymbol{\mu}_{a,i}\\}_{a=1,\ldots,r\,;\;\;i=1,\ldots,n_{a}}\;.$
(7.16)

One computes

${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}})=\sum_{\\{\overrightarrow{\boldsymbol{\mu}}\\}}\;\prod_{a=1}^{r}z_{a}^{\sum_{i=1}^{n_{a}}{\left|\boldsymbol{\mu}_{a,i}\right|}}\,{\boldsymbol{Z}}^{5d,vec}_{a}\cdot{\boldsymbol{Z}}^{5d,fund}_{a}\cdot{\boldsymbol{Z}}^{5d,CS}_{a}\cdot\prod^{n}_{b>a}{\boldsymbol{Z}}^{5d,bif}_{a,b}\,,$
(7.17)

which has the same functional form as the simply-laced index, but the various
factors now explicitly depend on root lengths in the $\mathfrak{g}$-quiver.

For example, the factor

$\displaystyle{\boldsymbol{Z}}^{5d,bif}_{a,b}=\prod_{i=1}^{n_{a}}\prod_{j=1}^{n_{b}}\left[\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}\left(f^{bif}_{a,b}\,\frac{e_{a,i}}{e_{b,j}}\,;q^{r_{ab}}\right)\right]^{\Delta_{a,b}}\;.$
(7.18)

stands for the contribution of a 5d bifundamental hypermultiplet between nodes
$a$ and $b$ in the $\mathfrak{g}$-quiver, with corresponding mass
$f^{bif}_{a,b}$. Note that the argument of the function
$\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}$ now takes into
account whether $\alpha_{a}$ and $\alpha_{b}$ are long or short roots:

${\mathcal{N}}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}(Q\,;q^{r_{ab}})\equiv\prod\limits_{k,k^{\prime}=1}^{\infty}\frac{\big{(}Q\,q^{r_{a}\boldsymbol{\mu}_{a,i,k}-r_{b}\boldsymbol{\mu}_{b,j,k^{\prime}}}\,t^{k^{\prime}-k+1}\,;q^{r_{ab}}\big{)}_{\infty}}{\big{(}Q\,q^{r_{a}\boldsymbol{\mu}_{a,i,k}-r_{b}\boldsymbol{\mu}_{b,j,k^{\prime}}}\,t^{k^{\prime}-k}\,;q^{r_{ab}}\big{)}_{\infty}}\,\frac{\big{(}Q\,t^{k^{\prime}-k}\,;q^{r_{ab}}\big{)}_{\infty}}{\big{(}Q\,t^{k^{\prime}-k+1}\,;q^{r_{ab}}\big{)}_{\infty}}\,.$
(7.19)

Our definition of the $\mathcal{W}_{q,t}(\mathfrak{g})$-algebras for
$\mathfrak{g}$ non simply-laced in Section 2 follows the conventions of [23,
58]; this fixes the bifundamental masses $f^{bif}_{a,b}$ between adjacent
nodes $a$ and $b$ to

$f^{bif}_{a,b}=\sqrt{q^{r_{ab}}/t}\;.$ (7.20)

For example, let $\mathfrak{g}_{\text{o}}=E_{6}$ and the outer automorphism
group be $A=\mathbb{Z}_{2}$. The simple roots in $\mathbb{Z}_{2}$-orbits of
length 1 become the long simple roots of $\mathfrak{g}=F_{4}$, which we label
as $\alpha_{1}$ and $\alpha_{2}$. in $\mathfrak{g}=F_{4}$. The remaining
$E_{6}$ simple roots are in $\mathbb{Z}_{2}$-orbits of length 2, they become
the short simple roots of $\mathfrak{g}=F_{4}$, which we label as $\alpha_{3}$
and $\alpha_{4}$. Then, in our conventions, the bifundamental masses in
$T^{5d}_{F_{4}}$ are fixed to be

$\displaystyle f^{bif}_{1,2}=\sqrt{q^{2}/t}\;,$ $\displaystyle
f^{bif}_{2,3}=\sqrt{q/t}\;,$ $\displaystyle f^{bif}_{3,4}=\sqrt{q/t}\;.$
(7.21)

The equality of 3d and 5d partition functions is a consequence of gauge-vortex
duality in the presence of the outer automorphism twist. A proof was presented
for handsaw quivers in [146], and is essentially the same for all Drinfeld
quivers, so we will not repeat it here. Instead, we will point out the
essential new features to make contact with the quantum algebras.

We start by freezing each 5d Coulomb modulus $e_{a,i}$ to a hypermultiplet
mass $x^{-1}_{d,s}$, and moreover turn on $N_{a,i}$ units of vortex flux as

$e_{a,i}=x^{-1}_{d,s}\;t^{\frac{\\#_{a,i}+1}{2}}q^{\frac{-\widetilde{\\#}_{a,i}-1}{2}}\,t^{N_{a,i}}\;.$
(7.22)

The factors $t^{\frac{\\#_{a,i}+1}{2}}$ and
$q^{\frac{-\widetilde{\\#}_{a,i}-1}{2}}$ are refinements due to the
$\Omega$-background, which is further corrected by
$q^{\frac{-\widetilde{\\#}_{a,i}}{2}}$. In our conventions, $\\#_{a,i}$ and
$\widetilde{\\#}_{a,i}$ are non-negative integers; $\\#_{a,i}$ records the
length of the “string” from node $a$ where the Coulomb modulus $e_{a,i}$ is
located, to the node where the hypermultiplet mass $x^{-1}_{d,s}$ is located.
In the simply-laced case, we had $\widetilde{\\#}_{a,i}=\\#_{a,i}$, but the
presence of the twist as we go around $S^{1}_{q}$ implies that $q$ and $t$ now
scale differently. From the 5d bulk perspective, the locus (7.22) describes
the Coulomb branch of $T^{5d}_{\mathfrak{g}}$, but only on an integer-valued
lattice in powers of $t$, due to the transverse $\Omega$-background on
$\mathbb{C}_{t}$.

The integers $\\#_{a,i}$ and $\widetilde{\\#}_{a,i}$ are uniquely determined
from the requirement that the instanton partition function of
$T^{5d}_{\mathfrak{g}}$ should truncate to the vortex partition function of
$T^{3d}_{\mathfrak{g}}$. Namely, for $N$ a positive integer,

$\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}\left(q^{r_{b}}\,t^{-N-1}\,;q^{r_{ab}}\right)=0\;\;\text{unless}\;\;l\left(\boldsymbol{\mu}_{b,j}\right)\leq
l\left(\boldsymbol{\mu}_{a,i}\right)+N$ (7.23)

where $l\left(\boldsymbol{\mu}_{b,j}\right)$ is the number of rows in a
partition $\boldsymbol{\mu}_{b,j}$, labeling the equivariant Coulomb modulus
$e_{b,j}$. But after specializing all the Coulomb moduli as (7.22), the
fundamental matter factor ${\boldsymbol{Z}}^{5d,fund}_{b}$ in the summand of
${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}})$ will necessarily contain a
factor

$N_{\boldsymbol{\emptyset}\,\boldsymbol{\mu}_{b,j}}\left(\sqrt{q^{2\,r_{b}}/t^{2}}\;t^{-N_{b,j}}\,;q^{r_{b}}\right)\;,$
(7.24)

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. A new subtlety unique to the non simply-laced case is that the
definition of the $\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}$
function depends on whether $a$ and $b$ are indices for long or short roots of
$\mathfrak{g}$. Making use of the trivial identity

$\mathcal{N}_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}(Q\,;q)=\prod_{k=0}^{r_{ab}-1}N_{\boldsymbol{\mu}_{a,i}\boldsymbol{\mu}_{b,j}}(q^{k}\,Q\,;q^{r_{ab}})\;,$
(7.25)

one proceeds to show that (7.23) is responsible for the truncation of _all_
partitions $\overrightarrow{\boldsymbol{\mu}}$, and find:

$\displaystyle{\mathcal{Z}}(T^{3d}_{\mathfrak{g},\bf
D_{EX}})=c_{3d}\cdot{\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}})_{e_{a,i}\propto\;x^{-1}_{d,s}\,t^{N_{a,i}}}\,.$
(7.26)

The normalization constant $c_{3d}$ stands for the empty partition
contribution to the partition function. In particular, the Higgsing procedure
implies the following conjecture:

######  Proposition 7.1.

Let $V_{a,\lambda_{s}}$ be a weight space appearing in the eigenspace
decomposition of a $U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ fundamental
representation $\widehat{V_{a}}=\bigoplus_{\lambda_{s}}\,V_{a,\lambda_{s}}$ as
$U_{\hbar}(^{L}\mathfrak{g})$-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 $U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$, for
$\mathfrak{g}=B_{r\leq 3}$, $\;\mathfrak{g}=C_{r\leq 3}$, and
$\mathfrak{g}=G_{2}$, meaning we identified the Drinfeld polynomials of all
weights in those cases. For $\mathfrak{g}=F_{4}$, we only performed a partial
check for some of the weights. Once again, we found that the weights are
typically realized in a different fashion compared to the Frenkel-Mukhin
algorithm.

##  8 Surface defects in $\mathcal{N}=4$ Super Yang-Mills and S-duality

Kapustin and Witten showed that the geometric Langlands correspondence can be
formulated at the level of supersymmetric gauge theory [30]. Namely, consider
maximally supersymmetric $\mathcal{N}=4$ Yang-Mills theory with a (simple and
connected) gauge group $G$ on a four-manifold $M_{4}$ that locally is a
product of Riemann surfaces, $M_{4}={\cal C}\times M_{2}$. Here, $\mathcal{C}$
is the Riemann surface on which the geometric Langlands program “lives,” it is
a fiber to the normal bundle to $M_{2}$. Among the various fields of the
theory, one finds a connection $A$ on a $G$-bundle $E\rightarrow\mathcal{C}$,
and a field $\Phi$ that is a one-form on $\mathcal{C}$ with values in
$\text{ad}(E)$.

More precisely, $\Phi$ arises from a Landau-Ginzburg-type twist on some of the
scalar fields of $\mathcal{N}=4$ super Yang-Mills. One finds that a necessary
condition for $A$ and $\Phi$ to preserve supersymmetry is that they should
solve Hitchin’s equations [344]:

$\displaystyle F-\phi\wedge\phi=0$ $\displaystyle D\Phi=0$ $\displaystyle
D\star\Phi=0$ (8.1)

Here, we denoted the gauge-covariant exterior derivative by $D=d+[A,\cdot]$,
and the Hodge star operator by $\star$. One can think of these equations as a
reduction of the self-dual Yang-Mills equations from four to two dimensions.
In this picture, $\Phi$ corresponds to the two components of the gauge field
lost in the dimensional reduction from four to two dimensions.

###  8.1 Tamely ramified Hitchin system: a short review

For a mathematical treatment, we refer the reader to earlier works by Simpson
[345, 346], Konno [347] and Nakajima [348]; see also the review by Donagi and
Pantev [349]. For conformal field theory considerations, see Frenkel [166,
350].

We are interested in a class of solutions to the above system with a
prescribed singularity behavior at punctures on $\mathcal{C}$, as studied by
Gukov and Witten [31]. Explicitly, we choose local Euclidean coordinates
$(x^{2},x^{3})\in\mathbb{R}^{2}$ near a point of ramification on
$\mathcal{C}$, say $x^{2}=x^{3}=0$, and further change to polar coordinates
$x^{2}+i\,x^{3}=r\,e^{i\theta}$. A surface defect is defined at
$x^{2}=x^{3}=0$ by specifying the singular profile of the fields
$A=A_{2}+i\,A_{3}$ and $\Phi=\Phi_{2}+i\,\Phi_{3}$ near there. A natural
ansatz is to require $\Phi$ to have a pole of some prescribed order at $r=0$.
One speaks of _tame_ ramification in the geometric Langlands program when
$\Phi$ has at most a _simple_ pole (of order 1) at each ramification point on
$\mathcal{C}$. All the defects of our paper are tamely ramified in that sense.
In contrast, when $\Phi$ is defined to have a pole _not_ of order 1 at some
ramification point, one speaks instead of _wild_ ramification [351].

Writing down an explicit ansatz is greatly simplified if one further demands
the defects to be superconformal and preserve as much supersymmetry as
possible. A conformal ansatz will be invariant under scaling and rotations in
the $\mathbb{R}^{2}$-plane, and will take the form

$\displaystyle A=a(r)\,d\theta+b(r)\,\frac{dr}{r}+\ldots$
$\displaystyle\Phi=c(r)\,\frac{dr}{r}+d(r)\,d\theta+\ldots\;,$ (8.2)

where the dots “$\ldots$” stand for regular terms that will not concern us.
One can always apply a gauge transformation to $A$ which sets $d(r)=0$.
Moreover, the invariance under scaling further constraints $a(r),b(r),$ and
$c(r)$ to be constants. We write these constants as $a(0)=\alpha$,
$b(0)=\beta$, and $c(0)=\gamma$. Hitchin’s equations imply that
$\alpha,\beta,\gamma$ all commute among themselves, so these constants are
naturally valued in the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$.
Near $r=0$, one can write

$\displaystyle A=\alpha\,d\theta+\ldots$
$\displaystyle\Phi=\beta\,\frac{dr}{r}-\gamma\,d\theta+\ldots\;.$ (8.3)

The surface defect breaks the gauge group $G$ to a maximal torus $T$, or
rather a subgroup $L$ which contains $T$, called a Levi subgroup. Which Levi
subgroup we get depends on the type of defect we insert; for instance, the
Levi subgroup $L$ coincides with $T$ whenever the defect is “generic”979797The
notion of generic here coincides with the conformal limit of a generic defect
from Section 2.1.5. This is not a coincidence, but a consequence of the
(local) Alday-Gaiotto-Tachikawa correspondence [57].. The global R-symmetry
group likewise breaks as $SO(6)\rightarrow SO(4)\times SO(2)$.

The moduli space of solutions to Hitchin’s equations (8) is called the Hitchin
moduli space. In the absence of singularities, it is a finite-dimensional
hyper-Kähler manifold. In particular, there exists a $\mathbb{CP}^{1}$ worth
of complex structures on it. In some complex structure we call $I$, a solution
to Hitchin’s equations on $\mathcal{C}$ can be viewed as a pair $(E,\varphi)$,
with $E$ a holomorphic $G$-bundle and $\varphi$ a holomorphic section of the
bundle $K\otimes\text{ad}(E)$, with K the canonical bundle on $\mathcal{C}$.
This data $(E,\varphi)$ goes by the name of Higgs bundle.

When we further impose that $A$ and $\phi$ have the singular behavior (8.1) at
points $\\{p_{1},\ldots,p_{L}\\}$, the moduli space of solutions is still
hyper-Kähler, and in complex structure $I$ it is known as the moduli space of
ramified Higgs bundle.

The holomorphic structure can be justified as follows: First consider the
complex-valued connection ${\mathbb{A}}=A+i\Phi$, which is flat by Hitchin’s
equations (8). The gauge-covariant exterior derivative can be written as
$d_{\mathbb{A}}=\partial_{\mathbb{A}}+\overline{\partial}_{\mathbb{A}}$, with
$\partial_{\mathbb{A}}$ a derivative of (1,0) type, and
$\overline{\partial}_{\mathbb{A}}$ a derivative of (0,1) type. It is this
operator $\overline{\partial}_{\mathbb{A}}$ which makes the bundle $E$
holomorphic, at least away from the points $\\{p_{1},\ldots,p_{L}\\}$. For
instance, consider a neighborhood of a point $p$ located, say, at $z=0$, where
$z=x^{1}+i\,x^{2}$; then the operator $\overline{\partial}_{\mathbb{A}}$
simplifies to $d\overline{z}\,\frac{\partial}{\partial\overline{z}}$. We
further decompose the Higgs field as $\Phi=\varphi^{(1,0)}+\varphi^{(0,1)}$,
with $\varphi\equiv\varphi^{(1,0)}$ of (1,0) type, and $\varphi^{(0,1)}$ of
(0,1) type. It follows that $(E,\varphi)$ is a Higgs bundle. In complex
structure $I$, we can rewrite the solution (8.1) near $z=0$ as

$\displaystyle A=\alpha\,d\theta+\ldots$
$\displaystyle\varphi^{(1,0)}=(\beta+i\,\gamma)\,\frac{dz}{z}+\ldots\;.$ (8.4)

The complex structure $I$ depends holomorphically on $\beta+i\,\gamma$, but
does not depend on $\alpha$. At the same time, in cohomology, the (1,1) Kähler
form $\omega_{I}$ depends linearly on $\alpha$, but does not depend on $\beta$
or $\gamma$.

Kähler moduli are complexified due to supersymmetry, so we expect additional
real parameters $\eta$ to be present besides the real Kähler moduli in
$\alpha$. The existence of $\eta$ can be qualitatively motivated by noting
that the 2-form field strength $F$ serves a dual purpose in four dimensions:
on the one-hand, $F$ must take the form

$F=2\pi\alpha\,\delta_{M_{2}}+\ldots\;,$ (8.5)

with $\delta_{M_{2}}$ is a 2-form delta-function Poincaré dual to the support
of the defect $M_{2}\subset M_{4}$. On the other-hand, we can consider
integrating the 2-form $F$ with support on the manifold $M_{2}$ itself,
resulting in a coupling

$\text{exp}\left(i\,\eta\int_{M_{2}}F\right)\;.$ (8.6)

An analogous phenomenon occurs in the study of line defects: (8.5) is the
surface defect analog of a magnetic ’t Hooft loop in $M_{4}$, while (8.6) is
the analog of an electric Wilson loop. In the case where the gauge group is
$G=U(1)$, we recognize the integrand in (8.6) as the first Chern class of the
$U(1)$ bundle over $M_{2}$, whose degree is measured by the integral.
Therefore, $\eta$ should be thought of as a two-dimensional theta angle, and
the exponential should be included as an additional factor inside the path
integral. If $G$ is nonabelian of rank $r$, the parameter $\eta$ instead
stands for a vector of $r$ two-dimensional theta-angles, one for each abelian
factor of the maximal torus $T\cong U(1)^{r}$.

The parameters $\alpha$, $\beta$ and $\gamma$ take values in $\mathfrak{h}$,
but a gauge transformation will shift $\alpha$ by an elements of the coroot
lattice, meaning $\alpha$ actually takes values in
$\mathfrak{h}/\Lambda_{cort}\cong T$, the maximal torus of $G$. The parameter
$\eta$ takes values in $\mathfrak{h}^{\vee}$, the Cartan subalgebra of the
Langlands dual algebra ${}^{L}\mathfrak{g}$; a gauge transformation will shift
$\eta$ by an element of the weight lattice, so it is valued in
$\mathfrak{h}/\Lambda_{wt}\cong\,^{L}T$, the maximal torus of ${}^{L}G$. All
parameters are furthermore invariant under Weyl group action, so in the end,

$(\alpha,\beta,\gamma,\eta)\in(T\times\mathfrak{h}\times\mathfrak{h}\times^{L}T)/W\;.$
(8.7)

####  8.1.1 The S-transformation

S-duality is a symmetry of $\mathcal{N}=4$ Yang-Mills which exchanges the
gauge group $G$ with its Langlands dual ${}^{L}G$. Let
$\tau^{4d}=\frac{\theta}{2\pi}+i\frac{4\pi}{g^{2}_{4d}}$ denote the gauge
coupling of $G$-type Yang-Mills, where $g_{4d}$ is the coupling constant and
$\theta$ the theta angle. Likewise, let ${}^{L}\tau^{4d}$ be the gauge
coupling of the S-dual ${}^{L}G$-type Yang-Mills theory. At the level of their
Lie algebras, S-duality acts as

$\text{S}:\;\;\;(\mathfrak{g},\tau^{4d})\longleftrightarrow(^{L}\mathfrak{g},^{L}\tau^{4d})\;,\qquad^{L}\tau=\frac{-1}{\mathfrak{n}_{\mathfrak{g}}\,\tau^{4d}}\;.$
(8.8)

Turning our attention to the surface defect, let $(\alpha,\beta,\gamma,\eta)$
be the set of defect parameters of the $G$-type theory, and
$(^{L}\alpha,^{L}\beta,^{L}\gamma,^{L}\eta)$ be the set of defect parameters
of the ${}^{L}G$-type theory. The Higgs field residues $(\beta,\gamma)$
transform almost trivially under S-duality: they are simply rescaled as

$(^{L}\beta,^{L}\gamma)=\text{Im}(\tau)\,(\beta^{*},\gamma^{*})\;.$ (8.9)

The notation $(\beta^{*},\gamma^{*})$ on the right-hand side is necessary
since the Higgs fields $\phi$ and ${}^{L}\phi$ are not valued in the same
space, so they cannot be directly compared. We therefore introduce a linear
map989898Our map is rescaled by a factor of $\mathfrak{n}_{\mathfrak{g}}$
compared to the map introduced in [31], because there the short roots have
length squared $2$, whereas for us they have length squared
$2/\mathfrak{n}_{\mathfrak{g}}$. It is perfectly fine to use the same map as
[31], at the expense of multiplying the right-hand side of (8.9) by an extra
factor of $\mathfrak{n}_{\mathfrak{g}}$, and understanding
$\alpha\mapsto\alpha^{*}$ as a map from the lattice $\Lambda_{rt}$ to the
lattice $\frac{1}{\mathfrak{n}_{\mathfrak{g}}}\cdot\Lambda_{cort}$ instead.
from the dual Cartan subalgebra $\mathfrak{h}^{\vee}$, where the roots of
$\mathfrak{g}$ are valued, to the Cartan subalgebra $\mathfrak{h}$, where the
coroots of $\mathfrak{g}$ are valued. The map acts on simple roots as
$\alpha\mapsto\alpha^{*}$, defined such that if $\alpha_{l}$ is a long root,
then $\alpha_{l}^{*}$ is $\mathfrak{n}_{\mathfrak{g}}$ times the short coroot
$\alpha^{\vee}_{s}$, and if $\alpha_{s}$ is a short root, then
$\alpha_{s}^{*}$ is the long coroot $\alpha^{\vee}_{l}$. This gives a well-
defined map from $\Lambda_{rt}$ to $\Lambda_{cort}$. The above map naturally
extends to a map of Higgs fields $\phi\mapsto\phi^{*}$ (and its residues at
the pole $(\beta,\gamma)\mapsto(\beta^{*},\gamma^{*})$), where the $a$-th
component $\phi_{a}$ is mapped to $\phi^{*}_{a}$ as above, according to
whether $a$ labels a long or a short root of $\mathfrak{g}$.

Note that when $\mathfrak{g}$ is simply-laced, roots and coroots coincide, and
the map $\alpha\mapsto\alpha^{*}$ is therefore the identity map, which
simplifies the discussion.

Meanwhile, the definition of the parameters $\alpha$ and $\eta$ reviewed in
last Section suggests that S-duality should act on them in a more nontrivial
way, as a proper electro-magnetic duality. In the abelian case $G=U(1)$, Gukov
and Witten prove [31] that this is indeed the case, and conjecture it should
still hold when $G$ is nonabelian. Explicitly, the conjecture reads

$(^{L}\alpha,^{L}\eta)=(\eta,-\alpha)$ (8.10)

######  Remark 8.1.

The works [352, 353] introduce explicitly a linear transformation $R$ which
maps the roots to the coroots of $\mathfrak{g}$. For instance, in the case
$\mathfrak{g}=G_{2}$, one can define $R$ as acting on the root lattice by an
angle $\pi/2$ and multiplication by a factor of $\sqrt{3}$ [353]; this results
in a “flip” in the Dynkin diagram of $G_{2}$. The transformation $R$ acts
nontrivially on the moduli space of the theory, on the lattice of electric and
magnetic charges, and therefore presumably also on the defects parameters
$(\alpha,\eta)$. In the case at hand, since our conventions for the root
lengths match those of [353], we would then conjecture the S-duality action
$\left({}^{L}\alpha,^{L}\eta\right)=\left(R\,\eta/\mathfrak{n}_{\mathfrak{g}},-R\,\alpha\right)$.
However, we choose instead to adopt the notational convenience of Appendix A
in [31] throughout our work, and write our expressions without explicitly
referring to $R$.

####  8.1.2 The T-transformation

Besides the non-perturbative S-duality we just reviewed, we also have a
classical T-symmetry from $G$ to itself, which acts on the gauge coupling
$\tau^{4d}=\frac{\theta}{2\pi}+i\,\frac{4\pi}{g^{2}_{4d}}$ as

$\text{T}:\;\;\;(\mathfrak{g},\tau^{4d})\longrightarrow(\mathfrak{g},\tau^{4d}-1)\;.$
(8.11)

This is a symmetry of $G$-Yang-Mills under a shift of the theta angle $\theta$
by $2\pi$. The surface defect parameters $(\beta,\gamma)$ are invariant under
the T-symmetry, while the parameters $(\alpha,\eta)$ transform as

$\text{T}:(\alpha,\eta)\longrightarrow(\alpha,\eta-\alpha)$ (8.12)

A gauge theory proof of this result is essentially the one found in the
“Witten effect” [354, 31]. Namely, $(\alpha,\eta)$ behaves just like the
charge of a dyon, which transforms precisely as above under the action of T.

####  8.1.3 The full duality group

For completeness, we briefly comment here on the symmetry group when both S
and T are present. The S and T symmetries generate a discrete infinite
subgroup of SL$(2,\mathbb{R})$. If $\mathfrak{g}$ is simply-laced, that group
is well-known to be the modular group SL$(2,\mathbb{Z})$.

If $\mathfrak{g}$ is non simply-laced, then S :
$\tau\mapsto-1/(\mathfrak{n}_{\mathfrak{g}}\,\tau)$, where
$\mathfrak{n}_{\mathfrak{g}}=2$ or $\mathfrak{n}_{\mathfrak{g}}=3$, is no
longer a modular transformation. Instead, the transformations T, STS, and the
central element S2 (which belongs in the Weyl group) generate a subgroup
$\Upsilon_{0}(\mathfrak{n}_{\mathfrak{g}})\subset\;$SL$(2,\mathbb{Z})$. In
that case, the full duality group is the $\mathbb{Z}_{2}$ quotient of the
semi-direct product
$\left(\Upsilon_{0}(\mathfrak{n}_{\mathfrak{g}})\rtimes\mathbb{Z}_{4}\right)/\mathbb{Z}_{2}$,
generated by S and T, with relations [352]. Here, the transformation S
generates the $\mathbb{Z}_{4}$ factor, and the reason for the quotient is that
the transformation S2 generates a $\mathbb{Z}_{2}$ which is already in
$\Upsilon_{0}(\mathfrak{n}_{\mathfrak{g}})$.

In the cases $\mathfrak{g}=B_{r}$ and $\mathfrak{g}=C_{r}$, the full duality
group is an equivalence between different theories, since the transformation S
maps the algebra $B_{r}$ to the algebra $C_{r}$, and vice-versa. It follows
that the self-duality group of each of these theories is only
$\Upsilon_{0}(2)$.

In the cases $\mathfrak{g}=G_{2}$ and $\mathfrak{g}=F_{4}$, the full duality
group we described above is a $\mathbb{Z}_{2}$ quotient of what is known as a
Hecke group [355]. For details on how these Hecke groups act on the moduli
space and on the electric and magnetic charges of these theories, a good
reference is [353].

###  8.2 4d S-duality from T-duality of little strings

We will now derive the S-duality of 4d Yang-Mills and its monodromy defects
straight from the T-duality of the little string. The argument in the absence
of D5 branes defects was first given by Vafa [42] (see also the last Section
of [353]). We will first review his argument and then proceed to add the D5
branes.

####  8.2.1 Warm-up: The Unramified Case

Consider the twisted background from Section 7:

$\frac{\mathbb{C}^{2}_{\Gamma}\times
S^{1}_{q}(R_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t}(R_{t})\times M_{4}\;,$ (8.13)

where the outer automorphism twist by
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$ acts by folding the Dynkin diagram
of $\mathfrak{g}_{\text{o}}$ after going once around $S^{1}_{q}$. If
$\mathfrak{g}_{\text{o}}$ doesn’t admit a nontrivial outer automorphism twist,
meaning whenever $\mathfrak{g}_{\text{o}}=A_{2p},E_{7},E_{8}$, then
$\mathbb{C}^{2}_{\Gamma}\times S^{1}_{q}(R_{q})$ will be understood without
the twist by $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$. We further take the
$g_{s}\rightarrow 0$ limit to obtain the $\mathfrak{g}_{\text{o}}$-type
$(2,0)$ little string theory on $S^{1}_{q}(R_{q})\times S^{1}_{t}(R_{t})\times
M_{4}$, in the presence of a $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist
along $S^{1}_{q}$. We now T-dualize the little string, in two different ways.

__

Figure 13: The two T-dualities from which one derives the S-duality of 4d
Super Yang-Mills. Adding D5 branes, one further derives the S-duality of the
Gukov-Witten monodromy defects.

First, we T-dualize the $(2,0)$ little string theory along $S^{1}_{t}$, and
obtain the (1,1) little string theory on $S^{1}_{q}(R_{q})\times
S^{1}_{t^{\prime}}(R^{\prime}_{t})\times M_{4}$, with
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist along $S^{1}_{q}$, and where
the circle $S^{1}_{t^{\prime}}$ has radius
$R^{\prime}_{t}=1/(m_{s}^{2}\,R_{t})$. In the full 10-dimensional superstring
background, this is a T-duality to type IIA string theory on

$\frac{\mathbb{C}^{2}_{\Gamma}\times
S^{1}_{q}(R_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t^{\prime}}(R^{\prime}_{t})\times M_{4}\;,$ (8.14)

Unlike its $(2,0)$ counterpart, the (1,1) little string theory has a gauge
symmetry. The gauge group has an $ADE$ Lie algebra $\mathfrak{g}_{\text{o}}$,
and inverse gauge coupling999999More precisely, the gauge theory description
of the (1,1) little string is valid at weak coupling. That description breaks
down when the energy scale becomes of order $m_{s}$, at which point new
stringy degrees of freedom need to be added. For a relatively recent
quantitative account of these matters, we refer the reader to the work [356].

$\frac{1}{(g^{6d})^{2}}=m^{2}_{s}\;.$ (8.15)

The $\mathfrak{h}$-valued gauge field has $r$ Cartan components
$A_{a}=\int_{S^{2}_{a}}C^{(3)}$, which is the Kaluza-Klein reduction of the
type IIA R-R 3-form over the $r$ compact 2-cycles of the resolved $ADE$
singularity.

In order to obtain a 4-dimensional gauge theory, we take the point particle
limit $m_{s}\rightarrow\infty$, and we further reduce the six-dimensional
theory on the torus $S^{1}_{q}\times S^{1}_{t^{\prime}}$ by taking
$R_{q}\rightarrow 0$ and $R^{\prime}_{t}\rightarrow 0$. Since
$R_{t}=1/(m_{s}^{2}\,R^{\prime}_{t})$, the radius of the initial $(2,0)$
circle $S^{1}_{t}$ stays finite in the limit. Then, the torus volume goes to
zero, $i\,R_{q}\,R^{\prime}_{t}\rightarrow 0$. The resulting theory is 4d
$\mathcal{N}=4$ Super Yang-Mills on $M_{4}$. The gauge group has a Lie algebra
of type $\mathfrak{g}$, and the (imaginary part of the) inverse gauge coupling
is given by the finite combination

$\tau^{4d}=m_{s}^{2}\,(i\,R_{q}\,R^{\prime}_{t})=i\,\frac{R_{q}}{R_{t}}\;.$
(8.16)

An alternate way to derive the above gauge coupling is to note that an
instanton in 4 dimensions is a fundamental string with Euclidean worldsheet
wrapping the torus, with action $2\pi m_{s}^{2}\,R_{q}\,R^{\prime}_{t}$. But
the action of a gauge theory instanton is $-2\pi i\tau^{4d}$, from which
(8.16) follows.

Alternatively, we can T-dualize the $(2,0)$ little string theory along
$S^{1}_{q}$, and obtain the (1,1) little string theory on
$S^{1}_{q^{\prime}}(R^{\prime}_{q})\times S^{1}_{t}(R_{t})\times M_{4}$, with
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist along $S^{1}_{q^{\prime}}$,
and where this circle $S^{1}_{q^{\prime}}$ has radius
$R^{\prime}_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,m_{s}^{2}\,R_{q})$. In the
full 10-dimensional type IIB background (8.13), this is a T-duality along
$S^{1}_{q}$ to type IIA string theory on

$\frac{\mathbb{C}^{2}_{\widetilde{\Gamma}}\times
S^{1}_{q^{\prime}}(R^{\prime}_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t}(R_{t})\times M_{4}\;.$ (8.17)

Above, $\mathbb{C}^{2}_{\widetilde{\Gamma}}$ is a resolved singularity labeled
by a $ADE$ Lie algebra $\widetilde{\mathfrak{g}_{\text{o}}}$ defined as
follows: if $\mathfrak{g}_{\text{o}}=A_{2p-1}$, then
$\widetilde{\mathfrak{g}_{\text{o}}}=D_{p+1}$, and vice versa (if
$\mathfrak{g}_{\text{o}}=D_{p+1}$, then
$\widetilde{\mathfrak{g}_{\text{o}}}=A_{2p-1}$), while
$\widetilde{\mathfrak{g}_{\text{o}}}=\mathfrak{g}_{\text{o}}$ in all other
cases. In particular,
$\widetilde{\mathfrak{g}_{\text{o}}}=\mathfrak{g}_{\text{o}}=E_{6}$ admits a
twist of order 2, while
$\widetilde{\mathfrak{g}_{\text{o}}}=\mathfrak{g}_{\text{o}}=D_{4}$ admits a
twist of order 3. As before, if
$\mathfrak{g}_{\text{o}}=\widetilde{\mathfrak{g}_{\text{o}}}=A_{2p},E_{7},E_{8}$,
then $\mathbb{C}^{2}_{\widetilde{\Gamma}}\times
S^{1}_{q^{\prime}}(R^{\prime}_{q})$ is understood without the twist by
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$. The algebra
$\widetilde{\mathfrak{g}_{\text{o}}}$ has an outer automorphism group
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$, and the subalgebra invariant under
the $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-action is the Lie algebra
${}^{L}\mathfrak{g}$. In order to correctly identify the radius
$R^{\prime}_{q}$ of $S^{1}_{q^{\prime}}$, note that in the type IIB setup
(8.13), the momentum along the original circle $S^{1}_{q}$ is quantized to be
an integer multiple of $1/(\mathfrak{n}_{\mathfrak{g}}\,R_{q})$, due of the
$\mathfrak{n}_{\mathfrak{g}}$-twist. Indeed, the fields are single-valued only
on the $\mathfrak{n}_{\mathfrak{g}}$-fold cover of $S^{1}_{q}$, meaning one
has to go around $S^{1}_{q}$ a total of $\mathfrak{n}_{\mathfrak{g}}$ times.
But then, by T-duality, a string which winds around $S^{1}_{q^{\prime}}$ will
have an energy quantized in the same units,
$m_{s}^{2}\,R^{\prime}_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,R_{q})$. We
conclude that the radius of the T-dual circle is as claimed,
$R^{\prime}_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,m_{s}^{2}\,R_{q})$.

In the limit $g_{s}\rightarrow 0$, this describes the (1,1) little string
theory on $S^{1}_{q^{\prime}}(R^{\prime}_{q})\times S^{1}_{t}(R_{t})\times
M_{4}$, with $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist along
$S^{1}_{q^{\prime}}$. The low energy effective theory is a gauge theory, with
$ADE$ Lie algebra $\widetilde{\mathfrak{g}_{\text{o}}}$, and inverse gauge
coupling

$\frac{1}{(\widetilde{g^{6d}})^{2}}=m^{2}_{s}\;.$ (8.18)

We once again take the point particle limit $m_{s}\rightarrow\infty$, and
further reduce on the torus $S^{1}_{q^{\prime}}\times S^{1}_{t}$ by taking
$R^{\prime}_{q}\rightarrow 0$, $R_{t}\rightarrow 0$100100100This time around,
it is $R_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,m_{s}^{2}\,R^{\prime}_{q})$, the
radius of the initial $(2,0)$ circle $S^{1}_{q}$, which stays finite in the
limit.. The torus volume $i\,R_{q^{\prime}}\,R_{t}\rightarrow 0$ then goes to
zero. The resulting theory is 4d $\mathcal{N}=4$ Super Yang-Mills on $M_{4}$.
The gauge group has a Lie algebra of type ${}^{L}\mathfrak{g}$, and the
(imaginary part of the) inverse gauge coupling is given by the finite
combination

$^{L}\tau^{4d}=m_{s}^{2}\,(i\,R_{q^{\prime}}\,R_{t})=i\,\frac{R_{t}}{\mathfrak{n}_{\mathfrak{g}}\,R_{q}}$
(8.19)

Comparing (8.16) and (8.19), we recover the transformation of the
4-dimensional gauge coupling under S-duality (8.8),

$\text{S}:\;\;\;(^{L}\mathfrak{g},^{L}\tau)\Longleftrightarrow(\mathfrak{g},\tau)\;,\qquad^{L}\tau=\frac{-1}{\mathfrak{n}_{\mathfrak{g}}\,\tau}\;.$
(8.20)

####  8.2.2 The Ramified Case

We consider again the twisted type IIB background (8.13), with
$M_{4}=\mathcal{C}\times M_{2}$, and add a collection of D5 branes wrapping

$\frac{S^{2}_{\\{a\\}}\times
S^{1}_{q}(R_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t}(R_{t})\times M_{2}\;.$ (8.21)

As before, we take the $(2,0)$ little string limit $g_{s}\rightarrow 0$, and
T-dualize in two different ways.

####  8.2.3 D-branes as monopoles, and the Hitchin integrable system

First, a T-duality along $S^{1}_{t}$ results in a configuration of D4 branes
in the (1,1) little string theory, wrapping

$\frac{S^{2}_{\\{a\\}}\times
S^{1}_{q}(R_{q})}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times M_{2}\;.$
(8.22)

In particular, the D4 branes are points on $\mathcal{C}\times
S^{1}_{t^{\prime}}(R^{\prime}_{t})=\mathbb{R}\times
S^{1}_{\mathcal{C}}(R_{\mathcal{C}})\times
S^{1}_{t^{\prime}}(R^{\prime}_{t})$, where
$R^{\prime}_{t}=1/(m_{s}^{2}\,R_{t})$. They are magnetically charged under the
gauge field of the (1,1) $\mathfrak{g}_{\text{o}}$-little string, whose $r$
Cartan components are $A_{a}=\int_{S^{2}_{a}}C^{(3)}$. As such, they are
$\mathfrak{g}_{\text{o}}$-monopoles, subjected to the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist along $S^{1}_{q}$. It follows
that the gauge field strength $F$ obeys the Bogomolny equations on
$\mathbb{R}\times S^{1}_{\mathcal{C}}\times S^{1}_{t^{\prime}}$,

$D\psi=*F\;,$ (8.23)

where $\psi$ is a real scalar field which approaches a constant value at the
end of the cylinder, far away from the D4 brane insertions on $\mathcal{C}$.
In the (1,1) little string, this constant scalar has Cartan components
$\psi_{a}=\lim_{g_{s}^{\prime}\rightarrow
0}m_{s}^{3}\int_{S^{2}_{a}}\frac{\omega_{I}}{g^{\prime}_{s}}$, for
$a=1,\ldots,r$ and $g_{s}^{\prime}$ the type IIA coupling.

__

Figure 14: Details of the first T-duality in the presence of D5 branes

If we instead T-dualize the D5 branes (8.21) along $S^{1}_{q}$, we obtain D4
branes in the (1,1) little string, wrapping

$\frac{\widetilde{S^{2}_{\\{a\\}}}}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times
S^{1}_{t}(R_{t})\times M_{2}\;.$ (8.24)

Above, $\widetilde{S^{2}_{\\{a\\}}}$ are the compact 2-cycles which span the
homology group $H_{2}({\mathbb{C}}^{2}_{\widetilde{\Gamma}},\mathbb{Z})$,
where the resolved singularity ${\mathbb{C}}^{2}_{\widetilde{\Gamma}}$ is the
one from (8.17). This time, the D4 branes are points on $\mathcal{C}\times
S^{1}_{q^{\prime}}(R^{\prime}_{q})=\mathbb{R}\times
S^{1}_{\mathcal{C}}(R)\times S^{1}_{q^{\prime}}(R^{\prime}_{q})$, where
$R^{\prime}_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,m_{s}^{2}\,R_{q})$. These are
$\widetilde{\mathfrak{g}_{\text{o}}}$-monopoles, subjected further to a
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist along $S^{1}_{q^{\prime}}$.
They are magnetically charged under the gauge field of the (1,1)
$\widetilde{\mathfrak{g}_{\text{o}}}$-little string, and the associated field
strength ${}^{L}F$ will obey the Bogomolny equations

$D^{L}\psi=*^{L}F\;,$ (8.25)

where ${}^{L}\psi$ is again a real scalar field originating from the (1,1)
little string.

__

Figure 15: Details of the second T-duality in the presence of D5 branes

When we say that the D4 branes are monopoles, we should be more specific,
since the branes wrap different types of 2-cycles in
$\mathbb{C}^{2}_{\Gamma}$: we have $n=\sum_{a=1}^{r}\,n_{a}$ of them wrap the
_compact_ 2-cycles, meaning their flux is identified with a class
$[S]=\sum_{a=1}^{r}\,n_{a}\,\alpha_{a}$ in
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\mathbb{Z})$, or the root lattice of
$\mathfrak{g}_{\text{o}}$. By Poincaré duality, this group is identified with
the cohomology group $H_{c}^{2}(\mathbb{C}^{2}_{\Gamma},\mathbb{Z})$, or the
coroot lattice of $\mathfrak{g}_{\text{o}}$. But elements of the coroot
lattice are in one-to-one correspondence with non-abelian monopole charges. We
conclude that the $n$ D4 branes are non-abelian monopoles.

Moreover, we have $n^{F}=\sum_{a=1}^{r}\,n^{F}_{a}$ branes wrapping the _non-
compact_ 2-cycles. The flux of such branes is identified with a class
$[S^{*}]=\sum_{a=1}^{r}\,n^{F}_{a}\,\lambda_{a}$ in
$H_{2}({\mathbb{C}}^{2}_{\Gamma},\partial\left({\mathbb{C}}^{2}_{\Gamma}\right),\mathbb{Z})$,
or the weight lattice of $\mathfrak{g}_{\text{o}}$. By Poincaré duality, we
further identify this group with the cohomology group
$H^{2}(\mathbb{C}^{2}_{\Gamma},\mathbb{Z})$, or the coweight lattice of
$\mathfrak{g}_{\text{o}}$. But elements of the coweight lattice are in one-to-
one correspondence with Dirac monopole charges. We conclude that the $n^{F}$
D4 branes are singular abelian Dirac monopoles. Both the non-abelian and Dirac
monopoles are subjected to the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist.

Consider first the T-duality along $S^{1}_{t}$, with the D4 branes in the type
IIA background (8.22). In order to make contact with surface defects of
4-dimensional $\mathfrak{g}_{\text{o}}$-type Yang Mills, we should reduce the
torus $S^{1}_{q}\times S^{1}_{t^{\prime}}$ to zero size and take the point
particle limit $m_{s}\rightarrow\infty$, just as we did in the unramified
case. Note that the circle $S^{1}_{q}$ is essentially a spectator here, since
the D4 branes wrapping $S^{1}_{q}$ are monopoles on $\mathcal{C}\times
S^{1}_{t^{\prime}}$, meaning the limit $R_{q}\rightarrow 0$ is a “trivial”
dimensional reduction. More interestingly, we should clarify what happens to
the branes as the circle $S^{1}_{t^{\prime}}$ becomes small. In that regime,
the Bogomolny equations lose the dependence on the
$S^{1}_{t^{\prime}}$-coordinate and we obtain:

$\displaystyle F-[\varphi,\overline{\varphi}]=0$
$\displaystyle\overline{D}_{x}\varphi=0$ $\displaystyle
D_{x}\overline{\varphi}=0\;,$ (8.26)

with $x$ a complex coordinate on $\mathcal{C}$, related to our previous
coordinate in (8.1) by $z=e^{x}$; the associated one-form on $\mathcal{C}$ is
$\varphi(x)\,dx$. This is not quite the Hitchin system (8), but rather a
“multiplicative” version of it on $\mathcal{C}$ [357]; the term multiplicative
here stands for the imaginary part of $\varphi(x)$ being a periodic scalar,
with period $2\pi/R^{\prime}_{t}$. This period is large, as we are taking
$S^{1}_{t^{\prime}}$ to be small.

In IIA variables, the Cartan components of the complex scalar field should be
written as $\varphi_{a}=\psi_{a}+i\,A^{\theta}_{a}$, with the imaginary part
as the holonomy $A^{\theta}_{a}$ of the (1,1) gauge field around
$S^{1}_{t^{\prime}}$. Its periodicity is a consequence of invariance under
large gauge transformations, $R^{\prime}_{t}\,A^{\theta}_{a}\rightarrow
R^{\prime}_{t}\,A^{\theta}_{a}+2\,\pi$.

In IIB variables, $\varphi_{a}$ takes a constant value
$(R_{t}\,m^{2}_{s})\,\tau^{6d}_{a}$ at the end of the cylinder. The holonomy
is rewritten as $A^{\theta}_{a}=R_{t}\,m^{2}_{s}\int_{S^{2}_{a}}C^{(2)}$, so
that $\tau^{6d}_{a}$ is a linear combination of $(2,0)$ little string moduli
in the bulk six-dimensions (6.14), hence nondynamical from the point of view
of the compactified theory101101101Alternatively, $m^{2}_{s}\,\tau^{6d}_{a}$
are the inverse gauge couplings of the effective theory on the D5 branes,
hence nondynamical.:

$\varphi_{a}=\lim_{g_{s}\rightarrow
0}(R_{t}\,m^{2}_{s})\int_{S^{2}_{a}}\left(\frac{m^{2}_{s}}{g_{s}}\,\omega_{I}+i\,C^{(2)}\right)\;,\;\qquad
a=1,\ldots,\text{rk}(\mathfrak{g})\;.$ (8.27)

The $n$ non-abelian monopoles and $n^{F}$ Dirac monopoles on
$\mathcal{C}\times S^{1}_{t^{\prime}}$ brought by the D4 branes will modify
the profile of the Higgs field $\varphi$, away from this constant.

In order to figure out the precise singularity profile of $\varphi$, it proves
useful to write the spectral curve of the multiplicative Hitchin system, which
can be derived from the Seiberg-Witten curve of the gauge theory on the
D-branes. For our quiver gauge theories $T^{5d}_{\mathfrak{g}_{\text{o}}}$,
the curve at a generic point on the Coulomb branch was derived by Nekrasov and
Pestun [83]. The method consists in writing down the instanton partition
function ${\mathcal{Z}}_{inst}(T^{5d}_{\mathfrak{g}})$ in a full
$\Omega$-background
$S^{1}_{\mathcal{C}^{\prime}}\times\mathbb{C}_{q}\times\mathbb{C}_{t}$ and
extracting the curve from a set of saddle point equations in the limit
$q,t\rightarrow 1$. Their analysis further implied that the Coulomb branch of
a 5d $\mathcal{N}=1$ $\mathfrak{g}_{\text{o}}$-type quiver gauge theory was
describing the moduli space of singular $\mathfrak{g}_{\text{o}}$-monopoles on
$\mathcal{C}\times S^{1}_{t^{\prime}}$, with $\mathcal{C}=\mathbb{C}^{\times}$
serving as the geometric Langlands curve102102102Soon after, Nekrasov-Pestun-
Shatashvili noted that the saddle point analysis can still be performed in the
limit $q\rightarrow 1$ while keeping $t$ arbitrary [84]. The saddle point
equations should now be understood as objects in “quantum geometry;”
physically, the “classical” Seiberg-Witten curve is upgraded to a Schrödinger-
type equation, and the curve coordinates $(p,x)$ are now better understood as
operators whose failure to commute is proportional to $\log(|t|)$. The moduli
space of monopoles likewise becomes quantized.. In order to make contact with
the quantum geometric Langlands program, the saddle point analysis should be
performed in a different limit: $t=q^{\beta}$, $q\rightarrow 1$ with $\beta$
fixed. The end result is a vanishing determinant, evaluated in a
representation $V$ of $\mathfrak{g}$:

$0=\text{Det}_{V}\left(e^{R^{\prime}_{t}\,p}-e^{R^{\prime}_{t}\,\varphi(x)}\right)\;.$
(8.28)

Here, the analysis is greatly simplified on the special point where the
Coulomb branch meets the Higgs branch103103103The curve at that locus is
sometimes called the UV curve; it should be understood as a multi-cover of the
punctured curve $\mathcal{C}$ [133]. The Seiberg-Witten at a generic point on
the Coulomb branch is recovered from the UV curve by turning on all possible
normalizable Coulomb moduli.. At that locus, the theory abelianizes and all
monopoles are Dirac monopoles. If we denote the multiplicative Higgs field
there as $\varphi^{\bf{S}}(x)$, then $e^{R^{\prime}_{t}\,\varphi^{\bf{S}}(x)}$
is diagonal and the determinant factorizes into $\text{dim}(V)$ factors, with

$e^{R^{\prime}_{t}\,\varphi^{\bf{S}}_{a}(x)}=e^{\tau^{6d}_{a}}\cdot\prod_{d=1}^{L}\prod_{s=1}^{|\\{{\lambda}\\}_{d}|}\prod_{b=1}^{\text{rk}(^{L}\mathfrak{g})}\left(1-e^{x}\,e^{-R^{\prime}_{t}\,\Upsilon_{d,s,b}}x^{-1}_{d,s}\right)^{\pm\deg(\mathcal{A}^{\mp}_{\lambda_{d,s},b})}\;.$
(8.29)

The factor $e^{\tau^{6d}_{a}}$ is the constant vacuum contribution, in the
absence of D-brane monopole insertions. The D4 branes located at $x_{d,s}$ on
$\mathcal{C}$ modify the right-hand side the product on the right-hand side,
where the D4 branes are , and we are following the $q$-primary vertex operator
notation of . In particular, $\\{{\lambda}\\}_{d}$ is a set of quantum affine
weights labeling the $d$-th primary operator, and
$\mathcal{A}^{\mp}_{\lambda_{d,s},b}$ are the associated Drinfeld polynomials,
whose roots are encoded in $\Upsilon_{d,s,b}$:

$\Upsilon_{d,s,b}=\;-\sum_{i=1}^{\deg(\mathcal{A}^{+}_{\lambda_{d,s},b})}\mathfrak{a}^{+}_{\lambda_{d,s},b,i}\left(1-\beta\right)+\sum_{j=1}^{\deg(\mathcal{A}^{-}_{\lambda_{d,s},b})}\mathfrak{a}^{-}_{\lambda_{d,s},b,j}\left(1-\beta\right)\;,$
(8.30)

as we defined it in (2.50), with $\beta=\log(t)/\log(q)$ encoding the central
charge of the $\mathcal{W}_{\beta}(\mathfrak{g})$ algebra.

######  Remark 8.2.

If the saddle point analysis of the instanton partition function is performed
in the fully classical limit $q,t\rightarrow 1$ without scaling by $\beta$,
one finds the curve (8.28) but with the Higgs field [83, 64]

$e^{R^{\prime}_{t}\,\varphi^{\bf{S}}_{a}(x)}=e^{\tau^{6d}_{a}}\cdot\prod_{d=1}^{L}\prod_{s=1}^{|\\{\vec{\underline{\lambda}}\\}_{d}|}\left(1-e^{x}\,x^{-1}_{d,s}\right)^{-\underline{\lambda^{\vee}_{d,s}}}\;.$
(8.31)

Most notably, the dependence on $\beta$ and on the Drinfeld polynomials of the
quantum algebra is lost: each
$\underline{\lambda^{\vee}_{d,s}}=2\log_{\hbar}\lambda^{+}_{a,0}$ is a
classical $\mathfrak{g}$-coweight, which only captures the 0-th mode
eigenvalue among the infinite number of modes in the quantum affine coweight
$\lambda^{\vee}_{d,s}=\lambda^{+}_{a,k}$, $k\geq 0$. For generic and semi-
degenerate defects, the Higgs fields (8.31) and (8.29) will coincide, but for
rigid and mixed defects, only (8.29) will correctly capture the singular
monopole profile; in particular, the Higgs field will explicitly depend on
$\beta$ for those punctures, see Section 2.1.5.

We are now ready to take the point particle limit, as well as reducing the
torus $S^{1}_{q}\times S^{1}_{t^{\prime}}$ to zero size:

$m_{s}\rightarrow\infty\;,\qquad R_{q}\rightarrow 0\;,\qquad
R^{\prime}_{t}\rightarrow 0\;.$ (8.32)

At the same time, we demand that the T-dual radii
$R^{\prime}_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,m_{s}^{2}\,R_{q})$ and
$R_{t}=1/(m_{s}^{2}\,R^{\prime}_{t})$. The geometric Langlands curve
$\mathcal{C}=\mathbb{R}\times S^{1}_{\mathcal{C}}(R_{\mathcal{C}})$ should
remain fixed as well. As in the unramified case, the (1,1)
$\mathfrak{g}_{\text{o}}$-type little string on $S^{1}_{q}\times
S^{1}_{t^{\prime}}\times\mathcal{C}\times\mathbb{C}$ with
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist reduces to $\mathcal{N}=4$
$\mathfrak{g}$-type Super Yang-Mills theory on $\mathcal{C}\times\mathbb{C}$
in the limit.

The D4 branes of the (1,1) little string wrapping $S^{2}_{\\{a\\}}\times
S^{1}_{q}\times\mathbb{C}$ flow to the monodromy defects (8.1) of Gukov and
Witten. One way to show this is to carefully take the limit while keeping the
Higgs field $\varphi^{\bf{S}}(x)$ fixed. In particular, as far as the constant
contribution is concerned, $\varphi^{\bf{S}}_{a}=\tau^{6d}_{a}/R^{\prime}_{t}$
must remain fixed, so the $\tau^{6d}_{a}$’s must go to 0. In the
$\mathcal{W}_{\beta}(\mathfrak{g})$-algebra, these $\text{rk}(\mathfrak{g})$
constants are the Cartan components of the Fock vacuum eigenvalue $\mu_{0}$
(2.1),

$\tau^{6d}_{a}/R^{\prime}_{t}=\langle\mu_{0},\alpha_{a}\rangle\;.$ (8.33)

For the D-brane contributions, introduce the usual center of mass coordinates
$x_{d,s}=\tilde{x}_{d}\,e^{R^{\prime}_{t}\,\sigma_{d,s}}$ for the $d$-th
primary operator insertion, and keep $\tilde{x}_{d}$ and $\sigma_{d,s}$ fixed
as $R^{\prime}_{t}\rightarrow 0$. This translates to the D4 branes
“coalescing” to $\tilde{x}_{d}$ on $\mathcal{C}$, keeping their relative
separation fixed. Note that the imaginary part of the Higgs field
$\varphi^{\bf{S}}(x)$ is no longer a periodic scalar in the limit. For
instance, the periods
$\text{lim}_{m_{s}\rightarrow\infty}m^{2}_{s}\int_{S^{2}_{a}}C^{(2)}$ are now
non-compact scalars in the moduli space of the $\mathfrak{g}_{\text{o}}$-type
$(2,0)$ SCFT, subjected to the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist. All in all, including the
brane defects, we obtain in the limit

$\varphi^{\bf{S}}(x)=\langle\mu_{0},\alpha_{a}\rangle+\sum_{d=1}^{L}\sum_{s=1}^{|\\{{\lambda}\\}_{d}|}\sum_{b=1}^{\text{rk}(^{L}\mathfrak{g})}\frac{\pm\left(\Upsilon_{d,s,b}+\sigma_{d,s}\right)\deg(\mathcal{A}^{\mp}_{\lambda_{d,s},b})}{e^{-x}\,\tilde{x}_{d}-1}\;.$
(8.34)

Switching to the coordinate $z=e^{x}$ on $\mathcal{C}$, the canonical one-form
is $\varphi^{\bf{S}}(x)\,dx=\varphi^{\bf{S}}(z)\,dz/z$, so we get

$\varphi^{\bf{S}}(z)=\frac{(\beta_{0}+i\,\gamma_{0})}{z}+\sum_{d=1}^{L}\frac{(\beta_{d}+i\,\gamma_{d})}{\tilde{x}_{d}-z}\;,$
(8.35)

with

$\beta_{0}+i\,\gamma_{0}=\langle\mu_{0},\alpha_{a}\rangle\;,\qquad\;\;\;\beta_{d}+i\,\gamma_{d}=\sum_{s=1}^{|\\{{\lambda}\\}_{d}|}\sum_{b=1}^{\text{rk}(^{L}\mathfrak{g})}\pm\left(\Upsilon_{d,s,b}+\sigma_{d,s}\right)\deg(\mathcal{A}^{\mp}_{\lambda_{d,s},b})\;.$
(8.36)

This is the Higgs field for a mass-deformed tamely ramified Hitchin system on
$\mathcal{C}$. The data is the residue $\beta_{0}+i\,\gamma_{0}$ of the
generic puncture at one end of the cylinder at $z=0$, and the residues at
$z=\tilde{x}_{d}$ for $d=1,\ldots,L$. There is another generic puncture at
$z=\infty$, but its data is entirely determined in terms of the previous
residues, as well as the number of normalizable deformations of the UV curve
(the number of Coulomb moduli). In the companion paper [130], we will classify
the residues (8.36) via the Drinfeld polynomial data, and show that it leads
to a new classification of mass deformations for tamely ramified Hitchin
systems. As a byproduct, we will obtain a description of all punctures in
terms of Verma modules in the $\mathcal{W}_{\beta}(\mathfrak{g})$-algebra,
thus giving a solution to the local Alday-Gaiotto-Tachikawa conjecture.

####  8.2.4 S-transformation of $(\beta,\gamma)$

Recall that the starting point of this analysis was a T-duality along
$S^{1}_{t}$. We now repeat the same arguments as in the previous section, but
with the T-duality along $S^{1}_{q}$ instead, with the D4 branes in the type
IIA background (8.24). In that case, the D4 branes are monopoles on
$\mathcal{C}\times S^{1}_{q^{\prime}}$. This time, it is the circle
$S^{1}_{t}$ which is essentially a spectator, and in the limit of small
$R^{\prime}_{q}$, the Bogomolny equations (8.25) reduce to a multiplicative
Hitching system, just as before.

The Higgs field Cartan components are
${}^{L}\varphi_{a}=^{L}\psi_{a}+i\;^{L}A^{\theta}_{a}$ in IIA notation, with
periodic imaginary part. They take constant values at the end of the cylinder
$\mathcal{C}$; in IIB notation,

$^{L}\varphi_{a}=(\mathfrak{n}_{\mathfrak{g}}\,R_{q}\,m^{2}_{s})\int_{\widetilde{S^{2}}_{a}}\left(\frac{m^{2}_{s}}{g_{s}}\,\omega_{I}+i\,C^{(2)}\right)\;,\;\qquad
a=1,\ldots,\text{rk}(^{L}\mathfrak{g})\;.$ (8.37)

where the compact 2-cycles $\widetilde{S^{2}}_{\\{a\\}}$ describe a resolved
singularity of type $\widetilde{\mathfrak{g}_{\text{o}}}$ (recall that
$\widetilde{\mathfrak{g}_{\text{o}}}=D_{p+1}$ if
$\mathfrak{g}_{\text{o}}=A_{2p-1}$, and vice-versa, while
$\widetilde{\mathfrak{g}_{\text{o}}}=\mathfrak{g}_{\text{o}}$ in all other
cases). Note the presence of the lacing number $\mathfrak{n}_{\mathfrak{g}}$
after rewriting $R^{\prime}_{q}$ in terms of $R_{q}$. The corresponding
multiplicative Hitchin system has spectral curve

$0=\text{Det}_{V}\left(e^{R^{\prime}_{q}\,p}-e^{R^{\prime}_{q}\,{}^{L}\varphi(x)}\right)\;.$
(8.38)

This time, the point-particle limit is

$m_{s}\rightarrow\infty\;,\qquad R^{\prime}_{q}\rightarrow 0\;,\qquad
R_{t}\rightarrow 0\;,$ (8.39)

and we demand that the T-dual radii
$R_{q}=1/(\mathfrak{n}_{\mathfrak{g}}\,m_{s}^{2}\,R^{\prime}_{q})$ and
$R^{\prime}_{t}=1/(m_{s}^{2}\,R_{t})$ remain fixed. In this limit, the (1,1)
$\widetilde{\mathfrak{g}_{\text{o}}}$-type little string on
$S^{1}_{q^{\prime}}\times S^{1}_{t}\times\mathcal{C}\times\mathbb{C}$ with
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist becomes the $\mathcal{N}=4$
${}^{L}\mathfrak{g}$-type Super Yang-Mills theory on
$\mathcal{C}\times\mathbb{C}$, and the D4 branes wrapping
$\widetilde{S^{2}}_{\\{a\\}}\times S^{1}_{t}\times\mathbb{C}$ become the
monodromy surface defects for the ${}^{L}\mathfrak{g}$ theory. In particular,
the Higgs field of 4-dimensional ${}^{L}\mathfrak{g}$-type Yang-Mills is
identified as

$^{L}\varphi^{\bf{S}}(z)=\frac{(^{L}\beta_{0}+i\,^{L}\gamma_{0})}{z}+\sum_{d=1}^{L}\frac{(^{L}\beta_{d}+i\,^{L}\gamma_{d})}{\tilde{x}_{d}-z}\;.$
(8.40)

Now, unless $\mathfrak{g}$ is simply-laced, we cannot directly compare
$({{}^{L}\beta},{{}^{L}\gamma})$ and $({\beta},{\gamma})$, since $\phi(x)$ is
valued in the Cartan subalgebra of $\mathfrak{g}$, whereas ${}^{L}\phi(x)$ is
valued in the Cartan subalgebra of ${}^{L}\mathfrak{g}$. We therefore use the
notation introduced in (8.9), namely $\alpha\mapsto\alpha^{*}$, where if
$\alpha_{l}$ is a long root, then $\alpha_{l}^{*}$ is
$\mathfrak{n}_{\mathfrak{g}}$ times the short coroot $\alpha^{\vee}_{s}$, and
if $\alpha_{s}$ is a short root, then $\alpha_{s}^{*}$ is the long coroot
$\alpha^{\vee}_{l}$. We note that the multiplicative Higgs fields enter as
$e^{R^{\prime}_{t}\,\varphi}$ and $e^{R^{\prime}_{q}\,{}^{L}\varphi}$. For
instance, for the puncture at $z=0$, this is written explicitly in (8.27) and
(8.37), respectively. Then, after correctly accounting for the factor of
$\mathfrak{n}_{\mathfrak{g}}$, we obtain:

$({{}^{L}\beta},{{}^{L}\gamma})=\frac{R_{q}}{R_{t}}\,(\beta^{*},\gamma^{*})\;.$
(8.41)

This relation holds for any of the $d$ punctures located at $z=\tilde{x}_{d}$,
with $d=0,1,\ldots,L$ (and $\tilde{x}_{0}=0$). Because the Yang-Mills gauge
coupling is $\tau^{4d}=i\,R_{q}/R_{t}$, this also reads

$({{}^{L}\beta},{{}^{L}\gamma})=\text{Im}(\tau^{4d})\,(\beta^{*},\gamma^{*})\;.$
(8.42)

This is precisely the 4-dimensional S-duality action on the Higgs field
parameters (8.9).

####  8.2.5 S-transformation of $(\alpha,\eta)$

Once again, we start with the type IIB background (8.13), in the presence of
the $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist. Recall that the D-branes
are at points on the Riemann surface $\mathcal{C}$, and these points are the
stringy definition of the punctures on $\mathcal{C}$. Let $S^{1}_{\\{p\\}}$ be
a small circle on $\mathcal{C}$ surrounding the location of the $d$-th
puncture, located at a point $p$. We define two $\mathfrak{h}$-valued periods,
whose Cartan components read

$\alpha_{a}^{\text{IIB}}=m_{s}^{4}\int_{S^{2}_{a}\times S^{1}_{\\{p\\}}\times
S^{1}_{q}}C^{(4)}\;,\qquad\;\;\eta_{a}^{\text{IIB}}=m_{s}^{4}\int_{S^{2}_{a}\times
S^{1}_{\\{p\\}}\times
S^{1}_{t}}C^{(4)}\;,\qquad\;\;a=1,\ldots,\text{rk}(\mathfrak{g})\;.$ (8.43)

Above, $C^{(4)}$ is the R-R 4-form of type IIB. Consider the usual T-duality
along $S^{1}_{t}$ to type IIA (8.14). The above periods get mapped to

$\alpha_{a}^{\text{IIA}}=m_{s}^{5}\int_{S^{2}_{a}\times S^{1}_{\\{p\\}}\times
S^{1}_{q}\times
S^{1}_{t^{\prime}}}\widetilde{C}^{(5)}\;,\qquad\;\;\eta_{a}^{\text{IIA}}=m_{s}^{3}\int_{S^{2}_{a}\times
S^{1}_{\\{p\\}}}C^{(3)}\;,\qquad\;\;a=1,\ldots,\text{rk}(\mathfrak{g})\;.$
(8.44)

Above, $C^{(3)}$ is the R-R 3-form of type IIA, and $\widetilde{C}^{(5)}$ is
the associated 5-form defined through Hodge duality
$d\widetilde{C}^{(5)}=\star dC^{(3)}$. In the limit $g_{s}\rightarrow 0$,
these are periods of the (1,1) $\mathfrak{g}_{\text{o}}$-type little string
subjected to a $\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist. In
particular, the integral of $C^{(3)}$ along $S^{2}_{a}$ is the gauge field
$A_{a}$ of the (1,1) little string, whose holonomy around the puncture $p$ on
$\mathcal{C}$ is precisely $\eta_{a}^{\text{IIA}}$. We further take the CFT
limit and reduce on the torus $S^{1}_{q}\times S^{1}_{t^{\prime}}$, requiring
the periods $\alpha_{a}^{\text{IIA}}$ and $\eta_{a}^{\text{IIA}}$ to remain
finite in the limit. We identify them with the surface defect parameters
$(\alpha,\eta)$ in $\mathfrak{g}$-type gauge theory:

$\lim_{\begin{subarray}{c}g_{s}\rightarrow 0\\\ m_{s}\rightarrow\infty\\\
R_{q},R^{\prime}_{t}\rightarrow
0\end{subarray}}\alpha_{a}^{\text{IIA}}=\alpha_{a}\;,\qquad\lim_{\begin{subarray}{c}g_{s}\rightarrow
0\\\ m_{s}\rightarrow\infty\\\ R_{q},R^{\prime}_{t}\rightarrow
0\end{subarray}}\eta_{a}^{\text{IIA}}=\eta_{a}\;,\qquad\;\;\;\;a=1,\ldots,\text{rk}(\mathfrak{g})\;.$
(8.45)

Alternatively, we perform T-duality from type IIB to type IIA along
$S^{1}_{q}$. The periods (8.43) get mapped to

$^{L}\alpha_{a}^{\text{IIA}}=m_{s}^{3}\int_{S^{2}_{a}\times
S^{1}_{\\{p\\}}}C^{(3)}\;,\qquad\;\;^{L}\eta_{a}^{\text{IIA}}=m_{s}^{5}\int_{S^{2}_{a}\times
S^{1}_{\\{p\\}}\times S^{1}_{t}\times
S^{1}_{q^{\prime}}}\widetilde{C}^{(5)}\;\qquad,\;\;a=1,\ldots,\text{rk}(^{L}\mathfrak{g})\;.$
(8.46)

In the limit $g_{s}\rightarrow 0$, these are periods of the (1,1)
$\widetilde{\mathfrak{g}_{\text{o}}}$-type little string subjected to a
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$-twist. We further take the CFT
limit and reduce on the torus $S^{1}_{q}\times S^{1}_{t^{\prime}}$. Requiring
the periods ${}^{L}\alpha_{a}^{\text{IIA}}$ and ${}^{L}\eta_{a}^{\text{IIA}}$
to remain finite in the limit, we identify them with the surface defect
parameters $(^{L}\alpha_{a},^{L}\eta_{a})$ in ${}^{L}\mathfrak{g}$-type gauge
theory:

$\lim_{\begin{subarray}{c}g_{s}\rightarrow 0\\\ m_{s}\rightarrow\infty\\\
R_{q},R^{\prime}_{t}\rightarrow
0\end{subarray}}{{}^{L}\alpha_{a}^{\text{IIA}}}={{}^{L}\alpha_{a}}\;,\qquad\lim_{\begin{subarray}{c}g_{s}\rightarrow
0\\\ m_{s}\rightarrow\infty\\\ R_{q},R^{\prime}_{t}\rightarrow
0\end{subarray}}{{}^{L}\eta_{a}^{\text{IIA}}}={{}^{L}\eta_{a}}\;\qquad,\;\;a=1,\ldots,\text{rk}(^{L}\mathfrak{g})\;.$
(8.47)

Then, from the IIA definition of the parameters, and taking into account the
reversal of orientation between $\alpha_{a}^{\text{IIA}}$ and
${}^{L}\eta_{a}^{\text{IIA}}$, we obtain in the limit:

$(^{L}\alpha,^{L}\eta)=(\eta,-\alpha)\;.$ (8.48)

This is precisely the 4-dimensional S-duality relation (8.10).

####  8.2.6 T-transformation of the parameters

So far, $\tau^{4d}=i\,\frac{4\pi}{g^{2}_{4d}}$ was purely imaginary. A non-
zero $\theta$-angle $\tau^{4d}=\frac{\theta}{2\pi}+i\,\frac{4\pi}{g^{2}_{4d}}$
can be incorporated by turning on an NS-NS $B^{(2)}$-field flux on the torus:

$\tau^{4d}=m_{s}^{2}\,(i\,R_{q}\,R^{\prime}_{t}-B^{(2)})\;.$ (8.49)

The S-duality of $\tau^{4d}$ is as before, (8.20). It is derived once again
from the T-duality of little strings, this time in a background with $B^{(2)}$
flux. This follows from a straightforward application of “Buscher rules”
[358], which we omit writing explicitly here not to overburden the
presentation; see for example Section 2 of [359]. Note that $B^{(2)}$ must go
to zero in the conformal limit, so that $m_{s}^{2}\,B^{(2)}$ remains fixed.

There is now a T-symmetry corresponding to shifting $B^{(2)}$ by $m_{s}^{-2}$,
which translates to the $2\pi$ shift of the $\theta$-angle:

$\text{T}:\;\;\;(\mathfrak{g},\tau^{4d})\Longleftrightarrow(\mathfrak{g},\tau^{4d}-1)\;.$
(8.50)

We now turn to the surface defect parameters: from the stringy definition of
$(\beta_{a},\gamma_{a})$, it is clear that those parameters are invariant
under $B^{(2)}$-shifts.

It follows from (8.44) that the parameters $\eta_{a}^{\text{IIA}}$ are
likewise invariant, while we find the following term in
$\alpha_{a}^{\text{IIA}}$:

$m_{s}^{5}\int_{S^{2}_{a}\times S^{1}_{\\{p\\}}\times S^{1}_{q}\times
S^{1}_{t^{\prime}}}\widetilde{C}^{(3)}\wedge B^{(2)}\;.$ (8.51)

Then, a shift of $B^{(2)}$ by $m_{s}^{-2}$ will map
$\alpha_{a}\rightarrow\alpha_{a}-\eta_{a}$. We have therefore recovered the
action of the T symmetry on the surface defect parameters:

$\text{T}:(\alpha,\eta)\longrightarrow(\alpha,\eta-\alpha)\;.$ (8.52)

###  8.3 On TS-duality versus S-duality

The quantum $q$-Langlands correspondence implies the dictionary

$q=\hbar^{-^{L}(\kappa+h^{\vee})}\,,\;\;\qquad
t=\hbar/q^{\mathfrak{n}_{\mathfrak{g}}}\,,$ (8.53)

between the parameters of $\mathcal{W}_{q,t}(\mathfrak{g})$ and
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}}_{{}^{L}\kappa})$. The physical origin
of this dictionary was first elucidated in the work [13], in the unramified
context. We will start by briefly reviewing those arguments, which are
unaffected by ramification, and then comment on the distinction between TS
duality and S duality in gauge theory.

First, we have shown that the partition function of the
$\mathfrak{g}_{\text{o}}$-type $(2,0)$ little string theory on the manifold

$M^{\times}_{6}=\frac{\mathbb{C}^{\times}_{q}}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times\mathbb{C}_{t}\times\mathcal{C}$
(8.54)

is a deformed conformal block of
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}_{\text{o}}})$, subjected to the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$ outer automorphism twist, with tame
ramification at points on $\mathcal{C}$. These are
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ blocks, by definition104104104As
previously mentioned, the construction of the folded qKZ equations in the non
simply-laced case is still not completely clear and deserves further study
[168].. In writing $\mathbb{C}^{\times}_{q}$, we removed the origin of
$\mathbb{C}_{q}$, consistent with the vector-valued nature of the conformal
blocks.

In the point particle limit $m_{s}\rightarrow\infty$, this is the chiral
conformal block of a $\widehat{{}^{L}\mathfrak{g}_{\text{o}}}$-type WZW model.
To make contact with 4d SYM, we consider once again $M^{\times}_{6}$ as a
$T^{2}=S^{1}_{q}(R_{q})\times S^{1}_{t}(R_{t})$ fibration:

$T^{2}\hookrightarrow M^{\times}_{6}\hookrightarrow
M_{4}=\mathcal{C}\times\mathbb{R}\times\mathbb{R}_{+}\;.$ (8.55)

As explained by Nekrasov and Witten [94], the reduction of the 6d theory on
this torus leads to 4d $\mathcal{N}=4$ SYM on $M_{4}$ with the topological
Landau-Ginzburg twist. Whether the 4d gauge group is of $\mathfrak{g}$-type or
${}^{L}\mathfrak{g}$-type depends on the order of the $T^{2}$ reduction, as we
saw throughout this paper. Far away from the origin of
$\mathbb{C}_{q}\times\mathbb{C}_{t}$ (the fixed point locus of the $U(1)^{2}$
equivariant action in the $\Omega$-background), the topological twist becomes
trivial, and $M^{\times}_{6}$ is flat. This makes it possible to identify the
circle radii in $T^{2}=S^{1}_{q}(R_{q})\times S^{1}_{t}(R_{t})$ with the
$\Omega$-background parameters:

$R_{q}=\frac{2\pi}{i\,\mathfrak{n}_{\mathfrak{g}}\,\epsilon_{q}}\;,\qquad\;R_{t}=\frac{2\pi}{\epsilon_{t}}\;.$
(8.56)

The dictionary relation (8.53) is a consequence of (8.56) and results of
Ooguri and Vafa [360]. Namely, consider the 6d $(2,0)$ SCFT on

$M^{\times}_{6}=\left(\frac{S^{1}_{q}}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times\mathbb{C}\right)\times
M_{3}\;,\qquad\;M_{3}=\mathbb{R}\times\mathcal{C}\;.$ (8.57)

This is the same manifold $M^{\times}_{6}$ as before, but now reinterpreted as
a product of two 3-manifolds instead of a product of a 2-manifold and a
4-manifold. The partition function of the 6d theory on $M^{\times}_{6}$
computes the partition function of ${}^{L}\mathfrak{g}$-type Chern-Simons
theory on $M_{3}$, at level ${}^{L}(\kappa+h^{\vee})$. The level itself is
determined by the $\Omega$-background parameter on
${S^{1}_{q}}\times\mathbb{C}$: as we go around ${S^{1}_{q}}$, the complex line
$\mathbb{C}$ is rotated by $q_{C.S.}=e^{2\,\pi\,i/^{L}(\kappa+h^{\vee})}$,
which needs to be compensated by an R-symmetry twist to preserve
supersymmetry. In our background, this is a twist not by $t^{-1}$ as one could
have naively expected, but instead by $t^{-1}\cdot
q^{\mathfrak{n}_{\mathfrak{g}}}=\hbar$; the appearance of $q$ is due to the
topological twist on the cylinder $\mathbb{C}^{\times}_{q}$ being trivial, and
the exponent $\mathfrak{n}_{\mathfrak{g}}$ accounts for the possible outer
automorphism twist.

In other words, had the origin of $\mathbb{C}_{q}$ not been removed, the twist
would have been by $e^{R_{q}\,(-\epsilon_{t})}$. Here, the R-symmetry twist as
we go around $S^{1}_{q}$ is instead
$q_{C.S.}=e^{\mathfrak{n}_{\mathfrak{g}}\,R_{q}\,\epsilon_{\hbar}}$, with
$\epsilon_{\hbar}=\mathfrak{n}_{\mathfrak{g}}\,\epsilon_{q}-\epsilon_{t}$.

It follows from the first relation in (8.56) that
$q_{C.S.}=e^{-2\,\pi\,i\,\epsilon_{\hbar}/\epsilon_{q}}$, from which we deduce

$\frac{\epsilon_{q}}{\epsilon_{\hbar}}=-^{L}(\kappa+h^{\vee})\;.$ (8.58)

After exponentiation, this is precisely the dictionary (8.53). If we perform
the reduction on $T^{2}$, this ratio of $\Omega$-background parameters is also
the gauge coupling of ${}^{L}\mathfrak{g}$-type 4d SYM:

$\frac{\epsilon_{q}}{\epsilon_{\hbar}}=\,^{L}\tau^{4d}\;.$ (8.59)

When we “fill the puncture” at the origin and replace
$\mathbb{C}^{\times}_{q}$ by $\mathbb{C}_{q}$, the partition function of the
$\mathfrak{g}_{\text{o}}$-type $(2,0)$ little string theory on the manifold

$M_{6}=\frac{\mathbb{C}_{q}}{\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}}\times\mathbb{C}_{t}\times\mathcal{C}\;.$
(8.60)

is instead a scalar-valued deformed conformal block of
$\mathcal{W}_{q,t}(\mathfrak{g}_{\text{o}})$, subjected to the
$\mathbb{Z}_{\mathfrak{n}_{\mathfrak{g}}}$ outer automorphism twist, with tame
ramification at points on $\mathcal{C}$. These are
$\mathcal{W}_{q,t}(\mathfrak{g})$ blocks, by definition. The parameters $q$
and $t$ are the 5d $\Omega$-background parameters
$q=e^{R_{\mathcal{C}^{\prime}}\,\epsilon_{q}}$ and
$t=e^{-R_{\mathcal{C}^{\prime}}\,\epsilon_{t}}$. After setting $t=q^{\beta}$,
this implies

$\beta=\frac{-\epsilon_{t}}{\epsilon_{q}}\;.$ (8.61)

This relation was first discovered as part of the AGT correspondence [56, 94],
in the conformal limit $m_{s}\rightarrow\infty$ to the
$\mathcal{W}_{\beta}(\mathfrak{g})$-algebra (this is the limit $q\rightarrow
1$ with $\beta$ fixed in the $\mathcal{W}_{q,t}(\mathfrak{g})$-algebra).

This time around, the $T^{2}=S^{1}_{q}(R_{q})\times S^{1}_{t}(R_{t})$
reduction yields $\mathfrak{g}$-type 4d SYM with gauge coupling

$\tau^{4d}=\frac{\epsilon_{t}}{\mathfrak{n}_{\mathfrak{g}}\,\epsilon_{q}}\;.$
(8.62)

In a frame where the 4d gauge coupling is $\tau^{4d}=i\,R_{q}/R_{t}$, it
follows that

$\tau^{4d}-1=\frac{-1}{\mathfrak{n}_{\mathfrak{g}}\,^{L}\tau^{4d}}\;.$ (8.63)

Then, the quantum $q$-Langlands correspondence is a TS-duality in gauge
theory. As we now argue, in order to realize Langlands duality as S-duality
proper, we need to engineer the 5d gauge theory
$T^{5d}_{\mathfrak{g}_{\text{o}}}$ in a spectral frame, or equivalently the 3d
Drinfeld gauge theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$ in a mirror frame.

####  8.3.1 Remarks on spectral duality

Sometimes, when $\mathfrak{g}_{\text{o}}=A_{r}$, it can happen that the 5d
gauge theory on $S^{1}_{\mathcal{C}^{\prime}}$ has more than one Lagrangian
description, related by spectral duality. This duality was first described in
the context of geometric engineering [147], and later discovered in integrable
systems [148, 361, 149, 362, 363]. In the multiplicative Hitchin system
associated to $T^{5d}_{A_{r}}$, the duality exchanges the roles of the
variables $e^{x}$ and $e^{p}$ in the spectral curve (8.38) (both are the
$\mathbb{C}^{\times}$-valued). In the 5d gauge theory, the gauge couplings are
exchanged with the masses and Coulomb moduli, and the $\Omega$-background
parameters are exchanged105105105After Higgsing to 3 dimensions, this is also
the mirror symmetry map. In particular, the $A$-twisted half-index we computed
in this paper would become $B$-twisted instead., as

$\displaystyle x^{\pm}_{d,a},\;e_{a,I}\;$
$\displaystyle\longleftrightarrow\;1/(g^{5d}_{b})^{2}\;,$ $\displaystyle t\;$
$\displaystyle\longleftrightarrow\;q/t\;.$ (8.64)

Our Hitchin system featured $L$ arbitrary punctures, the non-compact D5 branes
at points on the cylinder $\mathcal{C}$. Under certain conditions on $L$ and
on the effective Chern-Simons levels of the 5d theory, a spectral dual frame
may become manifest, with another Lagrangian.

We saw that the Lagrangian description of $T^{5d}_{A_{r}}$ is lost in the
conformal limit to 4 dimensions, as it becomes strongly coupled, but this is
not so for the spectral dual theory, which retains a weakly coupled
Lagrangian. One way to identify this Lagrangian in 4d is to conformally map
the cylinder $\mathcal{C}$ to a sphere with 2 generic punctures. The theory we
are after should then be spectral dual to a theory with 2 generic punctures
(at $\\{0\\}$ and $\\{\infty\\}$) and $L$ arbitrary ones on the sphere,
consistent with the Higgs field profile (8.35). Such a theory was
characterized by Gaiotto [133], and is given by _another_ quiver gauge theory.

Here, we will look at a simple example, a 5d quiver gauge theory

$T^{5d}_{A_{r-1}}\;:\;\;G_{5d}=\prod_{a=1}^{r-1}SU(n_{a})\;,\qquad
G^{F}_{5d}=SU(n^{F}_{1})\times SU(n^{F}_{r-1})$ (8.65)

with $n^{F}_{1}=n^{F}_{r-1}=n_{a}=2$. That is, the gauge group is $(r-1)$
$SU(2)$ factors, and the flavor group is $SU(2)$ on the first node and $SU(2)$
on the right node. The 5d spectral dual theory is readily identified, say from
the brane engineering of the theory in type IIB [150], as a
$\widetilde{T^{5d}_{A_{r-1}}}=T^{{}^{\prime}5d}_{A_{1}}$ gauge theory, that is

$T^{{}^{\prime}5d}_{A_{1}}\;:\;\;G^{\prime}_{5d}=SU(r)\;,\qquad
G^{{}^{\prime}F}_{5d}=SU(2r)$ (8.66)

In the conformal limit to 4 dimensions, $T^{5d}_{A_{r-1}}$ becomes strongly
coupled, but $T^{{}^{\prime}5d}_{A_{1}}$ doesn’t; instead, it becomes the same
gauge theory, but in the 4-dimensional sense:

$T^{{}^{\prime}4d}_{A_{1}}\;:\;\;G^{\prime}_{4d}=SU(r)\;,\qquad
G^{{}^{\prime}F}_{4d}=SU(2r)$ (8.67)

Recently, Nekrasov and Tsymbaliuk studied $T^{{}^{\prime}4d}_{A_{1}}$ in the
presence of generic codimension-2 monodromy defects [159]. These defects are
introduced by replacing the support manifold
$\mathbb{C}_{q}\times\mathbb{C}_{t}$ with the orbifold
$(\mathbb{C}_{q}/\mathbb{Z}_{r})\times\mathbb{C}_{t}$. Then, one can show that
the instanton partition function of $T^{{}^{\prime}4d}_{A_{1}}$ in the
presence of these defects defects takes the form of the following chiral
correlator

${\bf F}=\left\langle
v_{\nu^{\prime}_{\infty}},\widetilde{V}_{m}(z)\,\widetilde{V}_{\widetilde{m}}(1)\;v_{\nu^{\prime}_{0}}\right\rangle\;,$
(8.68)

which is a solution to the (rational) Knizhnik-Zamolodchikov
equations106106106Because the orbifold is along $\mathbb{C}_{q}$ in our
conventions, we exchanged $\epsilon_{q}\leftrightarrow\epsilon_{t}$ compared
to [159].:

$(\kappa^{\prime}+r)\,\frac{\partial{\bf F}}{\partial
z}=\left(\frac{\Omega_{z0}}{z}+\frac{\Omega_{z1}}{z-1}\right){\bf
F}\;,\qquad\;\;\kappa^{\prime}=\frac{\epsilon_{q}}{\epsilon_{t}}$ (8.69)

with $\Omega$ as in (2.106). In particular, note from the definition of
$\kappa^{\prime}$ that in this setup, it is S-duality and not TS-duality which
is more natural.

Just like in our setup, $v_{\nu^{\prime}_{0}}$ and $v_{\nu^{\prime}_{\infty}}$
are highest and lowest weight vectors of irreducible Verma modules,
respectively. Meanwhile, $\widetilde{V}_{m}$ and
$\widetilde{V}_{\widetilde{m}}$ are so-called twisted Heisenberg-Weyl modules
[364]. These modules encode the dependence on the masses, with the twist
encoding the dependence on the Coulomb moduli. Let us specialize to $r=2$,
meaning $SU(2)$ with 4 flavors. If we Higgs the theory by freezing the Coulomb
moduli to the masses, and further specialize to $r=2$, all modules become
Verma modules over $\widehat{A_{1}}$.

__

Figure 16: From the insertion of two $q$-primary operators on the cylinder to
the 4-point function on the sphere. The first arrow down denotes the CFT
limit.

Because $SU(2)$ with 4 flavors is known to be self-dual under spectral
duality, we should expect this description to agree with ours in that case.
And indeed, for $r=2$, and in the conformal limit, we are also describing a
4-point function on the sphere with vertex operator insertions labeled by
irreducible Verma modules over $\widehat{A_{1}}$. The rational limit of our
trigonometric KZ equations is

$(\kappa+2)\,\frac{\partial{\bf
F}}{\partial\tilde{x}}=\left(\frac{\Omega_{x0}}{\tilde{x}}+\frac{\Omega_{x1}}{\tilde{x}-1}\right){\bf
F}\;,\qquad\;\;\kappa=\frac{\epsilon_{q}}{\epsilon_{\hbar}}\;.$ (8.70)

Under the spectral duality map (8.3.1), this is precisely the Nekrasov-
Tsymbaliuk solution. It would be important to generalize the analysis to all
$r\geq 2$.

####  8.3.2 Instanton/Vortex counting and a theorem of Atiyah

A key part of this paper was the use of gauge-vortex duality as a means to
relate instanton counting in the 5d theory $T^{5d}_{\mathfrak{g}_{\text{o}}}$
to vortex counting in the 3d theory $T^{3d}_{\mathfrak{g}_{\text{o}}}$.

Mathematically, this is highly reminiscent of a theorem of Atiyah, formulated
in the 80’s [365]: given a classical group $G$, and a positive integer $k$,
the moduli space of $k$ $G$-instantons on $\mathbb{C}^{2}$ (suitably
regularized, and modulo based gauge transformations) is diffeomorphic to the
moduli space of (based) degree $k$ holomorphic maps
${\cal{M}}(\mathbb{P}^{1}\rightarrow\Omega G)$ to the (based) loop group of
$G$. For the instantons, based gauge transformations are those which are
trivial at $\\{\infty\\}\in\mathbb{C}^{2}$, and based holomorphic maps send
$\\{\infty\\}\in\mathbb{P}^{1}$ to the identity in $\Omega G$. The based loop
group $\Omega G$ is the group of maps $S^{1}\rightarrow G$ which send constant
maps to the identity in $G$.

When $\mathfrak{g}=A_{r-1}$, this theorem was used to identify the ground
states of the $(2,0)$ little string theory, defined through discrete light-
cone quantization (DLCQ) [366]. That is, the little string is compactified on
a light-like circle $S^{1}$ with $k$ units of discrete momentum along it,
obtained as a limit of compactification on a small space-like circle boosted
by a large amount [367]. The DLCQ of the $A_{r-1}$ little string theory with
$k$ units of momentum is a 2d $\mathcal{N}=(4,4)$ sigma model whose target
space is the moduli space of $k$ $SU(r)$ instantons on $\mathbb{C}^{2}$ [368].
The ground states of this sigma model are $L^{2}$-cohomology classes of this
instanton moduli space107107107$L^{2}$-cohomology arises here because the
moduli space is not compact., which by Atiyah’s theorem are $L^{2}$-cohomology
classes of ${\cal{M}}(\mathbb{P}^{1}\rightarrow\Omega SU(r))$. By studying the
$A$-twisted topological sigma model on $\Omega SU(r)$, it is therefore
possible to identify the ground states of the $(2,0)$ $A_{r-1}$ little string
theory; the analysis of [366] identifies these states as modules over an
affine Lie algebra $\widehat{\text{su}(r)}$, in agreement with a general
result of Braverman and Finkelberg [369].

In the context of our paper, it would be important to generalize these results
by first considering a K-theoretical count of the maps instead of a
cohomological one, and further introducing an $\Omega$-background. One should
expect the affine Lie algebra $\widehat{\text{su}(r)}$ to be replaced by the
Drinfeld quantization of its universal enveloping algebra, or rather a quantum
toroidal algebra108108108Indeed, there is growing evidence that for the little
string on $S^{1}$ and in the $\Omega$-background, the BPS strings wrapped on
$S^{1}$, as studied by [366], should be represented by the generators of the
quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{gl}_{1}}})$, and the
various D-branes, such as the ones studied in our paper, should be represented
by specific modules [214, 186, 174, 215, 220, 175]. These matters will also be
explored elsewhere [325]. [207, 208]. If this is true, then our results would
imply a spectral dual version of Atiyah’s theorem, at least for
$\mathfrak{g}=A_{r-1}$ and in the quantized K-theoretical setting: our
starting point was also the $A_{r-1}$ little string, but for us “$r$” labeled
the rank of a _quiver_ instead of the rank of a gauge group. The (quantum)
loop algebra symmetry we identified in the codimension-2 theory was likewise a
symmetry of a quiver. We leave this investigation to future work.

##  9 Case study: tamely ramified $A_{1}$ Langlands correspondence

Throughout this section, the Lie algebra will be $\mathfrak{g}=A_{1}$.

###  9.1 The $U_{\hbar}(\widehat{A_{1}})$-algebra

The quantum affine algebra $U_{\hbar}(\widehat{A_{1}})$ has a presentation in
terms of Chevalley generators, which following standard terminology goes by
the name of the Drinfeld-Jimbo realization [14, 15, 16, 17]. It consists of
generators $e_{0}^{\pm},e_{1}^{\pm},K_{0}^{\pm 1},K_{1}^{\pm 1}$, and
relations109109109We choose to match the notations of [23] rather than the
cited references, meaning our quantization parameter is defined as
$\hbar_{\text{here}}=\hbar^{2}_{\text{there}}=q/t$.:

$\displaystyle K_{0}\,K_{1}=K_{1}\,K_{0}\;,$ $\displaystyle
K_{i}\,K_{i}^{-1}=1=K_{i}^{-1}\,K_{i}\;,\qquad(i=0,1)$ $\displaystyle
K_{i}\,e_{i}^{\pm}=\hbar^{\pm 1}\,e_{i}^{\pm}\,K_{i}\;,\qquad(i=0,1)$
$\displaystyle K_{i}\,e_{j}^{\pm}=\hbar^{\mp
1}\,e_{j}^{\pm}\,K_{i}\;,\qquad(i\neq j)$ $\displaystyle
e_{i}^{+}\,e_{i}^{-}-e_{i}^{-}\,e_{i}^{+}=\frac{K_{i}-K_{i}^{-1}}{\hbar^{1/2}-\hbar^{-1/2}}\;,\qquad(i=0,1)$
$\displaystyle e_{0}^{\pm}\,e_{1}^{\mp}=e_{1}^{\mp}\,e_{0}^{\pm}\;,$
$\displaystyle(e_{i}^{\pm})^{3}e_{j}^{\pm}-\frac{\hbar^{3/2}-\hbar^{-3/2}}{\hbar^{1/2}-\hbar^{-1/2}}\,(e_{i}^{\pm})^{2}e_{j}^{\pm}e_{i}^{\pm}+\frac{\hbar^{3/2}-\hbar^{-3/2}}{\hbar^{1/2}-\hbar^{-1/2}}\,e_{i}^{\pm}e_{j}^{\pm}(e_{i}^{\pm})^{2}-e_{j}^{\pm}(e_{i}^{\pm})^{3}=0\;.\qquad(i\neq
j)$ (9.1)

The constraints on the last line are a quantized version of the familiar
$\widehat{A_{1}}$ Serre relations. The Hopf algebra 110110110For simplicity’s
sake, we skip writing down the action of the counit and antipode on the
generators. nature of $U_{\hbar}(\widehat{A_{1}})$ guarantees the existence of
a coproduct $\Delta:U_{\hbar}(\widehat{A_{1}})\rightarrow
U_{\hbar}(\widehat{A_{1}})\otimes U_{\hbar}(\widehat{A_{1}})$:

$\displaystyle\Delta(e_{i}^{+})=e_{i}^{+}\otimes K_{i}+1\otimes
e_{i}^{+}\;,\qquad(i=0,1)$ $\displaystyle\Delta(e_{i}^{-})=e_{i}^{-}\otimes
1+K_{i}^{-1}\otimes e_{i}^{-}\;,\qquad(i=0,1)$ $\displaystyle\Delta(K_{i}^{\pm
1})=K_{i}^{\pm 1}\otimes K_{i}^{\pm 1}\;.\qquad(i=0,1)$ (9.2)

Consider the subalgebra $U_{\hbar}(A_{1})$ of $U_{\hbar}(\widehat{A_{1}})$
generated by $e^{\pm}=e^{\pm}_{1}$ and $K^{\pm 1}=K^{\pm 1}_{1}$. For all
$x\in\mathbb{C}^{\times}$, evaluation representations are defined through the
(surjective) algebra homomorphism111111111We called this homomorphism
$\text{ev}\circ D_{x}$ in the main text, where $D_{x}$ denoted a shift
automorphism. $\text{ev}_{x}:U_{\hbar}(\widehat{A_{1}})\rightarrow
U_{\hbar}(A_{1})$, as

$\displaystyle\text{ev}_{x}(e^{\pm}_{0})=x^{\pm 1}\,\hbar^{\mp
1/2}\,e^{\mp}\;,\qquad\qquad\text{ev}_{x}(e^{\pm}_{1})=e^{\pm}\;,$
$\displaystyle\text{ev}_{x}(K^{\pm 1}_{0})=K^{\mp
1}\;,\qquad\qquad\text{ev}_{x}(K^{\pm 1}_{1})=K^{\pm 1}\;.$ (9.3)

Then, let ${\mathcal{V}}_{i}$ be a highest weight representation of
$U_{\hbar}(A_{1})$, with highest weight vector $v_{i}$ and corresponding
weight $\mu_{i}$, and let ${\mathcal{V}}_{i}(x)$ be the corresponding
evaluation representation in $U_{\hbar}(\widehat{A_{1}})$, obtained after
applying the homomorphism $\text{ev}_{x}$. The trigonometric quantum
$R$-matrix of $U_{\hbar}(\widehat{A_{1}})$ is an endomorphism
$R_{\mathcal{V}_{i}\,\mathcal{V}_{j}}(x):\text{End}(\mathcal{V}_{i}(x)\otimes\mathcal{V}_{j}(1))$,
defined so that it satisfies

$\Delta^{\prime}(y)\,R_{\mathcal{V}_{i}\,\mathcal{V}_{j}}(x)=R_{\mathcal{V}_{i}\,\mathcal{V}_{j}}(x)\,\Delta(y)\,,\qquad
y\in U_{\hbar}(\widehat{A_{1}})\;,$ (9.4)

with $\Delta^{\prime}=\sigma\circ\Delta$, and $\sigma(y\otimes z)=z\otimes y$.

The $R$-matrix is normalized so that

$R_{\mathcal{V}_{i}\,\mathcal{V}_{j}}(x)\,v_{i}\otimes v_{j}=v_{i}\otimes
v_{j}\;.$ (9.5)

It is sufficient to understand the action of the $R$-matrix on the two-
dimensional space with basis $\\{e^{-}v_{i}\otimes v_{j}\;,\;v_{i}\otimes
e^{-}v_{j}\\}$. One computes

$\displaystyle R_{\mathcal{V}_{i}\,\mathcal{V}_{j}}(x)(e^{-}v_{i}\otimes
v_{j})=\frac{x\,\hbar^{m_{j}}-\hbar^{m_{i}}}{x-\hbar^{m_{i}+m_{j}}}\,e^{-}v_{i}\otimes
v_{j}+\frac{1-\hbar^{2m_{j}}}{x-\hbar^{m_{i}+m_{j}}}\,v_{i}\otimes
e^{-}v_{j}\;,$ $\displaystyle
R_{\mathcal{V}_{i}\,\mathcal{V}_{j}}(x)(v_{i}\otimes
e^{-}v_{j})=\frac{x\,(1-\hbar^{2m_{j}})}{x-\hbar^{m_{i}+m_{j}}}\,e^{-}v_{i}\otimes
v_{j}+\frac{x\,\hbar^{m_{i}}-\hbar^{m_{j}}}{x-\hbar^{m_{i}+m_{j}}}\,v_{i}\otimes
e^{-}v_{j}\;,$ (9.6)

Above, we have defined complex parameters

$m_{i}=\frac{\langle\nu_{i},e^{+}\rangle}{2}\;,$ (9.7)

where $e^{+}=\alpha$ is the positive simple root of $A_{1}$. Integral
solutions to the $U_{\hbar}(\widehat{A_{1}})$ qKZ equations (2.2) have been
constructed by Matsuo [48, 49]. Such solutions are valued in an
$L$-dimensional space of weight

$\nu_{0}-\nu_{\infty}=\sum_{d=1}^{L}\nu_{d}-N\,e^{+}\;.$ (9.8)

All the features of tame ramification will already be manifest for $N=1$, so
we henceforth specialize to that case in order not to overburden the
presentation. Then, for all $i=1,\ldots,L$, define the integrals

${\bf
F}_{i}(\vec{x})=\oint_{\Gamma}\frac{dy}{y}\,y^{\eta}\,K_{i}(y,\vec{x})\,\prod_{d=1}^{L}\frac{(y/\tilde{x}_{d};q)_{\infty}}{(q^{\sigma_{d}}\,y/\tilde{x}_{d};q)_{\infty}}\;,$
(9.9)

where the complex numbers $\sigma_{d}$ and $\eta$ are determined in terms of
the highest weights $\nu_{d}$ and $\nu_{0}$ as

$q^{\sigma_{d}}=\hbar^{\langle\nu_{d},e^{+}\rangle}\;,\qquad\;q^{\eta}=\hbar^{-\langle\nu_{0},e^{+}\rangle}$
(9.10)

and

$K_{i}(y,\vec{x})=\frac{1}{1-y/\tilde{x}_{i}}\prod_{d=1}^{i-1}\frac{1-q^{\sigma_{d}}\,y/\tilde{x}_{d}}{1-y/\tilde{x}_{d}}\;.$
(9.11)

We renormalize the $F_{i}(\vec{x})$ as

$\varphi_{i}(\vec{x})=q^{(\sigma_{i+1}+\ldots+\sigma_{L})/2}\;x^{\sigma_{1}}_{1}\ldots
x^{\sigma_{L}}_{L}\,F_{i}(q^{\sigma_{1}/2}\,x_{1},\ldots,q^{\sigma_{L}/2}\,x_{L})\;.$
(9.12)

Then Matsuo showed that

$\Psi(\vec{x})=\sum_{i=1}^{L}\varphi_{i}(x_{1},\ldots,x_{L})\,v_{1}\otimes\ldots\otimes
e^{-}v_{i}\otimes\ldots\otimes v_{L}$ (9.13)

solves the $U_{\hbar}(\widehat{A_{1}})$ qKZ equations. More precisely, this is
a solution to the qKZ equations for any choice of contour for which the
integral remains invariant under $q$-shift. In particular, the contours are
such that $\int\frac{dy}{y}\,I(q\,y)=\int\frac{dy}{y}\,I(y)$, with $I(y)$ the
integrand in (9.9).

To make the appearance of K-theoretic stable envelopes explicit, it is useful
to rescale the equivariant parameters $\tilde{x}_{d}$ by $q^{\sigma_{d}}$ and
write

${\bf
F}_{i}(\vec{x})=\oint_{\Gamma}\frac{dy}{y}\,y^{\eta}\,K_{i}(y,\vec{x})\,\prod_{d=1}^{L}\frac{(q^{-\sigma_{d}}\,y/\tilde{x}_{d};q)_{\infty}}{(y/\tilde{x}_{d};q)_{\infty}}\;,$
(9.14)

with

$K_{i}(y,\vec{x})=\frac{1}{1-q^{-\sigma_{i}}\,y/\tilde{x}_{i}}\prod_{d=1}^{i-1}\frac{1-y/\tilde{x}_{d}}{1-q^{-\sigma_{d}}\,y/\tilde{x}_{d}}\;.$
(9.15)

But this is equal to

${\bf
F}_{i}(\vec{x})=\oint_{\Gamma}\frac{dy}{y}\,y^{\eta}\,\text{Stab}^{K}_{i}(y,\vec{x})\,\prod_{d=1}^{L}\frac{(q^{1-\sigma_{d}}\,y/\tilde{x}_{d};q)_{\infty}}{(y/\tilde{x}_{d};q)_{\infty}}\;,$
(9.16)

where

$\text{Stab}^{K}_{i}(y,\vec{x})=\prod_{d=1}^{i-1}(1-y/\tilde{x}_{d})\;\prod_{d=i+1}^{L}(1-q^{-\sigma_{d}}\,y/\tilde{x}_{d})$
(9.17)

is the stable envelope for the $i$-th fixed point.

###  9.2 The $\mathcal{W}_{q,t}(A_{1})$-algebra

The $q$-conformal blocks of $\mathcal{W}_{q,t}(A_{1})$ are the magnetic
blocks, which are components of the Vertex function

$\mathbb{V}=\left\langle
v_{\mu_{\infty}},{Q}^{\vee}\prod_{d=1}^{L}\mathcal{V}_{\\{\lambda\\}_{d}}(\tilde{x}_{d})\;v_{\mu_{0}}\right\rangle\,.$
(9.18)

The screening charge ${Q}^{\vee}$ is the $(-1)$-st Fourier coefficient of the
screening current,

$\displaystyle{Q}^{\vee}=\int dy\,S^{\vee}(y)\;,$ $\displaystyle
S^{\vee}(y)=y^{-\alpha[0]}\,:\exp\left(\sum_{k\neq 0}{\alpha[k]\over
q^{k\,\over 2}-q^{-\,{k\,\over 2}}}\,y^{k}\right):\;.$ (9.19)

The modes $\alpha[k]$ with $k\in\mathbb{Z}$ obey the deformed Heisenberg
algebra relations

$\displaystyle[\alpha[k],\alpha[n]]={1\over k}(q^{k\over 2}-q^{-{k\over
2}})(t^{{k\over 2}}-t^{-{k\over 2}})(q^{k\over 2}\,t^{-k\over 2}+q^{-k\over
2}\,t^{k\over 2})\delta_{k,-n}\;.$ (9.20)

The vector $v_{\mu_{0}}$ defines a Fock space representation as:

$\displaystyle\alpha[0]\,v_{\mu_{0}}$
$\displaystyle=\langle\mu_{0},\alpha\rangle\,v_{\mu_{0}}$
$\displaystyle\alpha[k]\,v_{\mu_{0}}$
$\displaystyle=0\,,\qquad\qquad\;\;\mbox{for}\;k>0\;.$ (9.21)

Our first task is to construct the deformed primary vertex operators
$V_{\\{{\lambda}\\}_{d}}$. According to our proposal, each such operator, say
the $d$-th one among $L$ of them, is defined via a set of $J_{d}$ weights
$\\{{\lambda}\\}_{d}=\\{\lambda_{d,1},\ldots,\lambda_{d,J_{d}}\\}$ chosen
among the fundamental representations of
$U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$, with a constraint on the degree of
their respective Drinfeld polynomials.

Explicitly, consider the Cartan eigenspace decomposition of the fundamental
representation $\widehat{V}$ in $U_{\hbar}(\widehat{A_{1}})$. This
representation is two-dimensional, and admits the following linear space
decomposition as a $U_{\hbar}({A_{1}})$-module:

$\widehat{V}=V_{Y}\oplus V_{Y^{-1}}\;.$ (9.22)

Consider the two weight spaces $V_{\lambda_{1}}=V_{Y}$ and
$V_{\lambda_{2}}=V_{Y^{-1}}$ in $U_{\hbar}(\widehat{A_{1}})$. As vector
spaces, $V_{Y}=\mathbb{C}v_{Y}$ with $v_{Y}$ the highest weight vector of the
fundamental representation, while $V_{Y^{-1}}=\mathbb{C}v_{Y^{-1}}$, with
$v_{Y^{-1}}$ the lowest weight vector of the fundamental representation. Let
us derive the Drinfeld polynomials for $V_{\lambda_{1}}$ and $V_{\lambda_{2}}$
explicitly.

—— _FIRST METHOD_ ——

First, we will work straight from the representation theory of
$Rep(U_{\hbar}(\widehat{A_{1}}))$. Instead of the Drinfeld-Jimbo presentation
we reviewed above (9.1), we will need instead to consider Drinfeld’s so-called
second realization of the quantum affine algebra [60]. It consists of
generators $x^{\pm}_{k},H_{r},K^{\pm 1}$, where $k\in\mathbb{Z}$ and
$r\in\mathbb{Z}^{*}$, and $c^{\pm 1}$ in the center, along with the relations:

$\displaystyle K\,K^{-1}=K^{-1}\,K=1\;,\qquad c\,c^{-1}=c^{-1}\,c=1\;,$
$\displaystyle[K,H_{r}]=0\;,\qquad\qquad\qquad\;\,K\,X^{\pm}_{k}=\hbar^{\pm
1}\,X^{\pm}_{k}\,K\;,$
$\displaystyle[H_{r},X^{\pm}_{k}]=\pm\frac{1}{r}\,\frac{\hbar^{r}-\hbar^{-r}}{\hbar^{1/2}-\hbar^{-1/2}}\,c^{\mp|r|}\,X^{\pm}_{r+k}\;,$
$\displaystyle[H_{r},H_{r^{\prime}}]=\delta_{r,-r^{\prime}}\frac{1}{r}\,\frac{\hbar^{r}-\hbar^{-r}}{\hbar^{1/2}-\hbar^{-1/2}}\,\frac{c^{r}-c^{-r}}{\hbar^{1/2}-\hbar^{-1/2}}\;,$
$\displaystyle X^{\pm}_{\kappa+1}\,X^{\pm}_{k^{\prime}}-\hbar^{\pm
1}\,X^{\pm}_{k^{\prime}}\,X^{\pm}_{\kappa+1}=\hbar^{\pm
1}\,X^{\pm}_{k}\,X^{\pm}_{k^{\prime}+1}-X^{\pm}_{k^{\prime}+1}\,X^{\pm}_{k}\;,$
$\displaystyle[X^{+}_{k},X^{-}_{k^{\prime}}]=\frac{1}{\hbar^{1/2}-\hbar^{-1/2}}(c^{k-k^{\prime}}\psi^{+}_{\kappa+k^{\prime}}-c^{k^{\prime}-k}\psi^{-}_{\kappa+k^{\prime}})\;.$
(9.23)

with

$\displaystyle\sum_{k=0}^{\infty}\psi^{\pm}_{\pm k}\,z^{\pm
k}=K^{\pm}\exp\left(\pm(\hbar-\hbar^{-1})\sum_{s=1}^{\infty}H_{\pm s}\,z^{\pm
s}\right)\;,$ (9.24)

and $\psi^{+}_{k}=0=\phi^{-}_{-k}$ for $k<0$. In this formalism, a highest
weight vector $v$ is defined so as to satisfy112112112From the action of $c$
on $v$ and the fourth line in (9.2), it follows that
$[H_{r},H_{r^{\prime}}]=0$ for highest weight modules, showing the existence
of a commutative “Cartan subalgebra” for those modules, like in the finite
case $Rep(U_{\hbar}({A_{1}}))$. When one further fixes the sign ambiguity by
imposing that $c$ act as the identity on $v$, the corresponding representation
is said to be of type 1. All the representations we consider here are of this
type.

$\displaystyle\psi^{+}_{k}\cdot v=\lambda^{+}_{k}\,v\;,\;\qquad k\geq
0,\;\;\lambda^{+}_{k}\in\mathbb{C}\;,$ $\displaystyle\psi^{-}_{k}\cdot
v=\lambda^{-}_{k}\,v\;,\;\qquad k\leq 0,\;\;\lambda^{-}_{k}\in\mathbb{C}\;,$
$\displaystyle X^{+}_{k}\cdot v=0\;,\;\;\;\;\;\,\qquad k\in\mathbb{Z}\;,$
$\displaystyle(c\cdot v=v)$ (9.25)

Our task is to identify the weights $\lambda^{+}_{k}$ explicitly. The weights
$\lambda^{-}_{k}$ will be redundant in this discussion, so we ignore them from
now on. The strategy is to invoke the evaluation representations from
$Rep(U_{\hbar}(\widehat{A_{1}}))$ to $Rep(U_{\hbar}({A_{1}}))$, and then pull
back the eigenvalue content of the latter to obtain the weights
$\lambda^{+}_{k}$. Explicitly, in Drinfeld’s second realization, the
surjective homomorphism $\text{ev}_{x}:U_{\hbar}(\widehat{A_{1}})\rightarrow
U_{\hbar}(A_{1})$ reads

$\displaystyle\widetilde{\text{ev}}_{x}(c^{\pm
1})=1\;,\qquad\qquad\qquad\qquad\;\;\;\;\;\widetilde{\text{ev}}_{x}(K)=K\;,$
$\displaystyle\widetilde{\text{ev}}_{x}(x^{+}_{k})=\hbar^{-k/2}\,x^{k}\,K^{k}\,e^{+}\;,\qquad\qquad\widetilde{\text{ev}}_{x}(x^{-}_{k})=\hbar^{-k/2}\,x^{k}\,e^{-}\,K^{k}\;,$
(9.26)

for $x\in\mathbb{C}^{\times}$. Then, consider a basis $(v_{Y},v_{Y^{-1}})$ of
the fundamental representation $V$ of $U_{\hbar}(A_{1})$, on which the
generators act as113113113This is simply the quantum analog of the familiar
action of the Pauli matrices $\sigma_{z}$, $\sigma_{+}$ and $\sigma_{-}$ on a
qubit in the fundamental representation of $A_{1}$.:

$\displaystyle K\cdot v_{Y}=\hbar^{1/2}\,v_{Y}\;,\qquad\;\;\;\;K\cdot
v_{Y^{-1}}=\hbar^{-1/2}v_{Y^{-1}}\;,$ $\displaystyle e^{+}\cdot
v_{Y^{-1}}=v_{Y}\;,\qquad\qquad e^{+}\cdot v_{Y}=0\;,$ $\displaystyle
e^{-}\cdot v_{Y}=v_{Y^{-1}}\;,\qquad\qquad e^{-}\cdot v_{Y^{-1}}=0\;.$ (9.27)

As a vector space, this fundamental representation $V$ coincides with the
fundamental representation $\widehat{V}=V_{Y}\oplus V_{Y^{-1}}$ in the quantum
affine case: $V_{Y}=\mathbb{C}v_{Y}$ with $v_{Y}$ the highest weight vector of
the fundamental representation, while $V_{Y^{-1}}=\mathbb{C}v_{Y^{-1}}$, with
$v_{Y^{-1}}$ the lowest weight vector of the fundamental representation.
Making use of the evaluation homomorphism $\text{ev}_{x}$, we derive the
action of the quantum affine raising and lowering generators on the basis
vectors:

$\displaystyle X^{+}_{k}\cdot v_{Y^{-1}}=x^{k}\,v_{Y}\;,\qquad\qquad
X^{+}_{k}\cdot v_{Y}=0\;,$ $\displaystyle X^{-}_{k}\cdot
v_{Y}=x^{k}\,v_{Y^{-1}}\;,\qquad\qquad X^{-}_{k}\cdot v_{Y^{-1}}=0\;.$ (9.28)

Using the commutation relation in the last line of (9.2), we easily deduce the
eigenvalues $\lambda^{+}_{k}$:

$\displaystyle\psi^{+}_{0}\cdot
v_{Y}=\hbar^{1/2}\,v_{Y}\;,\qquad\qquad\qquad\;\;\,\;\;\psi^{+}_{0}\cdot
v_{Y^{-1}}=\hbar^{-1/2}\,v_{Y}\;,$ $\displaystyle\psi^{+}_{k}\cdot
v_{Y}=x^{k}\,(\hbar^{1/2}-\hbar^{-1/2})\,v_{Y}\;,\qquad\psi^{+}_{k}\cdot
v_{Y^{-1}}=x^{k}\,(\hbar^{-1/2}-\hbar^{1/2})\,v_{Y^{-1}}\;,\;\;\;\;k>0\;.$
(9.29)

Following Chari and Pressley [59], the generating series of eigenvalues is a
ratio of polynomials. For the vector space $V_{Y}$, one finds114114114The
general statement is as follows: for any weight space $V_{\mu}$ present in the
decomposition of a $U_{\hbar}(\widehat{\mathfrak{g}})$ finite-dimensional
representation $\widehat{V}=\bigoplus_{\mu}V_{\mu}$, there exist polynomials
${\mathcal{A}}^{+}_{\mu}(z),\;\;{\mathcal{A}}^{-}_{\mu}(z)\in\mathbb{C}[z]$,
of constant term ${\mathcal{A}}^{+}_{\mu}(0)=1={\mathcal{A}}^{-}_{\mu}(0)$,
which satisfy
$\sum_{k=0}^{\infty}\lambda^{+}_{k,\mu}\,z^{k}=\hbar^{(\text{deg}({\mathcal{A}}^{+}_{\mu})-\text{deg}({\mathcal{A}}^{-}_{\mu}))/2}\,\frac{{\mathcal{A}}^{+}_{\mu}(\hbar^{-1/2}z)}{{\mathcal{A}}^{+}_{\mu}(\hbar^{1/2}z)}\,\frac{{\mathcal{A}}^{-}_{\mu}(\hbar^{1/2}z)}{{\mathcal{A}}^{-}_{\mu}(\hbar^{-1/2}\,z)}\;.$
(9.30)

$\sum_{k=0}^{\infty}\lambda^{+}_{k,Y}\,z^{k}=\hbar^{1/2}+(\hbar^{1/2}-\hbar^{-1/2})\sum_{k=1}^{\infty}x^{k}\,z^{k}=\hbar^{1/2}\,\frac{1-\hbar^{-1}\,x\,z}{1-x\,z}\equiv\hbar^{1/2}\,\frac{{\mathcal{A}}^{+}_{Y}(\hbar^{-1/2}z)}{{\mathcal{A}}^{+}_{Y}(\hbar^{1/2}z)}\;,$
(9.31)

where in the last step we defined the polynomial
${\mathcal{A}}^{+}_{Y}(z)=(1-x\,\hbar^{-1/2}z)$. Similarly for the vector
space $V_{Y^{-1}}$, one finds

$\sum_{k=0}^{\infty}\lambda^{+}_{k,Y^{-1}}\,z^{k}=\hbar^{-1/2}\,\frac{1-\hbar\,x\,z}{1-x\,z}\equiv\hbar^{-1/2}\,\frac{{\mathcal{A}}^{-}_{Y^{-1}}(\hbar^{1/2}\,z)}{{\mathcal{A}}^{-}_{Y^{-1}}(\hbar^{-1/2}\,z)}\;,$
(9.32)

where ${\mathcal{A}}^{-}_{Y^{-1}}(z)=(1-x\,\hbar^{1/2}z)$. Each weight space
is in bijection with its set of polynomials, so we are done. We normalize the
highest weight polynomial to be

${\mathcal{A}}^{+}_{Y}(z)|_{x=\hbar^{1/2}}=(1-z)$ (9.33)

so that

${\mathcal{A}}^{-}_{Y^{-1}}(z)|_{x=\hbar^{1/2}}=(1-\hbar\,z)\;.$ (9.34)

We call these normalized polynomials the Drinfeld polynomials.

Our definition of a deformed $\mathcal{W}_{q,t}(A_{1})$ primary operator, say
the $d$-th one, is

$\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d})=\;:\prod\limits_{s=1}^{J_{d}=2}\prod_{i=1}^{\deg(\mathcal{A}^{+}_{\lambda_{d,s}})}\Lambda\left(\hbar^{-\mathfrak{a}^{+}_{\lambda_{d,s},i}}\,x_{d,s}\right)\prod_{j=1}^{\deg(\mathcal{A}^{-}_{\lambda_{d,s}})}\Lambda^{-1}\left(\hbar^{-\mathfrak{a}^{-}_{\lambda_{d,s},i}}\,x_{d,s}\right):\;,$
(9.35)

with the arguments of the operators

$\Lambda^{\pm 1}(x)=\;:\exp\left(\pm\,\sum_{k\neq 0}{w[k]\over(q^{k\over
2}-q^{-\,{k\over 2}})(t^{k\over 2}-t^{-\,{k\over 2}})}\,x^{k}\right):\;,$
(9.36)

given by the Drinfeld roots. We find at once $\mathfrak{a}^{+}_{Y,1}=0$ and
$\mathfrak{a}^{-}_{Y^{-1},1}=1$ as the only nonzero integers in
${\mathcal{A}}^{+}_{Y}(z)=(1-\hbar^{\mathfrak{a}^{+}_{Y,1}}z)$ and
${\mathcal{A}}^{-}_{Y^{-1}}(z)=(1-\hbar^{\mathfrak{a}^{-}_{Y^{-1},1}}z)$,
respectively. In the end, we propose the following deformed primary operator
expression:

$\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d})=\;:\Lambda\left(x_{d,1}\right)\;\left[\Lambda\left(\hbar^{-1}\,x_{d,2}\right)\right]^{-1}:\;.$
(9.37)

######  Remark 9.1.

Since $A_{1}$ is simply-laced, the Drinfeld polynomials can simply be read off
the Frenkel-Reshetikhin $\hbar$-character map [61]:

$\displaystyle\chi_{\hbar}:Rep(U_{\hbar}(\widehat{A_{1}}))$
$\displaystyle\rightarrow\mathbb{Z}[Y^{\pm 1}(z)]$
$\displaystyle\widehat{V}\;\;\;$ $\displaystyle\mapsto
t^{\widehat{V}}(z)=Y(z)+Y^{-1}(\hbar^{-1}z)\,.$ (9.38)

The only nontrivial polynomials are ${\mathcal{A}}^{+}_{Y}(z)=(1-z)$, which
accounts for the first term $Y(z)$, and
${\mathcal{A}}^{-}_{Y^{-1}}(z)=(1-\hbar\,z)$, which accounts for the second
term $Y^{-1}(\hbar^{-1}z)$.

—— _SECOND METHOD:_ ——

We construct the generating currents of $\mathcal{W}_{q,t}(\mathfrak{g})$ as
the commutant of the screening charges, and then invoke the $(q,t)$-character
map (2.1.3) to make contact with the representation ring of the quantum affine
algebra. The algebra $\mathcal{W}_{q,t}(A_{1})$ has a single generator, the
deformed Virasoro stress tensor, constructed as the commutant of the $A_{1}$
screening charge (9.2) (and its dual). A computation shows that this commutant
has the following form:

$T^{\widehat{V}}(z)=\mathcal{Y}(z)+\left[\mathcal{Y}(\sqrt{t^{2}/q^{2}}\,z)\right]^{-1}\;.$
(9.39)

Here, $\mathcal{Y}$ is the $\mathfrak{g}=A_{1}$ version of the vertex
operators (2.23). But $T^{\widehat{V}}$ can also be understood as a
$(q,t)$-character homomorphism
$Rep(U_{q}(\widehat{A_{1}}))\rightarrow\mathcal{W}_{q,t}(A_{1})$, a
$t$-refinement of Frenkel and Reshetikhin’s $q$-character for the fundamental
representation $\widehat{V}$ of $U_{q}(\widehat{A_{1}})$. Note the argument of
the $\mathcal{Y}$ operators above does not depend on $q$ and $t$ separately,
but only on the ratio $q/t=\hbar$ (as is the case for all simply-laced
$\mathfrak{g}_{\text{o}}$). One reinterprets the $(q,t)$-character as a map
$Rep(U_{\hbar}(\widehat{A_{1}}))\rightarrow\mathcal{W}_{q,t}(A_{1})$, where
the current $T^{\widehat{V}}(x)$ stands for the image of the eigenspace
decomposition $\widehat{V}=V_{Y}\oplus V_{Y^{-1}}$ in
$Rep(U_{\hbar}(\widehat{A_{1}}))$, and the weight spaces $V_{Y}$ and
$V_{Y^{-1}}$ have respective Drinfeld polynomials
${\mathcal{A}}^{+}_{Y}(z)=(1-z)$ and
${\mathcal{A}}^{-}_{Y^{-1}}(z)=(1-\sqrt{q^{2}/t^{2}}\,z)$.

By our proposal, this data uniquely characterizes a deformed
$\mathcal{W}_{q,t}(A_{1})$ primary operator:

$\displaystyle\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d})=\;:\Lambda\left(x_{d,1}\right)\;\left[\Lambda\left(\sqrt{t^{2}/q^{2}}\,x_{d,2}\right)\right]^{-1}:\;.$
(9.40)

After rewriting the nontrivial Drinfeld root as
$\sqrt{t^{2}/q^{2}}=\hbar^{-1}$, this agrees nicely with the operator (9.37)
from the previous construction.

The deformed primary has a center of mass position $\tilde{x}_{d}$ on
$\mathcal{C}=\mathbb{C}^{\times}$. Each individual vertex operator
$\Lambda^{\pm}$ is itself located at a position $x_{d,j}$, which is
$q$-shifted relative to $\tilde{x}_{d}$ by some amount
$\sigma_{d,j}\in\mathbb{C}$:

$x_{d,j}\equiv\tilde{x}_{d}\,q^{\sigma_{d,j}}\;,\qquad\;\;j=1,2\;.$ (9.41)

Correspondingly, we define the “momentum” $\mathfrak{g}$-coweight of the
$d$-th deformed primary $\mathcal{V}_{\\{{\lambda}\\}_{d}}$ as

$\widetilde{\sigma}_{d}\equiv\sigma_{d,1}\,\underline{\lambda_{1}}+\sigma_{d,2}\,\underline{\lambda_{2}}\;,\qquad\text{with}\;\;\underline{\lambda_{1}}=[\phantom{-}1]\;,\;\;\;\underline{\lambda_{2}}=[-1]\;.$
(9.42)

In the rest of this section, we will also make repeated use of the notation

$\sigma_{d}\equiv\langle\widetilde{\sigma}_{d},e^{+}\rangle=\sigma_{d,1}-\sigma_{d,2}\;,$
(9.43)

Inside the chiral correlator (9.18), we insert $L$ such primaries, which are
simply $L$ copies of the above deformed primary,

$\prod_{d=1}^{L}\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d})\;,\qquad\;d=1,\ldots,L\;.$
(9.44)

We assume that the positions $\tilde{x}_{d}$ on $\mathcal{C}$ are distinct and
generic. Each of the $L$ operators
$\mathcal{V}_{\\{\lambda\\}_{d}}(\tilde{x}_{d})$ has a well-defined conformal
limit $t=q^{\beta}$, $q\rightarrow 1$, where it becomes the Virasoro primary
vertex operator $e^{\langle\widetilde{\sigma}_{d},\phi(x)\rangle}$, with
$\phi(x)$ the corresponding free boson on $\mathcal{C}$.

The vertex function $\mathbb{V}$ evaluates to

$\left\langle
v_{\mu_{\infty}},{Q}^{\vee}\;\prod_{d=1}^{L}\mathcal{V}_{\\{{\lambda}\\}_{d}}(\tilde{x}_{d})\;v_{\mu_{0}}\right\rangle=\oint_{\Gamma}\frac{dy}{y}\,y^{-\langle\mu_{0},\alpha\rangle}\,\prod_{d=1}^{L}\frac{\left(\sqrt{q/t}\;y/x_{d,1}\,;q\right)_{\infty}}{\left(\sqrt{t/q}\;y/x_{d,2}\,;q\right)_{\infty}}\;,$
(9.45)

in agreement with the general formula (5.2). After rescaling the equivariant
parameters $x_{d,s}$ by appropriate factors of $q$ and $t$ and switching to
the center of mass coordinates (9.41) and (9.43), this chiral correlator is
conveniently written as

$\oint_{\Gamma}\frac{dy}{y}\,y^{-\langle\mu_{0},\alpha\rangle}\,\prod_{d=1}^{L}\frac{(q^{1-\sigma_{d}}\,y/\tilde{x}_{d};q)_{\infty}}{(y/\tilde{x}_{d};q)_{\infty}}\;.$
(9.46)

Here, $\Gamma$ is a sum of contours enclosing all possible poles. For $|q|<1$,
we choose a contour $C_{k}$ which encloses “$\infty$” and the infinite number
of poles at

$C_{k}\;:\qquad y=q^{-n}\,\tilde{x}_{k}\,,\qquad\;n=0,1,2,\ldots$ (9.47)

This translates to choosing a particular component $\mathbb{V}_{k}$ of the
vertex function $\mathbb{V}$. The integral is readily evaluated by residues:

$\mathbb{V}_{k}=(\tilde{x}_{k})^{\eta}\frac{(q^{1-\sigma_{k}};q)_{\infty}}{(q;q)_{\infty}}\prod_{d\neq
k}^{L}\frac{(q^{1-\sigma_{d}}\tilde{x}_{k}/\tilde{x}_{d};q)_{\infty}}{(\tilde{x}_{k}/\tilde{x}_{d};q)_{\infty}}\sum_{r\geq
0}\left(\frac{z}{\prod_{i}q^{\sigma_{d}-1}}\right)^{r}\prod_{d=1}^{L}\frac{(q^{\sigma_{d}}\tilde{x}_{d}/\tilde{x}_{k};q)_{r}}{(q\,\tilde{x}_{d}/\tilde{x}_{k};q)_{r}}$
(9.48)

We have introduced the fugacity $z=q^{\langle\mu_{0},\alpha\rangle}$, and the
identification $\eta=-\langle\mu_{0},\alpha\rangle$. We also made use of the
finite $q$-Pochhammer symbol notation
$(x;q)_{r}=\prod_{k=0}^{r-1}(1-x\,q^{k})$, with $(x;q)_{0}=1$.

Up to normalization, this is a $q$-hypergeometric function:

$\mathbb{V}_{k}=(\tilde{x}_{k})^{\eta}\frac{(q^{1-\sigma_{k}};q)_{\infty}}{(q;q)_{\infty}}\prod_{i\neq
k}\frac{(q^{1-\sigma_{i}}\tilde{x}_{k}/\tilde{x}_{i};q)_{\infty}}{(\tilde{x}_{k}/\tilde{x}_{i};q)_{\infty}}\,{}_{n}\mathbb{F}_{n}{\left[\genfrac{.}{.}{0.0pt}{}{q^{\sigma_{i}}\tilde{x}_{k}/\tilde{x}_{i}}{q\,\tilde{x}_{k}/\tilde{x}_{i}};\frac{z}{\prod_{i}q^{\sigma_{i}-1}}\right]}\;,$
(9.49)

with

${}_{n}\mathbb{F}_{n}{\left[\genfrac{.}{.}{0.0pt}{}{a_{i}}{b_{i}};z\right]}=\sum_{r\geq
0}z^{r}\prod_{i=1}^{n}\frac{(a_{i};q)_{r}}{(b_{i};q)_{r}}\;.$ (9.50)

The vertex function $\mathbb{V}$, as well as the qKZ integral solutions (9.9),
are analytic in the chamber

$\mathfrak{C}_{C}=\\{|z|<1\\}\;,$ (9.51)

but they are not jointly analytic in any $\\{\tilde{x}_{d}\\}$-chamber. For
instance, looking at the components of the vertex function, it is clear from
our expressions that $\mathbb{V}_{1}$ is analytic when $|x_{1}|<|x_{2}|$,
while $\mathbb{V}_{2}$ is instead analytic when $|x_{2}|<|x_{1}|$ (note that
both components are $z$-analytic). Since bona fide $q$-conformal blocks of
$U_{\hbar}(\widehat{A_{1}})$ are correlation functions of chiral operators on
$\mathcal{C}$, they must be analytic in _some_ $\\{\tilde{x}_{d}\\}$-chamber.
To remedy this, we first fix a chamber

$\mathfrak{C}_{H}=\\{|x_{1}|<|x_{2}|<\ldots<|x_{L}|\\}\;,$ (9.52)

and then perform a linear change of basis between $z$-solutions and
$\\{\tilde{x}_{d}\\}$-solutions to obtain a new vertex function analytic in
chamber $\mathfrak{C}_{H}$:

$\mathbb{V}_{\mathfrak{C}_{H}}=\oint_{C}\frac{dy}{y}\,y^{\eta}\,\prod_{d=1}^{L}\frac{(q^{1-\sigma_{d}}\,y/\tilde{x}_{d};q)_{\infty}}{(y/\tilde{x}_{d};q)_{\infty}}\;\mathfrak{B}_{\mathfrak{C}_{H}}\;.$
(9.53)

The change of basis is implemented using a matrix
$\mathfrak{B}_{\mathfrak{C}_{H}}$ of $q$-periodic pseudo-constants:

$\mathfrak{B}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})=U_{\mathfrak{C}_{H},k}(z,\\{\tilde{x}_{d}\\})\;\frac{\text{Stab}^{Ell}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})}{\displaystyle\prod_{d=1}^{L}\Theta\left(q^{\sigma_{d}}\tilde{x}_{d}/y\right)}\;\text{e}(y,z)^{-1}\,,$
(9.54)

where $\text{Stab}^{Ell}_{\mathfrak{C}_{H},k}$ is the elliptic stable envelope
for the $k$-th fixed point of a quiver variety we will spell out in the next
section.

$\text{Stab}^{Ell}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})=\frac{\displaystyle\prod_{d<k}\Theta\left(\tilde{x}_{d}/y\right)\Theta\left(q^{\sum_{d=1}^{k}\sigma_{d}}\tilde{x}_{k}/yz\right)\prod_{d>k}\Theta\left(q^{\sigma_{d}}\tilde{x}_{d}/y\right)}{\displaystyle\Theta\left(q^{\sum_{d=1}^{k}\sigma_{d}}/z\right)}$
(9.55)

The additional factors present in $\mathfrak{B}_{\mathfrak{C}_{H},k}$ ensure
that the matrix is indeed $q$-periodic in the variables
$y,z,x_{1},x_{2},\ldots,x_{L}$; we have an overall normalization by the theta
function $\Theta\left(q^{\sum_{d=1}^{k}\sigma_{d}}/z\right)$, as well as

$\text{e}(y,z)^{-1}=\exp\left[\frac{\ln(y)\ln(z)}{\ln(q)}\right]$ (9.56)

and

$U_{\mathfrak{C}_{H},k}(z,\\{\tilde{x}_{d}\\})=\exp\left[\frac{\ln(\tilde{x}_{k})\ln(q^{\sum_{d=1}^{k}\sigma_{d}}/z)-\sum_{d\leq
k}\ln(\tilde{x}_{d})\ln(q^{\sigma_{d}})}{\ln(q)}\right]\;.$ (9.57)

The presence of the factor $\mathfrak{B}_{\mathfrak{C}_{H}}$ inside the
integrand of $\mathbb{V}_{\mathfrak{C}_{H}}$ (9.53) results in new poles,
which come in two series: they are either located at

$y=q^{-n}\,\tilde{x}_{d}\,,\qquad\;n=0,1,2,\ldots,\qquad d\geq k\;,$ (9.58)

in which case they accumulate at $y=\infty$ (we are assuming throughout that
$|q|<1$), or they are located at

$y=q^{n}\,\tilde{x}_{d}\,q^{\sigma_{d}}\,,\qquad\;n=0,1,2,\ldots,\qquad d\leq
k\;,$ (9.59)

in which case they accumulate at $y=0$. We define the contour $C$ in (9.53) to
enclose the poles (9.58), while the poles (9.59) near 0 are kept outside.

From the definition of the matrix

$\mathfrak{B}^{k^{\prime}}_{\mathfrak{C}_{H},k}=\mathfrak{B}_{\mathfrak{C}_{H},k}(x_{k^{\prime}})\;,$
(9.60)

it follows at once that

$\mathfrak{B}^{k^{\prime}}_{\mathfrak{C}_{H},k}=0\;\qquad\text{if}\;k>k^{\prime}\;,$
(9.61)

meaning $\mathfrak{B}^{k^{\prime}}_{\mathfrak{C}_{H},k}$ is triangular and the
elliptic stable envelope for $\mathfrak{C}_{H}$ has adequate support.

The regularity of the integral $\mathbb{V}_{\mathfrak{C}_{H}}$ depends on
whether or not the poles of the integrand pinch the contour; the poles (9.58)
and (9.59) on opposite sides of the contour coalesce if and only if

$\frac{\tilde{x}_{j}}{\tilde{x}_{i}}=q^{n+\sigma_{i}}\,,\qquad\;n=0,1,2,\ldots,\qquad
j\geq k\geq i\;.$ (9.62)

But the chamber $\mathfrak{C}_{H}$ was defined such that
$|\tilde{x}_{j}/\tilde{x}_{i}|>1$ for all $j>i$, ensuring no singularities. In
particular, $\mathbb{V}_{\mathfrak{C}_{H}}$ is holomorphic in a neighborhood
of 0, as long as we stay in the chamber $\mathfrak{C}_{H}$. Then, the change
of basis reads

$\mathbb{V}_{\mathfrak{C}_{H},k}=\sum_{k^{\prime}=1}^{L}\mathbb{V}_{\mathfrak{C}}\,\mathfrak{B}^{k^{\prime}}_{\mathfrak{C}_{H},k}\;.$
(9.63)

As an example, let us specialize to $L=2$, the insertion of two $A_{1}$
punctures, meaning two $q$-primary operators of $\mathcal{W}_{q,t}(A_{1})$,
labeled respectively by the momenta $\sigma_{1}$ and $\sigma_{2}$. In that
case, the vertex function $\mathbb{V}$ has two components,

$\displaystyle\mathbb{V}_{1}=(x_{1})^{\eta}\frac{(q^{1-\sigma_{1}})_{\infty}}{(q)_{\infty}}\,\frac{(q^{1-\sigma_{2}}x_{1}/x_{2})_{\infty}}{(x_{1}/x_{2})_{\infty}}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{q^{\sigma_{1}}\mskip
8.0mu\;\;q^{\sigma_{2}}x_{2}/x_{1}}{q\mskip 8.0mu\qquad
q\,x_{2}/x_{1}};\frac{z}{q^{\sigma_{1}+\sigma_{2}-2}}\right]}\;,$
$\displaystyle\mathbb{V}_{2}=(x_{2})^{\eta}\frac{(q^{1-\sigma_{2}})_{\infty}}{(q)_{\infty}}\,\frac{(q^{1-\sigma_{1}}x_{2}/x_{1})_{\infty}}{(x_{2}/x_{1})_{\infty}}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{q^{\sigma_{2}}\mskip
8.0mu\;\;q^{\sigma_{1}}x_{1}/x_{2}}{q\mskip 8.0mu\qquad
q\,x_{1}/x_{2}};\frac{z}{q^{\sigma_{1}+\sigma_{2}-2}}\right]}\;.$ (9.64)

Note that both components are analytic in $|z|<1$. Using Heine’s
transformation for $q$-hypergeometric series, it is easy to see that
$\mathbb{V}_{1}$ and $\mathbb{V}_{2}$ cannot both be analytic in any given
$\\{\tilde{x}_{i}\\}$-chamber _simultaneously_ :

$\displaystyle\mathbb{V}_{1}=(x_{1})^{\eta^{\prime}}\frac{(q^{1-\sigma_{1}})_{\infty}}{(q)_{\infty}}\,\frac{(q^{2-\sigma_{2}}z)_{\infty}\Theta(q^{\sigma_{2}}x_{2}/x_{1})}{(q^{2-\sigma_{1}-\sigma_{2}}z)_{\infty}\Theta(x_{1}/x_{2})}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{q^{1-\sigma_{2}}\mskip
8.0mu\;\;q^{2-\sigma_{1}-\sigma_{2}}\,z}{q\mskip 8.0mu\qquad
q^{2-\sigma_{2}}\,z};q^{\sigma_{2}}\frac{x_{2}}{x_{1}}\right]}\;,$
$\displaystyle\mathbb{V}_{2}=(x_{2})^{\eta^{\prime}}\frac{(q^{1-\sigma_{2}})_{\infty}}{(q)_{\infty}}\,\frac{(q^{2-\sigma_{1}}z)_{\infty}\Theta(q^{\sigma_{1}}x_{1}/x_{2})}{(q^{2-\sigma_{1}-\sigma_{2}}z)_{\infty}\Theta(x_{2}/x_{1})}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{q^{1-\sigma_{1}}\mskip
8.0mu\;\;q^{2-\sigma_{1}-\sigma_{2}}\,z}{q\mskip 8.0mu\qquad
q^{2-\sigma_{1}}\,z};q^{\sigma_{1}}\frac{x_{1}}{x_{2}}\right]}\;.$ (9.65)

In order to make the components of $\mathbb{V}$ analytic in the chamber
$\mathfrak{C}_{H}=\\{|x_{1}|<|x_{2}|\\}$, we apply the linear change of basis
(9.63), with the matrix

$\mathfrak{B}^{k^{\prime}}_{\mathfrak{C}_{H},k}=U_{\mathfrak{C}_{H},k}\;\begin{bmatrix}\dfrac{1}{\Theta(q^{\sigma_{1}})}&0\\\
\dfrac{\Theta(q^{\sigma_{1}}x_{1}/x_{2}\,z)}{\Theta(q^{\sigma_{1}}\,z)\Theta(q^{\sigma_{1}}x_{1}/x_{2})}&\dfrac{\Theta(x_{1}/x_{2})}{\Theta(q^{\sigma_{2}})\Theta(q^{\sigma_{1}}x_{1}/x_{2})}\\\
\end{bmatrix}\;\text{e}^{-1}\,.$ (9.66)

Above, both $\text{e}^{-1}$ and $U_{\mathfrak{C}_{H},k}$ are 2x2 diagonal
matrices with entries (9.56) and (9.57). The new vertex function
$\mathbb{V}_{\mathfrak{C}}$ has scalar components
$\mathbb{V}_{\mathfrak{C}_{H},1}$ and $\mathbb{V}_{\mathfrak{C}_{H},2}$. The
second component $\mathbb{V}_{\mathfrak{C}_{H},2}$ is simply a rescaling of
$\mathbb{V}_{2}$, since $\mathfrak{B}^{k^{\prime}}_{\mathfrak{C}_{H},k}$ is
lower-triangular. The computation of the first component
$\mathbb{V}_{\mathfrak{C}_{H},1}$ is more involved: at first sight, it is a
sum of two $q$-hypergeometric functions, but making use of the three-term
identity

$\displaystyle{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{a\mskip
8.0mub}{q\mskip 8.0muc};f\right]}$
$\displaystyle=\;\frac{(b)_{\infty}(c/a)_{\infty}\Theta(af)}{(c)_{\infty}(b/a)_{\infty}\Theta(f)}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{a\mskip
8.0mu\;qa/c}{q\mskip 8.0mu\;qa/b};\frac{qc}{abf}\right]}$
$\displaystyle\qquad\;\;+\frac{(a)_{\infty}(c/b)_{\infty}\Theta(bf)}{(c)_{\infty}(a/b)_{\infty}\Theta(f)}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{b\mskip
8.0mu\;qb/c}{q\mskip 8.0mu\;qb/a};\frac{qc}{abf}\right]}$ (9.67)

with $a=q^{1-\sigma_{2}}$, $b=q^{2-\sigma_{1}-\sigma_{2}}\,z$,
$c=q^{2-\sigma_{2}}\,z$, $f=q^{\sigma_{2}}\,x_{2}/x_{1}$, a cancellation
guarantees that only one $q$-hypergeometric series survives. All in all, we
find:

$\displaystyle\mathbb{V}_{\mathfrak{C}_{H},1}=(x_{1})^{\eta^{\prime}}\frac{(q^{1-\sigma_{1}})_{\infty}}{(q)_{\infty}\,\Theta(q^{\sigma_{1}})}\,\frac{(q\,z)_{\infty}}{(q^{1-\sigma_{1}}z)_{\infty}}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{q^{1-\sigma_{2}}\mskip
8.0mu\qquad\;\;1/z}{\;\;\;q\mskip 8.0mu\qquad
q^{1-\sigma_{1}}/z};\;q^{\sigma_{1}}\frac{x_{1}}{x_{2}}\right]}\;,$
$\displaystyle\mathbb{V}_{\mathfrak{C}_{H},2}=(x_{2})^{\eta^{\prime}}\frac{(q^{1-\sigma_{2}})_{\infty}}{(q)_{\infty}\,\Theta(q^{\sigma_{2}})}\,\frac{(q^{2-\sigma_{1}}z)_{\infty}}{(q^{2-\sigma_{1}-\sigma_{2}}z)_{\infty}}\;{}_{2}\mathbb{F}_{2}{\left[\genfrac{.}{.}{0.0pt}{}{q^{1-\sigma_{1}}\mskip
8.0mu\;\;q^{2-\sigma_{1}-\sigma_{2}}\,z}{q\mskip 8.0mu\qquad
q^{2-\sigma_{1}}\,z};\;q^{\sigma_{1}}\frac{x_{1}}{x_{2}}\right]}\;.$ (9.68)

In this new basis, the vertex function $\mathbb{V}_{\mathfrak{C}}$ is
manifestly analytic in chamber $\mathfrak{C}_{H}=\\{|x_{1}|<|x_{2}|\\}$.

Likewise, if we consider the electric blocks (9.16) analytic in chamber
$\mathfrak{C}_{C}$, then the actual $q$-conformal blocks of
$U_{\hbar}(\widehat{A_{1}})$ are obtained after a change basis to
$x$-solutions, using once again the matrix (9.54):

${\bf
F}_{\mathfrak{C}_{H},k,i}(\\{\tilde{x}_{d}\\})=\oint_{\Gamma}\frac{dy}{y}\,y^{\eta}\,\text{Stab}^{K}_{i}(y,\\{\tilde{x}_{d}\\})\prod_{d=1}^{L}\frac{(q^{1-\sigma_{d}}\,y/\tilde{x}_{d};q)_{\infty}}{(y/\tilde{x}_{d};q)_{\infty}}\;\mathfrak{B}_{\mathfrak{C}_{H},k}(y,z,\\{\tilde{x}_{d}\\})\;,$
(9.69)

for all $k=1,\ldots,n$.

Finally, the explicit relation between electric and magnetic blocks, whether
as $z$\- or $x$-solutions, is established via the existence of a pairing with
a certain covector $W$. For example, in the case of $z$-solutions, the pairing
reads

$\sum_{d=1}^{L}W_{d}\,{\bf F}^{d}_{k}=\mathbb{V}_{k}\;.$ (9.70)

This is a particular case of the general formula (3.55). We say that ${\bf
F}_{i}(\\{\tilde{x}_{d}\\})$ is a (vector) vertex function with descendant
insertions at 0. In particular, a trivial insertion at 0 gives a way to
identify the covector $W$ explicitly:

$\sum_{d=1}^{L}W_{d}\,\text{Stab}^{K}_{d}(y,\vec{x})=1\;.$ (9.71)

with $\text{Stab}^{K}_{d}(y,\vec{x})$ the K-theoretic stable envelope in
$\eqref{Ki3}$. The fact that the stable basis is upper triangular ensures that
all coefficients $W_{d}$ can be solved for recursively, precisely like in the
unramified case.

###  9.3 The 3d gauge theory

Our goal is to show that all of the previous $q$-conformal blocks are realized
as the half-indices or partition functions of a certain 3d $\mathcal{N}=2$
gauge theory, for different choices of 3d manifold and boundary conditions.

The Drinfeld gauge theory $T^{3d}_{A_{1}}$ in question has a single gauge
group $U(N)$ which we specialize to be abelian, $N=1$115115115The
generalization from the gauge group $U(1)$ to the maximal torus of $U(N)$
requires $N$ integration variables as described in the main text, see Section
3.3. Most notably, the 3d $\mathcal{N}=4$ vector multiplet would then contain
additional factors $\frac{1}{N!}\prod_{1\leq j\neq i\leq
N}\frac{\left(y_{i}/y_{j};q\right)_{\infty}}{\left(t\,y_{i}/y_{j};q\right)_{\infty}}$
which are responsible for new poles in the integrand. The poles are classified
by Young tableaux with at most $N$ rows [62]. In the case at hand, for the
gauge group $U(1)$, summing over Young tableaux with at most 1 row is just a
sum over all positive integers., and flavor symmetry $F_{H}\times U(1)_{t}$.
The bare Chern-Simons level is set to 0. According to Proposition 3.1, $F_{H}$
is uniquely defined via the $\mathcal{W}_{q,t}(A_{1})$ $q$-primary operator
content: here, we picked the weight $\lambda_{1}\in V_{Y}$ and the weight
$\lambda_{2}\in V_{Y^{-1}}$ a total of $L$ times each, so we have a flavor
group $F_{H}=U(L)\times U(L)$.

__

Figure 17: The 3d $\mathcal{N}=2$ Drinfeld gauge theory $T^{3d}_{A_{1}}$ in
the abelian case $N=1$.

The flavor symmetry maximal tori act by Hamiltonian isometries on the chiral
multiplets:

$T_{H}:\;(\Phi^{-}_{k},\Phi^{+}_{k})\;\rightarrow(e^{-i\theta_{k}}\Phi^{-}_{k},e^{+i\theta_{k}}\Phi^{+}_{k})\;,\qquad
T_{t}:\;(\Phi^{-}_{k},\Phi^{+}_{k})\;\rightarrow(e^{+i\theta_{k}}\Phi^{-}_{k},e^{+i\theta_{k}}\Phi^{+}_{k})\;.$
(9.72)

The superpotential is

$W=\sum_{d=1}^{L}\left(\sigma+m_{d,1}+\frac{\epsilon_{t}}{2}\right)|\Phi^{+}_{d}|^{2}+\sum_{d=1}^{L}\left(-\sigma-
m_{d,2}+\frac{\epsilon_{t}}{2}\right)|\Phi^{-}_{d}|^{2}-\xi\,\sigma\;,$ (9.73)

where $\sigma$ is the real scalar in the vector multiplet. From it, we deduce
the real moment map equation

$\sum_{d=1}^{L}\left(|\Phi^{+}_{d}|^{2}-|\Phi^{-}_{d}|^{2}\right)=\xi\;.$
(9.74)

For $k\in\\{1,\ldots,L\\}$, a Higgs vacuum arises when either $\Phi^{+}_{k}$
or $\Phi^{-}_{k}$ is given a nonzero vev. When $\xi>0$, the vev is
$|\Phi^{+}_{k}|^{2}=\xi$, with $\sigma=-m_{k,1}-\frac{\epsilon_{t}}{2}$. When
$\xi<0$, the vev is $|\Phi^{-}_{k}|^{2}=-\xi$, with
$\sigma=-m_{k,2}+\frac{\epsilon_{t}}{2}$. We work in the chamber
$\mathfrak{C}_{C}=\\{\xi>0\\}$ throughout.

Let us first analyze the magnetic blocks of $\mathcal{W}_{q,t}(A_{1})$