NITEP 92

OCU-PHYS 534

March 22, 2021

Theory space of one unitary matrix model
and its critical behavior associated with
Argyres-Douglas theory

H. Itoyamaa,b,c***e-mail<EMAIL_ADDRESS>and Katsuya
Yanob†††e-mail<EMAIL_ADDRESS>
aNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),
Osaka City University

bDepartment of Mathematics and Physics, Graduate School of Science,
Osaka City University

cOsaka City University Advanced Mathematical Institute (OCAMI)

3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Abstract

The lowest critical point of one unitary matrix model with cosine plus
logarithmic potential is known to correspond with the $(A_{1},A_{3})$ Argyres-
Douglas (AD) theory and its double scaling limit derives the Painlevé II
equation with parameter. Here, we consider the critical points associated with
all cosine potentials and determine the scaling operators, their vevs and
their scaling dimensions from perturbed string equations at planar level.
These dimensions agree with those of $(A_{1},A_{4k-1})$ AD theory.

##  1 Introduction

Matrix models are useful in the analysis of four dimensional supersymmetric
gauge theory. The instanton partition function [1, 2] of the four dimensional
$SU(2)$ linear quiver superconformal gauge theories can be identified with the
conformal block of regular vertex operators in two dimensional conformal field
theory by the AGT correspondence [3]. The integral representation of the
conformal block [4] is the multi-Penner type $\beta$-deformed matrix model
whose potential has logarithmic terms.

The simplest example is given by the four dimensional $\mathcal{N}=2$ $SU(2)$
supersymmetric gauge theory with four hypermultiplets. The instanton partition
function corresponds to a four point conformal block by the regular vertex
operators whose integral representation is the three-Penner type matrix model
[5, 6]. The instanton partition functions for the cases with less than four
flavors which are asymptotically free are obtained by taking a degeneration
limit of the regular vertex operators [7, 8]. As a result, an irregular
conformal block is formed. The emergent matrix model contains rational terms
[9, 10]. A similar limit can be taken for the four dimensional $\mathcal{N}=2$
$SU(2)$ linear quiver superconformal gauge theories and we can obtain
corresponding asymptotically free theories. The potentials of these matrix
models contain rational terms with higher powers [9, 11]. For reviews, see[12,
13, 14].

Connection with integrable systems is understood better by considering a
discrete Fourier transform of the instanton partition function at $\beta=1$
[15] with regard to the filling fraction, which is the Coulomb moduli111The
generating function of the $q$-deformed matrix model has been considered in
[16].. Connection between the tau function of Painlevé equations and the
generating function of the instanton partition function of four dimensional
$\mathcal{N}=2$ $SU(2)$ gauge theory with $N_{f}=0,\ldots,4$ and that of
Argyres-Douglas (AD) type theories [17] obtained in [18, 19] 222References in
the higher rank case include [20, 21, 22]. has been pointed out in [23, 24,
25, 26, 27]. Some of these cases can be derived by using matrix models [28,
29, 30]. In [28, 29, 31], it was shown that in the case of two flavors can be
represented by the unitary matrix model of potential $U+U^{\dagger}+\log U$
type. This is a Gross-Witten-Wadia (GWW) model [32, 33, 34] with a logarithmic
potential. The string equations, which are a set of difference equations
arising from the recursion relations among orthogonal polynomials, have been
shown to be the alternate discrete Painlevé II equations (alt-dPII)333It is
also called the discrete Painlevé equation
$\mathrm{d}$-$\mathrm{P}((2A_{1})^{(1)}/D_{6}^{(1)})$ in [35]. . We have also
shown that the partition function of this model is the tau function of alt-
dPII equation which is closely related to the Painlevé ${\rm III}_{1}$
equation. Using the partition function of this model, we have constructed the
tau function of Painlevé ${\rm III}_{1}$ equation. By taking the double
scaling limit, alt-dPII equation turns into Painlevé II equation with
accessory parameter444For an alternative approach based on the genus expansion
of two-cut Hermitian cubic model, see [36]..

Extension of GWW model plus a logarithmic potential is given by a potential
with higher powers of $U$ and those of $U^{\dagger}$. It was considered in
[37] without a logarithmic potential. (For the recent development about the
phase structures of a generalized GWW model, see [38]. ) It is natural to
identify the generating functional of this extended unitary matrix model with
that of $\hat{A}_{2m,2n}$ [39, 40] asymptotically free theory. In this view
point, the multicritical points are AD points of various type. In fact, in
[41], we have taken the double scaling limit at the next to the lowest order
critical point and derived the system of differential equations associated
with the $(A_{1},A_{7})$ AD theory.

In this paper we develop a further extension of this type. We will examine the
planar critical behavior at various critical points. It is known that the
critical points of the symmetric unitary matrix model are labeled by an
integer $k$ [42, 43]. The analysis of critical behavior and the construction
of the even type scaling operators which perturb these critical points are
given there.555See [44] for the hermitian matrix case. We take the viewpoint
that the Coulomb branch operators of some AD theory corresponds to these
scaling operators. Note that not all of the Coulomb branch operators are
reproduced by [42, 43]. We will construct scaling operators of odd type and
logarithmic type in addition to even type and derive their vevs, and determine
their scaling dimensions from these. By comparing their dimensions with those
of the Coulomb branch operator from AD theory, we give further evidence for
the correspondence between the $k$-th multicritical point of unitary matrix
model and the $(A_{1},A_{4k-1})$ AD point.

The paper is organized as follows. In section $2$, we first review the method
of orthogonal polynomial for a unitary matrix model and explain how planar
string equations are given. Then we derive an explicit form of these equations
at the $k$-th multicritical point. We determine the vevs and their dimensions
of scaling operators of various type. In the last section, we give a
dictionary between the scaling operators at the $k$-th multicritical point and
the Coulomb branch operators in the $(A_{1},A_{4k-1})$ AD theory by comparing
their scaling dimensions.

##  2 One unitary matrix model

The partition function of the one unitary matrix model is defined by

$\displaystyle Z_{U}=\frac{1}{N!}\left(\prod_{i=1}^{N}\oint\frac{dz_{i}}{2\pi
iz_{i}}\right)\Delta(z)\Delta(z^{-1})\exp\left(\sum_{i=1}^{N}W(z_{i})\right),$
(2.1)

where $\Delta(z)$ is the Vandermonde determinant
$\Delta(z)=\prod_{i<j}(z_{i}-z_{j})$ and $W(z)$ is the potential

$\displaystyle
W(z)=-\frac{1}{2\underline{g}_{s}}\left[\sum_{p=1}^{\infty}\frac{g^{+}_{p}}{p}\left(z^{p}+\frac{1}{z^{p}}\right)+\frac{g^{-}_{p}}{p}\left(z^{p}-\frac{1}{z^{p}}\right)\right].$
(2.2)

When all $g^{-}_{p}$ vanish, the model is called symmetric. We can perform the
large $N$ expansion for the free energy of (2.1):

$\displaystyle\mathcal{F}\equiv\log
Z_{U}=\sum_{g=0}^{\infty}N^{2-2g}F_{g}(\widetilde{S}).$ (2.3)

The coefficients in the expansion are the function of the parameters in the
potential $\\{g^{+}_{p},\,g^{-}_{p}\\}$ and the t’ Hooft coupling
$\widetilde{S}\equiv\underline{g}_{s}N$ which is fixed in the large $N$ limit.

A way to evaluate the free energy is to resort to the method of orthogonal
polynomials [42, 43, 45, 46, 47, 48, 49]. For a review in the hermitian case,
see [50] and other approaches, in particular, based on Virasoro constraints,
see [51, 52, 53, 54]. Let us introduce the set of monic orthogonal polynomials
$\\{p_{n}(z),\,\tilde{p}_{n}(1/z)\\}$. Their orthogonality condition with
regard to the measure is

$\displaystyle\oint d\mu(z)p_{n}(z)\tilde{p}_{m}(1/z)=h_{n}\delta_{n,m},\qquad
d\mu(z)\equiv\frac{dz}{2\pi iz}e^{W(z)}.$ (2.4)

Here, $p_{n}$$(\tilde{p}_{n})$ is the polynomial in $z$$(z^{-1})$ of degree
$n$

$\displaystyle
p_{n}(z)=z^{n}+\cdots+A_{n},\qquad\tilde{p}_{n}(1/z)=z^{-n}+\cdots+B_{n}.$
(2.5)

We have denoted the constant terms by $A_{n}\equiv
p_{n}(0),\,B_{n}\equiv\tilde{p}_{n}(0)$. They are related to $h_{n}$ by

$\displaystyle\frac{h_{n}}{h_{n-1}}=1-A_{n}B_{n}.$ (2.6)

From the orthogonality condition (2.4), one can show that these polynomials
obey the following recursion relations

$\displaystyle zp_{n}(z)=$ $\displaystyle
p_{n+1}(z)-\sum_{k=0}^{n}\frac{h_{n}}{h_{k}}A_{n+1}B_{k}p_{k}(z),$ (2.7)
$\displaystyle z^{-1}\tilde{p}_{n}(1/z)=$
$\displaystyle\tilde{p}_{n+1}(1/z)-\sum_{k=0}^{n}\frac{h_{n}}{h_{k}}A_{k}B_{n+1}\tilde{p}_{k}(1/z).$
(2.8)

Rewriting the Vandermonde determinant in (2.1) as

$\displaystyle\Delta(z)=\det(p_{j-1}(z_{i}))_{1\leq i,j\leq
N},\qquad\Delta(1/z)=\det(\tilde{p}_{j-1}(1/z_{i}))_{1\leq i,j\leq N},$ (2.9)

and using the orthogonality, we have

$\displaystyle
Z_{U}=\prod_{k=0}^{N-1}h_{k}=h_{0}^{N}\prod_{j=1}^{N-1}(1-A_{j}B_{j})^{N-j}.$
(2.10)

Then, in terms of the coefficients in orthogonal polynomials, the free energy
is written as

$\displaystyle\mathcal{F}=N\log
h_{0}+\sum_{j=1}^{N-1}(N-j)\log(1-A_{j}B_{j}).$ (2.11)

In particular, the planar free energy $F_{0}$ can be evaluated as

$\displaystyle
F_{0}\sim\int_{0}^{1}dx(1-x)\log(1-A(x)B(x)),\qquad\frac{n}{N}\rightarrow
x,\,A_{n}\rightarrow A(x),\,B_{n}\rightarrow B(x).$ (2.12)

Here $\sim$ means dropping the higher contributions in $1/N$.

###  2.1 string equation and $k$-th multicritical points

It is known that the parameter space of the symmetric unitary matrix model has
a set of critical points which are labeled by an integer $k$. At such points,
the coefficients in (2.3) behave as

$\displaystyle
F_{g}(S)\sim(\widetilde{S}^{(k)}_{c}-\widetilde{S})^{(2-\gamma)(2-2g)/2},$
(2.13)

where $\widetilde{S}^{(k)}_{c}$ is the $k$-th critical value of the t’ Hooft
coupling and $\gamma=-1/k$ is susceptibility. Therefore, around this point, we
can also expand the free energy by sending the parameters to their critical
values together with the large $N$ limit

$\displaystyle\mathcal{F}=\sum_{g=0}^{\infty}\kappa^{2g-2}f_{g}(c),\qquad
1-\widetilde{S}/\widetilde{S}^{(k)}_{c}=a^{2}c,\quad\kappa^{-1}\equiv
Na^{2-\gamma}.$ (2.14)

Here $a$ is the auxiliary parameter, and $\kappa$ is kept fixed under
$a\rightarrow 0$.

The free energy depends on the coefficients $A_{n}$ and $B_{n}$, which are
controlled by the recursion relations called string equations. They are given
by a set of identities

$\displaystyle 0=\oint dz\frac{\partial}{\partial z}\left\\{\frac{z^{k}}{2\pi
i}e^{W(z)}p_{\ell}(z)\tilde{p}_{m}(1/z)\right\\}.$ (2.15)

Therefore, the critical behavior of the free energy can be evaluated by
solving a set of string equations at the critical point. In particular, the
cases $(k,\,\ell,\,m)=(-1,\,n,\,n-1)$ and $(k,\,\ell,\,m)=(0,\,n,\,n)$ are
important:

$\displaystyle\oint d\mu(z)W^{\prime}(z)p_{n}(z)\tilde{p}_{n-1}(1/z)=$
$\displaystyle n(h_{n}-h_{n-1}),$ (2.16) $\displaystyle\oint
d\mu(z)zW^{\prime}(z)p_{n}(z)\tilde{p}_{n}(1/z)=$ $\displaystyle 0.$ (2.17)

For later convenience, we introduce the new variables $H_{n},\,R_{n}$ and
$G_{n}$ by

$\displaystyle H_{n}=\sqrt{\frac{h_{n}}{h_{n-1}}},\qquad
A_{n}=R_{n}D_{n},\qquad
B_{n}=\frac{R_{n}}{D_{n}},\qquad\frac{D_{n+1}}{D_{n}}=1+G_{n}.$ (2.18)

In terms of these variables, (2.6) reads

$\displaystyle H_{n}^{2}=1-R_{n}^{2}.$ (2.19)

Appropriate bases for (2.16) and (2.17) are respectively

$\displaystyle\mathcal{F}^{\pm}_{p}(H_{m},G_{n})\equiv$
$\displaystyle\frac{1}{h_{n}}\oint
d\mu(z)\left(z^{p-1}\mp\frac{1}{z^{p+1}}\right)p_{n}(z)\tilde{p}_{n-1}(1/z),$
(2.20) $\displaystyle\mathcal{G}^{\pm}_{p}(H_{m},G_{n})\equiv$
$\displaystyle\frac{1}{h_{n}}\oint
d\mu(z)\left(z^{p}\mp\frac{1}{z^{p}}\right)p_{n}(z)\tilde{p}_{n}(1/z).$ (2.21)

Using these equations, (2.16) and (2.17) can be written as

$\displaystyle\widetilde{S}\frac{n}{N}=$
$\displaystyle\frac{1-R_{n}^{2}}{2R_{n}^{2}}\sum_{p=1}^{\infty}\left(g^{+}_{p}\mathcal{F}^{+}_{p}(R,G)+g^{-}_{p}\mathcal{F}^{-}_{p}(R,G)\right),$
(2.22) $\displaystyle 0=$
$\displaystyle\frac{1-R_{n}^{2}}{2R_{n}^{2}}\sum_{p=1}^{\infty}\left(g^{+}_{p}\mathcal{G}^{+}_{p}(R,G)+g^{-}_{p}\mathcal{G}^{-}_{p}(R,G)\right).$
(2.23)

Here we have used (2.19) to write the functions $\mathcal{F},\mathcal{G}$ in
terms of $R_{n},G_{n}$.

To derive the explicit form of (2.20) and (2.21), we follow [43] with small
modification. Let us define the normalized orthogonal polynomial

$\displaystyle
P_{n}(z)\equiv\frac{1}{\sqrt{h_{n}}}p_{n}(z),\qquad\widetilde{P}_{n}(1/z)\equiv\frac{1}{\sqrt{h_{n}}}\tilde{p}(1/z).$
(2.24)

They are orthonormal with respect to $d\mu(z)$

$\displaystyle\oint d\mu(z)P_{n}(z)\widetilde{P}_{m}(1/z)=\delta_{n,m}.$
(2.25)

Let us introduce the operators $\hat{\ell}$ and $\hat{u}$ by

$\displaystyle\hat{\ell}P_{n}(z)=nP_{n}(z),\qquad\hat{u}P_{n}(z)=P_{n+1}(z).$
(2.26)

Let us write $z$ and $1/z$ in terms of $\hat{\ell}$ and $\hat{u}$. From (2.7)
and (2.8), we obtain

$\displaystyle\oint d\mu P_{m}(z)\widetilde{P}_{n}(1/z)z$
$\displaystyle=\left\\{\begin{array}[]{cc}H_{m+1}&n=m+1\\\
-R_{m+1}R_{m}(1+G_{m})&n=m\\\ -R_{m+1}H_{m}(1+G_{m})\cdots
H_{n+1}(1+G_{n+1})R_{n}(1+G_{n})&n<m\end{array}\right.,$ (2.30)
$\displaystyle\oint d\mu P_{m}(z)\widetilde{P}_{n}(1/z)z^{-1}$
$\displaystyle=\left\\{\begin{array}[]{cc}H_{m}&n=m-1\\\
-R_{m+1}R_{m}(1+G_{m})^{-1}&n=m\\\
-R_{m}(1+G_{m})^{-1}H_{m+1}(1+G_{m+1})^{-1}\cdots
H_{n}(1+G_{n})^{-1}R_{n+1}&n>m\end{array}\right..$ (2.34)

Therefore, denoting $H(n)=H_{n}$, … etc, we have

$\displaystyle z=$ $\displaystyle
H(\hat{\ell})\hat{u}-R(\hat{\ell})(1+G(\hat{\ell}))\hat{u}^{-1}\frac{1}{1-H(\hat{\ell})(1+G(\hat{\ell}))\hat{u}^{-1}}R(\hat{\ell})\hat{u},$
(2.35) $\displaystyle\frac{1}{z}=$
$\displaystyle\hat{u}^{-1}H(\hat{u})-\hat{u}^{-1}R(\hat{\ell})\hat{u}\frac{1}{1-H(\hat{\ell})(1+G(\hat{\ell}))^{-1}\hat{u}}\frac{R(\hat{\ell})}{1+G(\hat{\ell})}.$
(2.36)

We are interested in the planar limit where the coefficients $H_{n}$, $R_{n}$
and $G_{n}$ turn into the continuous function $H_{n}\rightarrow H(x)$,
$R_{n}\rightarrow R(x),\,G_{n}\rightarrow G(x),\,n/N\rightarrow x$. The
operators $z$ and $1/z$ in this limit become respectively

$\displaystyle
z=-\frac{1+G(x)-H(x)u}{1-H(x)(1+G(x))u^{-1}},\qquad\frac{1}{z}=-\frac{(1+G(x))^{-1}-H(x)u^{-1}}{1-H(x)(1+G(x))^{-1}u}.$
(2.37)

Substitute these representations into (2.20) and (2.21), we obtain the planar
form of $\mathcal{F}^{\pm}_{p}(R,G)$ and $\mathcal{G}^{\pm}_{p}(R,G)$ by
extracting appropriate coefficient in $u$:

$\displaystyle\mathcal{F}^{+}_{p}(H,G)=$ $\displaystyle\frac{1}{H_{n}}\oint
d\mu(z)\left(z^{p-1}-\frac{1}{z^{p+1}}\right)P_{n}(z)\widetilde{P}_{n-1}(1/z)$
$\displaystyle\rightarrow$
$\displaystyle\frac{1}{H}\left\\{\left.\left(-\frac{1+G-Hu}{1-H(1+G)u^{-1}}\right)^{p-1}\right|_{u^{-1}}-\left.\left(-\frac{(1+G)^{-1}-Hu^{-1}}{1-H(1+G)^{-1}u}\right)^{p+1}\right|_{u^{-1}}\right\\}$
$\displaystyle=$
$\displaystyle(1-H^{2})\left((1+G)^{p}\sum_{n=0}^{p-1}(-1)^{p+n-1}\frac{\Gamma(p+n+1)}{p\,\Gamma(n+1)\Gamma(n+2)\Gamma(p-n-1)}H^{2n}\right.$
$\displaystyle\qquad+\left.\frac{1}{(1+G)^{p}}\sum_{n=0}^{p-1}(-1)^{p+n-1}\frac{\Gamma(p+n+2)}{p\,\Gamma(n+1)\Gamma(n+2)\Gamma(p-n)}H^{2n}\right),$
(2.38) $\displaystyle\mathcal{F}^{-}_{p}(H,G)=$
$\displaystyle(1-H^{2})\left((1+G)^{p}\sum_{n=0}^{p-1}(-1)^{p+n-1}\frac{\Gamma(p+n+1)}{p\,\Gamma(n+1)\Gamma(n+2)\Gamma(p-n-1)}H^{2n}\right.$
$\displaystyle\qquad-\left.\frac{1}{(1+G)^{p}}\sum_{n=0}^{p-1}(-1)^{p+n-1}\frac{\Gamma(p+n+2)}{p\,\Gamma(n+1)\Gamma(n+2)\Gamma(p-n)}H^{2n}\right),$
(2.39)

and

$\displaystyle\mathcal{G}^{\pm}_{p}(H,G)=$
$\displaystyle\left\\{(1+G)^{p}\mp\frac{1}{(1+G)^{p}}\right\\}(1-H^{2})\sum_{n=0}^{p-1}(-1)^{p+n}\frac{\Gamma(p+n+1)}{p\,\Gamma(n+1)^{2}\Gamma(p-n)}H^{2n}.$
(2.40)

For later convenience, let us write the planar string equations as

$\displaystyle\vec{s}(x)=\left(\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{\infty}\sum_{i=+,-}g^{i}_{p}\mathcal{F}^{i}_{p}(R,G)\right),\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{\infty}\sum_{i=+,-}g^{i}_{p}\mathcal{G}^{i}_{p}(R,G)\right)\right)^{t},$
(2.41)

where $\vec{s}(x)=(\widetilde{S}x,0)^{t}$.

Let us move on to study critical points of this model. While we do not give
the derivation [43], the $k$-th multicritical potential is given by

$\displaystyle
W^{(k)}(z)=-\frac{1}{2\underline{g}_{s}}\sum_{p=1}^{k}\frac{t^{(k)}_{p}}{p}\left(z^{p}+\frac{1}{z^{p}}\right),$
(2.42)

where the critical values of the couplings at this critical point are

$\displaystyle\left.g^{+}_{p}\right|_{\rm
crit}=t^{(k)}_{p}=(-1)^{p+1}\frac{\Gamma(k)\Gamma(k+2)}{\Gamma(k-p+1)\Gamma(k+p+1)},\quad
1\leq p\leq k$ (2.43)

and zero otherwise.666Here, we normalize the critical coupling $t^{(k)}_{1}$
to be $1$. This normalization is different from [43].

The set of string equations can be expanded around $R=0$ and $G=0$ as

$\displaystyle\vec{s}(x)=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{r+s=n}\frac{R^{r}G^{s}}{r!s!}$
$\displaystyle\times\partial^{r}_{R}\partial^{s}_{G}\left(\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{\infty}\sum_{i=+,-}g^{i}_{p}\mathcal{F}^{i}_{p}(R,G)\right),\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{\infty}\sum_{i=+,-}g^{i}_{p}\mathcal{G}^{i}_{p}(R,G)\right)\right)^{t}.$
(2.44)

By a tedious but straightforward calculation, (2.44) is resummed to take the
following form:

$\displaystyle\vec{s}(x)=\vec{s}^{(k)}_{c}+a^{2-1/k}\sum_{n=1}^{k}\vec{a}^{(k)}_{n}r^{2(k-n)}g^{2n-1}+a^{2}\sum_{n=0}^{k}\vec{b}^{(k)}_{n}r^{2(k-n)}g^{2n}+\mathcal{O}(a^{2+1/k}).$
(2.45)

Here we have set $R=a^{1/k}r,\,G=a^{1/k}g$ and introduced

$\displaystyle\vec{s}^{(k)}_{c}=\left(\begin{array}[]{cc}\widetilde{S}^{(k)}_{c}\\\
0\end{array}\right)=\left(\begin{array}[]{cc}\dfrac{k+1}{2k}\\\
0\end{array}\right)$ (2.50)

as well as

$\displaystyle\vec{a}^{(k)}_{n}=$
$\displaystyle\dfrac{(-1)^{n}}{2}\dfrac{\Gamma(k)\Gamma(k+2)}{\Gamma(2n)\Gamma(k-n+1)\Gamma(k-n+2)}\left(\begin{array}[]{ccc}1\\\
1\end{array}\right),$ (2.53)

and

$\displaystyle\vec{b}^{(k)}_{n}=$
$\displaystyle\dfrac{(-1)^{n+1}}{2}\dfrac{\Gamma(k)\Gamma(k+2)}{\Gamma(2n+1)\Gamma(k-n+1)\Gamma(k-n+2)}\left(\begin{array}[]{ccc}(k-n+1)+n(2n-1)\\\
n(2n-1)\end{array}\right).$ (2.56)

Eq.(2.45) gets further simplified by multiplying by

$\displaystyle T=\left(\begin{array}[]{cc}1&-1\\\ 0&1\end{array}\right),$
(2.59)

as $T\vec{s}(x)=\vec{s}(x)$ and $T\vec{s}_{c}=\vec{s}_{c}$ :

$\displaystyle T\vec{a}^{(k)}_{n}=$
$\displaystyle\dfrac{(-1)^{n}}{2}\dfrac{\Gamma(k)\Gamma(k+2)}{\Gamma(2n)\Gamma(k-n+1)\Gamma(k-n+2)}\left(\begin{array}[]{ccc}0\\\
1\end{array}\right),$ (2.62) $\displaystyle T\vec{b}^{(k)}_{n}=$
$\displaystyle\dfrac{(-1)^{n+1}}{2}\dfrac{\Gamma(k)\Gamma(k+2)}{\Gamma(2n+1)\Gamma(k-n+1)^{2}}\left(\begin{array}[]{ccc}1\\\
\dfrac{n(2n-1)}{k-n+1}\end{array}\right).$ (2.65)

The first component of (2.45) in this form is $\mathcal{O}(a^{2})$ and the
second component is $\mathcal{O}(a^{2-1/k})$. Defining

$\displaystyle 1-\widetilde{S}x/\widetilde{S}^{(k)}_{c}=a^{2}t,\qquad
1-\widetilde{S}/\widetilde{S}^{(k)}_{c}=a^{2}c,$ (2.66)

we obtain a set of planar string equations by the scaling variables $t,r$ and
$g$

$\displaystyle t=$
$\displaystyle\sum_{n=0}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n+1)\Gamma(k-n+1)^{2}}r^{2(k-n)}g^{2n},$
(2.67) $\displaystyle 0=$
$\displaystyle\sum_{n=1}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n)\Gamma(k-n+1)\Gamma(k-n+2)}r^{2(k-n)}g^{2n-1}.$
(2.68)

Note that $g=0$ is always a solution to (2.68), where (2.67) reduces to

$\displaystyle t=r^{2k}.$ (2.69)

We can now compute the $k$-th critical behavior of the planar free
energy777Note that our definition of the free energy in (2.3) omits the minus
sign. This explains a few minus signs in what follows. from (2.12)

$\displaystyle F_{0}\sim$ $\displaystyle
a^{4+2/k}\int_{a^{-2}+c}^{c}dt\left\\{(t-c)t^{1/k}+\mathcal{O}(a^{2})\right\\}$
$\displaystyle\sim$
$\displaystyle-a^{4+2/k}\frac{k^{2}}{(2k+1)(k+1)}c^{2+1/k}.$ (2.70)

Then defining $\kappa^{-1}=Na^{2+1/k}$ which is kept fixed under the limit
$N\rightarrow\infty,\,a\rightarrow 0$, we obtain the planar free energy at the
$k$-th multicritical point

$\displaystyle\mathcal{F}=-\kappa^{-2}\frac{k^{2}}{(2k+1)(k+1)}c^{2+1/k}+\cdots,$
(2.71)

namely

$\displaystyle f_{0}=-\frac{k^{2}}{(2k+1)(k+1)}c^{2+1/k}.$ (2.72)

###  2.2 perturbation for critical points

In this section, we will construct the scaling operators which perturb the
$k$-th critical planar string equations (2.67) and (2.68). Let us first review
how to construct the scaling operators which preserve the $z\rightarrow 1/z$
invariance [43]. We will call them even type scaling operators. We also
construct the odd and logarithmic type scaling operators which are odd under
$z\rightarrow 1/z$.

####  2.2.1 even type perturbation

Let us consider the following perturbation of the $k$-th critical potential

$\displaystyle W^{(k)}(z)\rightarrow
W^{(k)}(z)+m^{+B}_{\ell}\sigma^{+B}_{\ell}(z),\qquad\sigma^{+B}_{\ell}(z)=W^{(\ell)}(z),$
(2.73)

where $m^{+B}_{\ell}$ is “bare” coupling and $\sigma^{+B}_{\ell}(z)$ is “bare”
scaling operator. It is clear that perturbed potential is invariant under
$z\rightarrow 1/z$. This perturbation changes the string equation (2.45) into

$\displaystyle\vec{s}(x)=$
$\displaystyle\vec{s}^{(k)}_{c}+a^{2-1/k}\sum_{n=1}^{k}T\vec{a}^{(k)}_{n}r^{2(k-n)}g^{2n-1}+a^{2}\sum_{n=0}^{k}T\vec{b}^{(k)}_{n}r^{2(k-n)}g^{2n}$
$\displaystyle+m^{+B}_{\ell}\left(\vec{s}^{(\ell)}_{c}+a^{(2\ell-1)/k}\sum_{n=1}^{\ell}T\vec{a}^{(\ell)}_{n}r^{2(\ell-n)}g^{2n-1}+a^{2\ell/k}\sum_{n=0}^{\ell}T\vec{b}^{(\ell)}_{n}r^{2(\ell-n)}g^{2n}\right).$
(2.74)

Since the critical value of $\widetilde{S}^{(k)}_{c}$ shifts by

$\displaystyle\widetilde{S}^{(k)}_{c}\rightarrow\widetilde{S}^{(k)\prime}_{c}=\widetilde{S}^{(k)}_{c}+m^{+B}_{\ell}\widetilde{S}^{(\ell)}_{c},$
(2.75)

we redefine the scaling variable for $\widetilde{S}$ by

$\displaystyle
1-\widetilde{S}x/\widetilde{S}^{(k)\prime}_{c}=a^{2}t^{\prime}.$ (2.76)

In order for the first component of (2.74) to be oder $a^{2}$, we should scale
$m^{+B}_{\ell}$ by

$\displaystyle
m^{+B}_{\ell}=a^{2(1-\ell/k)}\frac{\widetilde{S}^{(k)}_{c}}{\widetilde{S}^{(\ell)}_{c}}m^{+}_{\ell}.$
(2.77)

For $\ell=1,\cdots,k-1$, the critical value $\widetilde{S}^{(k)\prime}_{c}$
goes to $\widetilde{S}^{(k)}_{c}$ in the $a\rightarrow 0$ limit. It means that
the critical value of $\widetilde{S}^{(k)}_{c}$ does not change at the $k$-th
multicritical point. For $\ell=k$, the perturbation changes only the critical
values of $\widetilde{S}^{(k)}$. For $\ell>k$, we must set $m^{+}_{\ell}=0$ to
make $\widetilde{S}^{(k)\prime}_{c}$ into finite. We should therefore consider
the $\ell=1,\cdots,k-1$ cases, where the perturbed string equations read

$\displaystyle t=$
$\displaystyle\left(\sum_{n=0}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n+1)\Gamma(k-n+1)^{2}}r^{2(k-n)}g^{2n}\right)$
$\displaystyle\quad+m^{+}_{\ell}\left(\sum_{n=0}^{\ell}(-1)^{n}\dfrac{\Gamma(\ell+1)^{2}}{\Gamma(2n+1)\Gamma(\ell-n+1)^{2}}r^{2(\ell-n)}g^{2n}\right),$
(2.78) $\displaystyle 0=$
$\displaystyle\left(\sum_{n=1}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n)\Gamma(k-n+1)\Gamma(k-n+2)}r^{2(k-n)}g^{2n-1}\right)$
$\displaystyle\quad+m^{+}_{\ell}\left(\sum_{n=1}^{\ell}(-1)^{n}\dfrac{\Gamma(\ell+1)^{2}}{\Gamma(2n)\Gamma(\ell-n+1)\Gamma(\ell-n+2)}r^{2(\ell-n)}g^{2n-1}\right).$
(2.79)

Note that (2.79) has a solution $g=0$ for $\ell=1,\cdots,k-1$ and (2.78)
becomes

$\displaystyle t=m^{+}_{\ell}r^{2\ell}+r^{2k}.$ (2.80)

####  2.2.2 odd type perturbation

Let us now consider the following perturbation

$\displaystyle W^{(k)}(z)\rightarrow
W^{(k)}(z)+m^{-B}_{\ell}\sigma^{-B}_{\ell}(z),$ (2.81)

where $1\leq\ell\leq k-1$ and

$\displaystyle\sigma^{-B}_{\ell}(z)=-\frac{1}{2\underline{g}_{s}}\sum_{p=1}^{\ell}\bar{t}^{(\ell)}_{p}\left(z^{p}-\frac{1}{z^{p}}\right),\qquad\bar{t}^{(\ell)}_{p}=p\,t^{(\ell)}_{p}.$
(2.82)

Here $t^{(\ell)}_{p}$ is given by (2.43). The perturbed string equation is

$\displaystyle\vec{s}(x)=$
$\displaystyle\left(\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{k}t^{(k)}_{p}\mathcal{F}^{+}_{p}(R,G)\right),\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{k}t^{(k)}_{p}\mathcal{G}^{+}_{p}(R,G)\right)\right)^{t}$
$\displaystyle+m^{-B}_{\ell}\left(\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{\ell}\bar{t}^{(\ell)}_{p}\mathcal{F}^{-}_{p}(R,G)\right),\frac{1-R^{2}}{2R^{2}}\left(\sum_{p=1}^{\ell}\bar{t}^{(\ell)}_{p}\mathcal{G}^{-}_{p}(R,G)\right)\right)^{t}.$
(2.83)

The first line has been already computed in (2.45). The second line can be
resummed to give

$\displaystyle
m^{-B}_{\ell}a^{(2\ell-2)/k}\sum_{n=1}^{\ell}\vec{c}_{n}r^{2(\ell-n)}g^{2n-2}+m^{-B}_{\ell}a^{(2\ell-1)/k}\sum_{n=1}^{\ell}\vec{d}_{n}r^{2(\ell-n)}g^{2n-1}+\mathcal{O}(a^{2\ell/k}),$
(2.84)

where

$\displaystyle\vec{c}_{n}=$
$\displaystyle\frac{(-1)^{n}}{2}\dfrac{\Gamma(\ell)\Gamma(\ell+2)}{\Gamma(2n-1)\Gamma(\ell-n+1)\Gamma(\ell-n+2)}\left(\begin{array}[]{ccc}1\\\
1\end{array}\right),$ (2.87) $\displaystyle\vec{d}_{n}=$
$\displaystyle\frac{(-1)^{n+1}}{2}\dfrac{\Gamma(\ell)\Gamma(\ell+2)}{\Gamma(2n)\Gamma(\ell-n+1)\Gamma(\ell-n+2)}\left(\begin{array}[]{ccc}1+\ell-n+(n-1)(2n-1)\\\
(n-1)(2n-1)\end{array}\right).$ (2.90)

These can be also transformed into a simpler form by the multiplication by
$T$:

$\displaystyle T\vec{c}_{n}=$
$\displaystyle\frac{(-1)^{n}}{2}\dfrac{\Gamma(\ell)\Gamma(\ell+2)}{\Gamma(2n-1)\Gamma(\ell-n+1)\Gamma(\ell-n+2)}\left(\begin{array}[]{cc}0\\\
1\end{array}\right),$ (2.93) $\displaystyle T\vec{d}_{n}=$
$\displaystyle\frac{(-1)^{n}}{2}\frac{\Gamma(\ell)\Gamma(\ell+2)}{\Gamma(2n)\Gamma(\ell-n+1)^{2}}\left(\begin{array}[]{cc}1\\\
\dfrac{(n-1)(2n-1)}{\ell-n+1}\end{array}\right).$ (2.96)

Thus, this perturbation shifts the first string equation at
$\mathcal{O}(a^{(2\ell-1)/k})$ and the second string equation at
$\mathcal{O}(a^{(2\ell-2)/k})$. In order to make the first string equation
$\mathcal{O}(a^{2})$, we should rescale the coupling $m^{-B}_{\ell}$ as

$\displaystyle
m^{-B}_{\ell}=a^{2-(2\ell-1)/k}\frac{\widetilde{S}^{(k)}_{c}}{\widetilde{S}^{(\ell)}_{c}}m^{-}_{\ell}.$
(2.97)

The resulting perturbed string equations are

$\displaystyle t=$
$\displaystyle\sum_{n=0}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n+1)\Gamma(k-n+1)^{2}}r^{2(k-n)}g^{2n}$
$\displaystyle+m^{-}_{\ell}\sum_{n=1}^{\ell}(-1)^{n+1}\frac{\Gamma(\ell+1)^{2}}{\Gamma(2n)\Gamma(\ell-n+1)^{2}}r^{2(\ell-n)}g^{2n-1},$
(2.98) $\displaystyle 0=$
$\displaystyle\sum_{n=1}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n)\Gamma(k-n+1)\Gamma(k-n+2)}r^{2(k-n)}g^{2n-1}$
$\displaystyle+m^{-}_{\ell}\sum_{n=1}^{\ell}(-1)^{n}\frac{\Gamma(\ell+1)^{2}}{\Gamma(2n-1)\Gamma(\ell-n+1)\Gamma(\ell-n+2)}r^{2(\ell-n)}g^{2n-2}.$
(2.99)

####  2.2.3 logarithmic type perturbation

Let us finally consider the following perturbation

$\displaystyle W^{(k)}(z)\rightarrow W^{(k)}(z)+m^{B}_{\rm
log}\sigma^{B}_{0}(z),\qquad\sigma^{B}_{\rm
log}(z)=-\frac{1}{2\underline{g}_{s}}\log(z).$ (2.100)

Because

$\displaystyle\frac{1}{h_{n}}\oint
d\mu(z)\frac{1}{z}p_{n}(z)\tilde{p}_{n-1}(1/z)=\frac{1}{h_{n}}\oint
d\mu(z)p_{n}(z)\tilde{p}_{n}(1/z)=1,$ (2.101)

the net effect of this perturbation is just to shift r.h.s. of (2.45) by

$\displaystyle\frac{1-R^{2}}{R^{2}}(m^{B}_{\rm log},m^{B}_{\rm log})^{t}.$
(2.102)

The $k$-th critical string equations perturbed by (2.100) are

$\displaystyle\vec{s}(x)=\vec{s}_{c}+\left(a^{-2/k}r^{-2}-1\right)(0,\,m^{B}_{0})^{t}+a^{2-1/k}\sum_{n=1}^{k}T\vec{a}_{n}r^{2(k-n)}g^{2n-1}+a^{2}\sum_{n=0}^{k}T\vec{b}_{n}r^{2(k-n)}g^{2n}.$
(2.103)

This perturbation affects the second string equation only. For the second
component of (2.103) to be $\mathcal{O}(a^{2-1/k})$, we should scale
$m^{B}_{0}$ as

$\displaystyle m^{B}_{\rm log}=a^{2+1/k}m_{\rm log}.$ (2.104)

The perturbed string equations of the logarithmic type are

$\displaystyle t=$
$\displaystyle\sum_{n=0}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n+1)\Gamma(k-n+1)^{2}}r^{2(k-n)}g^{2n},$
(2.105) $\displaystyle 0=$ $\displaystyle m_{\rm
log}+\sum_{n=1}^{k}(-1)^{n}\dfrac{\Gamma(k+1)^{2}}{\Gamma(2n)\Gamma(k-n+1)\Gamma(k-n+2)}r^{2(k-n+1)}g^{2n-1}$
(2.106)

###  2.3 scaling behavior and the scaling dimension

####  2.3.1 scaling operators

In section 2.2, we derived the perturbed string equations obtained by the
various perturbations. The vevs of the scaling operators at the $k$-th
multicritical point are given by the derivative of free energy with respect to
the scaled perturbation parameter:

$\displaystyle\left<\sigma^{\pm}_{\ell}\right>=\frac{\partial\mathcal{F}}{\partial
m^{\pm}_{\ell}},\qquad\left<\sigma_{\rm
log}\right>=\frac{\partial\mathcal{F}}{\partial m_{\rm log}}.$ (2.107)

Since they can be also expanded in $\kappa$ as is the free energy, their
leading parts are written as

$\displaystyle\left<\sigma^{\pm}_{\ell,0}\right>=\frac{\partial
f_{0}}{\partial m^{\pm}_{\ell}},\qquad\left<\sigma_{{\rm
log},0}\right>=\frac{\partial f_{0}}{\partial m_{\rm log}}.$ (2.108)

We can evaluate $\left<\sigma^{\pm}_{\ell,0}\right>$ and $\left<\sigma_{{\rm
log},0}\right>$ by solving the perturbed string equations.

Let us consider the even type scaling operators which are studied in [43]:

$\displaystyle\left<\sigma^{+}_{\ell,0}\right>=\frac{\partial f_{0}}{\partial
m^{+}_{\ell}}\sim\int^{c}dt\,(t-c)\,\frac{\partial
r(t;\\{m^{+}_{\ell}\\})^{2}}{\partial m^{+}_{\ell}},$ (2.109)

where $r(t;\\{m^{+}_{\ell}\\})$ is the solution of

$\displaystyle t=\sum_{\ell=1}^{k-1}m^{+}_{\ell}r^{2\ell}+r^{2k},$ (2.110)

which can be obtained in the same way as in [55]:

$\displaystyle
r(t;\\{m^{+}_{\ell}\\})^{2}=t^{1/k}+\sum_{p=1}^{\infty}\frac{(-1)^{p}}{k\,p!}\left(\frac{\partial}{\partial
t}\right)^{p-1}\left\\{\left(\sum_{\ell=1}^{k-1}m^{+}_{\ell}t^{\ell/k}\right)^{p}t^{-1+1/k}\right\\}.$
(2.111)

We obtain

$\displaystyle\left<\sigma^{+}_{\ell,0}\right>\sim\frac{k}{(k+\ell+1)(\ell+1)}c^{1+(\ell+1)/k}+\mathcal{O}(m^{+}_{\ell}).$
(2.112)

To evaluate the odd type scaling operators, we solve the set of equations
(2.98) and (2.99). We assume that $m^{-}_{\ell}$ is small and $g$ can be
expanded in $m^{-}_{\ell}$ as

$\displaystyle
g(r;m^{-}_{\ell})=\sum_{n=1}^{\infty}g_{n}(r)(m^{-}_{\ell})^{2n-1}.$ (2.113)

We choose the solution of (2.99) to satisfy $g(r;m^{-}_{\ell}=0)=0$ so that it
reduces to (2.69) when there is no deformation. Similarly, we assume that $r$
has the following expansion

$\displaystyle
r(t;m^{-}_{\ell})^{2}=t^{1/k}\left(1+\sum_{n=1}^{\infty}r_{n}(t)(m^{-}_{\ell})^{2n}\right)$
(2.114)

such that $r=t^{1/(2k)}$ at $m^{-}_{\ell}=0$. Substituting (2.113) into
(2.98), and solving oder by oder, we obtain

$\displaystyle
r(t;m^{-}_{\ell})^{2}=t^{1/k}+\frac{\ell^{2}}{2k^{2}}(k+2\ell)t^{-2+2\ell/k}(m^{-}_{\ell})^{2}+\mathcal{O}((m^{-}_{\ell})^{4}),$
(2.115)

and

$\displaystyle\left<\sigma^{-}_{\ell,0}\right>=\frac{\partial f_{0}}{\partial
m^{-}_{\ell}}=$
$\displaystyle\frac{\ell^{2}}{k^{2}}(k+2\ell)m^{-}_{\ell}\int^{c}_{a^{-2}+c}dt(t-c)t^{-2+2\ell/k}+\mathcal{O}((m^{-}_{\ell})^{3})$
$\displaystyle=$
$\displaystyle\frac{\ell(k+2\ell)}{2(k-2\ell)}c^{2\ell/k}m^{-}_{\ell}+\mathcal{O}((m^{-}_{\ell})^{3}).$
(2.116)

Similarly, for the logarithmic type scaling operator, we obtain from the
solution of (2.105) and (2.106)

$\displaystyle r(m_{\rm log})^{2}=t^{1/k}+\frac{1}{2t^{2}}(m_{\rm
log})^{2}+\mathcal{O}((m_{\rm log})^{4}),$ (2.117)

and

$\displaystyle\left<\sigma_{{\rm log},0}\right>=\frac{\partial f_{0}}{\partial
m_{\rm log}}=$ $\displaystyle m_{\rm
log}\int^{c}_{a^{-2}+c}dt(t-c)t^{-2}+\mathcal{O}((m_{\rm log})^{3})$
$\displaystyle\sim$ $\displaystyle m_{\rm log}\left(\log
c+1\right)+\mathcal{O}((m_{\rm\log})^{3}).$ (2.118)

####  2.3.2 scaling dimensions

We search for the possibility that the free energy at the $k$-th multicritical
point is that of the AD theory of some type. Then, we define the scaling
dimension of the sphere contribution of the free energy $f_{0}$ to be
$\left[f_{0}\right]=2$, since it should be identified the Seiberg-Witten
prepotential of corresponding AD theory. The scaling dimensions of the other
variables are easily determined from the critical behavior of the free energy
(2.72) and the vevs of the scaling operators (2.112), (2.116) and (2.118):

$\displaystyle\left[c\right]=\frac{2k}{2k+1},$ (2.119)
$\displaystyle\left[\sigma^{+}_{\ell}\right]=\frac{2k+2\ell+2}{2k+1},\quad\left[\sigma^{-}_{\ell,0}\right]=\frac{2k+2\ell+1}{2k+1},\quad\left[\sigma_{{\rm
log},0}\right]=1,\quad\ell=1,\ldots,k-1,$ (2.120)

and

$\displaystyle\left[m^{+}_{\ell}\right]=\frac{2k-2\ell}{2k+1},\quad[m^{-}_{\ell}]=\frac{2k-2\ell+1}{2k+1},\quad\left[m_{\rm
log}\right]=1,\quad\ell=1,\ldots,k-1.$ (2.121)

##  3 Comparison with Argyres-Douglas theory

Let us compare the scaling operators which we constructed in section 2.3 with
those of the Coulomb branch operators in the AD theory. In this way, we can
easily see the relationship between the $k$-th multicritical point and the AD
theory. In fact, in the $(A_{1},A_{4k-1})$ theory, the scaling dimension for
the Coulomb branch operators $u_{2k+1+i}$ and their coupling constant
$m_{i}\equiv u_{2k+1-i}$ are respectively888Here, the Seiberg-Witten curve of
the $(A_{1},A_{4k-1})$ AD theory is given by
$x^{2}=z^{4k}+u_{2}z^{4k-2}+\cdots+u_{4k}$.

$\displaystyle[u_{2k+1+i}]=\frac{2k+1+i}{2k+1},\qquad[m_{i}]=\frac{2k+1-i}{2k+1},\qquad
i=1,\ldots,2k-1.$ (3.1)

We obtain a dictionary by setting $i=2\ell$ and $i=2\ell+1$

$\displaystyle[\sigma^{-}_{\ell,0}]$ $\displaystyle=[u_{2k+1+2\ell}],$
$\displaystyle[m^{-}_{\ell}]$
$\displaystyle=[m_{2\ell}],\qquad\ell=1,\ldots,k-1,$ (3.2)
$\displaystyle[\sigma^{+}_{\ell,0}]$ $\displaystyle=[u_{2k+2+2\ell}],$
$\displaystyle[m^{+}_{\ell}]$
$\displaystyle=[m_{2\ell+1}],\qquad\ell=0,\cdots,k-1,$ (3.3)

and

$\displaystyle[m_{\rm log}]=[u_{2k+1}].$ (3.4)

Here we introduced

$\displaystyle\sigma^{+}_{0,0}\equiv\frac{\partial f_{0}}{\partial c},\qquad
m^{+}_{0}\equiv c.$ (3.5)

The perturbations from the $k$-th multicritical point can capture all Coulomb
branch operators which are contained in the $(A_{1},A_{4k-1})$ theory. We
conclude that the $k$-th multicritical point at the even potential of the one
unitary matrix model corresponds to the $(A_{1},A_{4k-1})$ theory.

## Acknowledgment

We are indebted to Takeshi Oota for sharing understanding on various aspects
of this subject. We thank Kazumasa Okabayashi for helpful discussion. The work
of HI is supported in part by JSPS KAKENHI Grant Number 19K03828 and by the
Osaka City University (OCU) Strategic Research Grant 2020 for priority area
(OCU-SRG2019_TPR01).

## References

  * [1] N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting,” Adv. Theor. Math. Phys. 7, no.5, 831-864 (2003) [arXiv:hep-th/0206161 [hep-th]].
  * [2] H. Nakajima and K. Yoshioka, “Instanton counting on blowup. 1.,” Invent. Math. 162, 313-355 (2005) [arXiv:math/0306198 [math.AG]].
  * [3] L. F. Alday, D. Gaiotto and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. 91, 167-197 (2010) [arXiv:0906.3219 [hep-th]].
  * [4] V. S. Dotsenko and V. A. Fateev, “Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models,” Nucl. Phys. B 240, 312-348 (1984).
  * [5] 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 [hep-th]].
  * [6] 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 [hep-th]].
  * [7] D. Gaiotto, “Asymptotically free $\mathcal{N}=2$ theories and irregular conformal blocks,” J. Phys. Conf. Ser. 462, no.1, 012014 (2013) [arXiv:0908.0307 [hep-th]].
  * [8] D. Gaiotto and J. Teschner, “Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I,” JHEP 12, 050 (2012) [arXiv:1203.1052 [hep-th]].
  * [9] T. Eguchi and K. Maruyoshi, “Seiberg-Witten theory, matrix model and AGT relation,” JHEP 07, 081 (2010) [arXiv:1006.0828 [hep-th]].
  * [10] 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 [hep-th]].
  * [11] T. Nishinaka and C. Rim, “Matrix models for irregular conformal blocks and Argyres-Douglas theories,” JHEP 1210, 138 (2012) [arXiv:1207.4480 [hep-th]].
  * [12] K. Maruyoshi, “$\beta$-Deformed Matrix Models and 2d/4d Correspondence,” [arXiv:1412.7124 [hep-th]].
  * [13] H. Itoyama and R. Yoshioka, “Developments of theory of effective prepotential from extended Seiberg–Witten system and matrix models,” PTEP 2015, no.11, 11B103 (2015) [arXiv:1507.00260 [hep-th]].
  * [14] H. Itoyama, “Developments of supersymmetric gauge theory by matrices,” Nippon Butsuri Gakkai-Shi, 71, no.9, 607-616 (2016).
  * [15] A. Mironov and A. Morozov, “On determinant representation and integrability of Nekrasov functions,” Phys. Lett. B 773, 34-46 (2017) [arXiv:1707.02443 [hep-th]].
  * [16] A. Mironov, A. Morozov and Z. Zakirova, “Discrete Painlevé equation, Miwa variables and string equation in 5d matrix models,” JHEP 19, 227 (2020) [arXiv:1908.01278 [hep-th]].
  * [17] P. C. Argyres and M. R. Douglas, “New Phenomena in $SU(3)$ Supersymmetric Gauge Theory,” Nucl. Phys. B 448, 93-126 (1995) [arXiv:hep-th/9505062].
  * [18] 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) [arXiv:hep-th/9511154].
  * [19] 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) [arXiv:hep-th/9712054].
  * [20] T. Eguchi, K. Hori, K. Ito and S. K. Yang, “Study of N=2 superconformal field theories in four-dimensions,” Nucl. Phys. B 471, 430-444 (1996) [arXiv:hep-th/9603002 [hep-th]].
  * [21] K. Ito, S. Koizumi and T. Okubo, “Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories,” Phys. Lett. B 792, 29-34 (2019) [arXiv:1903.00168 [hep-th]].
  * [22] K. Ito, S. Koizumi and T. Okubo, “Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_{c})$ SQCD around the superconformal point,” Nucl. Phys. B 954, 115004 (2020) [arXiv:2001.08891 [hep-th]].
  * [23] O. Gamayun, N. Iorgov and O. Lisovyy, “Conformal field theory of Painlevé VI,” JHEP 1210, 038 (2012) [arXiv:1207.0787 [hep-th]].
  * [24] O. Gamayun, N. Iorgov and O. Lisovyy, “How instanton combinatorics solves Painlevé VI, V and IIIs,” J. Phys. A 46, 335203 (2013) [arXiv:1302.1832 [hep-th]].
  * [25] N. Iorgov, O. Lisovyy and Y. Tykhyy, “Painlevé VI connection problem and monodromy of $c=1$ conformal blocks,” JHEP 1312, 029 (2013) [arXiv:1308.4092 [hep-th]].
  * [26] H. Nagoya, “Conformal blocks and Painlevé functions,” arXiv:1611.08971 [math-ph].
  * [27] G. Bonelli, O. Lisovyy, K. Maruyoshi, A. Sciarappa and A. Tanzini, “On Painlevé/gauge theory correspondence,” Lett. Math. Phys. 107, 2359-2413 (2017) [arXiv:1612.06235 [hep-th]].
  * [28] H. Itoyama, T. Oota and K. Yano, “Discrete Painlevé 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 [hep-th]].
  * [29] H. Itoyama, T. Oota and K. Yano, “Discrete Painlevé system for the partition function of $N_{f}=2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit,” J. Math. Phys. A: Math. Theor. 52, 415401 (2019) [arXiv:1812.00811 [hep-th]].
  * [30] H. Itoyama, T. Oota and K. Yano, “Discrete Painlevé system associated with Unitary matrix model,” J. Phys. Conf. Ser. 1194, no.1, 012050 (2019)
  * [31] 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 [hep-th]].
  * [32] 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).
  * [33] S. R. Wadia, “A Study of $U(N)$ Lattice Gauge Theory in 2-dimensions,” U. Chicago preprint EFI 79/44, July 1979 [arXiv:1212.2906 [hep-th]].
  * [34] S. R. Wadia, “$N=\infty$ phase transition in a class of exactly soluble model lattice gauge theories,” Phys. Lett. 93B, 403-410 (1980).
  * [35] K. Kajiwara, M. Noumi and Y. Yamada, “Geometric aspects of Painlevé equations,” J. Phys. A: Math. Theor. 50, 073001(2017). [arXiv:1509.08186 [nlin.SI]]
  * [36] A. Grassi and J. Gu, “Argyres-Douglas theories, Painlevé II and quantum mechanics,” JHEP 02, 060 (2019) [arXiv:1803.02320 [hep-th]].
  * [37] M. J. Bowick, A. Morozov and D. Shevitz, “Reduced unitary matrix models and the hierarchy of tau functions,” Nucl. Phys. B 354, 496-530 (1991)
  * [38] L. Santilli and M. Tierz, “Multiple phases and meromorphic deformations of unitary matrix models,” [arXiv:2102.11305 [hep-th]].
  * [39] S. Cecotti and C. Vafa, “Classification of complete N=2 supersymmetric theories in 4 dimensions,” Surveys in differential geometry 18 (2013) [arXiv:1103.5832 [hep-th]].
  * [40] G. Bonelli, K. Maruyoshi and A. Tanzini, “Wild Quiver Gauge Theories,” JHEP 02, 031 (2012) [arXiv:1112.1691 [hep-th]].
  * [41] H. Itoyama, T. Oota and K. Yano, “Multicritical points of unitary matrix model with logarithmic potential identified with Argyres–Douglas points,” Int. J. Mod. Phys. A 35, no.24, 2050146 (2020) [arXiv:1909.10770 [hep-th]].
  * [42] V. Periwal and D. Shevitz, “Unitary Matrix Models As Exactly Solvable String Theories,” Phys. Rev. Lett. 64, 1326-1329 (1990).
  * [43] V. Periwal and D. Shevitz, “Exactly Solvable Unitary Matrix Models: Multicritical Potentials and Correlations,” Nucl. Phys. B 344, 731-746 (1990).
  * [44] V. A. Kazakov, “The Appearance of Matter Fields from Quantum Fluctuations of 2D Gravity,” Mod. Phys. Lett. A 4, 2125 (1989)
  * [45] R. C. Myers and V. Periwal, “Exact Solution of Critical Self-Dual Unitary-Matrix Models,” Phys. Rev. Lett. 65, 1088-1091 (1990).
  * [46] Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Translation 1954 (1954), no. 104.
  * [47] Ya. L. Geronimus, Polynomials Orthogonal on a Circle and Interval, Translated from the Russian by D. E. Brown, Ed. by I. N. Sneddon, International Series on Applied Mathematics, Vol. 18, Pergamon Press, London (1960).
  * [48] Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, Translations, Ser. 1, Vol. 3, Amer. Math. Soc. (1962).
  * [49] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Colloquium Publications (Amer. Math. Soc.), Vol. 54, Amer. Math. Soc., Province, Rhode Island (2005).
  * [50] L. Alvarez-Gaumé, “Random surfaces, statistical mechanics and string theory,” Helv. Phys. Acta 64, 359-526 (1991).
  * [51] H. Itoyama and Y. Matsuo, “Noncritical Virasoro algebra of d $<$ 1 matrix model and quantized string field,” Phys. Lett. B 255, 202-208 (1991)
  * [52] H. Itoyama, “Loop Equation, Matrix Model and Supersymmetry,” Soryushiron Kenkyu Electronics, 86, no.4, D8-D23 (1993)
  * [53] A. Mironov and A. Morozov, “On the origin of Virasoro constraints in matrix models: Lagrangian approach,” Phys. Lett. B 252, 47-52 (1990)
  * [54] F. David, “Loop Equations and Nonperturbative Effects in Two-dimensional Quantum Gravity,” Mod. Phys. Lett. A 5, 1019-1030 (1990)
  * [55] D. J. Gross and A. A. Migdal, “A Nonperturbative Treatment of Two-dimensional Quantum Gravity,” Nucl. Phys. B 340, 333-365 (1990)