NITEP 196

February, 2024

Large order behavior near the AD point:

the case of $\mathcal{N}=2$, $su(2)$, $N_{f}=2$

Chuan-Tsung Chana***e-mail<EMAIL_ADDRESS>H. Itoyamab,c†††e-mail:
<EMAIL_ADDRESS>and R. Yoshiokab,c‡‡‡e-mail<EMAIL_ADDRESS>
a Department of Applied Physics, Tunghai University,
Taichung, 40704, Taiwan

b Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),
Osaka Metropolitan University

cOsaka Central Advanced Mathematical Institute (OCAMI), Osaka Metropolitan
University

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

Abstract

A non-perturbative effect in $\kappa$ (renormalized string coupling) obtained
from the large order behavior in the vicinity of the prototypical Argyres-
Douglas critical point of $su(2)$, $N_{f}=2$, $\mathcal{N}=2$ susy gauge
theory can be studied in the GWW unitary matrix model with the log term: the
one as the work done against the barrier of the effective potential by a
single eigenvalue lifted from the sea and the other as a non-perturbative
function contained in the solutions of the nonlinear differential equation PII
that goes beyond the asymptotic series. The leading behaviors are of the form
$\exp(-\frac{4}{3}\frac{1}{\kappa}\,(1,\left(\frac{s}{K}\right)^{\frac{3}{2}}))$
respectively. We make comments on their agreement.

##  1 Introduction

Instanton expansion has continued to play a basic role in the development of
supersymmetric gauge theory ever since the exact determination of low energy
effective action (LEEA) of $\mathcal{N}=2$ SUSY gauge theory in four spacetime
dimensions [1, 2]. String theoretic corrections have been put in accord with
mathematics which enables a proper instanton counting, and which avoids volume
divergence [3, 4, 5]. The $\beta$-deformed matrix models of eigenvalue type
are seen to generate all these machineries in a concise manner [6, 7] as the
partition functions in general are identified with integral representation [8,
9, 10] of the regular and irregular $2d$ conformal blocks, which in turn
successfully generate [10, 11] the $4d$ instanton expansion [12]. In this
letter, we focus upon another well-known expansion, namely, that in $1/N$ in
the matrix model side which translates into that in $g_{s}$ for fixed flavor
masses (eq.(2.6)). We will search for a non-perturbative effect associated at
$\beta=1$ which is much less explored in gauge theory while, in the theory of
$2d$ gravity, this has been a central theme [13, 14, 15, 16]. A key ingredient
is the presence of a critical singularity in the large $N$ limit and the
attendant double scaling limit that enhances the contributions from higher
orders in $1/N$.111For non-perturbative effects other than this one, see, for
example, [17, 18]

In a series of papers [11, 19, 20, 21, 22, 23], we obtained, among other
things, the one-matrix model for irregular $2d$ conformal block that
corresponds to $4d$ $\mathcal{N}=2$, $N_{f}=2$, $su(2)$ instanton partition
function. See also [24, 25, 26, 27, 28]. In obtaining this from $2d$ conformal
block for $N_{f}=4$ by a limiting procedure with contour deformation involved,
we observed that the model naturally turned into a unitary matrix model of GWW
type [29, 30, 31] with the addition of the log potential. The notion of the
double scaling limit is effective here also as the moduli space of vacua is
known to contain an IR fixed point of Argyress-Douglas type [32, 33, 34, 35].
In the case of $\beta=1$, we have been successful in deriving the Painlevé II
(PII) equation with an accessory parameter in the double scaling limit from
this matrix model, using the method of orthogonal polynomials followed by a
set of recursion relations (string eqs.). We find it necessary to work on the
partition function without specifying the filling fraction, namely, the
Coulomb moduli $a$. This one is energetically favored and the preference for
this alternative over the instanton partition function has been the common
practice known in the study at $N_{f}=4$ [36, 37] that predates the current
one.

The above procedure serves as the prototypical study of the vicinity of the AD
point, using the matrix model. In the higher rank case where the multi-matrix
model of quiver type is relevant [7, 38], we have resorted to the symmetry
argument to identify the AD point [39, 40]. The “unitarization” made by the
same limiting procedure as in the $su(2)$ case has been crucial to make the
point of maximal symmetry in flavor mass moduli manifest.

In this paper, we continue our prototypical study at $N_{f}=2$. The equation
PII itself can not only generate $1/N$ expansion in the scaling limit as an
asymptotic one but also contains a non-perturbative piece on top of it. This
piece can be exhibited for large scaling variable denoted by $s$ in this
letter by a simple analysis in what follows. After giving a minimal account of
formulae for this introduction in section two (leaving the derivations to the
original papers), we extract from PII for the susceptibility function $u(s)$ a
non-perturbative piece contained beyond the asymptotic series in section
three. This is a well-known WKB analysis applied to PII. We pay a particular
attention to the parameter dependence of the scaling variable and that of the
scaling function. We make a brief digression on the dependency on parameters
associated with the canonical transformation from $(u,p_{u})$ to $(y,p_{y})$,
which puts PII in the standard form in mathematics literature [41].

In matrix models, the double scaling limit is taken in the vicinity of the
singularity associated with the maximum of a potential. In section four, we
study the work done against the barrier of the effective potential by a single
eigenvalue lifted from the sea of the filled ones in the continuum limit. A
closely related object is the vacuum to vacuum tunneling fluctuation where a
single eigenvalue hops from one edge of the eigenvalue distribution to the
other through the forbidden zone, namely outside the cut.

In section five, we compare the result on the non-perturbative piece in $u(s)$
in section three and that of section four. For that to be possible, this non-
perturbative function must be evaluated at a particular value. We see that the
evaluation at the point which is actually the end point of the free energy
integral in the double scaling limit makes the two objects agree.

##  2 Some preliminaries

As is stated in the introduction, the model which is relevant to our problem
is the unitary matrix model

$Z_{U(N)}=\frac{1}{\mathrm{Vol}(N)}\int[\mathrm{d}U]\exp\Big{(}\mathrm{Tr}\,W_{U}(U)\Big{)},$
(2.1)

with the potential222For an extension of the model and related aspects, see,
for example, [42, 43, 44, 45, 46, 47, 48, 49, 50].

$W_{U}(w)=-q_{02}\left(w+\frac{1}{w}\right)+M\log w.$ (2.2)

The relation of the parameters $N$, $M$ and $q_{02}$ with the 4d parameters
are listed below.

###  2.1 0d-4d connection [10, 11]

In the $\beta$ deformed matrix model for the $N_{f}=2$, there are four net 0d
parameters aside from $q_{02}$,

$\alpha_{1+2},~{}~{}\alpha_{3+4},~{}~{}\beta,~{}~{}N_{L},~{}~{}N_{R},~{}~{}(N\equiv
N_{L}+N_{R}),$ (2.3)

under the constraint:

$\alpha_{1+2}+\alpha_{3+4}+2\sqrt{\beta}N=2Q_{E},~{}~{}~{}~{}~{}c=1-6Q_{E}^{2},~{}~{}Q_{E}=\sqrt{\beta}-\frac{1}{\beta}.$
(2.4)

The number of free parameters reduces to two by setting $\beta=1$ and just
referring to $N$.

Likewise, there are four 4d parameters aside from $\Lambda_{2}$, the 4d scale
parameter:

$\frac{\epsilon_{1}}{g_{s}},~{}\frac{a}{g_{s}},~{}\frac{m_{1}}{g_{s}},~{}\frac{m_{2}}{g_{s}},~{}~{}~{}~{}~{}\epsilon_{1}=\sqrt{\beta}g_{s},~{}~{}\epsilon_{2}=-\frac{g_{s}}{\sqrt{\beta}},~{}~{}\epsilon=\epsilon_{1}+\epsilon_{2}.$
(2.5)

The number reduces to two by setting $\epsilon_{1}/g_{s}=1$ and not referring
to $a$.

These parameters are related by the 0d-4d dictionary:

$\begin{split}&\alpha_{1+2}=\frac{1}{g_{s}}\left(2m_{2}+\epsilon\right),\\\
&\alpha_{3+4}=\frac{1}{g_{s}}\left(2m_{1}+\epsilon\right),\\\
&N=\frac{a-m_{2}}{g_{s}}-\frac{a+m_{1}}{g_{s}}=-\frac{m_{1}+m_{2}}{g_{s}}.\end{split}$
(2.6)

Separately, the parameter for the 4d instanton expansion reads

$q_{02}=\frac{\Lambda_{2}}{2g_{s}}\equiv\frac{1}{2}\frac{1}{\underline{g}_{s}}.$
(2.7)

and $M$ in eq. (2.2) is related to the 4d parameter by

$M\equiv\alpha_{3+4}+N=\frac{m_{1}-m_{2}}{g_{s}}.$ (2.8)

We have seen in eq. (2.6) and eq. (2.8) that, by unitarization, 4d meaning of
the two independent 0d parameters $N$, $M$ has become manifest.

###  2.2 String equation from orthogonal polynomials [19, 20, 21]

The partition function with no specification of the filling fraction, which is
denoted by $\underline{Z}_{U(N)}$, can be expressed in terms of the orthogonal
polynomials [51, 52]:

$\underline{Z}_{U(N)}=h_{0}^{N}\prod_{j=1}^{N-1}(1-R_{j}^{2})^{N-j},$ (2.9)

where

$p_{n}(0)\equiv A_{n}=R_{n}D_{n},~{}~{}~{}~{}~{}~{}\tilde{p}_{n}(0)\equiv
B_{n}=\frac{R_{n}}{D_{n}},$ (2.10)

and $h_{0}$ is normalization coefficient. The product $R_{n}=A_{n}B_{n}$ is
going to be identified as a susceptibility.

Let us introduce $\xi_{n}\equiv R_{n}^{2}$, $\eta_{n}\equiv
n\underline{g}_{s}$, $\zeta\equiv M\underline{g}_{s}$. We obtain the string
equation:

$\displaystyle 0=$
$\displaystyle\eta_{n}^{2}\Bigl{[}\xi_{n}^{2}(1-\xi_{n})^{2}-\eta_{n}^{2}\,\xi_{n}^{2}+\zeta^{2}(1-\xi_{n})^{2}\Bigr{]}$
$\displaystyle+\frac{1}{2}\,\eta_{n}^{2}\,\xi_{n}\,(1-\xi_{n})^{2}(\xi_{n+1}-2\,\xi_{n}+\xi_{n-1})-\frac{1}{16}(1-\xi_{n})^{4}(\xi_{n+1}-\xi_{n-1})^{2}.$
(2.11)

###  2.3 The double scaling limit [19, 20, 21]

In the planar limit, the string equation (2.2) becomes quartic in $\xi$. The
three out of the four roots in $\xi$ degenerate to zero at $\eta=\pm
1,~{}\zeta=0$.

Let us rewrite the potential for the sake of taking a scaling limit:

$W_{N}(w)=\frac{N}{\tilde{S}}\left\\{-\frac{1}{2}\left(w+\frac{1}{w}\right)+\zeta\log
w\right\\}.$ (2.12)

Here, we have chosen to regard the two parameters as

$\tilde{S}\equiv N\underline{g}_{s}=-\frac{m_{1}+m_{2}}{\Lambda_{2}},$ (2.13)

and

$\zeta\equiv M\underline{g}_{s}=\frac{m_{1}-m_{2}}{\Lambda_{2}},$ (2.14)

while $1/N$ is the expansion parameter.

The double scaling limit is the limit where the fine tuning of $\tilde{S}$
(the critical scaling) and the large $N$ limit are simultaneously arranged so
as to turn the 2nd difference in eq.(2.2) into the 2nd derivative. Let

$x=\frac{n}{N},~{}~{}~{}\eta_{n}=\tilde{S}x=1-\frac{1}{2}a^{2}s,~{}~{}~{}\xi_{n}=\xi(n/N)\equiv
a^{2}u(s),~{}~{}~{}\zeta=\frac{M\tilde{S}}{N}.$ (2.15)

Setting $a^{3}=\frac{1}{N}$, we obtain the Painlevé II equation from eq.
(2.2):

$u^{\prime\prime}=\frac{(u^{\prime})^{2}}{2u}+u^{2}-\frac{1}{2}su-\frac{M^{2}}{2u}.$
(2.16)

##  3 WKB analysis to PII linearized

###  3.1 Non-perturbative function to $u$ asymptotic series

In this subsection, we consider a non-perturbative effect associated with PII
in the form of eq. (2.16) that goes beyond the asymptotic series.333For the
analysis of PI at 2d gravity, [53, 54, 55, 56]. The series begins with the
planar solution:

$u_{\rm asym}=\frac{1}{2}s+\sum_{n=1}^{\infty}u_{n}s^{-n+1}.$ (3.1)

Let $u$ and $\tilde{u}$ be the two solutions with the same asymptotics and

$\underline{z}\equiv\tilde{u}-u,~{}~{}~{}~{}~{}|\underline{z}|\ll|u|,~{}|\tilde{u}|.$
(3.2)

After linearization, we obtain

$\underline{z}^{\prime\prime}=\left(\frac{u^{\prime}}{u}\right)\underline{z}^{\prime}-\frac{1}{2}\frac{u^{\prime
2}}{u^{2}}\underline{z}+\left(2u-\frac{1}{2}s+\frac{M^{2}}{2}\frac{1}{u^{2}}\right)\underline{z}.$
(3.3)

Let the WKB ansatz be

$\frac{\underline{z}^{\prime}}{\underline{z}}\equiv h\sqrt{u}+c(\log
u)^{\prime}+\cdots.$ (3.4)

with $h$ and $c$ to be determined. Taking a derivative of eq. (3.4), using
eqs. (3.3), (3.4) and the planar solution

$u\approx\frac{1}{2}s,$ (3.5)

for large $s$, we obtain, to the leading order, $h^{2}=1$. In the next leading
order, we obtain $c=\frac{1}{4}$. Integrating (3.4) once more, we determine
the non-perturbative function up to the multiplicative factor

$\underline{z}\approx(\text{const})\,s^{\frac{1}{4}}\exp\left(-\frac{1}{\sqrt{2}}\frac{2}{3}s^{\frac{3}{2}}\right).$
(3.6)

###  3.2 Parameter dependence of PII

In order to make exploit our simple result eq. (3.6) later, we would like to
better understand the parameter dependence of PII and physical quantities upon
the double scaling limit. Let us recall that the finite $N$ free energy of the
matrix model is given by

$\displaystyle\tilde{W}_{N}\equiv N^{2}\log
F=-N\log\left(I_{M}(1/\underline{g_{s}})\right)+W_{N},$ (3.7) $\displaystyle
W_{N}=-N^{2}\sum_{n=1}^{N}\frac{1}{N}\left(1-\frac{n}{N}\right)\log(1-\xi_{n}),$
(3.8)

where $I_{M}(x)$ is the modified Bessel function [20]. Here, we work on a more
general parametrization of the double scaling limit than eq. (2.15):

$N=a^{-3},~{}~{}~{}~{}~{}\eta\equiv 1-\frac{1}{2}\alpha
a^{2}s,~{}~{}~{}~{}~{}\xi(\eta)\equiv\beta
a^{2}u(s),~{}~{}~{}~{}~{}\tilde{S}=1-\kappa^{-\frac{2}{3}}a^{2}.$ (3.9)

Here, we have introduced the two parameters $\alpha$ and $\beta$ as the
multiplicative factors of the scaling variable $s$ and the scaling function
$u$ respectively and $\kappa=\frac{1}{(1-\tilde{S})^{3/2}N}$ is the
renormalized string coupling. Note that the interval $[0,\tilde{S}]$ for
$\eta$ translates into the interval $[\infty,K]$ for $s$ with

$K=\frac{2\kappa^{-\frac{2}{3}}}{\alpha}.$ (3.10)

The free energy in the double scaling limit is given by

$W=\lim_{N\to\infty}W_{N}=\frac{1}{4}\alpha^{2}\beta\int_{K}^{\infty}\mathrm{d}s\,(s-K)u(s).$
(3.11)

Eq. (2.16) of PII gets scaled accordingly:

$u^{\prime\prime}=\frac{(u^{\prime})^{2}}{2u}+\alpha^{2}\beta
u^{2}-\frac{1}{2}\alpha^{3}su-\frac{\alpha^{2}M^{2}}{2\beta^{2}u}.$ (3.12)

We conclude that, for (3.12), eq. (3.6) is replaced by

$\underline{z}(s)\approx(\text{const})\,(\alpha
s)^{\frac{1}{4}}\exp\left(-\frac{1}{\sqrt{2}}\frac{2}{3}(\alpha
s)^{\frac{3}{2}}\right).$ (3.13)

###  3.3 Canonical transformation of PII

We have also investigated another form of (2.16) known in mathematics444See,
for example, [57, 41]. by a canonical transformation, paying a good attention
to its dependency on parameters. While we will not make exploit this later, we
include here the upshot briefly. There is one-parameter family of Hamiltonians
$H(u,p_{u})$ that derives (2.16):

$H(u,p_{u})=\frac{\tilde{\alpha}}{2}p_{u}^{2}u-\frac{1}{2\tilde{\alpha}}u^{2}+\frac{1}{2\tilde{\alpha}}su-\frac{M^{2}}{2u\tilde{\alpha}}.$
(3.14)

Next, we make a canonical transformation to the system which possesses a
standard kinetic term up to coefficient. This is done by

$u=-\frac{1}{\xi}p_{y},~{}~{}~{}~{}~{}p_{u}=\xi y+\eta\frac{M}{p_{y}}.$ (3.15)

We obtain

$\hat{H}(y,p_{y})=-\frac{1}{2\tilde{\alpha}\xi^{2}}p_{y}^{2}-\frac{\tilde{\alpha}\xi}{2}p_{y}y^{2}-\frac{1}{2\tilde{\alpha}\xi}sp_{y}-\tilde{\alpha}\eta
My.$ (3.16)

Here, we discarded the term proportional to $1/p_{y}$, setting

$\eta^{2}=\frac{\xi^{2}}{{\tilde{\alpha}}^{2}}.$ (3.17)

Eq. of motion for $y$ is

$y^{\prime\prime}=\frac{\tilde{\alpha}^{2}\xi^{2}}{2}y\left(y^{2}+\frac{s}{\tilde{\alpha}^{2}\xi^{2}}\right)-\frac{\eta}{\xi^{2}}M-\frac{1}{2\tilde{\alpha}\xi}.$
(3.18)

The Lagrangian for the $(u,p_{u})$ system and that of the $(y,p_{y})$ are
respectively,

$\displaystyle
L(u,u^{\prime})=\frac{1}{2\tilde{\alpha}}\frac{(u^{\prime})^{2}}{u}+\frac{1}{2\tilde{\alpha}}u^{2}-\frac{1}{2\tilde{\alpha}}su+\frac{M^{2}}{2\tilde{\alpha}u},$
(3.19)
$\displaystyle\hat{L}(y,y^{\prime})=-\frac{1}{2}\tilde{\alpha}\xi^{2}(y^{\prime})^{2}-\frac{\tilde{\alpha}^{2}\xi^{3}}{2}y^{\prime}\left(y^{2}+\frac{s}{\tilde{\alpha}^{2}\xi^{2}}\right)-\frac{1}{8}\tilde{\alpha}^{3}\xi^{4}\left(y^{2}+\frac{s}{\tilde{\alpha}^{2}\xi^{2}}\right)^{2}+\tilde{\alpha}\eta
My.$ (3.20)

###  3.4 Non-perturbative function to $y$ asymptotic series

In eq. (3.18) for $y$ variable, the asymptotic expansion goes like

$y_{\rm asym}=\frac{1}{|\tilde{\alpha}\xi|}\sqrt{-s}+\cdots.$ (3.21)

This is an asymptotic series distinct from $u_{\rm asym}$ of eq. (3.1) that
begins with the planar solution of our original problem, and probes another
corner of the solution space of PII. The way in which the non-perturbative
piece is investigated is completely analogues to that in sec 3.1. Let $y$ and
$\tilde{y}$ be two solutions with the same asymptotics,

$z\equiv\tilde{y}-y,~{}~{}~{}~{}~{}|z|\ll|y|,~{}|\tilde{y}|,$ (3.22)

and we linearize eq. (3.18) with respect to $z$. We just give our final result
here:

$z\approx(\text{const})(-s)^{-\frac{1}{4}}\exp\left\\{-\frac{2}{3|\tilde{\alpha}\xi|}(-s)^{\frac{3}{2}}\right\\}.$
(3.23)

##  4 Direct evaluation of the exponential factor

Let us turn our attention back to the original unitary matrix model and study
work done by a single eigenvalue as is stated in the introduction. The
computation of this object is given through that of the response of the finite
$N$ partition function $Z_{N}$ under a move of the single eigenvalue in the
presence of the remainder of $N-1$ eigenvalues lying inside the cut.

Let $Z_{N-1,\mathrm{1\,max}}$ be the partition function under the constraint
that the single eigenvalue lies on the maximum of the effective potential made
by the remainder of the $N-1$ eigenvalues. The object of our concern denoted
by $S_{N,\,{\rm s.e.v.}}$ is extracted as

$I_{N}\equiv\frac{Z_{N-1,\mathrm{1\,max}}}{Z_{N}}\equiv\mathrm{e}^{-S_{N},\,{\rm
s.e.v.}}.$ (4.1)

The square of eq. (4.1) may be regarded as a tunneling amplitude.

To evaluate this in the large $N$ limit, we find it more expedient to convert
the original distribution on the unit circle into the one on the real axis,
namely, to work actually on a Hermitian matrix model . This goes as follows.

Recall that the finite $N$ partition function of our original unitary matrix
model is

$Z_{U(N)}=\frac{1}{\mathrm{Vol}(N)}\int[\mathrm{d}U]\exp\Big{(}\mathrm{Tr}\,W_{U}(U)\Big{)},$
(4.2)

and

$(-W_{U})(z)=-\frac{1}{2\underline{g_{s}}}\left(z+\frac{1}{z}\right)-M\log z.$
(4.3)

(Here, we have made a change of variable $z\to-z$ for our convenience.) The
conversion into a Hermitian matrix model is simply given by the following
$SL(2,\mathbf{C})$ transformation:

$z=\frac{\mathrm{i}-w}{\mathrm{i}+w},~{}~{}~{}~{}~{}z=\mathrm{e}^{\mathrm{i}\theta},~{}~{}~{}w\in\mathbf{R},~{}~{}~{}w=\tan\frac{\theta}{2}.$
(4.4)

The resultant Hermitian matrix model takes the form

$(-W)(w)=-\frac{2(1-w^{2})}{1+w^{2}}-2M\underline{g_{s}}\log\frac{\mathrm{i}-w}{\mathrm{i}+w}+2\underline{g_{s}}N\log(1+w^{2}).$
(4.5)

Let us evaluate $I_{N}$ in the planar limit

$\displaystyle I\equiv\lim I_{N}\equiv\mathrm{e}^{-S_{\rm s.e.v.}},$ (4.6)
$\displaystyle S_{{\rm s.e.v.}}=N\int_{\text{end point}}^{\infty,\,\rm
maximum}\mathrm{d}x^{\prime}f(x^{\prime}),$ (4.7) $\displaystyle
f(x^{\prime})=\frac{1}{2\tilde{S}}(-W)^{\prime}(x^{\prime})-2\mathrm{Re}\,\omega_{\rm
pla}(x^{\prime}).$ (4.8)

Here $\omega_{\rm pla}(x^{\prime})$ is the resolvent in the planar limit:

$\omega_{\rm
pla}(x^{\prime})\equiv\lim_{N\to\infty}\frac{1}{N}\left\langle\\!\\!\\!\left\langle\sum_{j=1}^{N}\frac{1}{x^{\prime}-\lambda_{j}}\right\rangle\\!\\!\\!\right\rangle.$
(4.9)

In deriving eqs. (4.7), (4.8) we have used, for the hermitian matrix $H$,

$\left.\log\det(x^{\prime}-H)^{2}\right|_{\rm end\,point}^{\infty}=2\int_{\rm
end\,point}^{\infty}\mathrm{d}x^{\prime}\,\mathrm{tr}\left(\frac{1}{x^{\prime}-H}\right),$
(4.10)

and ignored the size of the matrix by one.

The evaluation of the planar resolvent is straightforward [58, 59] and we
obtain

$\omega_{\rm
pla}(x^{\prime})=\frac{1}{4\tilde{S}}\left((-W)^{\prime}(x^{\prime})-\frac{8\sqrt{x^{\prime
2}-b^{2}}}{\sqrt{1+b^{2}}(1+x^{\prime
2})^{2}}\right),~{}~{}~{}~{}~{}b=\sqrt{\frac{\tilde{S}}{1-\tilde{S}}},$ (4.11)

and eq. (4.8) is given by

$f(x^{\prime})=\frac{4}{\tilde{S}}\frac{\sqrt{x^{\prime
2}-b^{2}}}{\sqrt{1+b^{2}}(1+x^{\prime 2})^{2}},$ (4.12)

and

$S_{\rm
s.e.v.}=\frac{4N}{\tilde{S}}\int_{b}^{\infty}\mathrm{d}x^{\prime}\frac{\sqrt{x^{\prime
2}-b^{2}}}{\sqrt{1+b^{2}}(1+x^{\prime 2})^{2}}.$ (4.13)

Eq.(4.12) up to a factor $2\pi$ coincides with the density of the eigenvalues
extended to the forbidden zone, namely continued outside to the cut. In fact,
the eigenvalue density one-form agrees with that of the original Gross-Witten:

$\frac{1}{2\pi}|f(w)|\mathrm{d}w=\rho_{\rm
GW}(\theta)\mathrm{d}\theta,~{}~{}~{}~{}~{}\rho_{\rm
GW}(\theta)=\frac{1}{\pi\tilde{S}}\cos\frac{\theta}{2}\sqrt{\tilde{S}-\sin^{2}\frac{\theta}{2}}.$
(4.14)

Let us carry out the actual computation of the integral (4.13) in the
continuum limit. Let our scaling ansatz be $\tilde{S}=1-\beta a^{2}t$,
$a^{3}=\frac{1}{N}$ and accordingly we scale $x^{\prime}$ as
$x^{\prime}=\frac{1}{\sqrt{\beta}}a^{-1}\zeta$. We obtain

$S_{\text{s.e.v.}}=\frac{4}{3}(\beta t)^{\frac{3}{2}}.$ (4.15)

The apparent parameter dependence gets resolved by recalling the
renormalized(dressed) string coupling [22, 23]

$\kappa\equiv\frac{1}{N}\frac{1}{(1-\tilde{S})^{\frac{1}{2}(2-\gamma_{st})}}=(\beta
t)^{-\frac{3}{2}},$ (4.16)

and we obtain555For similar consideration and calculation in 2d noncritical
string theory, see [60, 61, 62, 63, 64].

$S_{\rm s.e.v.}=\frac{4}{3}\frac{1}{\kappa}.$ (4.17)

##  5 Comparison

Let us now compare the result eq. (3.13) in section three and that in section
four, eqs. (4.1), (4.6), (4.17). As is stated in the introduction, we will
evaluate $\underline{z}(s)$ at the end point $K$ of the free energy integral
eq. (3.11). We obtain

$\underline{z}(s)=(\text{const})^{\prime}\kappa^{-\frac{1}{6}}\left(\frac{s}{K}\right)^{\frac{1}{4}}\exp\left(-\frac{4}{3}\frac{1}{\kappa}\left(\frac{s}{K}\right)^{\frac{3}{2}}\right),$
(5.1)

and

$\underline{z}(K)=(\text{const})^{\prime}\kappa^{-\frac{1}{6}}\exp\left(-\frac{4}{3}\frac{1}{\kappa}\right).$
(5.2)

The exponential factor agrees with the value of $S_{\rm s.e.v.}$ in (4.17).

## Acknowledgments

We thank Asato Tsuchiya for helpful discussions and insightful remarks. We
also thank Kazunobu Maruyoshi and Jaewon Song for discussions of related
issues on this subject at Shuzenji workshop. The work of H.I. is supported in
part by JSPS KAKENHI (19K03828, 23K03393, 23K03394).

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