NITEP 144

May, 2023

Construction of irregular conformal/W block
and flavor mass relations of $\mathcal{N}=2$ SUSY gauge theory
from the $\bm{A_{n-1}}$ quiver matrix model

H. Itoyamaa,b,111e-mail<EMAIL_ADDRESS>T. Ootaa,b,222e-mail:
<EMAIL_ADDRESS>and R. Yoshiokaa,b,333e-mail<EMAIL_ADDRESS>
a Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),
Osaka Metropolitan University (formerly Osaka City University)
b Osaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan
University
aaaaaaaa 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

ABSTRACT

A sequence of massive scaling limits of the $\beta$-deformed $A_{n-1}$ quiver
matrix model that keeps the size of the matrices finite and that corresponds
to the $N_{f}=2n\rightarrow 2n-1,2n-2$ limits on the number of flavors at 4d
$su(n)$ ${\cal N}=2$ SUSY gauge theory side is carried out to provide us with
the integral representation of $su(n)$ irregular conformal/W block. The
original paths are naturally deformed into those in the complex plane,
permitting us to convert into an $su(n)$ extension of the unitary matrix model
of GWW type with a set of log potentials for all species of eigenvalues.
Looking at the region in the parameter space that enjoys the maximal symmetry
of the model, we derive a set of relations among the mass parameters which may
serve as evidence for the existence of the Argyres-Douglas critical
hypersurface.

##  1 Introduction

Matrix models have played several important roles in the development of gauge
theory, in particular, that of the ${\cal N}=2$ supersymmetric gauge theory
(for reviews, see, for example, [1, 2, 3]). “Beta” deformed matrix ensembles
derive the partition functions of such gauge theory [4] and assure the
existence of the attendant curve444by this we mean the following: the
Schwinger-Dyson eq. takes the form of Virasoro constraints acting on the
partition function. At the planar level, it implies the regularity of the
resolvent one-point function at $z=\infty$. This permits us to introduce a
singularity-free object in ${\bf C}^{2}$, namely, a curve. by the Schwinger-
Dyson equation [5, 6, 7], embodying the integrability: in the case of
$\beta=1$ (the case of free fermions), the partition function is by
construction a tau function of a certain integrable hierarchy.

Matrix models also permit us to make exploit a variety of methods as
statistical field theory (see, for example, textbooks [8]) and to probe
critical phenomena and phase structure. While largely unexplored, combination
of these two distinct disciplines appears to give us a fruitful avenue of
thought.

The case of 2d Virasoro (or CFT) block whose integral representation is
provided by one-matrix model of multi-log potential has been well-studied[9,
10, 11, 12, 13, 14, 15, 16, 17]. Much less studied are the cases of $W_{n}$
block ([18, 19, 11] and papers cited in therein) and for these cases, the
series of multi- matrix models of quiver type which obey the $W_{n}$
constraints by construction [20, 21, 22] is available.

Irregular limit [23, 24, 25, 26, 27, 28, 29, 30] of this series of models of
the multi-log type is an interesting place to look at to explore new critical
phenomena where an interplay between symmetry and phase structure ala Landau
is expected to play an essential role. In this letter, we wish to report on a
progress made in the construction of $A_{n-1}$ irregular block that is obeyed
by the $W_{n}$/Virasoro constraints and that is connected to susy gauge theory
via 0d-4d connection. We investigate broken symmetry structure of the model
enforced by the automorphism of the Dynkin diagram and its possible connection
to the critical hypersurface of the system, neither of which has been
uncovered before despite the well-known notion of the singularities of the
Argyres-Douglas type [31, 32, 33, 34]; these in principle can be obtained from
the Seiberg-Witten curve perse555QFT perse does not know the construction of
the Seiberg-Witten curve itself, which requires the introduction of ambient
space ${\bf C}^{2}$. The matrix model arising from the above-mentioned
2d-0d-4d connection realizes such curve as an eigenvalue distribution, which
is eventually recognized as a discontinuity of multi-sheeted complex plane.
The matrix model not only provides a construction of the curve but also
permits us to compute various observables represented by (multi-point)
resolvent correlators and to consider the problem of phases.. We see that
imposing the maximal symmetry leads us to a set of flavor mass relations.

In the series of developments on the irregular limit of the $A_{1}$(one-
matrix) case [35, 36, 37, 38, 39, 40, 41], we have been led to the procedure
of converting the hermitean matrix model into the unitary matrix model in
order to make such procedure well-defined. The upshot from the case of
$N_{f}=2$ is the well-known GWW model [42, 43, 44, 45] (see also [46, 47, 48,
49] for recent discussion) augmented by the log potential. The Painlevé II
equation with parameter has been derived in the double scaling limit [35, 36,
37]. Let us note that the procedure is a conversion process that keeps the
size of the matrix finite and is not just a simple universality argument [50].
Such “unitarization procedure” turns out to be responsible for making the
above mentioned maximal symmetry manifest.

In the next section, the $\beta$-deformed ADE quiver matrix model of three-log
potential is briefly recalled. In section three, we obtain $A_{n-1}$ irregular
conformal/W block that corresponds to $su(n)$, $N_{f}=2n-1,2n-2$, generalizing
the work of [26, 35, 36, 37, 38, 40]. In section four, after recasting the
model into the unitary type, we observe the region in the parameter space
which enjoys the maximal symmetry enhancement of the model. In this region, we
derive a set of interesting relations among the $2n-2$ flavor masses.
Furthermore, we can show that the Argyres-Douglas hypersurface lies within
this region of parameter space from Seiberg-Witten curve.

##  2 $\bm{\beta}$-deformed ADE quiver matrix model of three-log potential

Let us briefly recall the $\beta$-deformed ADE quiver matrix model
($\beta=-b^{2}$):

$Z\equiv\int\cdots\int\prod_{a=1}^{r}\left\\{\prod_{I=1}^{N_{a}}\mathrm{d}\lambda^{(a)}_{I}\right\\}\Bigl{(}\Delta_{\mathfrak{g}}(\lambda)\Bigr{)}^{-b^{2}}\exp\left(-\frac{\mathrm{i}b}{g_{s}}\sum_{a=1}^{r}\sum_{I=1}^{N_{a}}W_{a}(\lambda^{(a)}_{I})\right),$
(2.1)

where $W_{a}$ is a potential and

$\Delta_{\mathfrak{g}}(\lambda)=\prod_{a=1}^{r}\prod_{1\leq I<J\leq
N_{a}}(\lambda_{I}^{(a)}-\lambda_{J}^{(a)})^{2}\prod_{1\leq a<b\leq
r}\prod_{I=1}^{N_{a}}\prod_{J=1}^{N_{b}}(\lambda^{(a)}_{I}-\lambda^{(b)}_{J})^{(\bm{\alpha}_{a},\bm{\alpha}_{b})}.$
(2.2)

Here, $\mathfrak{g}$ is a finite dimensional Lie algebra of ADE type with rank
$r$, $\mathfrak{h}$ the Cartan subalgebra of $\mathfrak{g}$ and
$\mathfrak{h}^{*}$ its dual. We denote by $\bm{\alpha}_{a}\in\mathfrak{h}^{*}$
($a=1,2,\cdots,r$) the simple roots of $\mathfrak{g}$, $(\bullet,\bullet)$
being the inner product on $\mathfrak{h}^{*}$, which we normalize as
$(\bm{\alpha}_{a},\bm{\alpha}_{a})=2$.

For an application to the Virasoro/W block and the 2d-4d connection through
matrices, we choose the potential to be the three-Penner type. From now on, we
restrict our attention to the Lie algebra of $A_{n-1}$ type. Following [11],
let us introduce

$W_{a}(z)=\sum_{p=1}^{3}\left(\bm{\mu}_{p},\bm{\alpha}_{a}\right)\log(q_{p}-z),$
(2.3)

with $q_{1}=0$, $q_{2}=1$, $q_{3}=q$.

Eq.(2.1) represents a generic 4 point block of $su(n)$ Toda theory. Such block
has been originally introduced by $n-1$ independent free scalar fields in the
screening charge formalism, which is subsequently evaluated by the Wick
contractions to give rise to eq.(2.1). The $n-1$ species of screening charges
with arbitrary numbers $N_{1},N_{2},\cdots$, and $N_{n-1}$ give us the
momentum conservation:

$\bm{\mu}_{0}+\sum_{p=1}^{3}\bm{\mu}_{p}+\sum_{a=1}^{n-1}{S}_{a}\bm{\alpha}_{a}=0,$
(2.4)

where ${S}_{a}\equiv-\mathrm{i}bg_{s}N_{a}$. Following the recipe, we choose

$\bm{\mu}_{2}={\mu}_{2,1}\bm{\Lambda}^{1},~{}~{}~{}\bm{\mu}_{3}={\mu}_{3,n-1}\bm{\Lambda}^{n-1},$
(2.5)

in accordance with the “simple” puncture and $\bm{\mu}_{0}$, $\bm{\mu}_{1}$ to
be a generic type written in the bases of the fundamental weights
$\bm{\Lambda}^{a}$:

$\bm{\mu}_{p}=\sum_{a}\mu_{p,a}\bm{\Lambda}^{a}.$ (2.6)

##  3 Limit to irregular block: $\bm{su(3)}$ and $\bm{N_{f}=6\to 5,4}$ and the
$\bm{su(n)}$ extension

Let us first consider the sequence of limits of the model leading to the
irregular conformal block, taking the case of $n=3$, $N_{f}=5,4$.

In [11], the 0d-4d dictionary has been given already at $n=3,N_{f}=6$ by
comparing the Witten-Gaiotto curve with the matrix model curve:

$\begin{split}(\bm{\mu}_{0},\bm{\alpha}_{1})=-m_{1}+m_{2},~{}~{}~{}(\bm{\mu}_{1},\bm{\alpha}_{1})=m_{4}-m_{5},~{}~{}~{}(\bm{\mu}_{2},\bm{\alpha}_{1})=m_{1}+m_{2}+m_{3},~{}~{}~{}(\bm{\mu}_{3},\bm{\alpha}_{1})=0,\\\
(\bm{\mu}_{0},\bm{\alpha}_{2})=-m_{2}+m_{3},~{}~{}~{}(\bm{\mu}_{1},\bm{\alpha}_{2})=m_{5}-m_{6},~{}~{}~{}(\bm{\mu}_{2},\bm{\alpha}_{2})=0,~{}~{}~{}(\bm{\mu}_{3},\bm{\alpha}_{2})=m_{4}+m_{5}+m_{6}.\\\
\end{split}$ (3.1)

While this is at the planar level, it is adequate to tell that the appropriate
decoupling limits from $N_{f}=6$ to 5 and subsequently to 4 are obtained by
$m_{6}\to\infty$ and subsequently by $m_{1}\to\infty$. Other choices instead
of $m_{1}$ and $m_{6}$, such as $m_{2}$ and $m_{3}$, involve $N_{a}\to\infty$
limit (see eq. (4.3)), which spoils finiteness of a partial sum of masses
according to the 0d-4d dictionary. This is why we do not consider the further
limits $N_{f}=3,2,\cdots$ in this letter.

Following the recipe of [15] for the $su(2)$ case, we specify the integration
contours for $su(3)$, $N_{f}=6$ case to consist of $C_{L}=[0,q]$ and
$C_{R}=[1,\infty]$. We adopt the following obvious generalization for
$\mathrm{d}w_{I}^{a=1}{}_{I=1,\cdots,N_{1}}$ and for
$\mathrm{d}w_{J}^{a=2}{}_{J=1,\cdots,N_{2}}$:

$C_{I,a}^{(6)}=\begin{cases}C_{L}~{}~{}~{}\text{for }1\leq I\leq
N_{L}^{a},~{}a=1,2,\\\ C_{R}~{}~{}~{}\text{for }N_{L}^{a}+1\leq I\leq
N_{a}=N_{L}^{a}+N_{R}^{a},~{}a=1,2.\end{cases}$ (3.2)

The splitting of $N_{a}$ into $N_{L}^{a}$ and $N_{R}^{a}$ is related to the
Coulomb moduli parameters in the corresponding gauge theory and they remain
fixed in the limits $m_{1}\to\infty$ and $m_{6}\to\infty$.

Thus our partition function for $su(n=3)$ $N_{f}=6$ flavor case reads

$\displaystyle Z_{A_{n-1}=A_{2}}^{(N_{f}=6)}=\text{(const)}$
$\displaystyle\int\cdots\int\prod_{a=1}^{n-1=2}\left\\{\prod_{I=1}^{N_{a}}\mathrm{d}w_{I}^{(a)}\right\\}_{C_{I,a}^{(6)}}\Delta_{A_{n-1=2}}^{\beta}(w)$
$\displaystyle\prod_{I=1}^{N_{1}}|w_{I}^{(1)}|^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{1})}|1-w_{I}^{(1)}|^{\sqrt{\beta}(\bm{\mu}_{2},\bm{\alpha}_{1})}\prod_{J=1}^{N_{2}}|w_{J}^{(2)}|^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{2})}|q-w_{J}^{(2)}|^{\sqrt{\beta}(\bm{\mu}_{3},\bm{\alpha}_{2})}.$
(3.3)

We now turn to the procedure of taking the $N_{f}=6\to 5$ limit and
subsequently the $N_{f}=5\to 4$ limit. To handle infinities associated with
these limits, it is necessary to deform the original paths on the real axis at
$N_{f}=6$ to contours on the complex plane. In the case of the one-matrix
model, we have devised and resolved this technical problem in [26, 36]. Here
in the case of multi-matrix model, we adopt the same procedure. For detail, we
ask the reader interested to look at appendix F.2 and F.3 in [36]. Our
technique is, in fact, best summarized as the use of the following elementary
integrations:

$\displaystyle\frac{1}{(\mathrm{e}^{2\pi\mathrm{i}\alpha}-1)}\int_{C(q)}\mathrm{d}ww^{\alpha}\left(\sum_{n=0}^{\infty}c_{n}w^{n}\right)=\int_{0}^{q}\mathrm{d}ww^{\alpha}\left(\sum_{n=0}^{\infty}c_{n}w^{n}\right),~{}~{}~{}\text{for
}\mathrm{Re}\,\alpha>0,$ (3.4)
$\displaystyle\frac{1}{(1-\mathrm{e}^{2\pi\mathrm{i}\alpha})}\int_{C(1)}\mathrm{d}ww^{\alpha}\left(\sum_{n=0}^{\infty}c_{n}w^{-n}\right)=\int_{1}^{\infty}\mathrm{d}ww^{\alpha}\left(\sum_{n=0}^{\infty}c_{n}w^{-n}\right),~{}~{}~{}\text{for
}\mathrm{Re}\,\alpha<-1,$ (3.5)

where the contours $C(q)$ and $C(1)$ are depicted in Figure 1.

Figure 1: The integration contour $C(r)$.

The $N_{f}=6\to 5$ limit corresponds to
$\alpha=\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{2})\to\infty$ and
$\left(\sum_{n=0}^{\infty}c_{n}w^{n}\right)$ being essentially the expansion
of $(q-w)^{b}$ in (3.4), and the infinity associated in the right hand side
has been factorized in the left-hand side. This leads us to the expression (3)
in what follows. A similar procedure holds for the $N_{f}=5\to 4$ limit that
corresponds to $\alpha=\sqrt{\beta}(\bm{\mu}_{2},\bm{\alpha}_{1})\to\infty$
and $\left(\sum_{n=0}^{\infty}c_{n}w^{n}\right)$ being essentially the
expansion of $(1-1/w)^{a}$ in (3.5). This leads us to the expression (3.12) in
what follows.

The $N_{f}=6\to 5$ limit is specified by the following: $m_{6}\to\infty$,
$q\to 0$ with $\Lambda_{5}=\lim qm_{6}$ (up to constant) kept finite. Let us
introduce

$q_{05}\equiv\lim q(\bm{\mu}_{3},\bm{\alpha}_{2}),$ (3.6)

so that $q_{05}=\Lambda_{5}/g_{s}$ (up to constant).

The resulting partition function for $N_{f}=5$ is

$\displaystyle Z_{A_{2}}^{(N_{f}=5)}=\text{(const)}$
$\displaystyle\int\\!\\!\cdots\\!\\!\int\prod_{a=1}^{2}\left\\{\prod_{I=1}^{N_{a}}\mathrm{d}w_{I}^{(a)}\right\\}_{C_{I,a}^{(N_{f}=5)}}\Delta_{A_{2}}^{\beta}(w)\times$
$\displaystyle\times\prod_{I=1}^{N_{1}}(w_{I}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{1})}(1-w_{I}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{2},\bm{\alpha}_{1})}\prod_{J=1}^{N_{2}}(w_{J}^{(2)})^{\sqrt{\beta}(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{2})}\exp\left(-\sqrt{\beta}\frac{q_{05}}{w_{J}^{(2)}}\right),$
(3.7)

the potential being

$\displaystyle W_{1}^{(N_{f}=5)}(z)=(\bm{\mu}_{1},\bm{\alpha}_{1})\log
z+(\bm{\mu}_{2},\bm{\alpha}_{2})\log(1-z),$ (3.8) $\displaystyle
W_{2}^{(N_{f}=5)}(z)=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{2})\log
z-\frac{q_{05}}{z}.$ (3.9)

As for the complex contours $C_{I,a}^{(N_{f}=5)}$,

$C_{I,a}^{(N_{f}=5)}=\left\\{\\!\\!\begin{array}[]{ll}C_{0}(r_{0})&\text{for
}1\leq I\leq N_{L}^{a},~{}~{}~{}a=1,2,\\\ C(1)&\text{for }N_{L}^{a}\leq I\leq
N_{L}^{a}+N_{R}^{a},~{}~{}~{}a=1,2.\end{array}\right.$ (3.10)

$C_{0}(r_{0})$ is depicted in Figure 2. The radius of the arc is $r_{0}$,
which is an arbitrary real number satisfying $0<q<r_{0}<1$. We have $w>0$,
$1-w>0$ and $q-w>0$ for $w\in C_{L}$, and $w>0$, $1-w<0$ and $q-w<0$ for $w\in
C_{R}$ in the integrand of (3). Since the sign of the quantities inside the
modulus remain constant on the integration contour, the absolute value have
been dropped in (3).

Figure 2: The integration contour $C_{0}(r_{0})$.

Next, let us take the subsequent limit $N_{f}=5\to 4$: $m_{1}\to\infty$,
$q_{05}\to 0$ with $\Lambda_{4}\equiv\lim(m_{1}\Lambda)^{1/2}$ (up to
constant) kept finite.

We introduce

$q_{04}^{2}\equiv\lim q_{05}(\bm{\mu}_{2},\bm{\alpha}_{1}),$ (3.11)

so that $q_{04}=\Lambda_{4}/g_{s}$ up to some constant.

The resulting partition function for $N_{f}=4$ is

$\displaystyle Z_{A_{2}}^{(N_{f}=4)}=\text{(const)}$
$\displaystyle\int\\!\\!\cdots\\!\\!\int\prod_{a=1}^{2}\left\\{\prod_{I=1}^{N_{a}}\mathrm{d}w_{I}^{(a)}\right\\}_{C_{I,a}^{(N_{f}=4)}}\Delta_{A_{2}}^{\beta}(w)\times$
$\displaystyle\times\prod_{I=1}^{N_{1}}(w_{I}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{1})}\exp\left(-\sqrt{\beta}q_{04}w_{I}^{(1)}\right)\prod_{J=1}^{N_{2}}(w_{J}^{(2)})^{\sqrt{\beta}(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{2})}\exp\left(-\sqrt{\beta}\frac{q_{04}}{w_{J}^{(2)}}\right),$
(3.12)

the potential being

$\displaystyle W_{1}^{(N_{f}=4)}(z)=(\bm{\mu}_{1},\bm{\alpha}_{1})\log
z-q_{04}z,$ (3.13) $\displaystyle
W_{2}^{(N_{f}=4)}(z)=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{2})\log
z-\frac{q_{04}}{z}.$ (3.14)

As for the complex contours $C_{I,a}^{(N_{f}=4)}$,

$C_{I,a}^{(N_{f}=4)}=\left\\{\\!\\!\begin{array}[]{ll}C_{0}=C_{0}(1)&\text{for
}1\leq I\leq N_{L}^{a},~{}~{}~{}a=1,2,\\\ C_{\infty}&\text{for }N_{L}^{a}\leq
I\leq N_{L}^{a}+N_{R}^{a},~{}~{}~{}a=1,2,\end{array}\right.$ (3.15)

where $C_{\infty}$ is depicted in Figure 3.

Figure 3: The integration contour $C_{\infty}$.

It is completely straightforward to generalize the above discussion to
$A_{n-1}$ quiver matrix model for $n\geq 4$. The starting point is the case
$N_{f}=2n$:

$\displaystyle
Z_{A_{n-1}}^{(N_{f}=2n)}=(\text{const})\int\\!\\!\cdots\\!\\!\int$
$\displaystyle\prod_{a=1}^{n-1}\left\\{\prod_{I_{a}=1}^{N_{a}}\mathrm{d}w_{I_{a}}^{(a)}\right\\}_{C_{I,a}^{(N_{f}=2n)}}\left(\Delta_{A_{n-1}}(w)\right)^{\beta}\times$
$\displaystyle\times\left(\prod_{I_{1}=1}^{N_{1}}(w_{I_{1}}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{1})}(1-w_{I_{1}}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{2},\bm{\alpha}_{1})}\right)\prod_{a=2}^{n-2}\prod_{I_{a}=1}^{N_{a}}(w_{I_{a}}^{(a)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{a})}\times$
$\displaystyle\times\left(\prod_{I_{n-1}=1}^{N_{n-1}}(w_{I_{n-1}}^{(n-1)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{n-1})}(q-w_{I_{n-1}}^{(n-1)})^{\sqrt{\beta}(\bm{\mu}_{3},\bm{\alpha}_{n-1})}\right).$
(3.16)

After the limit $N_{f}=2n\to 2n-1$, we obtain

$\displaystyle
Z_{A_{n-1}}^{(N_{f}=2n-1)}=(\text{const})\int\\!\\!\cdots\\!\\!\int$
$\displaystyle\prod_{a=1}^{n-1}\left\\{\prod_{I_{a}=1}^{N_{a}}\mathrm{d}w_{I_{a}}^{(a)}\right\\}_{C_{I,a}^{(N_{f}=2n-1)}}\left(\Delta_{A_{n-1}}(w)\right)^{\beta}\times$
$\displaystyle\times\left(\prod_{I_{1}=1}^{N_{1}}(w_{I_{1}}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{1})}(1-w_{I_{1}}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{2},\bm{\alpha}_{1})}\right)\prod_{a=2}^{n-2}\prod_{I_{a}=1}^{N_{a}}(w_{I_{a}}^{(a)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{a})}\times$
$\displaystyle\times\left(\prod_{I_{n-1}=1}^{N_{n-1}}(w_{I_{n-1}}^{(n-1)})^{\sqrt{\beta}(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{n-1})}\exp\left(-\sqrt{\beta}\frac{q_{05}}{w_{I_{n-1}}^{(n-1)}}\right)\right),$
(3.17)

where $W_{1}(z)$, $W_{2}(z)$, $a=2\cdots,n-2$ are the same as those at
$N_{f}=2n$ and

$\displaystyle W_{n-1}(z)=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{n-1})\log
z-\frac{q_{05}}{z},$ (3.18) $\displaystyle q_{05}\equiv\lim
q(\bm{\mu}_{3},\bm{\alpha}_{n-1}).$ (3.19)

After the limit $N_{f}=2n-1\to 2n-2$, we obtain

$\displaystyle
Z_{A_{n-1}}^{(N_{f}=2n-2)}=(\text{const})\int\\!\\!\cdots\\!\\!\int$
$\displaystyle\prod_{a=1}^{n-1}\left\\{\prod_{I_{a}=1}^{N_{a}}\mathrm{d}w_{I_{a}}^{(a)}\right\\}_{C_{I,a}^{(N_{f}=2n-2)}}\left(\Delta_{A_{n-1}}(w)\right)^{\beta}\times$
$\displaystyle\times\left(\prod_{I_{1}=1}^{N_{1}}(w_{I_{1}}^{(1)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{1})}\exp\left(-\sqrt{\beta}{q_{04}}{w_{I_{1}}^{(1)}}\right)\right)\prod_{a=2}^{n-2}\prod_{I_{a}=1}^{N_{a}}(w_{I_{a}}^{(a)})^{\sqrt{\beta}(\bm{\mu}_{1},\bm{\alpha}_{a})}\times$
$\displaystyle\times\left(\prod_{I_{n-1}=1}^{N_{n-1}}(w_{I_{n-1}}^{(n-1)})^{\sqrt{\beta}(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{n-1})}\exp\left(-\sqrt{\beta}\frac{q_{04}}{w_{I_{n-1}}^{(n-1)}}\right)\right),$
(3.20)

where $q_{04}^{2}=\lim q_{05}(\bm{\mu}_{2},\bm{\alpha}_{1})$. The potentials
are

$\begin{split}&W_{1}(w^{(1)})=(\bm{\mu}_{1},\bm{\alpha}_{1})\log
w^{(1)}-q_{04}w^{(1)},\\\ &W_{a}(w^{(a)})=(\bm{\mu}_{1},\bm{\alpha}_{a})\log
w^{(a)},~{}~{}~{}~{}~{}~{}~{}~{}~{}a=2,\cdots,n-2,\\\
&W_{n-1}(w^{(n-1)})=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{n-1})\log
w^{(n-1)}-\frac{q_{04}}{w^{(n-1)}}.\end{split}$ (3.21)

In this form, the (broken) symmetry of the model based on the even parity of
the potential and the automorphism of the Dynkin diagram is still not
manifest.

The 0d-4d dictionary worked out in [11] for the $A_{n-1}$ case reads

$\begin{split}&\bm{\mu}_{0}=\sum_{a=1}^{n-1}(-m_{a}+m_{a+1})\bm{\Lambda}^{a},~{}~{}~{}\bm{\mu}_{1}=\sum_{a=1}^{n-1}(m_{n+a}-m_{n+a+1})\bm{\Lambda}^{a},\\\
&\bm{\mu}_{2}=\left(\sum_{i=1}^{n}m_{i}\right)\bm{\Lambda}^{1},~{}~{}~{}\bm{\mu}_{3}=\left(\sum_{i=1}^{n}m_{i+n}\right)\bm{\Lambda}^{n-1}.\end{split}$
(3.22)

Further limit to $N_{f}=2n-3$ and the lower ones are not available at the
current approach by the same reason as was mentioned before at (3.1).

##  4 The coefficients after “unitarization”

As was stated in the introduction, we would like to recast the above $A_{n-1}$
quiver matrix model for the irregular Virasoro/W block into the unitary form:

$\displaystyle
Z_{A_{n-1},U}^{N_{f}=2n-2}\equiv\int\\!\\!\cdots\\!\\!\int\prod_{a=1}^{n-1}\left\\{\prod_{I_{a}=1}^{N_{a}}\frac{\mathrm{d}w_{I_{a}}^{(a)}}{w_{I_{a}}}\right\\}_{C_{I,a}^{N_{f}=2n-2}}\left(\Delta_{A_{n-1}}(w)\right)^{\frac{\beta}{2}}\left(\Delta_{A_{n-1}}\left(\frac{1}{w}\right)\right)^{\frac{\beta}{2}}\times$
$\displaystyle\times\prod_{I_{1}=1}^{N_{1}}(w_{I_{1}}^{(1)})^{\sqrt{\beta}c_{1}}\exp\left(-\sqrt{\beta}q_{04}w_{I_{1}}^{(1)}\right)\prod_{a=2}^{n-2}\prod_{I_{a}=1}^{N_{a}}(w_{I_{a}}^{(a)})^{\sqrt{\beta}c_{a}}\prod_{I_{n-1}=1}^{N_{n-1}}(w_{I_{n}-1}^{(n-1)})^{\sqrt{\beta}c_{n-1}}\exp\left(-\sqrt{\beta}\frac{q_{04}}{w_{I_{n-1}}^{(n-1)}}\right).$
(4.1)

The potentials have the following form:

$\begin{split}&W_{U,1}(w^{(1)})=c_{1}\log w^{(1)}-q_{04}w^{(1)},\\\
&W_{U,a}(w^{(a)})=c_{a}\log
w^{(a)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a=2,\cdots,n-2,\\\
&W_{U,n-1}(w^{(n-1)})=c_{n-1}\log
w^{(n-1)}-\frac{q_{04}}{w^{(n-1)}}.\end{split}$ (4.2)

Let us turn to the evaluation of $c_{a}$. Recasting the partition function in
the form (4) brings the extra contribution to the potential, from which we
deduce the coefficients $c_{a}$. For definiteness, let us again first consider
the case of $su(3)$. Combining (2.4) with (3.1), we can readily obtain

$-\mathrm{i}bg_{s}N_{1}={S}_{1}=-m_{2}-m_{3}-m_{4},~{}~{}~{}~{}~{}-\mathrm{i}bg_{s}N_{2}={S}_{2}=-m_{3}-m_{4}-m_{5}.$
(4.3)

Hence,

$\displaystyle c_{1}$
$\displaystyle=(\bm{\mu}_{1},\bm{\alpha}_{1})+{S}_{1}+\frac{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{2}{S}_{2}$
$\displaystyle=(m_{4}-m_{5})-(m_{2}+m_{3}+m_{4})+\frac{1}{2}(m_{3}+m_{4}+m_{5})$
$\displaystyle=-\left(m_{2}-m_{4}+\frac{1}{2}(m_{3}+m_{4}+m_{5})\right),$
(4.4) $\displaystyle c_{2}$
$\displaystyle=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{2})+\frac{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{2}{S}_{1}+{S}_{2}$
$\displaystyle=(m_{3+2}-m_{3+2+1})+\sum_{i=1}^{3}m_{i+3}+\frac{1}{2}(m_{2}+m_{3}+m_{4})-(m_{3}+m_{4}+m_{5})$
$\displaystyle=m_{5}-m_{3}+\frac{1}{2}(m_{2}+m_{3}+m_{4}).$ (4.5)

Let us define $\check{c}_{a}$ by replacing $\bm{\mu}_{1}+\bm{\mu}_{3}$ in
$c_{a}$ with $\bm{\mu}_{0}+\bm{\mu}_{2}$. Note that
$(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{1})=(\bm{\mu}_{1},\bm{\alpha}_{1})$.
We observe that

$\displaystyle\check{c}_{1}$
$\displaystyle\equiv(\bm{\mu}_{0}+\bm{\mu}_{2},\bm{\alpha}_{1})+{S}_{1}+\frac{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{2}{S}_{2}$
$\displaystyle=-m_{1}+m_{2}+\sum_{i=1}^{3}m_{i}-(m_{2}+m_{3}+m_{4})+\frac{1}{2}(m_{3}+m_{4}+m_{5})$
$\displaystyle=m_{2}-m_{4}+\frac{1}{2}(m_{3}+m_{4}+m_{5})=-c_{1}.$ (4.6)

This is actually a relation valid for arbitrary $n$:

$\displaystyle\check{c}_{1}+c_{1}$
$\displaystyle=(\bm{\mu}_{0}+\bm{\mu}_{2}+\bm{{\mu}_{1},\bm{\alpha}_{1}})+2{S}_{1}+{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{S}_{2}$
$\displaystyle\underset{\eqref{mom-
con}}{=}-(\bm{\mu}_{3}+{S}_{1}\bm{\alpha}_{1}+{S}_{2}\bm{\alpha}_{2},\bm{\alpha}_{1})+2{S}_{1}+{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{S}_{2}$
$\displaystyle\underset{\eqref{dictionary:n=3}}{=}0.$ (4.7)

So, by the “unitarization” procedure, we have restored the left-right symmetry
of the original irregular 4 point block, namely, the symmetry under
$\bm{\mu}_{0}+\bm{\mu}_{2}\leftrightarrow\bm{\mu}_{1}+\bm{\mu}_{3}$ which has
become invisible by the $SL(2,\mathbb{C})$ fixing. Let us note that, in the
known case $su(2)$, $N_{1}=-m_{2}-m_{3}$ and

$\displaystyle\check{c}_{1}=-m_{1}+m_{2}+\sum_{i=1}^{2}m_{i}+{S}_{1}=m_{2}-m_{3},$
(4.8) $\displaystyle
c_{n-1}=c_{1}=m_{3}-m_{4}+\sum_{i=1}^{2}m_{i+2}+{S}_{1}=m_{3}-m_{2}.$ (4.9)

One can proceed further to work out the coefficients
$c_{1}=-\check{c}_{1},c_{2},\cdots,c_{n-1}$ for the general case $su(n)$ from
$N_{1},\cdots,N_{n-1}$ and the dictionary (3.22). Note that eq.(2.4) is
inverted to give

$\begin{pmatrix}{S}_{1}\\\ {S}_{2}\\\ \vdots\\\
{S}_{n-1}\end{pmatrix}=-\mathcal{C}^{-1}\begin{pmatrix}(\bm{\alpha}_{1},\sum_{i}\hat{\bm{\mu}}_{i})\\\
(\bm{\alpha}_{2},\sum_{i}\hat{\bm{\mu}}_{i})\\\ \vdots\\\
(\bm{\alpha}_{n-1},\sum_{i}\hat{\bm{\mu}}_{i})\end{pmatrix},$ (4.10)

where

$\mathcal{C}^{-1}_{a,a^{\prime}}=(\bm{\Lambda}_{a},\bm{\Lambda}_{a^{\prime}})=\frac{1}{n}\min(a,a^{\prime})\\{n-\max(a,a^{\prime})\\},$
(4.11)

is the inverse of the Cartan matrix.

For $n=4$, we obtain

${S}_{1}=-(m_{2}+m_{3}+m_{4}+m_{5}),~{}~{}~{}{S}_{2}=-(m_{3}+m_{4}+m_{5}+m_{6}),~{}~{}~{}{S}_{3}=-(m_{4}+m_{5}+m_{6}+m_{7}),$
(4.12) $\displaystyle\check{c}_{1}$
$\displaystyle=m_{2}-m_{5}+\frac{1}{2}(m_{3}+m_{4}+m_{5}+m_{6})$
$\displaystyle=m_{2}+\frac{1}{2}(m_{3}+m_{4}-m_{5}+m_{6}),$ (4.13)
$\displaystyle\check{c}_{2}$
$\displaystyle\equiv(\bm{\mu}_{0}+\bm{\mu}_{2},\bm{\alpha}_{2})-\frac{1}{2}{S}_{1}+{S}_{2}-\frac{1}{2}{S}_{3}$
$\displaystyle=-m_{2}-m_{6}+\frac{1}{2}(m_{2}+m_{3})+\frac{1}{2}(m_{6}+m_{7})$
$\displaystyle=\frac{1}{2}(-m_{2}+m_{7})+\frac{1}{2}(m_{3}-m_{6}),$ (4.14)
$\displaystyle c_{3}$
$\displaystyle=m_{7}-m_{4}+\frac{1}{2}(m_{3}+m_{4}+m_{5}+m_{6})$
$\displaystyle=m_{7}+\frac{1}{2}(m_{3}-m_{4}+m_{5}+m_{6}).$ (4.15)

For general $n$,

$\displaystyle c_{1}$
$\displaystyle=(\bm{\mu}_{1},\bm{\alpha}_{1})+{S}_{1}+\frac{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{2}{S}_{2},$
(4.16) $\displaystyle\vdots$ $\displaystyle c_{a}$
$\displaystyle=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{a})+\frac{(\bm{\alpha}_{a},\bm{\alpha}_{a-1})}{2}{S}_{a-1}+{S}_{a}+\frac{(\bm{\alpha}_{a},\bm{\alpha}_{a+1})}{2}{S}_{a+1},$
$\displaystyle~{}~{}~{}~{}~{}\text{with
}(\bm{\mu}_{3},\bm{\alpha}_{a})=0,~{}~{}~{}a=2,\cdots,n-2,$ (4.17)
$\displaystyle\vdots$ $\displaystyle c_{n-1}$
$\displaystyle=(\bm{\mu}_{1}+\bm{\mu}_{3},\bm{\alpha}_{n-1})+\frac{(\bm{\alpha}_{n-1},\bm{\alpha}_{n-2})}{2}{S}_{n-2}+{S}_{n-1}.$
(4.18)

Let

$\displaystyle\check{c}_{1}$
$\displaystyle\equiv(\bm{\mu}_{0}+\bm{\mu}_{2},\bm{\alpha}_{1})+{S}_{1}+\frac{(\bm{\alpha}_{1},\bm{\alpha}_{2})}{2}{S}_{2},$
(4.19) $\displaystyle\vdots$ $\displaystyle\check{c}_{a}$
$\displaystyle\equiv(\bm{\mu}_{0}+\bm{\mu}_{2},\bm{\alpha}_{a})+\frac{(\bm{\alpha}_{a},\bm{\alpha}_{a-1})}{2}{S}_{a-1}+{S}_{a}+\frac{(\bm{\alpha}_{a},\bm{\alpha}_{a+1})}{2}{S}_{a+1},,~{}~{}~{}a=2,\cdots,n-2,$
(4.20) $\displaystyle\vdots$ $\displaystyle\check{c}_{n-1}$
$\displaystyle\equiv(\bm{\mu}_{0}+\bm{\mu}_{2},\bm{\alpha}_{n-1})+\frac{(\bm{\alpha}_{n-1},\bm{\alpha}_{n-2})}{2}{S}_{n-2}+{S}_{n-1}.$
(4.21)

We have checked

$c_{a}=-\check{c}_{a},~{}~{}~{}~{}~{}\text{for }a=1,\cdots,n-1,$ (4.22)

namely, that $c_{a}$ has a dual expression $-\check{c}_{a}$. We obtain

$\displaystyle{S}_{1}=-(m_{2}+\cdots+m_{n+1}),~{}~{}~{}{S}_{2}=-(m_{3}+\cdots+m_{n+2}),~{}~{}~{}\cdots,~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}{S}_{n-2}=-(m_{n-1}+\cdots+m_{2n-2}),~{}~{}~{}{S}_{n-1}=-(m_{n}+\cdots+m_{2n-1}),$
(4.23)

and

$\displaystyle\check{c}_{1}$
$\displaystyle=m_{2}-m_{n+1}+\frac{1}{2}\sum_{i=1}^{n}m_{2+i},$ (4.24)
$\displaystyle\vdots$ $\displaystyle\check{c}_{a}$
$\displaystyle=(-m_{a}+m_{a+1})+\frac{1}{2}(m_{a}-m_{a+1})-\frac{1}{2}(m_{a+n}-m_{a+n+1})$
$\displaystyle=-\frac{1}{2}(m_{a}-m_{a+1})-\frac{1}{2}(m_{a+n}-m_{a+n+1}),~{}~{}~{}a=2,\cdots,n-2,$
(4.25) $\displaystyle\vdots$ $\displaystyle\check{c}_{n-1}$
$\displaystyle=m_{2n-1}-m_{n}+\frac{1}{2}\sum_{j=1}^{n}m_{2n-1-j}.$ (4.26)

##  5 Constraints of maximal symmetry

We would now like to impose the maximal symmetry which the system can have in
some region of the parameter space and to derive a set of relations among the
mass parameters $m_{2},m_{3},\cdots$ and $m_{2n-1}$.

Let us recall that, in $n=2$, the case of $su(2)$, $c_{1}\log w^{(1)}$ in
eq.(4.2) is the term that breaks even parity of the potential
$W_{U,1}(w^{(1)})$ under $w^{(1)}=1/w^{(1)\prime}$. Demanding the invariance
under this inversion, we obtain

$c_{1}=-\check{c}_{1}=0,$ (5.1)

which means $m_{2}=m_{3}$, namely, the Argyres-Douglas point.

For $n=3$, the case of $su(3)$, let us demand the symmetry under a generalized
inversion that reflects the combination of even parity and folding of the
Dynkin diagram:

$w^{(1)}=\frac{1}{w^{(2)\prime}},~{}~{}~{}~{}~{}w^{(2)}=\frac{1}{w^{(1)\prime}}.$
(5.2)

This gives us

${S}_{1}={S}_{2}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}\check{c}_{1}=c_{2},$
(5.3)

which imply

$m_{2}=m_{5},~{}~{}~{}~{}~{}m_{3}=m_{4}.$ (5.4)

Likewise, for $n=4$, the case of $su(4)$, demand the invariance under

$w^{(1)}=\frac{1}{w^{(3)\prime}},~{}~{}~{}~{}~{}w^{(3)}=\frac{1}{w^{(1)\prime}},~{}~{}~{}~{}~{}w^{(2)}=\frac{1}{w^{(2)\prime}}.$
(5.5)

This gives us

${S}_{1}={S}_{3},~{}~{}~{}~{}~{}\check{c}_{1}=c_{3},~{}~{}~{}~{}~{}\check{c}_{2}=c_{2}=0.$
(5.6)

These respectively imply

$m_{2}-m_{7}=-(m_{3}-m_{6}),~{}~{}~{}~{}~{}m_{2}-m_{7}=-(m_{4}-m_{5}),~{}~{}~{}~{}~{}m_{2}-m_{7}=m_{3}-m_{6}.$
(5.7)

We conclude

$m_{2}-m_{7}=m_{3}-m_{6}=m_{4}-m_{7}=0.$ (5.8)

This interesting pattern persists at $n=5$. Demanding the invariance, we
obtain

${S}_{1}={S}_{4},~{}~{}~{}~{}~{}{S}_{2}={S}_{3},~{}~{}~{}~{}~{}\check{c}_{1}=c_{4},~{}~{}~{}~{}~{}\check{c}_{2}=c_{3}.$
(5.9)

These respectively imply

$\begin{split}&(m_{2}-m_{4})+(m_{3}-m_{8})+(m_{4}-m_{7})=0,\\\
&m_{3}-m_{8}=0,~{}~{}~{}~{}~{}m_{2}-m_{9}=-(m_{5}-m_{6}),~{}~{}~{}~{}~{}m_{2}-m_{9}=m_{3}-m_{8}.\end{split}$
(5.10)

We conclude

$m_{2}-m_{9}=m_{3}-m_{8}=m_{4}-m_{7}=m_{5}-m_{6}=0.$ (5.11)

At general $n$, there are $2n-2$ mass parameters, and the conditions to be
imposed are the invariance under $w^{(a)}=1/w^{(n-a)\prime},a=1,\cdots,n-1$.
For $n$ odd, there are $\frac{n-1}{2}$ conditions of ${S}_{a}={S}_{n-a}$,
$a=1,\cdots,\frac{n-1}{2}$, and $\frac{n-1}{2}$ conditions of
$\check{c}_{a}=c_{n-a}$, $a=1,\cdots,\frac{n-1}{2}$. For $n$ even, there are
$\frac{n}{2}-1$ conditions of ${S}_{a}={S}_{n-a}$, $a=1,\cdots,\frac{n}{2}-1$,
and $\frac{n}{2}$ conditions of $\check{c}_{a}=c_{n-a}$,
$a=1,\cdots,\frac{n}{2}-1$ and $\check{c}_{n/2}=c_{n/2}=0$. In both cases, the
number of net free parameters is $n-1$. Doing the same analysis as that in
$n=4$ for $n$ even, and that in $n=5$ for $n$ odd, we conclude

$m_{i}=m_{2n+1-i},~{}~{}~{}~{}~{}i=2,\cdots,n.$ (5.12)

It is interesting to see that, after imposing the maximal symmetry of the
system, we still do not reach the point of all masses being equal, which is
sometimes assumed in the analysis of the Seiberg-Witten curve in general. As
the matrix model curve of this model and the corresponding Witten-Gaiotto
curve are known to be isomorphic [11], the same conclusion should be drawn at
the level of the Seiberg-Witten curve perse.

For example, combining the Seiberg-Witten curve for $su(n)$, $N_{f}=2n-2$ case
presented in [51] with (5.12), we can show (for detail, see [52]) that, under
the appropriate conditions on the parameters, the Seiberg-Witten curve can be
written as

$y^{2}=x^{n+1}\left\\{x^{n-1}\pm
2\Lambda\sum_{a=0}^{n-2}e_{a}x^{n-2-a}\right\\}.$ (5.13)

Here $\Lambda$ is the scale parameter and
$e_{a}=e_{a}(\tilde{m}_{2},\cdots,\tilde{m}_{n-1})$ is the elementary
symmetric polynomial in $\tilde{m}_{i}=m_{i}-m_{n}$. The curve is maximally
degenerate with multiplicity $n+1$. The original Seiberg-Witten curve has
$2n-1$ parameters, which consist of $n-1$ moduli parameters, $n-1$ mass
parameters and the scale parameter. In order to derive (5.13), we need $n$
conditions and these define Argyres-Douglas critical hypersurface in the
parameter space.

## Acknowledgments

We thank Katsuya Yano for the series of collaborations on this subject. The
work of HI is supported in part by JSPS KAKENHI Grant Number 19K03828 and by
the Osaka City University (OCU) Strategic Research Grant 2021 for priority
area (OMU-SRPP2022_SU04).

## References

  * [1] L. Alvarez-Gaume, “Random surfaces, statistical mechanics and string theory,” Helv. Phys. Acta 64, 359–526 (1991).
  * [2] P. H. Ginsparg and G. W. Moore,  “Lectures on 2-D gravity and 2-D string theory,”,  In Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles, pages 277–469 (10 1993), hep-th/9304011.
  * [3] H. Itoyama and R. Yoshioka, “Developments of theory of effective prepotential from extended Seiberg–Witten system and matrix models,” PTEP 2015(11), 11B103 (2015), arXiv:1507.00260; H. Itoyama, “Developments of supersymmetric gauge theory by matrices,” Nippon Butsuri Gakkai-Shi 71(9), 607–616 (2016).
  * [4] N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting,” Adv. Theor. Math. Phys. 7(5), 831–864 (2003), hep-th/0206161.
  * [5] F. David, “Loop Equations and Nonperturbative Effects in Two-dimensional Quantum Gravity,” Mod. Phys. Lett. A 5, 1019–1030 (1990).
  * [6] A. Mironov and A. Morozov, “On the origin of Virasoro constraints in matrix models: Lagrangian approach,” Phys. Lett. B 252, 47–52 (1990).
  * [7] H. Itoyama and Y. Matsuo, “Noncritical Virasoro algebra of $d<1$ matrix model and quantized string field,” Phys. Lett. B 255, 202–208 (1991); H. Itoyama and Y. Matsuo, “$W_{1+\infty}$ type constraints in matrix models at finite $N$,” Phys. Lett. B 262, 233–239 (1991)
  * [8] J. Zinn-Justin,  Quantum Field Theory and Critical Phenomena, volume 171,  (Oxford University Press, 2021); C. Itzykson and J. B. Zuber,  Quantum field theory,  (Courier Corporation, 2012).
  * [9] L. F. Alday and D. Gaiotto and Tachikawa “Liouville Correlation Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. 91 (2010) 167-197 (2010), arXiv:0906.3219.
  * [10] R. Dijkgraaf and C. Vafa, “Toda Theories, Matrix Models, Topological Strings, and $\mathcal{N}=2$ Gauge Systems,” (2009), arXiv:0909.2453.
  * [11] H. Itoyama, K. Maruyoshi, and T. Oota, “The Quiver Matrix Model and 2d-4d Conformal Connection,” Prog. Theor. Phys. 123, 957–987 (2010), arXiv:0911.4244.
  * [12] T. Eguchi and K. Maruyoshi, “Penner Type Matrix Model and Seiberg-Witten Theory,” JHEP 02, 022 (2010), arXiv:0911.4797.
  * [13] A. Mironov, A. Morozov, and Sh. Shakirov, “Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions,” JHEP 02, 030 (2010), arXiv:0911.5721.
  * [14] A. Mironov, A. Morozov, and Sh. Shakirov, “Conformal blocks as Dotsenko-Fateev Integral Discriminants,” Int. J. Mod. Phys. A 25, 3173–3207 (2010), arXiv:1001.0563.
  * [15] H. Itoyama and T. Oota, “Method of Generating q-Expansion Coefficients for Conformal Block and $\mathcal{N}=2$ Nekrasov Function by beta-Deformed Matrix Model,” Nucl. Phys. B 838, 298–330 (2010), arXiv:1003.2929.
  * [16] A. Mironov, A. Morozov, and And. Morozov, “Conformal blocks and generalized Selberg integrals,” Nucl. Phys. B 843, 534–557 (2011), arXiv:1003.5752.
  * [17] A. Morozov and S. Shakirov, “The matrix model version of AGT conjecture and CIV-DV prepotential,” JHEP 08, 066 (2010), arXiv:1004.2917.
  * [18] N. Wyllard, “$A_{N-1}$ conformal Toda field theory correlation functions from conformal $\mathcal{N}=2$ $SU(N)$ quiver gauge theories,” JHEP 11, 002 (2009), arXiv:0907.2189.
  * [19] A. Mironov and A. Morozov, “On AGT relation in the case of $U(3)$,” Nucl. Phys. B 825, 1–37 (2010), arXiv:0908.2569.
  * [20] A. Marshakov, A. Mironov, and A. Morozov, “Generalized matrix models as conformal field theories: Discrete case,” Phys. Lett. B 265, 99–107 (1991).
  * [21] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and S. Pakuliak, “Conformal matrix models as an alternative to conventional multimatrix models,” Nucl. Phys. B 404, 717–750 (1993), hep-th/9208044.
  * [22] I. K. Kostov, “Gauge invariant matrix model for the A-D-E closed strings,” Phys. Lett. B 297, 74–81 (1992), hep-th/9208053.
  * [23] D. Gaiotto, “Asymptotically free $\mathcal{N}=2$ theories and irregular conformal blocks,” J. Phys. Conf. Ser. 462(1), 012014 (2013), arXiv:0908.0307.
  * [24] D. Gaiotto and J. Teschner, “Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I,” JHEP 12, 050 (2012), arXiv:1203.1052.
  * [25] A. Marshakov, A. Mironov, and A. Morozov, “On non-conformal limit of the AGT relations,” Phys. Lett. B 682, 125–129 (2009), arXiv:0909.2052.
  * [26] H. Itoyama, T. Oota, and N. Yonezawa, “Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type,” Phys. Rev. D 82, 085031 (2010), arXiv:1008.1861.
  * [27] G. Bonelli, K. Maruyoshi, and A. Tanzini, “Wild Quiver Gauge Theories,” JHEP 02, 031 (2012), arXiv:1112.1691.
  * [28] T. Nishinaka and C. Rim, “$\beta$-Deformed Matrix Model and Nekrasov Partition Function,” JHEP 02, 114 (2012), arXiv:1112.3545.
  * [29] T. Nishinaka and C. Rim, “Matrix models for irregular conformal blocks and Argyres-Douglas theories,” JHEP 10, 138 (2012), arXiv:1207.4480.
  * [30] S.-K. Choi and C. Rim, “Parametric dependence of irregular conformal block,” JHEP 04, 106 (2014), arXiv:1312.5535.
  * [31] P. C. Argyres and M. R. Douglas, “New phenomena in $SU(3)$ supersymmetric gauge theory,” Nucl. Phys. B 448, 93–126 (1995), hep-th/9505062.
  * [32] P. C. Argyres, M. R. Plesser, N. Seiberg, and E. Witten, “New $\mathcal{N}=2$ superconformal field theories in four-dimensions,” Nucl. Phys. B 461, 71–84 (1996), hep-th/9511154.
  * [33] T. Kubota and N. Yokoi, “Renormalization group flow near the superconformal points in $\mathcal{N}=2$ supersymmetric gauge theories,” Prog. Theor. Phys. 100, 423–436 (1998), hep-th/9712054.
  * [34] D. Xie, “General Argyres-Douglas Theory,” JHEP 01, 100 (2013), arXiv:1204.2270.
  * [35] H. Itoyama, T. Oota, and Katsuya Yano, “Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory,” Phys. Lett. B 789, 605–609 (2019), arXiv:1805.05057.
  * [36] H. Itoyama, T. Oota, and Katsuya Yano, “Discrete Painlevé system for the partition function of $N_{f}=2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit,” J. Phys. A 52(41), 415401 (2019), arXiv:1812.00811.
  * [37] H. Itoyama, T. Oota, and Katsuya Yano, “Discrete Painlevé system associated with Unitary matrix model,” J. Phys. Conf. Ser. 1194(1), 012050 (2019).
  * [38] H. Itoyama, T. Oota, and Katsuya Yano, “Multicritical points of unitary matrix model with logarithmic potential identified with Argyres–Douglas points,” Int. J. Mod. Phys. A 35(24), 2050146 (2020), arXiv:1909.10770.
  * [39] S. Mizoguchi, H. Otsuka, and H. Tashiro, “Unitary matrix with a Penner-like potential also yields $N_{f}$ = 2,” Phys. Lett. B 800, 135075 (2020), arXiv:1909.09041.
  * [40] H. Itoyama and Katsuya Yano, “Theory space of one unitary matrix model and its critical behavior associated with Argyres–Douglas theory,” Int. J. Mod. Phys. A 36(30), 2150227 (2021), arXiv:2103.11428.
  * [41] T. Oota, “Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories,” Nucl. Phys. B 976, 115718 (2022), arXiv:2112.14441.
  * [42] D. J. Gross and E. Witten, “Possible Third Order Phase Transition in the Large $N$ Lattice Gauge Theory,” Phys. Rev. D 21, 446–453 (1980).
  * [43] S. R. Wadia, “$N=\infty$ Phase Transition in a Class of Exactly Soluble Model Lattice Gauge Theories,” Phys. Lett. B 93, 403–410 (1980).
  * [44] S. R. Wadia,  “A Study of $U(N)$ Lattice Gauge Theory in 2-dimensions,” (2012), arXiv:1212.2906.
  * [45] S. Chaudhuri, H. Itoyama and T. Ooshita  “Universal and nonperturbative behavior in the one plaquette model of two-dimensional string theory,” Nucl. Phys. B 409, 397-416 (1993), hep-th/9305126.
  * [46] K. Okuyama, “Wilson loops in unitary matrix models at finite $N$,” JHEP 07, 030 (2017), arXiv:1705.06542.
  * [47] J. G. Russo and M. Tierz, “Multiple phases in a generalized Gross-Witten-Wadia matrix model,” JHEP 09, 081 (2020), arXiv:2007.08515.
  * [48] J. G. Russo, “Phases of unitary matrix models and lattice QCD2,” Phys. Rev. D 102(10), 105019 (2020), arXiv:2010.02950.
  * [49] L. Santilli and M. Tierz, “Multiple phases and meromorphic deformations of unitary matrix models,” Nucl. Phys. B 976, 115694 (2022), arXiv:2102.11305.
  * [50] S. Mizoguchi, “On unitary/hermitian duality in matrix models,” Nucl. Phys. B 716, 462–486 (2005), hep-th/0411049.
  * [51] A. Hanany and Y. Oz, “On the quantum moduli space of vacua of $\mathcal{N}=2$ supersymmetric $SU(N_{c})$ gauge theories,” Nucl. Phys. B 452, 283–312 (1995), hep-th/9505075.
  * [52] H. Itoyama, T. Oota and R. Yoshioka, “A-D hypersurface of $su(n)$ $\mathcal{N}=2$ supersymmetric gauge theory with $N_{f}=2n-2$ flavors”, Int. J. Mod. Phys. A 38 (2023) 02, 2350017, arXiv:2212.06590.