to the brane world-volume gives rise to the gauge field and (real) scalar
degrees of freedom: A D$p$ brane has a $(p+1)$-dimensional world-volume, and
its codimension is given by $10-(p+1)=9-p$. Let us take $p=3$ to realize
$\mathsf{d}=4$ gauge theory. Then, there appear six real scalar fields, which
are combined into $\mathcal{N}=4$ vector multiplet together with the gauge
fields (and also the corresponding fermionic partners). This is the maximally
supersymmetric gauge theory in four dimensions with 16 supercharges. In this
case, the R-symmetry $\mathrm{SU}(4)=\mathrm{SO}(6)$ has an interpretation as
the rotation symmetry of the transverse directions to the D3 branes.

645789(a)NS5NS5$L\propto\frac{1}{g^{2}}$$\mathsf{a}_{1}$$n$
D4$\mathsf{a}_{2}$$\mathsf{a}_{3}$$\vdots$$\vdots$$\mathsf{a}_{n}$$n^{\text{f}}$
D4$m_{1}$$m_{2}$$m_{n^{\text{f}}}$$\vdots$$\vdots$(b)NS5NS5$\vdots$$\vdots$$\vdots$D6(c)NS5NS5$\vdots$
Figure 4.3: Type IIA brane configurations of four-dimensional $\mathcal{N}=2$
$\mathrm{SU}(n)$ gauge theory with $n^{\text{f}}$ fundamental hypermultiplets
with (a) semi-infinite D4 branes, (b) external D4 branes terminated by D6
branes, and (c) D6 branes inside the NS5 branes via the Hanany–Witten move.

The $\mathcal{N}=2$ vector multiplet in four dimensions, on the other hand,
contains a single complex scalar field (two real components). Hence, we should
freeze some of the scalar fields to reduce the supersymmetry from
$\mathcal{N}=4$ to $\mathcal{N}=2$ theory. A standard way to do this is to
consider D4 branes suspended between NS5 branes, which is known as the
Hanany–Witten construction of gauge theory with eight supercharges in string
theory [HW97]. See Tab. 4.1.

(IIA) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
NS5 | - | - | - | - | - | - |  |  |  |
D4 | - | - | - | - |  |  | - |  |  |
D6 | - | - | - | - |  |  |  | - | - | -
Table 4.1: The brane configuration of four-dimensional $\mathcal{N}=2$ gauge
theory in Type IIA string theory.

As shown in Fig. 4.3, D4 branes are now extending in five dimensions (01236),
but with a finite interval $L$ in 6-direction. Therefore, if the interval is
sufficiently small, it would effectively behave as a four-dimensional theory.
In fact, the interval $L$ is interpreted as the gauge coupling up to a
constant factor, $L\propto 1/g^{2}$.

In this setup, the transverse modes of D4 branes in 789-directions are fixed
by the Dirichlet boundary condition with NS5 branes, which are not dynamical
any longer. There remain two real components in the scalar field, which
correspond to 45-directions, combined into a single scalar field in the
$\mathcal{N}=2$ vector multiplet, $\phi=\phi_{4}+\mathrm{i}\phi_{5}$. As
discussed before, the scalar field takes a value in the Cartan subalgebra of
the gauge algebra, $\phi\in\mathfrak{h}\subset\mathfrak{g}$. In this picture,
the diagonal values of the scalar field,
$\phi=\mathop{\mathrm{diag}}(\mathsf{a}_{1},\ldots,\mathsf{a}_{n})$ for
$G=\mathrm{SU}(n)$, are interpreted as the positions of D4 branes in
45-directions.

In Fig. 4.3(a), there are also external D4 branes outside the NS5 branes. In
fact, the open strings connecting these internal and external D4 branes
transform under the bifundamental representation of
$\mathrm{SU}(n)\times\mathrm{SU}(n^{\text{f}})$. Since the external ones are
infinitely long, their transverse modes are not dynamical. Similarly, we may
incorporate the antifundamental matter by adding the external D4 branes going
left of the NS5 brane. Hence, a stack of external D4 branes gives rise to the
flavor symmetry $\mathrm{SU}(n^{\text{f}})$, and thus the configuration shown
in Fig. 4.3 realizes $\mathcal{N}=2$ gauge theory with $n^{\text{f}}$
fundamental hypermultiplets. It is also possible to terminate the external D4
branes with D6 branes (0123789), keeping the dynamics of D4 branes in
45-directions frozen by D6 branes (Fig. 4.3(b)). Taking the D6 branes inside
the NS5 branes, the suspended D4 branes are eliminated due to the
Hanany–Witten effect. This shows that the same matter content is realized only
with the internal D4 and D6 branes (Fig. 4.3(c)).

#####  4.5.2 Seiberg–Witten curve from M-theory

Let us remark the relation between the brane description of $\mathcal{N}=2$
theory and the Seiberg–Witten geometry. First of all, since the complex scalar
describes the positions of D4 branes in 45-directions, the $x$-variable in the
Seiberg–Witten curve is identified with $x\simeq x_{4}+\mathrm{i}x_{5}$. Then,
the $y$-variable is instead for the positions of NS5 branes. (Recall the
distance between NS5 branes corresponds to the gauge coupling constant.)

Furthermore, the brane configuration discussed above, which consists of NS5
and D4 branes, is promoted to a single M5 brane after lifting up from IIA
string theory to M-theory. Let $x_{\text{M}}$ be the eleventh dimension with
the periodicity $x_{\text{M}}\simeq x_{\text{M}}+2\pi R$. Then, the M5 brane
is extended in 0123-directions and a two-dimensional subspace of 456M, which
are six dimensions in total. It has been shown in [Wit97] that the latter two-
dimensional subspace is precisely given as the Seiberg–Witten curve of
$\mathrm{SU}(n)$ gauge theory (4.2.26) with the identification:

$\displaystyle(x,y)=\left(x_{4}+\mathrm{i}x_{5},\,\exp\left(\frac{x_{6}+\mathrm{i}x_{\text{M}}}{R}\right)\right)\in\mathbb{C}\times\mathbb{C}^{\times}\,.$
(4.5.1)

We remark that the $y$-variable is now given as an exponential form to
manifest the periodicity in M-direction.

#####  4.5.3 Quiver gauge theory

We have discussed the brane construction of a single $\mathrm{SU}(n)$ gauge
theory. Let us show how to generalize this construction to quiver gauge theory
involving multiple gauge nodes.

###### Linear quiver: $A_{p}$

We start with $A_{p}$ quiver, which is a linear quiver with $p$ gauge nodes.
In this case, we consider $p+1$ parallel NS5 branes along 6-direction with
suspended D4 branes. An example with $p=3$ is given as

(4.5.2)

The number of D4 branes in each interval between NS5 branes provides the rank
of gauge groups $(n_{i})_{i=1,\ldots,p}$. In this configuration, we can add
the (anti)fundamental hypermultiplet only to the first and the last nodes with
the external D4 branes, where the corresponding rank of the flavor symmetries
are denoted by $n_{0}$ and $n_{p+1}$. In order to add the fundamental matter
to the middle nodes, we have to use internal D6 branes as in Fig. 4.3.

The open string degrees of freedom between $i$-th and $(i+1)$-st intervals
give rise to the hypermultiplet in bifundamental representation of
$\mathrm{SU}(n_{i})\times\mathrm{SU}(n_{i+1})$. The corresponding
bifundamental mass is given by the relative difference between the D4 brane
positions in 45-directions. In this sense, it is natural to regard the
bifundamental mass as a complex parameter, which can be compensated by the
center-of-mass shift of D4 branes, namely the $\mathrm{U}(1)$ gauge degrees of
freedom of each gauge node.

###### Cyclic quiver: $\widehat{A}_{p-1}$

We can similarly formulate the cyclic quiver with $p$ nodes
($\widehat{A}_{p-1}$ quiver). In this case, we align $p$ NS5 branes in
periodic 6-direction:

(4.5.3)

Now the external D4 branes with red signs are identified with each other. We
remark that we cannot impose the fundamental hypermultiplet to this
configuration using the external D4 branes.

Since 6-direction is periodic in this setup, we may consider T-dual
transformation in this direction. The configuration obtained after T-duality
is given as follows:

(IIB) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
TNp |  |  |  |  |  |  | - | - | - | -
D3 | - | - | - | - |  |  |  |  |  |
Table 4.2: The brane configuration of four-dimensional $\mathcal{N}=2$ gauge
theory in Type IIB string theory. We denote the $p$-centered Taub-NUT geometry
by TNp in 6789 directions.

Compared to the previous one (See Tab. 4.1), NS5 branes are converted to the
so-called H-monopoles encoded in the Taub-NUT space (TN), which is locally
asymptotic to $\mathbb{R}^{3}\times S^{1}$ at infinity: The positions of NS5
branes in 789 directions correspond to the multiple centers of the Taub-NUT
space. See, for example, [GHM97, Ton02, Wit09] for details. From this point of
view, the supersymmetry is reduced from $\mathcal{N}=4$ to $\mathcal{N}=2$ due
to the H-flux of TN. We remark that, by taking the size of $S^{1}$ to
infinity, we obtain the ALE space $\mathbb{R}^{4}/\mathbb{Z}_{p}$ from TNp.
This is the shrinking limit of 6-direction on Type IIA side, and is the strong
coupling limit from the gauge theory perspective,
$\mathfrak{q}_{\text{tot}}:=\mathfrak{q}_{0}\mathfrak{q}_{1}\cdots\mathfrak{q}_{p-1}\to
1$, where $\mathfrak{q}_{i}=\exp\left(2\pi\mathrm{i}\tau_{i}\right)$ for
$i=0,\ldots,p-1$. On the other hand, in the large radius limit on Type IIA
side, we may lose the periodicity in 6-direction. This means the reduction
from cyclic to linear quiver by freezing (at least) one of the gauge nodes of
the quiver, $\mathfrak{q}_{\text{tot}}\to 0$.

###### D-type quiver

The brane construction discussed above has a natural generalization to $D_{p}$
and $\widehat{D}_{p}$ quivers, by introducing the ON0 plane to the
configuration [Kap98, HZ99, HKL+16], which is related to a combination of O5-
plane and a D5 brane through the duality chain. For $D_{p}$ quiver, the brane
configuration is given as follows:

(4.5.4)

Here the ON- plane is S-dual to an O5- plane, which would be combined into the
ON0 plane together with the NS5 brane. Furthermore, we can add the fundamental
hypermultiplet, but only to the left most node. We can similarly deal with
$\widehat{D}_{p}$ quiver by imposing ON0 planes also on the left end of the
configuration, so that we cannot incorporate the fundamental mattes for the
affine case similarly to $\widehat{A}_{p-1}$ quiver.

###### Generic quiver

Although we can consider generic quiver in gauge theory, it seems not
straightforward to construct such a generic quiver in string theory. Let us
just mention a geometric realization for $\Gamma=ADE$: We may consider a stack
of D3 branes with the transverse Taub-NUT space TNΓ, which is reduced to the
ALE space $\mathbb{C}^{2}/\Gamma$ in the large radius limit as well. Then, in
this case, we would obtain the affine quiver gauge theory
$\widehat{\Gamma}=\widehat{ADE}$ similarly to the discussion for the cyclic
quiver. See also §2.3 for a related discussion.

For $\Gamma\not\in ADE$, we should apply the formalism of fractional quiver
discussed in §2.4. Based on the algebraic construction of the brane web
formalism, we can construct arbitrary quiver gauge theory for $\Gamma=ABCDEFG$
and their affinization (including twisted versions). See [KZ19] for details.

#####  4.5.4 Higgsing and vortices

In §1.9.2, we have discussed a specific property of the (anti)fundamental
matter part of the instanton partition function: The partition function is
truncated if one tunes the fundamental mass parameter related to the Coulomb
moduli (1.9.13). We now discuss its brane perspective.

Turning off the $\Omega$-background parameter $\epsilon_{1,2}\to 0$, the root
of Higgs branch condition (1.9.13) is simply given by equating the fundamental
mass and the Coulomb moduli parameter [Dor98, DHT99]:

$\displaystyle m_{f}=\mathsf{a}_{\alpha}\,.$ (4.5.5)

Since they are interpreted as positions of the internal and external D4 branes
in Type IIA configuration (Fig. 4.3), under this condition, they come to the
same place in 45-directions to be a single semi-infinite D4 brane.
Furthermore, from this configuration, one can remove the right NS5 brane in
9-direction, and D2 branes suspended in 9-direction may emerge as shown below:

(4.5.6)

(IIA) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
NS5 | - | - | - | - | - | - |  |  |  |
(D6) | - | - | - | - |  |  |  | - | - | -
D4 | - | - | - | - |  |  | - |  |  |
D2 | - | - |  |  |  |  |  |  |  | -
Table 4.3: The brane configuration of four-dimensional $\mathcal{N}=2$ gauge
theory in Higgs branch.

See also Tab. 4.3. A similar manipulation is possible with the internal D6
branes describing the fundamental matter related through the Hanany–Witten
move.

After removing the right NS5 brane, it is not in the Coulomb branch of the
moduli space, but in the Higgs branch, where the gauge symmetry is locked with
the flavor symmetry by the condition (4.5.5). In this case, D2 branes are
interpreted as BPS vortex strings in 4d theory, and their length in
9-direction $L_{9}$ is identified with the Fayet–Iliopoulos (FI) parameter in
four-dimensions [HT04].

This configuration has an alternative description as the world-volume theory
of D2 branes, which is two-dimensional $\mathcal{N}=(2,2)$ gauge theory in
01-directions. From 4d $\mathcal{N}=2$ $\mathrm{SU}(n)$ gauge theory with
$n^{\text{f}}=n$ fundamental and $n^{\text{af}}$ antifundamental matters, we
obtain the 2d $\mathcal{N}=(2,2)$ theory with $n$ fundamental, $n^{\text{af}}$
antifundamental, and a single adjoint chiral multiplets:5558 SUSY (4d
$\mathcal{N}=2$) vector multiplet splits into 4 SUSY (2d $\mathcal{N}=(2,2)$;
4d $\mathcal{N}$ = 1) vector and a adjoint chiral multiplets.

(4.5.7)

where the blue circle  stands for the gauge node in two dimensions, while
white nodes are for four dimensions. The gauge group rank in two dimensions is
given by the number of vortices in 4d theory in the Higgs branch. In addition,
the gauge coupling and FI parameter for this 2d theory are identified with
$L_{9}$ and $L_{6}$, respectively. In this picture, the instanton in 4d
realized using D0 branes is interpreted as an instanton also in the 2d theory,
i.e., a vortex in a vortex (a trapped vortex). Based on such a relation
between 4d and 2d theories, one can obtain the vortex partition function of 2d
$\mathcal{N}=(2,2)$ theory [DGH10, BTZ12]. See also [FKNO12] for another, but
direct derivation of the vortex partition function.

(a)NS5NS5NS5(b)NS5NS5NS5(c)NS5NS5NS5(d)NS5NS5NS5 Figure 4.4: Higgsing process
for $A_{2}$ quiver. (a) Coulomb branch, (b) Higgsing the second node, (c)
Higgsing both nodes, similarly described as the configuration (d). The
intervals of NS5 branes in 9-direction are identified with the FI parameters.

###### Quiver gauge theory

This manipulation is applied similarly in quiver gauge theory. We describe the
Higgsing process for $A_{2}$ quiver in Fig. 4.4, and the corresponding quiver
diagrams are given as follows:

(4.5.8)

In this way, we obtain the 4d-2d coupled quiver description from the Higgsing
process. See also [GLF16] for generic linear quiver theory.

#####  4.5.5 Higgsing in Seiberg–Witten geometry

######  $A_{1}$ quiver

We discuss the Seiberg–Witten geometric point of view of the Higgsing process.
We write the Seiberg–Witten curve for $\mathrm{SU}(n)$ gauge theory with
$(n^{\text{f}},n^{\text{af}})$ flavors (4.2.30) as follows:

$\displaystyle y^{2}-\mathsf{T}_{1,x}\,y-P_{1,x}\widetilde{P}_{1,x}=0$ (4.5.9)

where $(P_{1,x},\widetilde{P}_{1,x})$ are the matter polynomials of the degree
$(n^{\text{f}},n^{\text{af}})$.666Compared to the previous definition
(2.2.12b), we simply denote them by $(P_{i,x},\widetilde{P}_{i,x})$ as long as
no confusion.  Since, from the brane picture (4.5.6), there remains a single
NS5 brane after the Higgsing process, the algebraic curve would be given by a
linear relation for the $y$-variable. Such a situation is realized by imposing
the condition

$\displaystyle\mathsf{T}_{1,x}=P_{1,x}+\widetilde{P}_{1,x}$ (4.5.10)

such that,

$\displaystyle
y^{2}-\mathsf{T}_{1,x}\,y-P_{1,x}\widetilde{P}_{1,x}=(y-P_{1,x})(y-\widetilde{P}_{1,x})=0\,.$
(4.5.11)

Then, the factorized curve implies the relation

$\displaystyle y=P_{1,x}\quad\text{or}\quad y=\widetilde{P}_{1,x}\,,$ (4.5.12)

which describes the Higgsed configuration with respect to the fundamental or
the antifundamental matter.

We remark that the condition (4.5.10) has a natural interpretation in the
context of the integrable system: According to the correspondence between
gauge theory and integrable system, $\mathsf{T}$-function is identified with
the (eigenvalue of) transfer matrix, which is given by taking a trace of the
monodromy matrix

$\displaystyle\mathsf{T}_{x}=\operatorname{tr}\mathcal{T}_{x}\,,\qquad\mathcal{T}_{x}=\begin{pmatrix}a(x)&b(x)\\\
c(x)&d(x)\end{pmatrix}\,.$ (4.5.13)

In fact, the expression (4.5.10) implies that the diagonal elements
$(a(x),d(x))$ are identified with the matter polynomials
$(P_{1,x},\widetilde{P}_{1,x})$ as shown in §5.8.

######  $A_{2}$ quiver

This argument is straightforwardly generalized to quiver gauge theory. We
consider $A_{2}$ quiver with fundamental matters. The Seiberg–Witten curve is
described by a pair of fundamental characters:

$\displaystyle\mathsf{T}_{1,x}=y_{1}+\frac{y_{0}y_{2}}{y_{1}}+\frac{y_{0}y_{3}}{y_{2}}\,,$
(4.5.14a)
$\displaystyle\mathsf{T}_{2,x}=y_{2}+\frac{y_{1}y_{3}}{y_{2}}+\frac{y_{0}y_{3}}{y_{1}}\,,$
(4.5.14b)

where $(y_{0},y_{3})$ are the matter polynomials of the first and second gauge
nodes. They are combined into a single equation to define the algebraic curve:

$\displaystyle
y_{1}^{3}-\mathsf{T}_{1,x}\,y_{1}^{2}+\mathsf{T}_{2,x}\,y_{0}y_{1}-y_{0}^{2}y_{3}=0\quad\text{or}\quad
y_{2}^{3}-\mathsf{T}_{2,x}\,y_{2}^{2}+\mathsf{T}_{1,x}\,y_{2}y_{3}-y_{0}y_{3}^{2}=0\,.$
(4.5.15)

Similarly to $A_{1}$ quiver, let us find a factorization condition for them.
In this case, we impose the condition

$\displaystyle\mathsf{T}_{1,x}-y_{0}=\mathsf{T}_{2,x}-y_{3}=:\widetilde{\mathsf{T}}_{1,x}\,,$
(4.5.16)

which leads to the factorization:

$\displaystyle(y_{1}-y_{0})(y_{1}^{2}-\widetilde{\mathsf{T}}_{1,x}\,y_{1}+y_{0}y_{3})=0\,,\qquad(y_{2}-y_{3})(y_{2}^{2}-\widetilde{\mathsf{T}}_{1,x}\,y_{2}+y_{0}y_{3})=0\,.$
(4.5.17)

This shows that the $A_{2}$ curve is factorized into the linear term and the
$A_{1}$ (quadratic) curve. Imposing a further condition,
$\widetilde{\mathsf{T}}_{1,x}=y_{0}+y_{3}$, the curve is completely Higgsed:

$\displaystyle(y_{1}-y_{0})^{2}(y_{1}-y_{3})=0\,,\qquad(y_{2}-y_{0})(y_{2}-y_{3})^{2}=0\,.$
(4.5.18)

Generalization to generic quiver theory is straightforward.

#####  4.5.6 Supergroup gauge theory

###### Positive and negative branes

Following the discussion in §4.5.1, $\mathcal{N}=2$ SU$(n_{0}|n_{1})$ SYM
theory in four dimensions is realized as a world-volume theory of positive and
negative branes, D4+ and D4-, suspended between two separated NS5 branes
[OT06]:

(4.5.19)

where D4+ and D4- branes are depicted as horizontal solid and dotted lines. It
has been pointed out in [DHJV18] that the negative branes are removed through
gauging process:

(4.5.20)

and the resulting configuration is equivalent to SU$(n_{0})$ SYM theory with
$n^{\text{f}}=2n_{1}$ flavors. More precisely, there are $n_{1}$ fundamental
and $n_{1}$ anti-fundamental hypermultiplets having the same masses because we
imposed horizontal D4+ branes before gauging. This is consistent with the
decoupling trick argument in §3.2. We remark that such a reduction is possible
only at the special locus of the Coulomb branch of the moduli space of vacua,
and this does not mean the agreement of the moduli spaces themselves.

###### One-to-many correspondence

The non-Abelian gauge theory is realized as a stack of D-branes. In
particular, we should use both the positive and negative branes to realize the
supergroup gauge theory. Since, in this case, there are two different branes,
we should be careful of the ordering of the branes. For example, there are
three possible realizations of SU$(2|1)$ gauge theory as follows:

(4.5.21)

This one-to-many correspondence between the gauge theory and the brane
configurations is a peculiar property to the supergroup theory, which is
essentially related to the ambiguity of the simple root decomposition of the
supergroup. Besides the brane configurations, we also show the corresponding
Dynkin diagrams of $\mathrm{SU}(2|1)$: (a) , (b) , and (c) , where the node
denoted by  is the fermionic node [Kac77]. The correspondence is given as
follows: We assign the ordinary node to the neighboring pair of D4+-D4+ or D4
--D4- branes, and the fermionic node is assigned to the neighboring pair of
D4+-D4- branes. This argument is also applicable to the external flavor
branes. Even though there are several different brane realizations, we can see
that the partition function does not depend on the ordering of positive and
negative branes from the topological string analysis [KS20, CKL20b].

###### Affine quiver realization

As discussed in §3.1.3, supergroup gauge theory has a realization in the
unphysical parameter regime of quiver gauge theory ($\widehat{A}_{1}$ quiver).
This fact is also understood from the brane configuration. We start with
$\widehat{A}_{1}$ quiver gauge theory, which is a cyclic quiver with two gauge
nodes, $\mathrm{SU}(n_{1})\times\mathrm{SU}(n_{2})$:

(4.5.22)

In order to obtain the supergroup gauge theory, we apply the analytic
continuation for the coupling constant $1/g_{2}^{2}\to-1/g_{1}^{2}$,
corresponding to $L_{2}\to-L_{1}$ in the brane configuration. Then, the
external branes are now interpreted as the internal negative branes after the
analytic continuation, which is consistent with the brane configuration of the
supergroup gauge theory based on the positive and negative branes.

###### Quiver gauge theory

This argument is easily generalized to quiver theory. The brane configuration
of $A_{p}$ quiver with gauge group SU$(n_{0}|n_{1})$ is shown in Fig. 4.5.

SU$(n_{0}|n_{1})$SU$(n_{0}|n_{1})$SU$(n_{0}|n_{1})$SU$(n_{0})$SU$(n_{0})$SU$(n_{0})$SU$(n_{1})$SU$(n_{1})$
Figure 4.5: Reduction of supergroup quiver gauge theory via gauging.

For the moment, we assign the same super gauge group to all the gauge node for
simplicity. The negative branes are annihilated by gauging, and the resulting
theory is the linear quiver theory with gauge groups SU$(n_{0})$ and flavor
nodes SU$(n_{1})$ attached to the left and right most nodes. We can consider
the situation with different super gauge groups assigned to each node.
However, in such a case, it’s not possible to annihilate all the negative
branes at the same time in general. See also §3.2.

NS5NS5NS5ON-
SU$(2n_{0}|2n_{1})$SU$(2n_{0}|2n_{1})$SU$(n_{0}|n_{1})$SU$(n_{0}|n_{1})$SU$(2n_{0})$SU$(2n_{0})$SU$(n_{0})$SU$(n_{0})$SU$(2n_{1})$
Figure 4.6: Reduction of supergroup gauge theory via gauging for $D$-type
quiver.

Similarly, we can consider the D-type quiver theory. Applying the previous
argument, D-type quiver gauge theory is realized using the ON0 plane as in
Fig. 4.6. Thus the configuration is reduced to that for $D_{r}$ quiver with
non-supergroup gauge nodes through the gauging process. Such a reduction is
consistent with another approach discussed in §3.2.3. We can similarly deal
with $\widehat{D}_{r}$ quiver by imposing ON0 planes on the both ends of the
configuration.

####  4.6 Eight supercharge theory in higher dimensions

In this Section, we consider a higher dimensional analog of the geometric
analysis discussed in this Chapter [Nek98, Nek04, HIV08]. In fact, 6d
$\mathcal{N}=(1,0)$ theory has eight supercharges, and is reduced to 4d
$\mathcal{N}=2$ theory via toroidal compactification. The same argument is
applicable to 5d $\mathcal{N}=1$ theory. In these cases, the corresponding
R-symmetry is given by $\mathrm{Sp}(1)=\mathrm{SU}(2)$.

The bosonic components in the vector multiplet in $\mathsf{d}=4,5,6$ are in
the following: In $\mathsf{d}=6$, it contains the gauge field
$A_{0,1,2,3,4,5}$. In $\mathsf{d}=5$, it contains the gauge field
$A_{0,1,2,3,4}$ and also the real scalar $\phi$. In $\mathsf{d}=4$, it
contains the gauge field $A_{0,1,2,3}$ and the complex scalar
$(\phi,\bar{\phi})$. The scalar fields in $\mathsf{d}=4,5$ are obtained
through the dimensional reduction of the fourth and fifth components of the
gauge field $A_{4,5}$ in $\mathsf{d}=6$:

(6d)(5d)(4d)${A_{4,5}}$${A_{4},\phi}$${\phi,\bar{\phi}}$ (4.6.1)

We explore the Seiberg–Witten approach to these higher dimensional theories
based on the toroidal compactification.

#####  4.6.1 5d $\mathcal{N}=1$ theory

We now consider 5d $\mathcal{N}=1$ gauge theory compactified on a circle
$S^{1}$. In this case, the fourth component of the gauge field $A_{4}$ and the
real scalar play a similar role to the complex scalar in 4d $\mathcal{N}=2$
gauge theory, which are combined into the Wilson loop operator along the
compactification circle $S^{1}$:777In supersymmetric gauge theory, one should
combine the gauge field and the corresponding scalar field to preserve (a part
of) supersymmetry. See, for example, [GY07] for details.

$\displaystyle\Phi:=\operatorname{P}\exp\left(\oint_{S^{1}}\left(A_{4}+\mathrm{i}\phi\,dx_{4}\right)\right)\in
G\,,$ (4.6.2)

where P stands for the path-ordering product. We remark that $\Phi$ is a
$G$-valued chiral ring operator (multiplicative), while the complex scalar in
four dimensions takes a value in $\mathfrak{g}=\operatorname{Lie}G$
(additive). Imposing the supersymmetric vacuum condition as discussed in
§4.1.1, $\Phi$ takes a value in the Cartan subgroup of the gauge group,
$H\subset G$, which parametrizes the Coulomb branch of the moduli space of the
vacua.

For $G=\mathrm{SU}(n)$ with $(n^{\text{f}},n^{\text{af}})$ flavors, the
Seiberg–Witten curve is given by the multiplicative analog of the curve
presented in (4.2.30):

$\displaystyle\Sigma:\quad
y+\Lambda_{\text{5d}}^{b}\,\frac{x^{n-\kappa}}{y}\,P(x)\widetilde{P}(x)=\det(1-x\,\Phi^{-1})\,,\qquad(x,y)\in\mathbb{C}^{\times}\times\mathbb{C}^{\times}\,,$
(4.6.3)

where $b=2n-n^{\text{f}}-n^{\text{af}}$, and $\Lambda_{\text{5d}}$ is the
dynamical parameter for 5d theory. There are several specific points to 5d
theory. $P(x)$ and $\widetilde{P}(x)$ are the multiplicative (K-theory)
analogs of the matter polynomial for the fundamental and antifundamental
hypermultiplets (2.2.12b):

$\displaystyle
P(x)=\prod_{f=1}^{n^{\text{f}}}\left(1-\mathrm{e}^{m_{f}}/x\right)\,,\qquad\widetilde{P}(x)=\prod_{f=1}^{n^{\text{af}}}(1-x/\mathrm{e}^{\widetilde{m}_{f}})\,.$
(4.6.4)

In fact, their asymptotic behaviors are different at $x\to\infty,0$, and we
should distinguish the fundamental and antifundamental matters in five
dimensions.

We denote the Chern–Simons level by $\kappa\in\mathbb{Z}$, a coefficient of
the topological term $\displaystyle\int\operatorname{tr}\left(A\wedge F\wedge
F\right)$, which is reduced to the Wilson loop on $S^{1}$ with the instanton
background. There is an upper bound for the level $\kappa\leq n$ for
$\mathrm{SU}(n)$ theory, concerning the asymptotic freedom.

In this case, the $x$-variable is a multiplicative variable,
$x\in\mathbb{C}^{\times}$. The one-form and the associated symplectic two-form
on the curve are now given by

$\displaystyle\lambda$ $\displaystyle=\log x\,\frac{dy}{y}=\log x\,d\log y\,,$
(4.6.5a) $\displaystyle\omega$ $\displaystyle=d\lambda=d\log x\wedge d\log
y\,.$ (4.6.5b)

Hence we can swap $x$ and $y$ as a symplectic transform. As discussed in §4.3,
the $x$ and $y$ variables are corresponding to the gauge and quiver structure
on the Seiberg–Witten curve. This $x\leftrightarrow y$ symmetry implies an
interesting duality between the gauge and quiver structures. This argument
will be also justified using the brane description later.

###### Reduction to 4d theory

Let us discuss the reduction of the 5d Seiberg–Witten curve to the 4d curve.
In order to take this scaling limit, we put the scaling parameter $R$, s.t.,
$x=\mathrm{e}^{R\,z}$,
$\Phi=\mathop{\mathrm{diag}}(\mathrm{e}^{R\,\mathsf{a}_{\alpha}})_{\alpha=1,\ldots,n}$.
In addition, we impose the special unitary condition
$\displaystyle\sum_{\alpha=1}^{n}\mathsf{a}_{\alpha}=0$, so that $\det\Phi=1$,
for simplicity. Then, one can rewrite the curve for 5d pure $\mathrm{SU}(n)$
SYM theory in the form of

$\displaystyle
y+\left(\frac{\Lambda_{\text{5d}}}{R}\right)^{2n}\frac{x^{-\kappa}}{y}=\prod_{\alpha=1}^{n}\frac{2}{R}\sinh\left(\frac{R}{2}\left(z-\mathsf{a}_{\alpha}\right)\right)$
(4.6.6)

where we shift the $y$ variable as $y\to(-1)^{n}R^{n}x^{n/2}y$. Recalling
$\displaystyle\lim_{R\to 0}\frac{2}{R}\sinh\frac{R}{2}z=z$ and
$\displaystyle\lim_{R\to 0}\mathrm{e}^{R\,z}=1$, it is reduced to the curve
for 4d theory (4.2.26) in the limit $R\to 0$ under the identification

$\displaystyle\lim_{R\to
0}\frac{\Lambda_{\text{5d}}}{R}=\Lambda_{\text{4d}}\,.$ (4.6.7)

In fact, the scaling parameter $R$ is interpreted as the size of the
compactification circle. On the other hand, denoting $x=\mathrm{e}^{R\,z}$, it
shows the periodicity $z\simeq z+2\pi\mathrm{i}/R$. Namely, $x$ is a
coordinate of the cylinder,
$x\in\mathbb{C}^{\times}=\check{S}^{1}\times\mathbb{R}$, where $\check{S}^{1}$
is the dual circle with the size $\check{R}=1/R$. In general, for gauge theory
defined on $\mathcal{S}\times\mathcal{C}$, the $x$-variable takes a value in
$\check{\mathcal{C}}$, which is dual to $\mathcal{C}$:

$\displaystyle\mathcal{C}=\begin{cases}\text{pt}=\text{pt}\times\text{pt}&(4d)\\\
S^{1}=S^{1}\times\text{pt}&(5d)\\\ T^{2}=S^{1}\times
S^{1}&(6d)\end{cases}\quad\longleftrightarrow\quad\check{\mathcal{C}}=\begin{cases}\mathbb{C}=\mathbb{R}\times\mathbb{R}&(4d)\\\
\mathbb{C}^{\times}=\check{S}^{1}\times\mathbb{R}&(5d)\\\
\check{T}^{2}=\check{S}^{1}\times\check{S}^{1}&(6d)\end{cases}$ (4.6.8)

This relation is explained as follows [SW97, NP12]: We take a further
compactification from 4d $\Gamma$-quiver gauge theory to 3d,
$\mathcal{S}=\mathbb{R}^{4}\to\mathbb{R}^{3}\times S^{1}$. Then the resulting
three-dimensional theory is $\mathcal{N}=4$ sigma model, and the target space
is given by the moduli space of the periodic $G_{\Gamma}$-monopole on
$S^{1}\times\check{\mathcal{C}}$ as a result of the Nahm dual transformation
(3d mirror symmetry). In fact, the Seiberg–Witten curve is identified with the
spectral curve of the $G_{\Gamma}$-monopole, and thus the $x$-variable plays a
role of the spectral parameter that takes a value in
$x\in\check{\mathcal{C}}$. See also §5.8.3 for a related discussion.

###### Brane configuration

We can consider the brane configuration for $\mathcal{N}=1$ gauge theory in
five dimensions similarly to the discussion in §4.5. Suppose we compactify
4-direction on a circle $S^{1}$ in Type IIA setup as in Tab. 4.1. Then, we can
consider the T-dual transformation in this direction to obtain Type IIB
configuration [AHK98]:

(IIB) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
NS5 | - | - | - | - | - | - |  |  |  |
D5 | - | - | - | - | - |  | - |  |  |
D7 | - | - | - | - | - |  |  | - | - | -
Table 4.4: The brane configuration of five-dimensional $\mathcal{N}=1$ gauge
theory in Type IIB string theory obtained through the T-dual transformation
from the IIA setup.

Now NS5 and D5 branes are extended in 5 and 6 directions in addition to the
common world-volume, 01234. In this case, these branes are converted to each
other through S-duality in Type IIB string theory, and form the $(p,q)$
5-brane web in 56-directions:

(4.6.9)

where $(0,1)$ and $(1,0)$ stand for NS5 and D5 branes, and $(p,q)$-brane is a
composition of $p$ D5 and $q$ NS5 branes. In fact, $(p,q)$-brane goes to
$(p,q)$ direction in 56-plane in order to balance tensions between the
5-branes. In this context, D7 branes are used to terminate these 5-branes
similarly to D6 branes in Type IIA setup.

The simplest example is pure $\mathrm{SU}(2)$ gauge theory realized with two
NS5 and two D5 branes. In this case, there are three possibilities providing
different Chern–Simons level $\kappa=0,1,2$:

(4.6.10)

We remark that we cannot consider higher Chern–Simons level beyond $\kappa=2$
as mentioned in the context of the Seiberg–Witten curve. Furthermore, this
$(p,q)$-brane web diagram is interpreted as a dual to the toric diagram of the
non-compact toric Calabi–Yau three-fold. Such a connection between gauge
theory and Calabi–Yau geometry is known as the geometric engineering [KKV97,
KMV98, DV02], and one can see the agreement of 5d $\mathcal{N}=1$ instanton
partition function and the topological string amplitude on the corresponding
Calabi–Yau three-fold [AKMV05, AK05, IKV09].

This construction is straightforwardly generalized to quiver gauge theory. Let
us consider $A_{3}$ quiver as an example,

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\pgfsys@invoke{\lxSVG@closescope
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (4.6.11)

The brane descriptions in Type IIA and IIB theories are given as follows:

(4.6.12)

Here we also incorporate the flavor nodes to the left most and the right most
gauge nodes. One can similarly consider the $AD$-type affine quiver by
imposing the periodicity in 6-direction, and taking into account O-planes as
discussed in §4.5.3.

Since S-duality exchanges NS5 and D5 branes in Type IIB string theory, the IIB
web diagram in (4.6.12) also has another equivalent description:

(4.6.13)

These web diagrams provide

  * •

Theory (a): $A_{3}$ quiver with $G_{A_{1}}=\mathrm{SU}(2)$ gauge symmetry

  * •

Theory (b): $A_{1}$ quiver with $G_{A_{3}}=\mathrm{SU}(4)$ gauge symmetry

Namely, the gauge and quiver structures are exchanged through the S-duality
[KMV98, AHK98]. The agreement between the partition functions of both theories
are explicitly checked [BPTY12]. This duality is specific to 5d
$\mathcal{N}=1$ gauge theory, and from geometric point of view, it is
interpreted as a consequence of the symplectic transform on the Seiberg–Witten
curve, $x\leftrightarrow y$, as discussed before. In addition, it is also
possible to discuss the S-duality in the presence of the defect operators. See
[AS18, NPZ18b] for details.

Let us briefly mention the brane description for supergroup gauge theory. As
shown in [OT06], supergroup gauge theory is engineered using the negative
brane, also known as the ghost brane. Based on this argument, the 5-brane
description of 5d $\mathcal{N}=1$ supergroup gauge theory has been considered
[KS20], and the partition function is reproduced by a generalized version of
the topological vertex formalism [AKMV05, AK05, IKV09].

#####  4.6.2 6d $\mathcal{N}=(1,0)$ theory

We then consider 6d $\mathcal{N}=(1,0)$ theory compactified on a torus
$T^{2}$. Since we have two periodic directions, $(x_{4},x_{5})\in T^{2}$, the
chiral ring operator is given as a combination of the loop operators as
follows:

(4d)(5d)(6d)${\phi,\bar{\phi}}$${\displaystyle\oint_{x_{4}}\left(A_{4}+\mathrm{i}\phi\,dx_{4}\right)}$${\displaystyle\oint_{x_{4}}A_{4}+\tau\oint_{x_{5}}A_{5}}$
(4.6.14)

where $\tau$ is the modulus of the torus $T^{2}$. See [BEHT15] for a similar
argument in two dimensions. In this case, the Seiberg–Witten curve for
$\mathrm{SU}(n)$ gauge theory with $n^{\text{f}}=2n$ flavors is accordingly
written with a doubly periodic variable [HIV08]:

$\displaystyle\Sigma:\quad
y+\frac{\mathfrak{q}}{y}\prod_{f=1}^{2n}\theta(x/\mathrm{e}^{R\,m_{f}};p)=\prod_{\alpha=1}^{n}\theta(x/\mathrm{e}^{R\,\mathsf{a}_{\alpha}};p)\,,\qquad(x,y)\in\mathbb{C}^{\times}\times\mathbb{C}^{\times}\,,$
(4.6.15)

where $\theta(x;p)$ is the theta function defined in (A.3.1) with the elliptic
nome $p=\exp\left(2\pi\mathrm{i}\tau\right)\in\mathbb{C}^{\times}$, and
$\mathfrak{q}\in\mathbb{C}^{\times}$ is the (dimensionless) coupling constant.
$R$ is the size of the circle $S^{1}$ in 4-direction as before. In this case,
we use the same one-form $\lambda$ and the symplectic two-form
$\omega=d\lambda$ on the curve as in 5d theory (4.6.5). The theta function
shows a (quasi) modular property under the shift $x\to px$. Hence, denoting
$x=\mathrm{e}^{R\,z}$, the variable $z$ turns out to be doubly periodic under
the shift:

$\displaystyle z\ \simeq\ z+\frac{2\pi\mathrm{i}}{R}\,,\
z+\frac{2\pi\mathrm{i}}{R}\tau\qquad\left(x\ \simeq\ px\right)\,.$ (4.6.16)

Namely, it takes a value in the elliptic curve

$\displaystyle\mathcal{E}=\mathbb{C}^{\times}/p^{\mathbb{Z}}=\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)\,.$
(4.6.17)

We remark that, since 6d $\mathcal{N}=(1,0)$ theory is a chiral theory, we
should impose the anomaly free condition for the matter content. For
$\mathrm{SU}(n)$ gauge theory, $n^{\text{af}}=2n$ is the unique possibility
for the fundamental hypermultiplet.888$\mathrm{SU}(n)$ gauge theory with a
single adjoint matter is also possible in six dimensions. In addition, we
should also introduce the tensor multiplet to cancel the anomaly, similarly to
the Green–Schwarz mechanism [GSW85]. See, for example, [Ohm16, Tom20] for
detailed reviews.  From the algebraic geometric point of view, all the terms
in the curve (4.6.15) should be elliptic functions with the same degree.
Assuming the $y$-variable has the degree $n$, the second term on the left hand
side should be also of degree $n$, which is possible only for
$n^{\text{f}}=2n$.

The same argument is applicable to quiver gauge theory. In this case, the
anomaly free condition is given by

$\displaystyle n^{\text{f}}_{i}=\sum_{j\in\Gamma_{0}}n_{j}\,c_{ji}$ (4.6.18)

where $(c_{ij})_{i,j\in\Gamma_{0}}$ is the classical quiver Cartan matrix
(2.2.37). We remark that this is equivalent to the conformal condition in 4d
$\mathcal{N}=2$ gauge theory, where the coupling constant becomes a
dimensionless parameter.

###### Branes and dualities

Let us discuss the brane description of 6d $\mathcal{N}=(1,0)$ theory with the
toroidal compactification. Compared to 5d $\mathcal{N}=1$ theory, 5-direction
is now periodic as well as 4-direction. Therefore, we can apply the T-duality
along 5-direction to obtain Type IIA description [BK98, HZ98]:

(IIA) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
NS5 | - | - | - | - | - | - |  |  |  |
D6 | - | - | - | - | - | - | - |  |  |
D8 | - | - | - | - | - | - |  | - | - | -
Table 4.5: The brane configuration of six-dimensional $\mathcal{N}=(1,0)$
gauge theory in Type IIA string theory.

The corresponding brane diagram is given as

(4.6.19)

for the chain of $\mathrm{SU}(n)$ gauge theories. Although we have such a Type
IIA description, we can also describe 6d $\mathcal{N}=(1,0)$ theory using a
$(p,q)$-brane configuration by imposing the periodicity in 5-direction:

(4.6.20)

where $(0,1)$ 5-branes with red signs are identified along the periodic
5-direction. We remark that other 5-branes than $(0,1)$-brane cannot be
compatible with the periodicity. For example, the configuration for the pure
$\mathrm{SU}(2)$ theory depicted in (4.6.10) cannot be promoted to 6d theory
since the directions of external legs are not compatible with the periodicity
in 5-direction. This is also consistent with the anomaly condition in 6d
theory.

Since our description is based on $(p,q)$ 5-branes, we can consider the
S-duality for this configuration as well. In this case, 6d gauge theory is
dual to 5d theory with affine quiver structure:

(4.6.21)

More precisely, the matter contents obtained from these web diagrams are as
follows:

  * •

Theory (a): $\widehat{A}_{3}$ quiver with $G_{A_{1}}=\mathrm{SU}(2)$ gauge
symmetry in 5d

  * •

Theory (b): $A_{1}$ quiver with $G_{A_{3}}=\mathrm{SU}(4)$ gauge symmetry in
6d

This implies that 5d theory with $\widehat{G}$ gauge symmetry is interpreted
as 6d theory with $G$ gauge symmetry, and the center of mass of the
bifundamental mass parameters on 5d theory side is converted to the elliptic
modulus on 6d side through the duality.

There exist similar dualities for $DE$\- and $\widehat{DE}$-type quiver gauge
theories. Let us briefly comment on it. The dual theory of $D$-type quiver
theory in 5d is $A$-type quiver with SO-Sp gauge symmetries in 5d (also known
as SO-Sp alternating quiver), and its affinization, $\widehat{D}$-type quiver
theory in 5d, is dual to the SO-Sp alternating quiver in 6d [HKL+16]. For
$E$-type theory, the dual theory is $(E,E)$-type conformal matter theory in
6d. See [DZHTV15] for details.

Finally, let us consider affine quiver theory in six dimensions. In this case,
the web diagram becomes doubly periodic as follows:

(4.6.22)

The matter contents obtained from these diagrams are

  * •

Theory (a): $\widehat{A}_{3}$ quiver with $G_{A_{1}}=\mathrm{SU}(2)$ gauge
symmetry in 6d

  * •

Theory (b): $\widehat{A}_{1}$ quiver with $G_{A_{3}}=\mathrm{SU}(4)$ gauge
symmetry in 6d

In this case, the duality connects two affine quiver gauge theories both in
six dimensions, which are described as the little string theory. Under this
duality, the (center of) bifundamental mass and the elliptic modulus are
exchanged. This duality is also interpreted as the T-duality in the context of
the little string theory [BDZH+16].

In addition to the S-duality exchanging $(1,0)$ and $(0,1)$ 5-branes, we can
also discuss the triality including $(1,1)$ 5-brane, which is specific to
doubly periodic situation, namely 6d affine quiver gauge theory. See [HIR17,
BHIR18b, BHIR18c, BHIR18a, BHIR19] for details. This triality is expected to
be specific to the $A$-type theories [DZO20].

###  Chapter 5 Quantization of geometry

The low energy effective behavior of $\mathcal{N}=2$ gauge theory has the
geometric characterization due to the Seiberg–Witten theory. In particular,
the algebraic curve, called the Seiberg–Witten curve, geometrically encodes
the information about the prepotential of $\mathcal{N}=2$ theory. In this
Chapter, we show how to obtain such an algebraic object from the microscopic
path integral formalism together with the instanton counting described in Part
I. As discussed there, the partition function obtained from the instanton
counting involves additional parameters, that we call the $\Omega$-background
parameters, compared to the Seiberg–Witten geometry. We will see that these
additional parameters play a role of quantum deformation parameter in the
Seiberg–Witten theory, and also point out the correspondence to quantum
integrable systems. In particular, we will see that the quantization of
Seiberg–Witten curve gives rise to the TQ-relation, and the saddle point
equation with respect to the instanton partition function is identified with
the Bethe equation of the corresponding integrable system.

####  5.1 Non-perturbative Schwinger–Dyson equation

The Schwinger–Dyson equation is a functional relation between the correlation
functions in QFT. Its derivation is based on the invariance of the partition
function in the path integral formalism under infinitesimal deformation of the
field. As discussed in Chapter 1, the gauge theory partition function is given
as a superposition of different topological sectors ($\theta$-vacua), so that
it would be covariant under the process of adding/removing an instanton, which
is similar to the coherent state in quantum mechanics. Since changing the
topological sector is a non-perturbative effect in gauge theory, the
functional relation obtained from such an operation is called the non-
perturbative Schwinger–Dyson equation [Nek16]. For this purpose, we consider
the adding/removing-instanton operation for the instanton partition function,
and discuss its geometric interpretation.

#####  5.1.1 Add/remove instantons

The instanton partition function localizes on a fixed point locus on the
instanton moduli space $\mathfrak{M}$ under the equivariant actions. In this
context, the instanton configuration is combinatorially labeled by a partition
$\lambda$, so that the adding/removing-instanton operation gives rise to shift
of its specific component, $\lambda_{i,\alpha,k}\to\lambda_{i,\alpha,k}\pm 1$.
We study the behavior of the instanton partition function under this partition
shift.

###### Vector multiplet

We start with the vector multiplet bundle and its character given as111Here we
omit the node index $i\in\Gamma_{0}$ for simplicity.

$\displaystyle\mathbf{V}=\frac{\wedge\mathbf{Q}_{1}^{\vee}}{\wedge\mathbf{Q}_{2}}\mathbf{X}^{\vee}\mathbf{X}\
\xrightarrow{\operatorname{ch}_{\mathsf{T}}}\
\frac{1-q_{1}^{-1}}{1-q_{2}}\sum_{\begin{subarray}{c}(x,x^{\prime})\in\mathcal{X}\times\mathcal{X}\\\
x\neq x^{\prime}\end{subarray}}\frac{x^{\prime}}{x}\,.$ (5.1.1)

Then, we take the variation

$\displaystyle\delta\mathbf{V}:=\mathbf{V}[\mathcal{X}_{\text{ad}:(\alpha,k)}]-\mathbf{V}[\mathcal{X}]$
(5.1.2)

where we denote the configuration obtained by adding/removing an instanton by

$\displaystyle\mathcal{X}_{\text{ad}:(i,\alpha,k)}=(\mathcal{X}\backslash\\{x_{i,\alpha,k}\\})\sqcup\\{q_{2}x_{i,\alpha,k}\\}\,,\qquad\mathcal{X}_{\text{rm}:(i,\alpha,k)}=(\mathcal{X}\backslash\\{x_{i,\alpha,k}\\})\sqcup\\{q_{2}^{-1}x_{i,\alpha,k}\\}\,.$
(5.1.3)

We simply denote $x=x_{\alpha,k}$, hence the character of the variation is
given by

$\displaystyle\operatorname{ch}_{\mathsf{T}}\delta\mathbf{V}$
$\displaystyle=\frac{1-q_{1}^{-1}}{1-q_{2}}\sum_{x^{\prime}\in\mathcal{X}\backslash\\{x\\}}\left((q_{2}^{-1}-1)\frac{x^{\prime}}{x}+(q_{2}-1)\frac{x}{x^{\prime}}\right)$
$\displaystyle=(1-q_{1}^{-1})\sum_{x^{\prime}\in\mathcal{X}\backslash\\{x\\}}\left(q_{2}^{-1}\frac{x^{\prime}}{x}-\frac{x}{x^{\prime}}\right)$
$\displaystyle=(1-q_{1}^{-1})\sum_{x^{\prime}\in\mathcal{X}}\left(q_{2}^{-1}\frac{x^{\prime}}{x}-\frac{x}{x^{\prime}}\right)-(1-q_{1}^{-1})\lim_{x^{\prime\prime}\to
x}\left(q_{2}^{-1}\frac{x^{\prime\prime}}{x}-\frac{x}{x^{\prime\prime}}\right)\,.$
(5.1.4)

We have to carefully deal with the last term since, in the limit
$x^{\prime\prime}\to x$, it gives rise to the zero mode:

$\displaystyle\lim_{x^{\prime\prime}\to
x}\mathbb{I}\left[\frac{x^{\prime\prime}}{x}\right]=\lim_{x^{\prime\prime}\to
x}[\log x^{\prime\prime}-\log x]=0\,.$ (5.1.5)

In order to remove this zero mode, we replace the summation over the set
$\mathcal{X}$ with $\mathcal{X}_{\text{ad}}$,

$\displaystyle(1-q_{1}^{-1})\left(\sum_{x^{\prime}\in\mathcal{X}_{\text{ad}}}q_{2}^{-1}\frac{x^{\prime}}{x}-\sum_{x^{\prime}\in\mathcal{X}}\frac{x}{x^{\prime}}\right)+(1-q_{1}^{-1})\lim_{x^{\prime\prime}\to
x}\left(\frac{x}{x^{\prime\prime}}-\frac{x^{\prime\prime}}{x}\right)$
$\displaystyle=-(1-q_{1})\sum_{x^{\prime}\in\mathcal{X}_{\text{ad}}}q^{-1}\frac{x^{\prime}}{x}-(1-q_{1}^{-1})\sum_{x^{\prime}\in\mathcal{X}_{\text{ad}}}\frac{x}{x^{\prime}}+(1-q_{1}^{-1})\lim_{x^{\prime\prime}\to
x}\left(\frac{x}{x^{\prime\prime}}-\frac{x^{\prime\prime}}{x}\right)$
$\displaystyle=-\operatorname{ch}_{\mathsf{T}}\left(\det\mathbf{Q}^{\vee}\mathbf{x}^{\vee}\mathbf{Y}[\mathcal{X}_{\text{ad}}]+\mathbf{Y}[\mathcal{X}]^{\vee}\mathbf{x}\right)+(1-q_{1}^{-1})\lim_{x^{\prime\prime}\to
x}\left(\frac{x}{x^{\prime\prime}}-\frac{x^{\prime\prime}}{x}\right)$ (5.1.6)

where we define a bundle $\mathbf{x}$, s.t.,
$\operatorname{ch}_{\mathsf{T}}\mathbf{x}=x$, and the last term is now given
by

$\displaystyle\lim_{x^{\prime\prime}\to
x}\mathbb{I}\left[(1-q_{1}^{-1})\left(\frac{x}{x^{\prime\prime}}-\frac{x^{\prime\prime}}{x}\right)\right]$
$\displaystyle=\lim_{x^{\prime\prime}\to x}\frac{[\log x-\log
x^{\prime\prime}][\log x^{\prime\prime}-\log x-\epsilon_{1}]}{[\log
x^{\prime\prime}-\log x][\log x-\log x^{\prime\prime}-\epsilon_{1}]}$
$\displaystyle=-1\,.$ (5.1.7)

Applying the index formula, we then obtain the ratio of the vector multiplet
contributions to the partition function evaluated with $\mathcal{X}$ and
$\mathcal{X}_{\text{ad}}$:

$\displaystyle\mathbb{I}[\delta\mathbf{V}]=\frac{Z^{\text{vec}}[\mathcal{X}_{\text{ad}}]}{Z^{\text{vec}}[\mathcal{X}]}=-\frac{1}{\mathsf{Y}_{qx}^{\vee}[\mathcal{X}_{\text{ad}}]\mathsf{Y}_{x}[\mathcal{X}]}$
(5.1.8)

where we define $\mathsf{Y}$-functions:

$\displaystyle\mathsf{Y}_{i,x}=\mathbb{I}[\mathbf{Y}_{i}^{\vee}\mathbf{x}]\,,\qquad\mathsf{Y}^{\vee}_{i,x}=\mathbb{I}[\mathbf{x}^{\vee}\mathbf{Y}_{i}]\,.$
(5.1.9)

We will discuss properties of the $\mathsf{Y}$-function shortly.

###### Vector multiplet: another derivation

The variation of the vector multiplet bundle is similarly considered in terms
of the framing and the instanton bundles. Adding an instanton corresponds to
the shift of the instanton bundle:

$\displaystyle\mathbf{K}\ \longrightarrow\ \mathbf{K}+\mathbf{x}\,.$ (5.1.10)

Recalling the expression of the observable bundle $\mathbf{Y}$ in terms of
$(\mathbf{N},\mathbf{K})$ as in (1.8.14), and the instanton part of the vector
multiplet bundle is given as (1.9.4), the variation is computed as follows:

$\displaystyle\delta\mathbf{V}=\delta\mathbf{V}^{\text{inst}}$
$\displaystyle=-\det\mathbf{Q}^{\vee}\mathbf{x}^{\vee}\mathbf{Y}-\mathbf{Y}^{\vee}\mathbf{x}+\wedge\mathbf{Q}^{\vee}\,.$
(5.1.11)

The last term will give a zero mode as well as the previous computation.

In order to regularize the zero mode term, we define the shifted observable
bundle:

$\displaystyle\mathbf{Y}_{\text{ad}}$
$\displaystyle:=\mathbf{N}-\wedge\mathbf{Q}\cdot\left(\mathbf{K}+\mathbf{x}\right)\,,$
(5.1.12a) $\displaystyle\mathbf{Y}_{\text{rm}}$
$\displaystyle:=\mathbf{N}-\wedge\mathbf{Q}\cdot\left(\mathbf{K}-\mathbf{x}\right)\,.$
(5.1.12b)

Then, the variation of the vector multiplet bundle is given as

$\displaystyle\delta\mathbf{V}$
$\displaystyle=-\det\mathbf{Q}^{\vee}\mathbf{x}^{\vee}\mathbf{Y}_{\text{ad}}-\mathbf{Y}^{\vee}\mathbf{x}=-\det\mathbf{Q}^{\vee}\mathbf{x}^{\vee}\mathbf{Y}-\mathbf{Y}_{\text{ad}}^{\vee}\mathbf{x}\,,$
(5.1.13)

which is consistent with the previous argument.

######  $\mathsf{Y}$-function: observable generating function

Let us discuss properties of the $\mathsf{Y}$-function. Since the observable
bundle $(\mathbf{Y}_{i})_{i\in\Gamma_{0}}$ has several expressions,
correspondingly the $\mathsf{Y}$-function also has several forms:

$\displaystyle\mathsf{Y}_{i,x}[\mathcal{X}]$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:quiver_univ_bundle}}}{{=}}\prod_{\alpha=1}^{n_{i}}\left[\left(1-\frac{\mathrm{e}^{\mathsf{a}_{i,\alpha}}}{x}\right)\prod_{(s_{1},s_{2})\in\lambda_{i,\alpha}}\mathscr{S}\left(\frac{\mathrm{e}^{\mathsf{a}_{\alpha}}q_{1}^{s_{1}-1}q_{2}^{s_{2}-1}}{x}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:quiver_partial_red_univ_bundle}}}{{=}}\prod_{x^{\prime}\in\mathcal{X}_{i}}\frac{1-x^{\prime}/x}{1-q_{1}x^{\prime}/x}$
(5.1.14a) $\displaystyle\mathsf{Y}_{i,x}[\mathcal{X}]^{\vee}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:quiver_univ_bundle}}}{{=}}\prod_{\alpha=1}^{n_{i}}\left[\left(1-\frac{x}{\mathrm{e}^{\mathsf{a}_{i,\alpha}}}\right)\prod_{(s_{1},s_{2})\in\lambda_{i,\alpha}}\mathscr{S}\left(\frac{\mathrm{e}^{\mathsf{a}_{\alpha}}q_{1}^{s_{1}-1}q_{2}^{s_{2}-1}}{x}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:quiver_partial_red_univ_bundle}}}{{=}}\prod_{x^{\prime}\in\mathcal{X}_{i}}\frac{1-x/x^{\prime}}{1-q_{1}^{-1}x/x^{\prime}}$
(5.1.14b)

where we apply the K-theory convention, and the $\mathscr{S}$-function is
defined in (1.9.27). They also have the following expressions:

$\displaystyle\mathsf{Y}_{i,x}[\mathcal{X}]$
$\displaystyle=\prod_{\alpha=1}^{n_{i}}\left(\prod_{s\in\partial_{+}\lambda_{\alpha}}\left(1-\mathrm{e}^{\mathsf{a}_{i,\alpha}}q_{1}^{s_{1}-1}q_{2}^{s_{2}-1}/x\right)\prod_{s\in\partial_{-}\lambda_{\alpha}}\left(1-\mathrm{e}^{\mathsf{a}_{i,\alpha}}q_{1}^{s_{1}}q_{2}^{s_{2}}/x\right)^{-1}\right)\,,$
(5.1.15a) $\displaystyle\mathsf{Y}_{i,x}[\mathcal{X}]^{\vee}$
$\displaystyle=\prod_{\alpha=1}^{n_{i}}\left(\prod_{s\in\partial_{+}\lambda_{\alpha}}\left(1-x/\mathrm{e}^{\mathsf{a}_{i,\alpha}}q_{1}^{s_{1}-1}q_{2}^{s_{2}-1}\right)\prod_{s\in\partial_{-}\lambda_{\alpha}}\left(1-x/\mathrm{e}^{\mathsf{a}_{i,\alpha}}q_{1}^{s_{1}}q_{2}^{s_{2}}\right)^{-1}\right)\,,$
(5.1.15b)

where $\partial_{\pm}\lambda$ is the outer/inner boundary of the partition:

(5.1.16)

One can add a box to the outer boundary  $\in\partial_{+}\lambda$, and one can
remove a box from the inner boundary  $\in\partial_{-}\lambda$ of the
partition $\lambda$. We remark $|\partial_{+}\lambda|-|\partial_{-}\lambda|=1$
for $\forall\lambda$. Furthermore, we may express the $\mathsf{Y}$-function in
terms of the observable bundle:

$\displaystyle\mathsf{Y}_{i,x}=\exp\left(-\sum_{n=1}^{\infty}\frac{x^{-n}}{n}\operatorname{ch}_{\mathsf{T}}\mathbf{Y}_{i}^{[n]}\right)\,.$
(5.1.17)

Since the character of the observable bundle is given as the chiral ring
operator, the $\mathsf{Y}$-function plays a role of the chiral ring generating
function.

Based on the finite product formula of the $\mathsf{Y}$-function, we can see
its asymptotic behavior:

$\displaystyle\mathsf{Y}_{i,x}$ $\displaystyle\ \longrightarrow\
\begin{cases}\displaystyle(-x)^{-n}\prod_{\alpha=1}^{n_{i}}\mathrm{e}^{\mathsf{a}_{i,\alpha}}&(x\to
0)\\\ 1&(x\to\infty)\end{cases}$ (5.1.18a)
$\displaystyle\mathsf{Y}_{i,x}^{\vee}$ $\displaystyle\ \longrightarrow\
\begin{cases}1&(x\to 0)\\\
\displaystyle(-x)^{+n}\prod_{\alpha=1}^{n_{i}}\mathrm{e}^{-\mathsf{a}_{i,\alpha}}&(x\to\infty)\end{cases}$
(5.1.18b)

In addition, $\mathsf{Y}_{i,x}$ and $\mathsf{Y}^{\vee}_{i,x}$ have the same
zeros and poles depending on the instanton configuration $\mathcal{X}$:

$\displaystyle\text{zeros}:\ x=x^{\prime}\,,\qquad\text{poles}:\
x=q_{1}x^{\prime}\qquad(x^{\prime}\in\mathcal{X}_{i})\,.$ (5.1.19)

Hence they are related to each other up to an over all non-singular factor:

$\displaystyle\mathsf{Y}_{i,x}[\mathcal{X}]$
$\displaystyle=\prod_{\alpha=1}^{n_{i}}\left[\left(-\frac{\mathrm{e}^{\mathsf{a}_{i,\alpha}}}{x}\right)\left(1-\frac{x}{\mathrm{e}^{\mathsf{a}_{i,\alpha}}}\right)\prod_{(s_{1},s_{2})\in\lambda_{i,\alpha}}\mathscr{S}\left(\frac{\mathrm{e}^{\mathsf{a}_{\alpha}}q_{1}^{s_{1}-1}q_{2}^{s_{2}-1}}{x}\right)\right]$
$\displaystyle=\left((-x)^{-n}\prod_{\alpha=1}^{n_{i}}\mathrm{e}^{\mathsf{a}_{i,\alpha}}\right)\mathsf{Y}_{i,x}[\mathcal{X}]^{\vee}\,.$
(5.1.20)

From this point of view, $\mathsf{Y}$ and $\mathsf{Y}^{\vee}$ are interpreted
to give the expansion around $x=\infty$ and $x=0$ of the same function, and
the factor which converts from $\mathsf{Y}$ to $\mathsf{Y}^{\vee}$ does not
depend on the instanton configuration $\mathcal{X}$. See also its operator
realization discussed in §6.2.5.

We also remark that the expression (5.1.8) may be singular because of
$\mathsf{Y}_{x}[\mathcal{X}]=0$ for $x\in\mathcal{X}$. However, this
singularity is cancelled by the pole of
$\mathsf{Y}_{qx}[\mathcal{X}_{\text{ad}}]^{\vee}$, then the product
$\mathsf{Y}_{qx}^{\vee}[\mathcal{X}_{\text{ad}}]\mathsf{Y}_{x}[\mathcal{X}]$
itself remains regular. This regularity argument shall play an important role
in the derivation of the Schwinger–Dyson equation.

###### (Bi)fundamental hypermultiplet

Let us then consider the bifundamental hypermultiplet contribution. From
(2.2.1), the bifundamental hypermultiplet bundle is given as

$\displaystyle\mathbf{H}_{e:i\to
j}=-\mathbf{M}_{e}\frac{\wedge\mathbf{Q}_{1}^{\vee}}{\wedge\mathbf{Q}_{2}}\mathbf{X}_{i}^{\vee}\mathbf{X}_{j}\,.$
(5.1.21)

The variation is similarly computed as follows:

$\displaystyle\delta_{i}\mathbf{H}_{e:i\to j}$
$\displaystyle:=\mathbf{H}_{e:i\to
j}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]-\mathbf{H}_{e:i\to
j}[\mathcal{X}]=\mathbf{M}_{e}\det\mathbf{Q}^{\vee}\mathbf{x}^{\vee}\mathbf{Y}_{j}[\mathcal{X}]\,,$
(5.1.22a) $\displaystyle\delta_{i}\mathbf{H}_{e:j\to i}$
$\displaystyle:=\mathbf{H}_{e:j\to
i}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]-\mathbf{H}_{e:j\to
i}[\mathcal{X}]=\mathbf{M}_{e}\mathbf{Y}_{j}[\mathcal{X}]^{\vee}\mathbf{x}\,,$
(5.1.22b)

with $\operatorname{ch}\mathbf{x}=x_{i,\alpha,k}=:x$. In this case, we do not
need to take care of the zero mode. Hence, applying the index formula, we
obtain the ratio of the bifundamental matter contributions to the partition
function:

$\displaystyle\mathbb{I}[\delta_{i}\mathbf{H}_{e:i\to j}]$
$\displaystyle=\frac{Z^{\text{bf}}_{e:i\to
j}[\mathcal{X}_{\text{ad}:(i,:a,k)}]}{Z^{\text{bf}}_{e:i\to
j}[\mathcal{X}]}=\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}\,,$ (5.1.23a)
$\displaystyle\mathbb{I}[\delta_{i}\mathbf{H}_{e:j\to i}]$
$\displaystyle=\frac{Z^{\text{bf}}_{e:j\to
i}[\mathcal{X}_{\text{ad}:(i,:a,k)}]}{Z^{\text{bf}}_{e:j\to
i}[\mathcal{X}]}=\mathsf{Y}_{j,\mu_{e}x}\,.$ (5.1.23b)

The (anti)fundamental hypermultiplet contribution is obtained from the
bifundamental matter by freezing the gauge node:

$\displaystyle\mathbb{I}[\delta\mathbf{H}_{i}^{\text{f}}]$
$\displaystyle=\frac{Z^{\text{f}}_{i}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z^{\text{f}}_{i}[\mathcal{X}]}=P_{i,x}\,,$
(5.1.24a) $\displaystyle\mathbb{I}[\delta\mathbf{H}_{i}^{\text{af}}]$
$\displaystyle=\frac{Z^{\text{af}}_{i}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z^{\text{af}}_{i}[\mathcal{X}]}=\widetilde{P}^{\vee}_{i,qx}\,,$
(5.1.24b)

where $(P_{i,x},\widetilde{P}^{\vee}_{i,x})$ are the matter polynomials
(2.2.12b) in the K-theory convention:222We omit the symbol, (a, af), for the
matter polynomials as long as no confusion.

$\displaystyle
P_{i,x}=\prod_{f=1}^{n_{i}^{\text{f}}}\left(1-x^{-1}\,\mathrm{e}^{m_{i,f}}\right)\,,\qquad\widetilde{P}_{i,x}^{\vee}=\prod_{f=1}^{n_{i}^{\text{af}}}\left(1-x\,\mathrm{e}^{-\widetilde{m}_{i,f}}\right)\,.$
(5.1.25)

These are polynomials in $x^{-1}$ and $x$, respectively, which are converted
to each other as follows:

$\displaystyle
P_{i,x}=\left((-x)^{-n_{i}^{\text{f}}}\prod_{f=1}^{n_{i}^{\text{f}}}\mathrm{e}^{m_{i,f}}\right)P_{i,x}^{\vee}\,,\qquad\widetilde{P}_{i,x}^{\vee}=\left((-x)^{n_{i}^{\text{af}}}\prod_{f=1}^{n_{i}^{\text{af}}}\mathrm{e}^{-\widetilde{m}_{i,f}}\right)\widetilde{P}_{i,x}\,.$
(5.1.26)

###### Adjoint hypermultiplet

In the case of the adjoint hypermultiplet, the computation is much similar to
the vector multiplet. For the adjoint hypermultiplet bundle given by

$\displaystyle\mathbf{H}^{\text{adj}}=-\mathbf{M}_{\text{adj}}\frac{\wedge\mathbf{Q}_{1}^{\vee}}{\wedge\mathbf{Q}_{2}}\mathbf{X}^{\vee}\mathbf{X}\,,$
(5.1.27)

we take the variation:

$\displaystyle\delta\mathbf{H}^{\text{adj}}$
$\displaystyle:=\mathbf{H}^{\text{adj}}[\mathcal{X}_{\text{ad}:(\alpha,k)}]-\mathbf{H}^{\text{adj}}[\mathcal{X}]$
$\displaystyle=\mathbf{M}_{\text{adj}}\det\mathbf{Q}^{\vee}\mathbf{x}^{\vee}\mathbf{Y}+\mathbf{M}_{\text{adj}}\mathbf{Y}^{\vee}\mathbf{x}-\mathbf{M}_{\text{adj}}\wedge\mathbf{Q}^{\vee}\,.$
(5.1.28)

Then, the ratio of the adjoint matter contributions to the partition function
is given by

$\displaystyle\mathbb{I}[\delta\mathbf{H}^{\text{adj}}]=\frac{Z^{\text{adj}}[\mathcal{X}_{\text{ad}:(\alpha,k)}]}{Z^{\text{adj}}[\mathcal{X}]}=\mathscr{S}(\mu^{-1})\,\mathsf{Y}_{\mu
x}\mathsf{Y}^{\vee}_{\mu^{-1}qx}$ (5.1.29)

where
$\operatorname{ch}_{\mathsf{T}}\mathbf{M}_{\text{adj}}=\mu\in\mathbb{C}^{\times}$
is the adjoint mass parameter. Appearance of the $\mathscr{S}$-function is
specific to the adjoint matter contribution.

###### Topological term and Chern–Simons term

The coupling constant part, namely the topological term, simply behaves as

$\displaystyle\frac{Z_{i}^{\text{top}}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z_{i}^{\text{top}}[\mathcal{X}]}=\mathfrak{q}_{i}$
(5.1.30)

under the adding-instanton operation.

In addition, for 5d gauge theory, we may have the Chern–Simons term. Based on
the expression in §1.9.4, we simply obtain the behavior of the partition
functions under the adding-instanton operation:

$\displaystyle\frac{Z_{i}^{\text{CS}}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z_{i}^{\text{CS}}[\mathcal{X}]}=x^{-\kappa_{i}}\,,\qquad
x=x_{i,\alpha,k}\,.$ (5.1.31)

####  5.2 $qq$-character

Gathering all the contributions, the total partition function (except for the
adjoint matter term) behaves under the adding-instanton operation as follows:

$\displaystyle\frac{Z[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z[\mathcal{X}]}$
$\displaystyle=-\mathfrak{q}_{i}\,x^{-\kappa_{i}}\frac{P_{i,x}\widetilde{P}^{\vee}_{i,qx}}{\mathsf{Y}_{i,qx}^{\vee}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]\mathsf{Y}_{i,x}[\mathcal{X}]}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}[\mathcal{X}]\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}x}[\mathcal{X}]$ (5.2.1)

for $x=x_{i,\alpha,k}$. This is also written in the following form:

$\displaystyle\mathop{\mathrm{Res}}_{x\,=\,x_{i,\alpha,k}}\Bigg{[}Z[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]\mathsf{Y}_{i,qx}^{\vee}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]$
$\displaystyle\hskip
50.00008pt+Z[\mathcal{X}]\left(\mathfrak{q}_{i}\,x^{-\kappa_{i}}\frac{P_{i,x}\widetilde{P}^{\vee}_{i,qx}}{\mathsf{Y}_{i,x}[\mathcal{X}]}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}[\mathcal{X}]\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}x}[\mathcal{X}]\right)\Bigg{]}=0\,,$ (5.2.2)

which means that the pole singularity of
$\mathsf{Y}_{i,qx}^{\vee}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]$ at
$x=x_{i,\alpha,k}$ in the first term is cancelled by the second term. Hence,
summing up all the instanton configuration, we obtain

$\displaystyle\Big{<}\,\mathsf{Y}_{i,qx}^{\vee}\,\Big{>}+\mathfrak{q}_{i}\,x^{-\kappa_{i}}\,P_{i,x}\widetilde{P}^{\vee}_{i,qx}\left<\,\frac{1}{\mathsf{Y}_{i,x}}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}x}\,\right>\,.$ (5.2.3)

In this expression, it is guaranteed that all the pole singularities in the
first term are cancelled by the second contribution, which will play an
important role to discuss the analytic behavior.

#####  5.2.1 iWeyl reflection

The procedure to add a term which compensates the pole singularity of the
previous term is called the iWeyl reflection.333Here $i$ is for instanton
[NP12].  The reason why we can interpret this as a deformation of the Weyl
reflection is described in the following.

We denote the iWeyl reflection action:

$\displaystyle\text{iWeyl}:\ \mathsf{Y}_{i,qx}^{\vee}$
$\displaystyle\longmapsto\
\mathsf{Y}_{i,qx}^{\vee}\times\left(\mathfrak{q}_{i}\,x^{-\kappa_{i}}\frac{P_{i,x}\widetilde{P}^{\vee}_{i,qx}}{\mathsf{Y}_{i,qx}^{\vee}\mathsf{Y}_{i,x}}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}x}\right)$
$\displaystyle=:\mathsf{Y}_{i,qx}^{\vee}\times\left(\mathfrak{q}_{i}\,x^{-\kappa_{i}}\frac{P_{i,x}\widetilde{P}^{\vee}_{i,qx}}{\mathsf{A}_{i,x}}\right)\,,$
(5.2.4)

where we define the $\mathsf{A}$-function

$\displaystyle\mathsf{A}_{i,x}=\mathsf{Y}_{i,qx}^{\vee}\mathsf{Y}_{i,x}\left(\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}x}\right)^{-1}\,.$ (5.2.5)

It is also possible to write the reflection formula (5.2.4) only in terms of
either $(\mathsf{Y}_{i,x})$ or $(\mathsf{Y}^{\vee}_{i,x})$ as follows:

$\displaystyle\mathsf{Y}^{\vee}_{i,x}\ \longmapsto\
\left((-1)^{n_{i}+\sum_{j\to
i}n_{j}+n_{i}^{\text{f}}}\mathrm{e}^{\sum_{j}c_{ij}^{-[\log]}n_{j}}q^{n_{i}^{\text{f}}+\kappa_{i}}\right)\mathfrak{q}_{i}\,x^{\sum_{j}c_{ij}^{-[0]}n_{j}-n_{i}^{\text{f}}-\kappa_{i}}$
$\displaystyle\hskip
130.0002pt\times\frac{P_{i,q^{-1}x}^{\vee}\widetilde{P}^{\vee}_{i,x}}{\mathsf{Y}^{\vee}_{i,q^{-1}x}}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}x}^{\vee}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}q^{-1}x}^{\vee}\,$ (5.2.6a)
$\displaystyle\mathsf{Y}_{i,x}\ \longmapsto\ \left((-1)^{n_{i}+\sum_{i\to
j}n_{j}+n_{i}^{\text{f}}}\mathrm{e}^{\sum_{j}c_{ij}^{+[\log]}n_{j}}q^{\kappa_{i}}\right)\mathfrak{q}_{i}\,x^{\sum_{j}c_{ij}^{+[0]}n_{j}+n_{i}^{\text{f}}-\kappa_{i}}$
$\displaystyle\hskip
130.0002pt\times\frac{P_{i,q^{-1}x}\widetilde{P}_{i,x}}{\mathsf{Y}_{i,q^{-1}x}}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}x}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}q^{-1}x}\,,$ (5.2.6b)

where $(c_{ij}^{\pm[0]})$ is the classical version of the half Cartan matrices
(See §2.2.3), and we define the logarithm analog:

$\displaystyle c_{ij}^{+[\log]}$ $\displaystyle=\delta_{ij}-\sum_{e:i\to
j}\log\mu_{e}^{-1}\,,$ (5.2.7a) $\displaystyle c_{ij}^{-[\log]}$
$\displaystyle=\log q^{-1}\delta_{ij}-\sum_{e:j\to i}\log(\mu_{e}q^{-1})\,.$
(5.2.7b)

We also impose the special unitary condition for simplicity:

$\displaystyle\sum_{\alpha=1}^{n_{i}}\mathsf{a}_{i,\alpha}=0\,\qquad\sum_{f=1}^{n_{i}^{\text{f}}}m_{i,f}=0\,,\qquad\sum_{f=1}^{n_{i}^{\text{af}}}\widetilde{m}_{i,f}=0\,.$
(5.2.8)

In particular, we can concisely (also schematically) express the
$\mathsf{A}$-function using the (classical) quiver Cartan matrix (2.2.37):

$\displaystyle\log\mathsf{A}_{i}=\log\mathsf{Y}_{i}^{2}-\log\left(\prod_{e:i\to
j}\mathsf{Y}_{j}\prod_{e:j\to
i}\mathsf{Y}_{j}\right)=\sum_{j\in\Gamma_{0}}\log\mathsf{Y}_{j}\,c_{ji}^{[0]}\,.$
(5.2.9)

In order to precisely deal with the argument shift of the
$\mathsf{Y}$-functions, we should use the $q$-deformation of quiver Cartan
matrix. See §6.2.6 for more precise relation. Hence, identifying
$\mathsf{Y}_{i}$ as a fundamental weight corresponding to the node
$i\in\Gamma_{0}$, it is converted to the simple root $\mathsf{A}_{i}$
corresponding to the node $i$, and thus this process is interpreted as the
Weyl reflection associated with a representation constructed on the quiver.

Then, in order to compensate the possible pole singularities from
$\mathsf{Y}_{j,\mu_{e}^{-1}qx}^{\vee}$ and $\mathsf{Y}_{j,\mu_{e}x}$ in
(5.2.3), we may add further terms. In this way, the iWeyl reflection generates
the highest weight module on quiver $\Gamma$ where the first term
$\mathsf{Y}_{i}^{\vee}$ plays a role of the highest weight. Summing up all the
contributions generated by the iWeyl reflection, we shall obtain a pole-free
function, which turns out to be a polynomial in the variable $x$, that is
called (the vev of) the $qq$-character [Nek16]:

$\displaystyle\mathsf{T}_{i,x}=\mathsf{Y}_{i,x}^{\vee}+\mathfrak{q}_{i}\left(\frac{x}{q}\right)^{-\kappa_{i}}\frac{P_{i,q^{-1}x}\widetilde{P}^{\vee}_{i,x}}{\mathsf{Y}_{i,q^{-1}x}}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}x}^{\vee}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}q^{-1}x}+\cdots\,.$ (5.2.10)

Such a relation for the $\mathsf{Y}$-functions is interpreted as the non-
perturbative version of the Schwinger–Dyson equation in this context. Due to
the Dynkin–Cartan classification, the highest weight module is finite
dimensional for the finite-type quiver $(\det c>0)$, so that the iWeyl
reflection terminates within finite times. For the affine and hyperbolic
quivers $(\det c\leq 0)$, it becomes infinite dimensional, and the
corresponding $qq$-character is given by an infinite series expansion.

The $qq$-character has a physical interpretation as the codimension-four
defect operator in 4d gauge theory, which generates the chiral ring operators
in the full $\Omega$-background [Nek16, Nek18]. This interpretation is also
possible for 5d and 6d setups by imposing the defect branes [Kim16, KMS18,
AKKS18, AS18].

#####  5.2.2 Supergroup gauge theory

We can apply a parallel analysis for supergroup gauge theory based on the
instanton partition function discussed in §3.4. Since we have two sets of
partitions characterizing the equivariant fixed point, we consider the
partition shift for the positive and negative ones:444See [KP19a] for details
of the derivation

$\displaystyle\frac{Z[\mathcal{X}_{\text{ad:}(i,\alpha,k,0)}]}{Z[\mathcal{X}]}$
$\displaystyle=-\mathfrak{q}_{i}\,x^{-\kappa_{i}^{0}}\,\frac{P_{i,x}\widetilde{P}_{i,qx}^{\vee}}{\mathsf{Y}_{i,qx}[\mathcal{X}_{\text{ad}}]\mathsf{Y}_{i,x}^{\vee}[\mathcal{X}]}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}qx}[\mathcal{X}]\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}x}^{\vee}[\mathcal{X}]\Bigg{|}_{x=x_{i,\alpha,k}^{0}}$
(5.2.11a)
$\displaystyle\frac{Z[\mathcal{X}_{\text{ad:}(i,\alpha,k,1)}]}{Z[\mathcal{X}]}$
$\displaystyle=-\mathfrak{q}_{i}^{-1}\,(q^{-1}x)^{+\kappa_{i}^{1}}\,\frac{\mathsf{Y}_{i,x}[\mathcal{X}]\mathsf{Y}_{i,q^{-1}x}^{\vee}[\mathcal{X}_{\text{ad}}]}{P_{i,q^{-1}x}\widetilde{P}_{i,x}^{\vee}}\prod_{e:i\to
j}\mathsf{Y}_{j,\mu_{e}^{-1}x}[\mathcal{X}]^{-1}\prod_{e:j\to
i}\mathsf{Y}_{j,\mu_{e}q^{-1}x}^{\vee}[\mathcal{X}]^{-1}\Bigg{|}_{x=x_{i,\alpha,k}^{1}}$
(5.2.11b)

where we define the $\mathsf{Y}$-functions

$\displaystyle\mathsf{Y}_{i,x}=\frac{\mathsf{Y}_{i,x}^{0}}{\mathsf{Y}_{i,x}^{1}}\,,\qquad\mathsf{Y}_{i,x}^{\vee}=\frac{\mathsf{Y}_{i,x}^{0\vee}}{\mathsf{Y}_{i,x}^{1\vee}}\,,$
(5.2.12)

and the matter functions

$\displaystyle
P_{i,x}=\frac{P_{i,x}^{0}}{P_{i,x}^{1}}\,,\qquad\widetilde{P}_{i,x}=\frac{\widetilde{P}_{i,x}^{0}}{\widetilde{P}_{i,x}^{1}}\,,\qquad
P_{i,x}^{\vee}=\frac{P_{i,x}^{0\vee}}{P_{i,x}^{1\vee}}\,,\qquad\widetilde{P}_{i,x}^{\vee}=\frac{\widetilde{P}_{i,x}^{0\vee}}{\widetilde{P}_{i,x}^{1\vee}}\,,$
(5.2.13)

with

$\displaystyle\mathsf{Y}_{i,x}^{\sigma}=\prod_{x^{\prime}\in\mathcal{X}_{i}^{\sigma}}\frac{1-x^{\prime}/x}{1-q_{1}x^{\prime}/x}\,,\qquad\mathsf{Y}_{i,x}^{\sigma\vee}=\prod_{x^{\prime}\in\mathcal{X}_{i}^{\sigma}}\frac{1-x/x^{\prime}}{1-q_{1}^{-1}x/x^{\prime}}\,,$
(5.2.14a) $\displaystyle P_{i,x}^{\sigma}$
$\displaystyle=\prod_{\mu\in\mathcal{M}_{i}^{\sigma}}\left(1-\frac{\mu}{x}\right)\,,\qquad\widetilde{P}_{i,x}^{\sigma}=\prod_{\mu\in\widetilde{\mathcal{M}}_{i}^{\sigma}}\left(1-\frac{\mu}{x}\right)\,,$
(5.2.14b) $\displaystyle P_{i,x}^{\sigma\vee}$
$\displaystyle=\prod_{\mu\in\mathcal{M}_{i}^{\sigma}}\left(1-\frac{x}{\mu}\right)\,,\qquad\widetilde{P}_{i,x}^{\sigma\vee}=\prod_{\mu\in\widetilde{\mathcal{M}}_{i}^{\sigma}}\left(1-\frac{x}{\mu}\right)\,.$
(5.2.14c)

Then, we see that adding/removing for the positive/negative node is equivalent
to removing/adding for the negative/positive node if the two Chern–Simons
levels coincide $\kappa_{i}^{0}=\kappa_{i}^{1}$, as required for the
supergroup gauge invariance. Hence, (5.2.11a) and (5.2.11b) are essentially
inverse operations. The analysis above shows that, using the full
$\mathsf{Y}$-functions (5.2.12) and the matter functions (5.2.13), consisting
of both the positive and negative ones, we can apply the same argument to
construct the $qq$-character with the supergroup gauge and flavor nodes as the
ordinary (non-supergroup) gauge theory.

#####  5.2.3 Higher weight current

In general, it is possible to consider the $qq$-character starting with a
generic product of the $\mathsf{Y}$-functions as the highest weight of the
corresponding module:

$\displaystyle\mathsf{Y}_{\underline{w},\underline{x}}=\prod_{i\in\Gamma_{0}}\prod_{k=1}^{w_{i}}\mathsf{Y}_{i,x_{i,k}}$
(5.2.15)

where $\underline{w}=(w_{i})_{i\in\Gamma_{0}}\in\mathbb{Z}_{\geq
0}^{\mathop{\mathrm{rk}}\Gamma}$ are the Dynkin labels of the weight, and
$\underline{x}=(x_{i,k})_{i\in\Gamma_{0},k=1,\ldots,w_{i}}$ are the associated
parameters. Given Dynkin labels, one can construct a highest weight
representation on the quiver $\Gamma$. In particular, we call it a fundamental
representation if $w_{j}=\delta_{ij}$: Only one of the Dynkin labels is one,
and the others are zero,
$\underline{w}=(0,\ldots,0,\underbrace{1}_{i\text{-th}},0,\ldots,0)$. Then,
the corresponding $qq$-character consists of the highest weight
$\displaystyle\mathsf{Y}_{\underline{w},\underline{x}}$ with the remaining
terms generated by the iWeyl reflections:

$\displaystyle\mathsf{T}_{\underline{w},\underline{x}}=\mathsf{Y}_{\underline{w},\underline{x}}+\cdots\,.$
(5.2.16)

Let us consider the iWeyl reflection for such a generic weight. We take the
highest weight,
$\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}$.
Then, from the pole cancellation relation (5.2.2), we obtain

$\displaystyle\mathop{\mathrm{Res}}_{x\,=\,x_{i,\alpha,k}}\Bigg{[}Z\cdot\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}\Bigg{|}_{\mathcal{X}_{\text{ad}:(i,\alpha,k)}}$
$\displaystyle\hskip
50.00008pt+Z\cdot\left(\mathfrak{q}_{i}\,x^{-\kappa_{i}}P_{i,x}\widetilde{P}^{\vee}_{i,qx}\frac{\mathsf{Y}^{\vee}_{i,qx}}{\mathsf{A}_{i,x}}\right)\Bigg{|}_{\mathcal{X}}\times\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}\Bigg{|}_{\mathcal{X}_{\text{ad}:(i,\alpha,k)}}\Bigg{]}=0\,.$
(5.2.17)

One can rewrite the $\mathsf{Y}$-functions evaluated with the configuration
$\mathcal{X}_{\text{ad}:(i,\alpha,k)}$ using the relation

$\displaystyle\mathsf{Y}^{\vee}_{i,x^{\prime}}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]=\mathscr{S}\left(\frac{x_{i,\alpha,k}}{x^{\prime}}\right)\
\mathsf{Y}^{\vee}_{i,x^{\prime}}[\mathcal{X}]\stackrel{{\scriptstyle\eqref{eq:S_func_reflection}}}{{=}}\mathscr{S}\left(\frac{q^{-1}x^{\prime}}{x_{i,\alpha,k}}\right)\
\mathsf{Y}^{\vee}_{i,x^{\prime}}[\mathcal{X}]\,.$ (5.2.18)

which follows from the expression (5.1.14). Summing up the instanton
configuration, we obtain the reflection with respect to the first
$\mathsf{Y}$-function:

$\displaystyle\Big{<}\,\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}\,\Big{>}+\mathfrak{q}_{i}\,x^{-\kappa_{i}}\,P_{i,x}\widetilde{P}^{\vee}_{i,qx}\,\mathscr{S}\left(\frac{x_{1}}{x}\right)\cdots\mathscr{S}\left(\frac{x_{k}}{x}\right)\left<\,\frac{\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}}{\mathsf{A}_{i,x}}\,\right>\,.$
(5.2.19)

Applying this process recursively, the iWeyl reflection for the higher weight
is given by

$\displaystyle\text{iWeyl}:\
\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}\ \longmapsto\
\sum_{I\sqcup J=\\{1,\ldots,k\\}}\prod_{\mathsf{i}\in I,\,\mathsf{j}\in
J}\mathscr{S}\left(\frac{x_{\mathsf{i}}}{x_{\mathsf{j}}}\right)\mathsf{Y}^{\vee}_{i,qx_{1}}\cdots\mathsf{Y}^{\vee}_{i,qx_{k}}\prod_{\mathsf{j}\in
J}\mathsf{A}_{i,x_{\mathsf{j}}}^{-1}$ (5.2.20)

where we only focus on the $\mathsf{Y}$ and $\mathsf{A}$-function structure,
and omit the coupling constant and the matter polynomials, for simplicity.

#####  5.2.4 Collision limit

Let us consider the collision limit of the higher weight current. The degree-
two weight has the following iWeyl reflection structure:

$\displaystyle\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx^{\prime}}+\mathscr{S}\left(\frac{x^{\prime}}{x}\right)\frac{\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx^{\prime}}}{\mathsf{A}_{i,x}}+\mathscr{S}\left(\frac{x}{x^{\prime}}\right)\frac{\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx^{\prime}}}{\mathsf{A}_{i,x^{\prime}}}+\frac{\mathsf{Y}^{\vee}_{i,qx}\mathsf{Y}^{\vee}_{i,qx^{\prime}}}{\mathsf{A}_{i,x}\mathsf{A}_{i,x^{\prime}}}\,.$
(5.2.21)

Since $\mathscr{S}(z)$ has a pole at $z=1$ (and also at $z=q^{-1}$), we have
to carefully deal with the collision limit, $x^{\prime}\to x$. In fact, in
this limit, we have

$\displaystyle\mathsf{Y}_{i,qx}^{\vee
2}+\left(\mathfrak{c}(q_{1,2})-\frac{(1-q_{1})(1-q_{2})}{1-q}\partial_{\log
x}\log\mathsf{A}_{i,x}\right)\frac{\mathsf{Y}_{i,qx}^{\vee
2}}{\mathsf{A}_{i,x}}+\frac{\mathsf{Y}_{i,qx}^{\vee 2}}{\mathsf{A}^{2}_{i,x}}$
(5.2.22)

where the constant $\mathfrak{c}(q_{1,2})$ is given by

$\displaystyle\mathfrak{c}(q_{1,2})=\lim_{z\to
1}\left(\mathscr{S}(z)+\mathscr{S}(z^{-1})\right)=\frac{1-6q+q^{2}+(q_{1}+q_{2})(1+q)}{(1-q)^{2}}\
\xrightarrow{q_{1(2)}\to 1}\ 2\,.$ (5.2.23)

We can similarly consider the collision limit for the higher weight with
degree greater than two.

####  5.3 Classical limit

#####  5.3.1 (Very) classical limit: $\epsilon_{1,2}\to 0$

The $qq$-character has two deformation parameters $(q_{1},q_{2})$, and depends
also on the bifundamental mass parameters $(\mu_{e})_{e\in\Gamma_{1}}$. In
general (except for cyclic quivers), the bifundamental mass can be gauged away
using the $\mathrm{U}(1)$ gauge degrees of freedom of each node,555See also
the argument in §4.5.3 on the relation between the $\mathrm{U}(1)$ factor and
the bifundamental mass parameters.  so that we just put $\mu_{e}\to 1$ for the
moment. Then, in the classical limit $q_{1,2}\to 1$ ($\epsilon_{1,2}\to 0$),
there is no argument shift of $\mathsf{Y}$-functions in the $qq$-character. It
is reduced to the ordinary character of the representation associated with the
quiver [NP12], which reproduces the Seiberg–Witten geometry of $\Gamma$-quiver
gauge theory as discussed in §4.3.

###### Partition function and saddle point analysis

As shown in §1.6.3, the $\Omega$-background parameters $(q_{1},q_{2})$ are
introduced as the equivariant parameters with respect to the spacetime
rotation, and the path integral localizes on the fixed point under the
corresponding equivariant action, which regularizes the non-compactness of the
spacetime manifold (IR divergence). If the $\Omega$-background is turned off,
the partition function defined through the path integral is expected to
diverge as a consequence of the IR singularity. Nekrasov’s proposal was that,
although it diverges, the asymptotic behavior of the partition function
provides an important information, which would be identified with the
Seiberg–Witten prepotential [Nek04]:

$\displaystyle Z\ \xrightarrow{\epsilon_{1,2}\to 0}\
\exp\left(\frac{1}{\epsilon_{1}\epsilon_{2}}\mathscr{F}+\cdots\right)$ (5.3.1)

where the subleading terms are less singular in the limit $\epsilon_{1,2}\to
0$. This proposal has been then confirmed by Nekrasov–Okounkov [NO06],
Nakajima–Yoshioka [NY05], and Braverman–Etingof [BE06].

The asymptotic behavior of the partition function (5.3.1) suggests that one
can apply the saddle point analysis in the limit $\epsilon_{1,2}\to 0$ as
follows: The partition function $Z$ is given as a summation over all the
possible instanton configurations, but in the limit $\epsilon_{1,2}\to 0$, the
saddle point configuration denoted by $\mathcal{X}_{*}$ dominates in the
instanton sum:

$\displaystyle Z$
$\displaystyle=\sum_{\mathcal{X}\in\mathfrak{M}^{\mathsf{T}}}Z[\mathcal{X}]=:\sum_{\mathcal{X}\in\mathfrak{M}^{\mathsf{T}}}\exp\left(\frac{1}{\epsilon_{1}\epsilon_{2}}\mathscr{F}[\mathcal{X}]\right)\
\xrightarrow{\epsilon_{1,2}\to 0}\
\exp\left(\frac{1}{\epsilon_{1}\epsilon_{2}}\mathscr{F}[\mathcal{X}_{*}]+\cdots\right)\,.$
(5.3.2)

We see that the on-shell value of the prepotential
$\mathscr{F}[\mathcal{X}_{*}]$ derived in this way agrees with the
Seiberg–Witten prepotential. Similarly, under this limit, the expectation
value of the gauge theory observable is given by the on-shell value:

$\displaystyle\langle\,\mathcal{O}\,\rangle$
$\displaystyle=\frac{1}{Z}\sum_{\mathcal{X}\in\mathfrak{M}^{\mathsf{T}}}Z[\mathcal{X}]\,\mathcal{O}[\mathcal{X}]\
\xrightarrow{\epsilon_{1,2}\to 0}\ \mathcal{O}[\mathcal{X}_{*}]\,.$ (5.3.3)

Hence, the multi-point function is simply factorized

$\displaystyle\langle\,\mathcal{O}_{1}^{m_{1}}\cdots\mathcal{O}_{n}^{m_{n}}\,\rangle\
\longrightarrow\
\left(\mathcal{O}_{1}^{m_{1}}\cdots\mathcal{O}_{n}^{m_{n}}\right)[\mathcal{X}_{*}]\,.$
(5.3.4)

Let us discuss how to consider the saddle point analysis associated with the
instanton partition function. The partition function is given as a discrete
instanton sum, and thus the dynamical variable in this case is the discrete
$x$-variable, $\mathcal{X}=(x_{i,\alpha,k})$, corresponding to two-dimensional
partitions $\lambda=(\lambda_{i,\alpha,k})$. Recalling the definition of the
$x$-variable (2.1.15), it will behave as a continuous variable in the limit
$\epsilon_{1,2}\to 0$. Therefore, the saddle point configuration
$\mathcal{X}_{*}$ is given as a solution to the saddle point equation with
respect to the $x$-variable,

$\displaystyle x\frac{\partial\mathscr{F}[\mathcal{X}]}{\partial
x}=0\qquad\text{for}\qquad\forall x\in\mathcal{X}\,.$ (5.3.5)

We remark that the $x$-variable is multiplicative $x\in\mathbb{C}^{\times}$,
so that the derivative is taken with respect to $\log x$, $\partial_{\log
x}=x\partial_{x}$.

We then point out the relation between the saddle point equation and the iWeyl
reflection discussed in §5.2. The behavior of the partition function under the
adding-instanton operation (5.2.1) is also written as follows:

$\displaystyle\frac{Z[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z[\mathcal{X}]}$
$\displaystyle=\exp\left(\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\mathscr{F}[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]-\mathscr{F}[\mathcal{X}]\right)\right)$
$\displaystyle\xrightarrow{\epsilon_{2}\to 0}\
\exp\left(\frac{1}{\epsilon_{1}}x\frac{\partial\mathscr{F}[\mathcal{X}]}{\partial
x}\right)\,.$ (5.3.6)

We remark that the configuration $\mathcal{X}_{\text{ad}:(i,\alpha,k)}$
involves the $q_{2}$-shift of the variable, $x\to q_{2}x$. Therefore, the
saddle point equation (5.3.5) is rephrased as the invariance of the partition
function under the adding-instanton operation:

$\displaystyle\frac{Z[\mathcal{X}_{\text{ad}:(i,\alpha,k)}]}{Z[\mathcal{X}]}=1\,.$
(5.3.7)

In the limit $\epsilon_{1,2}\to 0$, the pole singularities of
$\mathsf{Y}$-function are promoted to the cut singularities, thus the saddle
point equation provides the cut-crossing relation for $\mathsf{Y}$-functions
[NP12].

#####  5.3.2 Nekrasov–Shatashvili limit: $\epsilon_{2}\to 0$

We can also consider a partially classical limit, $q_{1}=$ finite, $q_{2}\to
1$ ($\epsilon_{1}=$ finite, $\epsilon_{2}\to 0$), that is called the
Nekrasov–Shatashvili (NS) limit in the gauge theory context [NS09b]. In this
case, as shown in [NPS18], the $qq$-character is reduced to the $q$-character
of the corresponding affine Yangian/quantum affine algebra [Kni95, FR99],
which plays a central role in the connection to quantum integrable systems
(§5.7).

As well as the (very) classical case $\epsilon_{1,2}\to 0$, in the NS limit,
we turn off the $\Omega$-background parameter $\epsilon_{2}\to 0$, and the
second complex plane $\mathbb{C}_{2}$ becomes non-compact. Hence we similarly
expect the diverging behavior of the partition function

$\displaystyle Z\ \xrightarrow{\epsilon_{2}\to 0}\
\exp\left(\frac{1}{\epsilon_{2}}\widetilde{\mathscr{W}}(\epsilon_{1})+\cdots\right)\,,$
(5.3.8)

where $\widetilde{\mathscr{W}}(\epsilon_{1})$ is identified with the effective
twisted superpotential of the corresponding two-dimensional
$\mathcal{N}=(2,2)$ gauge theory on $\mathbb{C}_{2}$, and $\epsilon_{1}$ plays
a role of the adjoint mass parameter in 2d theory.666In many cases, the
adjoint mass parameter behaves as the equivariant parameter for the transverse
rotation symmetry. See §1.8.7.

Since we still have a small parameter $\epsilon_{2}$, we can apply the saddle
point analysis as well in the NS limit:

$\displaystyle
Z=:\sum_{\mathcal{X}}\exp\left(\frac{1}{\epsilon_{2}}\widetilde{\mathscr{W}}[\mathcal{X}]\right)\
\xrightarrow{\epsilon_{2}\to 0}\
\exp\left(\frac{1}{\epsilon_{2}}\widetilde{\mathscr{W}}[\mathcal{X}_{*}]+\cdots\right)\,,$
(5.3.9)

where
$\epsilon_{1}\widetilde{\mathscr{W}}[\mathcal{X}]=\mathscr{F}[\mathcal{X}]$.
Since the $x$-variable is still continuous, we can apply the same argument to
the limit $\epsilon_{1,2}\to 0$. Then, the invariance of the partition
function under the adding-instanton (5.3.7) ends up with the saddle point
equation with respect to the twisted superpotential

$\displaystyle\eqref{eq:Z/Z=1}\ \iff\
x\frac{\partial\widetilde{\mathscr{W}}[\mathcal{X}]}{\partial
x}=2\pi\mathrm{i}\,\mathbb{Z}\qquad\text{for}\qquad\forall x\in\mathcal{X}\,.$
(5.3.10)

From the 2d theory point of view, this equation provides the supersymmetric
vacuum condition, that is called the twisted F-term condition.

####  5.4 Examples

#####  5.4.1 $A_{1}$ quiver

We then focus on a specific example, 5d $\mathrm{SU}(n)$ gauge theory with
$(n^{\text{f}},n^{\text{af}})$ fundamental flavors, which is classified into
$A_{1}$ quiver. Let us discuss details of the $qq$-character in this case.

###### Fundamental character

Since $A_{1}$ quiver consists of a single gauge node, denoted by $i=1$, the
iWeyl reflection (5.2.4) is reduced to

$\displaystyle\text{iWeyl}:\ \mathsf{Y}_{1,qx}^{\vee}$
$\displaystyle\longmapsto\
\mathsf{Y}_{1,qx}^{\vee}\times\left(\mathfrak{q}_{1}\,x^{-\kappa_{1}}\frac{P_{1,x}\widetilde{P}^{\vee}_{1,qx}}{\mathsf{Y}_{1,qx}^{\vee}\mathsf{Y}_{1,x}}\right)$
$\displaystyle=\mathfrak{q}_{1}\,x^{-\kappa_{1}}\frac{P_{1,x}\widetilde{P}^{\vee}_{1,qx}}{\mathsf{Y}_{1,x}}\stackrel{{\scriptstyle\eqref{eq:YtoYdual}}}{{=}}(-1)^{n}\,\mathfrak{q}_{1}\,x^{n-\kappa_{1}}\frac{P_{1,x}\widetilde{P}^{\vee}_{1,qx}}{\mathsf{Y}_{1,x}^{\vee}}$
(5.4.1)

where we impose the special unitary condition

$\displaystyle\sum_{\alpha=1}^{n}\mathsf{a}_{\alpha}=0\,.$ (5.4.2)

In this case, there is no $\mathsf{Y}$-function in the numerator after the
reflection: No further pole singularities are generated, and the reflection
terminates here. Therefore, the fundamental $qq$-character for $\Gamma=A_{1}$
is given as follows:[Nek16]:

$\displaystyle\mathsf{T}_{1,x}=\mathsf{Y}_{1,x}^{\vee}+\mathfrak{q}_{1}\,x^{n-\kappa_{1}}\,\frac{P_{1,q^{-1}x}\widetilde{P}^{\vee}_{1,x}}{\mathsf{Y}_{1,q^{-1}x}^{\vee}}$
(5.4.3)

with the shift of the coupling constant777Since the coupling constant
$\mathfrak{q}\in\mathbb{C}^{\times}$ is given by (1.3.4) together with
(1.3.5), the sign change $\mathfrak{q}\to-\mathfrak{q}$ corresponds to the
$\theta$-angle shift, $\theta\to\theta+\pi$. This shift of the $\theta$-angle
may provide a subtle difference of the global structure [Hao20].

$\displaystyle\frac{(-1)^{n}}{q^{n-\kappa_{1}}}\mathfrak{q}_{1}\
\longrightarrow\ \mathfrak{q}_{1}\,.$ (5.4.4)

Concerning the asymptotic behavior of $\mathsf{Y}^{\vee}$-function (5.1.18),
the instanton average of the $qq$-character should be a polynomial in $x$ of
degree $n$ (a regular function), which can be written in the form:

$\displaystyle\Big{<}\,\mathsf{T}_{1,x}\,\Big{>}=\left<\,\mathsf{Y}_{1,x}^{\vee}+\mathfrak{q}_{1}\,x^{n-\kappa_{1}}\,\frac{P_{1,q^{-1}x}\widetilde{P}^{\vee}_{1,x}}{\mathsf{Y}_{1,q^{-1}x}^{\vee}}\,\right>=\det\left(1-x\,\Phi_{1}^{-1}\right)$
(5.4.5)

for $\Phi_{1}\in\mathrm{SU}(n)$.

Let us discuss the classical limit of this $qq$-character. As shown in §5.3,
the gauge theory observable is simply replaced with its on-shell value with
respect to the saddle point configuration $\mathcal{X}_{*}$, which we denote

$\displaystyle y_{x}:=\lim_{\epsilon_{1,2}\to
0}\langle\,\mathsf{Y}_{1,x}^{\vee}\,\rangle=\mathsf{Y}_{1,x}^{\vee}[\mathcal{X}_{*}]\,.$
(5.4.6)

We remark that, due to the factorization property (5.3.4), we have

$\displaystyle\left<\,\frac{1}{\mathsf{Y}_{1,q^{-1}x}^{\vee}}\,\right>\
\xrightarrow{\epsilon_{1,2}\to 0}\ \frac{1}{y_{x}}\,.$ (5.4.7)

Simply denoting $y=y_{x}$, we reproduce the Seiberg–Witten curve for 5d
$\mathrm{SU}(n)$ gauge theory with $(n^{\text{f}},n^{\text{af}})$ flavors
(4.6.3).

###### Higher character

We then consider generic higher character for $A_{1}$ quiver. Applying the
higher reflection formula (5.2.20), the degree-$w$ character is given as
follows:

$\displaystyle\mathsf{T}_{w,\underline{x}}$
$\displaystyle=\mathsf{Y}_{1,x_{1}}^{\vee}\cdots\mathsf{Y}_{1,x_{w}}^{\vee}+\cdots$
$\displaystyle=\sum_{I\sqcup J=\\{1,\ldots,w\\}}\prod_{\mathsf{i}\in
I,\,\mathsf{j}\in
J}\mathscr{S}\left(\frac{x_{\mathsf{i}}}{x_{\mathsf{j}}}\right)\prod_{\mathsf{i}\in
I}\mathsf{Y}_{1,x_{\mathsf{i}}}^{\vee}\prod_{\mathsf{j}\in
J}\mathsf{Y}_{1,q^{-1}x_{\mathsf{j}}}^{-1}\,,\quad\underline{x}=(x_{\mathsf{i}})_{\mathsf{i}=1,\ldots,w}\,,$
(5.4.8)

where we do not write the coupling constant and the matter polynomials for
simplicity. We remark that $\mathsf{A}$-function (5.2.9) is given for $A_{1}$
quiver by

$\displaystyle\mathsf{A}_{1,x}=\mathsf{Y}_{1,qx}^{\vee}\mathsf{Y}_{1,x}\,.$
(5.4.9)

For example, the degree-two current $w=2$ is given by

$\displaystyle\mathsf{T}_{(2),(x_{1},x_{2})}=\mathsf{Y}^{\vee}_{1,x_{1}}\mathsf{Y}^{\vee}_{1,x_{2}}+\mathscr{S}\left(\frac{x_{2}}{x_{1}}\right)\frac{\mathsf{Y}^{\vee}_{1,x_{2}}}{\mathsf{Y}_{1,q^{-1}x_{1}}}+\mathscr{S}\left(\frac{x_{1}}{x_{2}}\right)\frac{\mathsf{Y}^{\vee}_{1,x_{1}}}{\mathsf{Y}_{1,q^{-1}x_{2}}}+\mathsf{Y}^{-1}_{1,q^{-1}x_{1}}\mathsf{Y}^{-1}_{1,q^{-1}x_{2}}\,.$
(5.4.10)

This expression corresponds to the tensor product of two-dimensional
representations of $G_{A_{1}}=\mathrm{SL}(2)$, which is irreducible for
generic $(x_{1},x_{2})$ in the doubly quantum situation. Recalling
$\mathscr{S}$-function has zeros at $\mathscr{S}(z=q_{1,2}^{-1})=0$, one can
eliminate one of the zero weight terms in the degree-two $qq$-character, for
example:

$\displaystyle\mathsf{T}_{(2),(x,q_{1}^{-1}x)}=\mathsf{Y}^{\vee}_{1,x}\mathsf{Y}^{\vee}_{1,q_{1}^{-1}x}+\mathscr{S}\left(q_{1}\right)\frac{\mathsf{Y}^{\vee}_{1,x}}{\mathsf{Y}_{1,q_{1}^{-2}q_{2}^{-1}x}}+\mathsf{Y}^{-1}_{1,q^{-1}x}\mathsf{Y}^{-1}_{1,q_{1}^{-2}q_{2}^{-1}x}\,.$
(5.4.11)

Let us take further limit to see the relation to the three-dimensional
representation of $G_{A_{1}}=\mathrm{SL}(2)$. Since $\mathscr{S}$-function
behaves

$\displaystyle\mathscr{S}(q_{1})=\frac{(1-q_{1}^{2})(1-q)}{(1-q_{1})(1-q_{1}^{2}q_{2})}=\frac{1-q}{1-q_{1}^{2}q_{2}}(1+q_{1})\
\longrightarrow\ \begin{cases}2&(q_{1}\to 1)\\\ 1&(q_{2}\to 1)\end{cases}\,,$
(5.4.12)

we obtain two different results in the limit:888We remark that a similar
computation is also found in the twisted reduction (the root of unity limit)
of the $qq$-character [KP19b].

$\displaystyle\mathsf{T}_{(2),(x,x)}^{(q_{1}\to 1)}$
$\displaystyle=\mathsf{Y}^{\vee}_{1,x}\mathsf{Y}^{\vee}_{1,x}+2\,\frac{\mathsf{Y}^{\vee}_{1,x}}{\mathsf{Y}_{1,q_{2}^{-1}x}}+\mathsf{Y}^{-1}_{1,q_{2}^{-1}x}\mathsf{Y}^{-1}_{1,q_{2}^{-1}x}=\left(\mathsf{T}_{1,x}^{(q_{1}\to
1)}\right)^{2}\,$ (5.4.13a)
$\displaystyle\mathsf{T}_{(2),(x,q_{1}^{-1}x)}^{(q_{2}\to 1)}$
$\displaystyle=\mathsf{Y}^{\vee}_{1,x}\mathsf{Y}^{\vee}_{1,q_{1}^{-1}x}+\frac{\mathsf{Y}^{\vee}_{1,x}}{\mathsf{Y}_{1,q_{1}^{-2}x}}+\mathsf{Y}^{-1}_{1,q^{-1}x}\mathsf{Y}^{-1}_{1,q_{1}^{-2}x}=\mathsf{T}_{1,x}^{(q_{2}\to
1)}\mathsf{T}^{(q_{2}\to 1)}_{1,q_{1}^{-1}}-1\,,$ (5.4.13b)

where $\mathsf{T}_{w,\underline{x}}^{(q_{1,2}\to 1)}$ is called the
$q$-character (more precisely, $q_{2,1}$-character) of $A_{1}$ quiver theory.
In these expressions, the first limit corresponds to the tensor product,
$\scriptsize\yng(1)\otimes\yng(1)$, and the second one is the degree-two
symmetric representation,
$\scriptsize\yng(2)=\yng(1)\otimes\yng(1)-\emptyset$. In this way, the
$q$-character obeys a functional relation, called the T-system, that
originates from the tensor product of the representations associated with
quiver.999Originally the T-system was introduced as a functional relation
between the T-functions obtained from the transfer matrices of the
corresponding quantum integrable system. Afterward, it has been realized that
the T-function is identified with the associated $q$-character, and one can
interpret it as a decomposition of the tensor product between them. See a
review article for details [KNS11].  See also §7.4.10 for a related argument.

Let us briefly mention a gauge theory interpretation of the T-system. The
$q$-character is obtained in the NS limit of the $qq$-character, hence it is
interpreted as a 2d reduction of the codimension-four defect, which is namely
a codimension-two defect (also called the surface defect). See §4.5.4 for a
related argument. Since the T-system is a functional relation for the
$q$-characters, it is interpreted as a fusion rule for the surface defects in
this context. This would be a dual description of the bootstrap approach to
the surface defect based on the class $\mathcal{S}$ perspective [GRR13,
ABFH13, BFHR14].

###### Supergroup gauge theory

Let us briefly mention the $qq$-character for supergroup gauge theory. As
shown in §5.2.2, the iWeyl reflection for supergroup theory is equivalent to
the ordinary one, if we replace the $\mathsf{Y}$-function with its super
analog (5.2.12). Recalling the super $\mathsf{Y}$-function is given as a ratio
of $\mathsf{Y}^{0,1}$, the vev of the $qq$-character is not a polynomial, but
a rational function:

$\displaystyle\Big{<}\,\mathsf{T}_{1,x}\,\Big{>}=\left<\,\mathsf{Y}_{1,x}^{\vee}+\mathfrak{q}_{1}\,x^{n_{0}+n_{1}-\kappa_{1}}\,\frac{P_{1,q^{-1}x}\widetilde{P}^{\vee}_{1,x}}{\mathsf{Y}_{1,q^{-1}x}^{\vee}}\,\right>=\operatorname{sdet}\left(1-x\,\Phi_{1}^{-1}\right)$
(5.4.14)

for $\Phi_{1}\in\mathrm{SU}(n_{0}|n_{1})$. Taking the classical limit
$\epsilon_{1,2}\to 0$, this reproduces the Seiberg–Witten curve for supergroup
gauge theory discussed in §4.4.

#####  5.4.2 $A_{2}$ quiver

Let us consider $A_{2}$ quiver theory. We first rewrite the iWeyl reflection
(5.2.4) in terms of $\mathsf{Y}^{\vee}$-functions,

$\displaystyle\mathsf{Y}_{1,x}^{\vee}\ \longmapsto\ $ $\displaystyle
q^{\kappa_{1}}\mathfrak{q}_{1}\,x^{-\kappa_{1}}\,P_{1,q^{-1}x}\widetilde{P}^{\vee}_{1,x}\frac{\mathsf{Y}_{2,\mu^{-1}x}^{\vee}}{\mathsf{Y}_{1,q^{-1}x}}$
$\displaystyle=(-1)^{n_{1}}q^{\kappa_{1}-n_{1}}\mathfrak{q}_{1}\,x^{n_{1}-\kappa_{1}}\,P_{1,q^{-1}x}\widetilde{P}^{\vee}_{1,x}\frac{\mathsf{Y}_{2,\mu^{-1}x}^{\vee}}{\mathsf{Y}^{\vee}_{1,q^{-1}x}}$
$\displaystyle\xrightarrow{\eqref{eq:parameter_shift_A2}}\
\mathfrak{q}_{1}\,x^{-\kappa_{1}}\,P_{1,q^{-1}x}\widetilde{P}^{\vee}_{1,x}\frac{\mathsf{Y}_{2,\mu^{-1}x}^{\vee}}{\mathsf{Y}^{\vee}_{1,q^{-1}x}}$
(5.4.15a) $\displaystyle\mathsf{Y}_{2,x}^{\vee}\ \longmapsto\ $ $\displaystyle
q^{\kappa_{2}}\mathfrak{q}_{2}\,x^{-\kappa_{2}}\,P_{2,q^{-1}x}\widetilde{P}^{\vee}_{2,x}\frac{\mathsf{Y}_{1,\mu
q^{-1}x}}{\mathsf{Y}_{2,q^{-1}x}}$
$\displaystyle=(-1)^{n_{1}+n_{2}}q^{\kappa_{2}-n_{2}+n_{1}}\mu^{-n_{1}}\mathfrak{q}_{2}\,x^{n_{2}-n_{1}-\kappa_{2}}\,P_{2,q^{-1}x}\widetilde{P}^{\vee}_{2,x}\frac{\mathsf{Y}^{\vee}_{1,\mu
q^{-1}x}}{\mathsf{Y}^{\vee}_{2,q^{-1}x}}$
$\displaystyle\xrightarrow{\eqref{eq:parameter_shift_A2}}\
\mathfrak{q}_{2}\,x^{-\kappa_{2}}\,P_{2,q^{-1}x}\widetilde{P}^{\vee}_{2,x}\frac{\mathsf{Y}^{\vee}_{1,\mu
q^{-1}x}}{\mathsf{Y}^{\vee}_{2,q^{-1}x}}$ (5.4.15b)

where we denote the bifundamental mass by $\mu=\mu_{1\to 2}=\mu_{2\to
1}^{-1}q$, and we apply the special unitary condition (5.2.8). We also shift
the coupling constant and the Chern–Simons level
$(\mathfrak{q}_{i},\kappa_{i})_{i=1,2}$ as follows: