Discrete Painleve system for the partition function of $N_f =2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit

TL;DR

This paper explores a discrete Painleve system linked to the partition function of a specific supersymmetric gauge theory, revealing new connections through a matrix model extension and its double scaling limit to study Argyres-Douglas theories.

Contribution

It introduces a novel discrete Painleve system derived from a matrix model extension with a logarithmic potential, advancing the understanding of supersymmetric gauge theories and their critical behavior.

Findings

Identifies a Painleve system associated with the $N_f=2$ $SU(2)$ gauge theory.

Develops a method to analyze the double scaling limit for critical behavior.

Extends orthogonal polynomial techniques to complex matrix models.

Abstract

We continue to study the matrix model of the $N_f =2$ $SU(2)$ case that represents the irregular conformal block. What provides us with the Painlev\'{e} system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross-Witten-Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres-Douglas type theory from IR. We elaborate upon the method of orthogonal polynomial and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential.

Figures

Equations

$$Z(N_L,N_R)=\mathcal{C}\left(\prod _{i=1}^{N_L}{\int }_{C_L}\mathrm{d}{w}_i\right)\left(\prod _{j=1}^{N_R}{\int }_{C_R}\mathrm{d}{w}_{N_L+j}\right){\Delta }^{2\beta }(w)\exp \left(\sqrt{\beta }\sum _{I=1}^{N_L+N_R}W(w_I)\right),$$

$$\Delta (w)=\prod _{1\le I<J\le N_L+N_R}(w_I-w_J).$$

$$\begin{array}{rl}\underline{Z}(N;{\mu }_L,{\mu }_R)& =\sum _{N_L+N_R=N}\frac{{\mu }_L^{N_L}}{N_L!}\frac{{\mu }_R^{N_R}}{N_R!}Z(N_L,N_R)\\ & =\frac{\mathcal{C}}{N!}{\int }_C{\mathrm{d}}^{N}w{\Delta }^{2\beta }(w)\exp \left(\sqrt{\beta }\sum _{I=1}^{N}W(w_I)\right),\end{array}$$

$${\int }_C={\mu }_L{\int }_{C_L}+{\mu }_R{\int }_{C_R}.$$

$$Z_{U(N)}:=\frac{1}{\mathrm{vol}(U(N))}\int[\mathrm{d}U]\,\exp\Bigl{(}\mathrm{Tr}\,W_{U}(U)\Bigr{)},$$

$$\mathrm{d}s^{2}=\mathrm{Tr}\bigl{(}\mathrm{d}U^{{\dagger}}\,\mathrm{d}U\bigr{)}=-\mathrm{Tr}\bigl{(}U^{-1}\mathrm{d}U\bigr{)}^{2}.$$

$$\mathrm{vol}(U(N))=\int [\mathrm{d}U]=\frac{(2\pi {)}^{(1/2)N(N+1)}}{G_2(N+1)},$$

$$G_2(N+1)=\prod _{j=1}^{N-1}j!=\prod _{k=1}^{N-1}{k}^{N-k}.$$

$$U=V^{-1}{U}_{D}V,U_D=\mathrm{diag}(z_1,z_2,\cdots ,z_N),|z_i|=1,$$

$$Z_{U(N)}=\frac{1}{N!}\left(\prod _{i=1}^N\oint \frac{\mathrm{d}{z}_i}{2\pi \mathrm{i}{z}_i}\right)\Delta (z)\Delta (z^{-1})\exp \left(\sum _{i=1}^N{W}_U(z_i)\right),$$

$$\Delta (z)=\prod _{1\le i<j\le N}(z_i-z_j),\Delta (z^{-1})=\prod _{1\le i<j\le N}(z_i^{-1}-z_j^{-1}).$$

$$\mathrm{d}\mu(z):=\frac{\mathrm{d}z}{2\pi\mathrm{i}\,z}\exp\Bigl{(}W_{U}(z)\Bigr{)}.$$

$$Z_{U(N)}=\frac{1}{N!}\int \prod _{i=1}^N\mathrm{d}\mu (z_i)\Delta (z)\Delta (z^{-1}).$$

$$\int \mathrm{d}\mu (z)p_n(z){\tilde{p}}_m(1/z)=h_n{\delta }_{n,m},$$

$$p_n(z)=z^n+\sum _{k=0}^{n-1}{A}_k^{(n)}{z}^k,{\tilde{p}}_n(1/z)=z^{-n}+\sum _{k=0}^{n-1}{B}_k^{(n)}{z}^{-k}.$$

$${\mu }_n:=\int \mathrm{d}\mu (z)z^n,(n\in \mathbb{Z}).$$

$$\mathcal{K}^{(n)}_{k}:=\det\bigl{(}\mu_{j-i+k}\bigr{)}_{1\leq i,j\leq n},\qquad(n\geq 0,k\in\mathbb{Z}).$$

$$\int \mathrm{d}\mu (z)p_n(z)z^{-k}=0,(k=0,1,\cdots ,n-1),$$

$$\int \mathrm{d}\mu (z)z^k{\tilde{p}}_n(1/z)=0,(k=0,1,\cdots ,n-1).$$

$$p_n(z)=\frac{1}{{\tau }_n}\left|\begin{array}{ccccc}{\mu }_0& {\mu }_1& {\mu }_2& \cdots & {\mu }_n\\ {\mu }_{-1}& {\mu }_0& {\mu }_1& \cdots & {\mu }_{n-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mu }_{-n+1}& {\mu }_{-n+2}& {\mu }_{-n+3}& \cdots & {\mu }_1\\ 1& z& z^2& \cdots & z^n\end{array}\right|,$$

$${\tilde{p}}_n(1/z)=\frac{1}{{\tau }_n}\left|\begin{array}{ccccc}{\mu }_0& {\mu }_{-1}& {\mu }_{-2}& \cdots & {\mu }_{-n}\\ {\mu }_1& {\mu }_0& {\mu }_{-1}& \cdots & {\mu }_{-n+1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mu }_{n-1}& {\mu }_{n-2}& {\mu }_{n-3}& \cdots & {\mu }_{-1}\\ 1& z^{-1}& z^{-2}& \cdots & z^{-n}\end{array}\right|,$$

$$\tau_{n}:=\mathcal{K}^{(n)}_{0}=\det\bigl{(}\mu_{j-i}\bigr{)}_{1\leq i,j\leq n}.$$

$$\int \mathrm{d}\mu (z)p_n(z)z^{-k}=\frac{1}{{\tau }_n}\left|\begin{array}{ccccc}{\mu }_0& {\mu }_1& {\mu }_2& \cdots & {\mu }_n\\ {\mu }_{-1}& {\mu }_0& {\mu }_1& \cdots & {\mu }_{n-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mu }_{-n+1}& {\mu }_{-n+2}& {\mu }_{-n+3}& \cdots & {\mu }_1\\ {\mu }_{-k}& {\mu }_{1-k}& {\mu }_{2-k}& \cdots & {\mu }_{n-k}\end{array}\right|=0,$$

$$h_n=\frac{{\tau }_{n+1}}{{\tau }_n}=\frac{{\mathcal{K}}_0^{(n+1)}}{{\mathcal{K}}_0^{(n)}}.$$

$$A_n:=p_n(0)=A_0^{(n)}=(-1{)}^n\frac{{\mathcal{K}}_1^{(n)}}{{\mathcal{K}}_0^{(n)}},B_n:={\tilde{p}}_n(0)=B_0^{(n)}=(-1{)}^n\frac{{\mathcal{K}}_{-1}^{(n)}}{{\mathcal{K}}_0^{(n)}}.$$

$$\frac{h_n}{h_{n-1}}=\frac{{\tau }_{n+1}{\tau }_{n-1}}{{\tau }_n^2},1-\frac{h_n}{h_{n-1}}=\frac{{\tau }_n^2-{\tau }_{n+1}{\tau }_{n-1}}{{\tau }_n^2}.$$

$$\tau_{n}^{2}-\tau_{n+1}\tau_{n-1}=\bigl{(}\mathcal{K}^{(n)}_{0}\bigr{)}^{2}-\mathcal{K}^{(n+1)}_{0}\mathcal{K}^{(n-1)}_{0}=\mathcal{K}^{(n)}_{1}\mathcal{K}^{(n)}_{-1},$$

$$1-\frac{h_n}{h_{n-1}}=\frac{{\mathcal{K}}_1^{(n)}{\mathcal{K}}_{-1}^{(n)}}{({\mathcal{K}}_0^{(n)}{)}^2}=A_n{B}_n.$$

$$\frac{h_n}{h_{n-1}}=1-A_n{B}_n.$$

$$\prod_{1\leq i<j\leq N}(z_{j}-z_{i})=\det\bigl{(}p_{j-1}(z_{i})\bigr{)}_{1\leq i,j\leq N}=\sum_{\sigma\in S_{N}}(-1)^{\varepsilon(\sigma)}\prod_{k=1}^{N}p_{\sigma(k)-1}(z_{k}),$$

Full text

NITEP 1

OCU-PHYS 488

December, 2018

Discrete Painlevé system for the partition function

of $N_{f}=2$ $SU(2)$ supersymmetric gauge theory

and its double scaling limit

H. Itoyamaa,b,c**e-mail: itoyama@sci.osaka-cu.ac.jp, T. Ootaa,b,c*†††e-mail: toota@sci.osaka-cu.ac.jp and Katsuya Yanob‡‡‡e-mail: yanok@sci.osaka-cu.ac.jp

*a**Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),

Osaka City University

*b**Department of Mathematics and Physics, Graduate School of Science,

Osaka City University

c**Osaka City University Advanced Mathematical Institute (OCAMI)

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

Abstract

We continue to study the matrix model of the $N_{f}=2$ $SU(2)$ case that represents the irregular conformal block. What provides us with the Painlevé system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross-Witten-Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres-Douglas type theory from IR. We elaborate upon the method of orthogonal polynomials and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential.

1 Introduction

This is a continuation of the paper [1] by three of us and is preferably read in sequel. In the previous letter, we considered the partition function of $N_{f}=2$, $SU(2)$, $\mathcal{N}=2$ supersymmetric gauge theory in four dimensions by the one-matrix model obtained from a generic irregular conformal block [2, 3, 4] as an integral representation [5]. See also [6, 7, 8] for $N_{f}=4$ conformal case. We derived the discrete Painlevé system from a set of recursion relations for $R_{n}$ which is directly related to the finite $N$ partition function. (See eq. (3.45) of this paper.) Here, $N$ is the size of the matrix and, at the same time, the sum of the masses of the two flavors $-(m_{1}+m_{2})/g_{s}$. Taking the double scaling limit [9, 10, 11] to the critical point of this system, we succeeded in deriving the Painlevé II differential equation containing the parameter $M\equiv\alpha_{1+2}+N=(m_{2}-m_{1})/g_{s}$. See [12, 8, 5, 1] and Appendix F of this paper for detail of these transcriptions or matrix model/4d gauge theory dictionary. This critical point is the simplest example of the Argyres-Douglas type points [13, 14, 15, 1] well studied by using the Seiberg-Witten curve of cubic or quartic type [16, 17]. From the point of view of matrix models per se, this is the extension of the Gross-Witten-Wadia model [18, 19, 20] by the addition of a logarithmic potential that turns out to keep the planar critical behavior intact. Here in this paper, we fill in the details of these derivations by elaborating upon the method of orthogonal polynomials [21, 22, 23, 24] which we further develop here in the case of a generic unitary matrix model and that of a special case relevant to the problem here where the potential contains a logarithm.

The original gauge theory partition function is an instanton partition function [25, 26] of ${\cal N}=2$ $SU(2)$ theory which contains a Coulomb-moduli $=$ filling fraction parameter. It is this partition function which is precisely identified with the irregular conformal block by the $2$d-$4$d connection via matrices. The partition function which is represented as a discrete Painlevé system is, however, NOT of this kind, but instead given by a finite $N$ analog of the Fourier transform [1]. This important difference appears to provide a new arena of escaping from instanton vacua to move to the construction of a more general generating function. This last point has been discussed in [1], taking the corresponding structure [27, 28, 29, 30, 31, 32, 33, 34] and the reasoning [35] at $N_{f}=4$ as a hint. We elaborate upon this in Appendix F of this paper as well.

Most generically, let us consider the following type of the $\beta$-deformed “matrix model” depending on two integration contours $C_{L}$ and $C_{R}$:

[TABLE]

where $\mathcal{C}$ is a normalization constant and $\Delta(w)$ is the Vandermonde determinant:

[TABLE]

Following [35], we introduce their generating function

[TABLE]

where $C=\mu_{L}C_{L}+\mu_{R}C_{R}$, i.e.,

[TABLE]

As we recall from [1] and elaborate further in Appendix F, in the case of $\beta=1$ which we consider in the body of this paper, the $N_{f}=2$ matrix model of the above form with $\alpha_{1+2}\in\mathbb{Z}$ in fact reduces to the unitary matrix model with cosine $+$ logarithmic potential in section 3.

In the next section, we consider a generic unitary one-matrix model and utilize the method of orthogonal polynomials. In section 3, we study properties of the unitary matrix model with the above potential at finite $N$ in depth: these are string equations as a set of recursion relations, small $\underline{g}{}_{s}$ expansion, and the free energy. In section 4, we consider the planar and the double scaling limit of the model, which leads us to the derivation of the Painlevé II equation. In Appendix A$\sim$H, we elaborate upon several contents in the text, and review a few of the past work of relevance.

It is instructive to count the number of free parameters, starting from the $N_{f}=4$ case. Letting aside the coupling constant (or the cross ratio), we have six parameters in the original [math]d-$4$d dictionary [8]. After sending two of the four mass parameters to infinity, we have four parameters. By the transition from the instanton partition function to the generating function denoted in this paper by $\underline{Z}$, we get rid of the Coulomb moduli parameter. Finally setting $\beta=1$, the number of free parameters reduces to two. These are nothing but the sum and the difference of the two remaining mass parameters, letting aside the QCD like scale parameter obtained by the dimensional transmutation.

2 Unitary matrix model

In this section, we briefly review the unitary matrix model, the method of orthogonal polynomials and the string equations to explain our notation.

The partition function of the unitary matrix model is defined by

[TABLE]

where $U$ is an $N\times N$ unitary matrix and $W_{U}(U)$ is a potential. We define a unitary Haar measure $[\mathrm{d}U]$ from the metric

[TABLE]

With this normalization of the measure, the volume of the unitary group $U(N)$ is given by (see Appendix A for derivation)

[TABLE]

where $G_{2}(z)$ is the Barnes function. Explicitly, $G_{2}(N+1)$ is given by

[TABLE]

If we diagonalize the unitary matrix $U$ as

[TABLE]

we have

[TABLE]

where

[TABLE]

Let

[TABLE]

Then

[TABLE]

The partition function (2.6) expressed in eigenvalue integrals may be generalized to the form of the two-contour model (1.1). A natural choice of the two contours $C_{L}$ and $C_{R}$ is to take them as circles of radius $r_{L}$ and $r_{R}$ respectively. Suppose, $r_{L}<r_{R}$ and there is no singularity in the region $r_{L}\leq|w|\leq r_{R}$. Then the contours can be smoothly deformed to circles of the same radius, i.e., to the same contour: $C_{L}=C_{R}$. Then, for the two-contour unitary matrix model, $Z_{U(N)}(N_{L},N_{R})$ depends only on $N=N_{L}+N_{R}$, and the generating function $\underline{Z}{}_{U(N)}$ is essentially $Z_{U(N)}(N,0)$. Because $\underline{Z}{}_{U(N)}=(\mu_{L}+\mu_{R})^{N}Z_{U(N)}$, we can set $\mu_{L}+\mu_{R}=1$ without loss of generality. Hence $\underline{Z}{}_{U(N)}=Z_{U(N)}$.

2.1 Orthogonal polynomials

The unitary matrix model can be solved [23, 24, 36] by the method of orthogonal polynomials [21, 22]. For orthogonal polynomials on a unit circle, see, for example, [37, 38, 39], and references therein111The raising and lowering operators for orthogonal polynomials on the unit circle are considered in [40].. We will use the monic orthogonal polynomials [23, 24]222In [36], orthogonal polynomials of different type have been introduced to solve the unitary matrix model.. (See also [39].)

2.1.1 Definitions and properties

Let $p_{n}$ and $\tilde{p}_{n}$ ($n\geq 0$) be monic polynomials satisfying orthogonality conditions with respect to the measure (2.8)

[TABLE]

where

[TABLE]

Let us introduce the moments $\mu_{n}$ for the measure (2.8) by

[TABLE]

For later convenience, we define $\mathcal{K}^{(n)}_{k}$ by

[TABLE]

From the definition, the orthogonal polynomials have the following properties:

[TABLE]

[TABLE]

Using these and the monic properties, the orthogonal polynomials are determined as

[TABLE]

[TABLE]

where

[TABLE]

(We set $\tau_{0}=1$). We can easily see that these polynomials obey (2.14) or (2.15). For example,

[TABLE]

for $0\leq k\leq n-1$.

The normalization constants $h_{n}$ defined by (2.10) are given by

[TABLE]

The constant terms of these polynomials will play important roles.

[TABLE]

Note that

[TABLE]

Using an identity

[TABLE]

we can show that

[TABLE]

Thus we have the following relations:

[TABLE]

2.1.2 Partition function and the orthogonal polynomials

Note that

[TABLE]

[TABLE]

Using these relations, the partition function (2.6) is evaluated as

[TABLE]

Also, it can be written as

[TABLE]

2.1.3 Recursion relations for orthogonal polynomials

If we expand $z\,p_{n}(z)$ in the $\{p_{k}(z)\}$ basis, all of the lower degree polynomials are generated:

[TABLE]

where

[TABLE]

Similarly, we have for $\tilde{p}_{n}$

[TABLE]

where

[TABLE]

The above relations (2.30) and (2.32) can be rewritten as three-term relations:

[TABLE]

[TABLE]

These are called Szegő recursion equations in [39]. The inverse Szegő recursion equations are given by

[TABLE]

[TABLE]

Combining these recursion equations, we have the following three-term relations [39]:

[TABLE]

[TABLE]

2.2 String equations

Recall that

[TABLE]

Using the following constraints for $k\in\mathbb{Z}$ and $\ell,m\geq 0$,

[TABLE]

we can obtain various polynomial equations for $A_{n}$ and $B_{n}$.

In particular, let us consider the following three cases of (2.41): (i) $(k,\ell,m)=(-1,n,n-1)$, (ii) $(k,\ell,m)=(0,n,n)$ and (iii) $(k,\ell,m)=(1,n-1,n)$. They lead to the “string equations”

[TABLE]

Here we have used

[TABLE]

For derivation of these equations, see Appendix B.1.

3 Unitary matrix model with logarithmic potential

Let us consider the unitary matrix model with the following potential333The matrix model with this potential was considered in [41] where the integrations over the eigenvalues are taken along the real axis.

[TABLE]

This model is studied in [42, 43, 44]. The connection with the Painlevé III equation is shown in [43]. In the gauge theory parameters, $\underline{g}{}_{s}=g_{s}/\Lambda_{2}$ and $M=\alpha_{1+2}+N=(m_{2}-m_{1})/g_{s}$. See Appendix F for our notation. We assume that $M$ is an integer. Note that $1/(2\underline{g}{}_{s})=\Lambda_{2}/(2g_{s})=q_{02}$.

When $M=0$, this is the Gross-Witten-Wadia model [18, 19, 20]444 See, for example, [45] and references therein. It is also known that the GWW model has connection with the Painlevé III equation [46, 47].. In this paper, we think $M$ as a finite parameter. Then, in the large $N$ limit, the planar free energy of this model does not depend on $M$ and equals that of the GWW model. Hence this model in the large $N$ also has a third order phase transition at $\widetilde{S}:=N\,\underline{g}{}_{s}=1$555 Another type of generalization of the GWW model can be found, for example, in [48]. The phase transition of the asymmetric GWW model is studied in [49]. The GWW model with boundary terms is examined in [50, 51]..

3.1 Moments and related quantities

The moments of this potential are given by

[TABLE]

where $I_{\nu}(z)$ is the modified Bessel function of the first kind:

[TABLE]

Note that

[TABLE]

where

[TABLE]

For an integer $k$, it holds that $I_{-k}(z)=I_{k}(z)$. Therefore, for $j-i+M+k\in\mathbb{Z}$, we have $I_{j-i+M+k}(1/\underline{g}{}_{s})=I_{|j-i+M+k|}(1/\underline{g}{}_{s})$. Also, we have $K^{(n)}_{-k}=K^{(n)}_{k}$ ($k\in\mathbb{Z}$). For later convenience, we have defined $K^{(n)}_{\nu}$ (3.5) as a determinant of $I_{j-i+\nu}(1/\underline{g}{}_{s})$ such that the index $M$ in $K^{(n)}_{M+k}$ can be analytically continued from an integer to any complex number.

Note that

[TABLE]

The normalization constants of the orthogonal polynomials are given by

[TABLE]

In particular, $h_{0}=(-1)^{M}\,I_{M}(1/\underline{g}{}_{s})$.

The constant terms of the orthogonal polynomials are written in terms of $K^{(n)}_{k}$ as follows:

[TABLE]

[TABLE]

The partition function (2.29) can be written in terms of these objects:

[TABLE]

3.2 The partition function as the tau function of PIII*′*

The partition function (3.10) is essentially the tau function of the Painlevé III*′* equation666For the appearance of PIII in Ising system, see [52, 53, 54]. .

For $q=q(s)$ and $p=p(s)$, let us choose the Hamiltonian for the PIII*′* as

[TABLE]

The Hamiltonian system

[TABLE]

is equivalent to the PIII*′* equation

[TABLE]

with

[TABLE]

The tau function $\tau(s)=\tau_{\mathrm{III}^{\prime}}(s)$ for PIII*′* is introduced by

[TABLE]

Let

[TABLE]

This function satisfies the $\sigma$-form of PIII*′*:

[TABLE]

Combining eq.(4.8) of [55] and Corollary 4.5 of [55]777 The referee of our previous paper [1] called our attention to this Corollary 4.5 of [55]. , we can show that

[TABLE]

is the tau function of PIII*′* with

[TABLE]

Hence, if we set

[TABLE]

the partition function $\underline{Z}{}_{U(N)}(M)=(-1)^{MN}K^{(N)}_{M}$ is the tau function of PIII*′* (up to an overall factor $s^{(1/2)MN}$).

It is well-known that PIII*′* equation is equivalent to the PIII equation

[TABLE]

The corresponding Hamiltonian for $y=y(\mathfrak{t})$ and its conjugate momentum $p_{y}=p_{y}(\mathfrak{t})$ is given by

[TABLE]

The variables and the Hamiltonian of PIII*′* are related to those of PIII by

[TABLE]

Hence, we find the connection between the coupling constant $\underline{g}{}_{s}$ and the time variable $\mathfrak{t}$ of PIII equation: $\mathfrak{t}=1/(2\underline{g}{}_{s})$.

The Bäcklund transformations of the PIII1 (or PIII($D_{6}^{(1)}$) ) form the affine Weyl group of type $(2A_{1})^{(1)}$. The translation subgroup generates the alt-dPII equation. The translations lead to integer shifts of parameters $v_{1,2}$. In terms of the gauge theory parameters, they correspond to constant shifts of mass parameters $m_{1,2}$.

3.3 String equations

Let us write the string equations (2.42), (2.43) and (2.44) explicitly for the case of the potential (3.1). Since

[TABLE]

we have

[TABLE]

[TABLE]

[TABLE]

Here we have used (2.30) and (2.32). (See Appendix B.2 for details.)

Then the string equations (2.42), (2.43), (2.44) for this potential become

[TABLE]

Using

[TABLE]

the string equations lead to the following recursion relations for $A_{n}$ and $B_{n}$:

[TABLE]

[TABLE]

With the initial conditions $A_{0}=B_{0}=1$, and

[TABLE]

the remaining constants $A_{n}$ and $B_{n}$ are characterized by the recursion relations (3.30), (3.31). We remark that one of recursion relations (3.30) can be obtained by combining the other of (3.30) with (3.31).

Recall that the modified Bessel function satisfies the following recursion relation:

[TABLE]

By examining (3.31) for $n=0$, we can see that the range of the parameter $M$ in the initial conditions (3.32) can be extended from the integers to any complex numbers. Furthermore,

[TABLE]

indeed solve the string equations (3.30) and (3.31). Here $K^{(n)}_{\nu}$ is defined by (3.5).

3.3.1 Singularity confinement

We show that the string equations (3.30) and (3.31) pass the “singularity confinement criterion” [56] which may be the discrete version of the Painlevé property.

The string equation (3.30) develops a singularity when $A_{n}B_{n}=1$. Let us assume that this occurs for $n=n_{0}$. Using the invariance of the recursion relations under

[TABLE]

we consider the case

[TABLE]

where $\varepsilon$ is a small parameter. The “initial data” $A_{n_{0}-1}$ and $B_{n_{0}-1}$ are finite and obey the relation

[TABLE]

For simplicity we assume that $A_{n_{0}-1}+B_{n_{0}-1}\neq 0$. Then the singularities appear at $n=n_{0}+1$:

[TABLE]

[TABLE]

At $n=n_{0}+2$,

[TABLE]

[TABLE]

hence it again holds that $A_{n_{0}+2}B_{n_{0}+2}=1+O(\varepsilon)$, but there is no singularity at the next level $n=n_{0}+3$:

[TABLE]

[TABLE]

The singularity is confined to the level $n=n_{0}+1$ and the memory of the initial data is kept in crossing this singularity. Therefore, the string equations (3.30) and (3.31) pass the criterion.

We remark that if we replace the recursion relations (3.30) by

[TABLE]

where $a$ and $b$ are arbitrary constants, these equations with (3.31) also pass the singularity confinement test.

3.3.2 Equations for $R_{n}^{2}=A_{n}B_{n}$

Note that the partition function (3.10) depends on $A_{j}$ and $B_{j}$ only through their product $A_{j}B_{j}$. Let $A_{n}=R_{n}D_{n}$ and $B_{n}=R_{n}/D_{n}$. Then the partition function (3.10) becomes

[TABLE]

The equation (3.31) turns into

[TABLE]

This leads to

[TABLE]

[TABLE]

By substituting these relations into the remaining relations (3.30), we find

[TABLE]

This is equivalent to

[TABLE]

where $\xi_{n}\equiv R_{n}^{2}$, $\eta_{n}\equiv n\,\underline{g}{}_{s}$, $\zeta\equiv M\,\underline{g}{}_{s}$.

When $M=0$ (i.e., with no logarithmic potential), (3.49) reduces to the string equation considered in [23]

[TABLE]

3.3.3 Relation with alternate discrete Painlevé II equation

Let us introduce variables $x_{n}$ and $y_{n}$ by

[TABLE]

They respectively obey the alternate discrete Painlevé II equation [57, 58] with different values of the parameter $\tilde{\mu}$. (See Appendix C for details.) With the initial conditions $A_{0}=1$ and $B_{0}=1$, $A_{n}$ and $B_{n}$ can be expressed by these variables:

[TABLE]

3.4 Small $\underline{g}{}_{s}$ expansion

In this subsection, we comment on a perturbative solution for the string equations (3.30), (3.31). We consider the perturbation in $\underline{g}{}_{s}$. We remark that in the time variable $\mathfrak{t}$

[TABLE]

for the Painlevé III1, the small $\underline{g}{}_{s}$ corresponds to the large $\mathfrak{t}$ regime.

Recall that $A_{n}(M)$ and $B_{n}(M)$ are expressed by the modified Bessel functions $I_{\nu}(1/\underline{g}{}_{s})$. The large $z$ behavior of the modified Bessel function is given by

[TABLE]

for $-\pi/2<\mathrm{arg}\,z<3\pi/2$. Here

[TABLE]

Note that $\widetilde{\mathcal{C}}_{k}(-\nu)=\widetilde{\mathcal{C}}_{k}(\nu)$.

We assume that $\mathrm{Re}\,\underline{g}{}_{s}>0$. Then the small $\underline{g}{}_{s}$ asymptotic expansion yields

[TABLE]

3.4.1 Perturbative contribution

The asymptotic expansion (3.57) has a part which is exponentially small term. We interpret that this part yields the “instanton contribution”. As an approximation, we omit this instanton part:

[TABLE]

With the truncated initial conditions

[TABLE]

[TABLE]

[TABLE]

we can solve the string equations and determine $A_{n}$ and $B_{n}$ as the power series in $\underline{g}{}_{s}$. Using

[TABLE]

we can check that the above truncated initial conditions are consistent with

[TABLE]

Note that for the perturbative case, it holds that $B_{1}(M)=A_{1}(-M)$ and subsequently $B_{n}(M)=A_{n}(-M)$. (When the “instanton contributions” are taken into account, $B_{n}(M)=A_{n}(-M)$ does not hold for generic $M$.)

We write the coefficients in the $\underline{g}{}_{s}$ expansion as follows:

[TABLE]

We have determined the coefficients $A_{n,k}(M)$ for $0\leq k\leq 11$. First few of them are

[TABLE]

Explicit form of $A_{n,k}(M)$ for $5\leq k\leq 11$ are given in Appendix D.

Then, we find the small $\underline{g}{}_{s}$ expansion of $R_{n}^{2}(M)$ and $D_{n}^{2}(M)$:

[TABLE]

[TABLE]

We know the exact expressions of $R_{n}^{2}(M)$ and $D_{n}^{2}(M)$ up to order $\underline{g}{}_{s}^{11}$ terms, but they are too lengthy. So we truncated them to the above orders.

3.4.2 Properties of $R_{n}^{2}(M)$

By inspecting the small $\underline{g}{}_{s}$ expansion (3.66), we may parametrize the $M$ dependence of $R_{n}^{2}(M)$ as follows:

[TABLE]

where $r_{n,k}$ are independent on $M$ and $\widetilde{\mathcal{C}}_{k}(M)$ are introduced in (3.56).

In the string equation, we can think $M$ as a free parameter. Hence it can be set to a half-integer. From (3.68), we have following conditions:

[TABLE]

etc. In general, we have

[TABLE]

By solving these equations for $r_{n,k}$, we can express $r_{n,k}$ as linear combination of $R_{n}^{2}(j+1/2)$’s. For example,

[TABLE]

The general solutions are given by

[TABLE]

By substituting this result into (3.68), we can express $R_{n}^{2}(M)$ in terms of $R_{n}^{2}(j+1/2)$’s:

[TABLE]

where

[TABLE]

In Appendix E, we show that $\widetilde{J}_{j}(M)$ is simplified to

[TABLE]

Note that

[TABLE]

Now we have the following expansion:

[TABLE]

When $M$ is a half-integer, the initial function $A_{1}$ (3.60) and $B_{1}$ (3.61) become rational functions of $\underline{g}{}_{s}$, hence $A_{n}=A_{n}(M)$ and $B_{n}=A_{n}(-M)$ also become rational functions. For example,

[TABLE]

etc. Let

[TABLE]

The second equation of (3.30) can be rewritten as

[TABLE]

By substituting (3.78) into this equation, we find the following recursion relations for $V_{n,j}$:

[TABLE]

With the initial conditions,

[TABLE]

we can determine $V_{n,j}$. The recursion relation (3.31) yields the following consistency condition:

[TABLE]

From this condition, $V_{n,j}$ is seen to have an expression which looks like a continued fraction.

For example, for $j=2$,

[TABLE]

We also have the following relations:

[TABLE]

For $j=0,1$, the explicit forms of $R_{n}^{2}(j+1/2)$ are given by

[TABLE]

[TABLE]

We finally have

[TABLE]

3.5 Free energy

By discarding an irrelevant phase factor $(-1)^{MN}$, let us define the free energy of the partition function (3.45) by

[TABLE]

Let us examine the $\underline{g}{}_{s}$ expansion of this free energy.

From (3.66), we can see that the perturbative contribution of $1-R_{j}^{2}$ is given by

[TABLE]

Also, by inspecting the asymptotic behavior (3.57), we can see that it is convenient to rearrange the free energy (3.88) as follows

[TABLE]

where

[TABLE]

and

[TABLE]

Here we stress that (3.90) is an exact rearrangement of terms with no approximation. Note that $\mathcal{F}^{(\mathrm{G})}$ is the free energy of the $N\times N$ Hermitian matrix model with Gaussian potential

[TABLE]

and $\mathcal{F}^{(\mathrm{n})}=\mathcal{F}-\mathcal{F}^{(\mathrm{G})}$ is the “normalized free energy” (see, for example, [59]).

Now let us consider the $\underline{g}{}_{s}$ perturbation of $\mathcal{F}^{(\mathrm{n})}$ ignoring the “instanton contributions”, i.e., exponentially small terms. Let us denote the expansion as follows:

[TABLE]

Obviously, we have

[TABLE]

We have determined $\widetilde{\mathcal{F}}^{(\mathrm{n})}_{k}$ up to $k=10$. The explicit form of $\widetilde{\mathcal{F}}^{(\mathrm{n})}_{k}$ are given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Note that for $k\geq 1$, the leading and subleading terms are guessed as follows

[TABLE]

Let $\widetilde{S}:=N\underline{g}{}_{s}$. The normalized free energy (3.94) can be rearranged as follows:

[TABLE]

Then, we have determined $\mathcal{F}^{(\mathrm{n})}_{g}$ up to $g=3$ by extrapolating our perturbative calculation:

[TABLE]

When $M=0$, these results coincide with the free energy of the GWW model in the weak coupling phase ($\widetilde{S}<1$) [60, 61].

We expect that for $g\geq 2$, the $\mathcal{F}_{g}^{(\mathrm{n})}$ takes the following form:

[TABLE]

where $3(g-1)=(2-\gamma_{\mathrm{st}})(g-1)$ with $\gamma_{\mathrm{st}}=-1$ is the susceptibility of this model.

We remark that by setting $M$ to a half-integer, the expression of the free energy takes a simpler form. In particular at $M=\pm 1/2$, it seems that $\mathcal{F}^{(\mathrm{n})}=\widetilde{S}\,\underline{g}{}_{s}^{-2}$. Hence, it may be useful to rewrite the $M$-dependence of $\mathcal{F}^{(\mathrm{n})}$ similar to (3.77):

[TABLE]

4 Various limits of $N_{f}=2$ matrix model

4.1 Planar limit of (3.50) and critical points

The planar limit is taken by sending $N\rightarrow\infty$, $\underline{g}{}_{s}\rightarrow 0$ with $\widetilde{S}=N\tilde{g}{}_{s}$ kept finite. In the planar limit, $\eta_{n}=n\,\tilde{g}{}_{s}$ turns into a continuous function of $x\equiv n/N$: $\eta_{n}\rightarrow\eta(x)=\widetilde{S}\,x$ with $0\leq x\leq 1$. We assume that $\xi_{n}$ also becomes a continuous function of $x$:

[TABLE]

In the planar limit, the second line of (3.50) is ignored because

[TABLE]

Hence the following quartic equation for $\xi$ is obtained from (3.50):

[TABLE]

The discriminant of this equation with respect to $\xi$ is given by

[TABLE]

where $\omega\equiv\exp(2\pi\mathrm{i}/3)$. At $\eta=\pm 1$, $\zeta=0$, the quartic equation (4.3) turns into

[TABLE]

and the three roots out of four degenerate to $\xi=0$. Therefore, we choose the critical values of $(\xi,\eta,\zeta)$ as

[TABLE]

where we take the continuum limit.

4.1.1 Planar free energy and susceptibility

Using (4.2), the planar free energy can be written as

[TABLE]

Note that this planar free energy is related to the planar part of the free energy $\mathcal{F}$ (3.88) as $\mathcal{F}_{0}=-\widetilde{S}^{2}\,F$, where

[TABLE]

Let $a\equiv N^{-1/3}$. Setting $\xi=a^{2}\,U$, $\eta=\pm(1-a^{2}t)$, $\zeta=a^{3}z$, (3.50) leads to the following equation at $\mathcal{O}(a^{6})$:

[TABLE]

For simplicity, we consider the case $\eta=1-a^{2}t=\widetilde{S}\,x$. Then, for small $a>0$, (4.9) is equivalent to

[TABLE]

This leads to a cubic equation for $\xi$. Let $\xi(x;a)$ be one of three solutions to the cubic equation. Note that

[TABLE]

Hence, in the planar limit, we have

[TABLE]

Therefore, the planar free energy (4.7) is evaluated as

[TABLE]

Equivalently, the planar part of the free energy (4.8) is equal to

[TABLE]

Recall that the planar free energy of the GWW model is given by

[TABLE]

Hence the planar free energy (4.14) coincides with that of GWW model in the weak coupling phase $(\widetilde{S}\leq 1)$ and is independent on $z$. (The parameter $z$ is proportional to $M$).

In the planar limit, we have $\xi=1-\widetilde{S}\,x$, hence

[TABLE]

where

[TABLE]

[TABLE]

Here we have fixed the ambiguity in the integration constants by taking them to be zero. Note that

[TABLE]

Then, the contribution to the free energy around $x=1$ (i.e., around $\xi=1-\widetilde{S}$) is

[TABLE]

Thinking the critical value of the $\widetilde{S}$ as $\widetilde{S}_{\mathrm{c}}=1$, we can read off the susceptibility $\gamma_{\mathrm{st}}$ from the leading term (see, for example, [62, 63]):

[TABLE]

The susceptibility $\gamma_{\mathrm{st}}$ is $-1$ and independent of $z$.

Let us denote the planar free energy of GWW model in the strong and weak coupling phases respectively by

[TABLE]

[TABLE]

Note that

[TABLE]

So, at least in the leading order in small $(1-\widetilde{S})$, (4.20) is equal to the discontinuity of the planar free energy:

[TABLE]

4.1.2 Singular K3 surface

In the planar limit, eq.(3.50) also becomes the defining relation of an algebraic variety. With the introduction of the homogeneous coordinates $(\xi:\eta:\zeta:1)=(\mathcal{X}:\mathcal{Y}:\mathcal{Z}:\mathcal{W})$ of the three-dimensional complex projective space $\mathbb{P}^{3}$, this algebraic variety is the union of the hyperplane $\mathcal{Y}=0$ (with multiplicity two) and the singular K3 surface

[TABLE]

The singular loci of this surface consist of three spheres:

[TABLE]

The intersections of these spheres are given by

[TABLE]

The appearance of the singular K3 surface is just an observation. At this moment, we have no idea about its geometrical or physical meaning. The alternate discrete Painlevé II equation is closely related to a rational surface, called the space of initial values [64, 65], characterized by the affine root system $D^{(1)}_{6}$. The relation between these two surfaces is still not known to us.

We remark that the critical point (4.6) is on the first sphere $S^{2}_{(1)}$:

[TABLE]

4.2 Double scaling limit

Note that

[TABLE]

This is the parameter we fine tune to $\pm 1$, and is the counterpart of the bare cosmological constant in $2$d gravity. Also note that

[TABLE]

and the two masses are fine tuned to be equal in this limit.

4.2.1 Double scaling limit and Painlevé II equation

Let us consider the double scaling limit of (3.50). Let $x\equiv n/N$, $a^{3}\equiv 1/N$ and

[TABLE]

[TABLE]

With these scaling ansatz, the double scaling limit is defined as the $N\rightarrow\infty$ ($a\rightarrow 0$) limit while simultaneously sending $\widetilde{S}$ to its critical value $1$ by (4.32). The original ’t Hooft expansion parameter $1/N$ gets dressed by the combination which is kept finite in this limit:

[TABLE]

with $\gamma_{\mathrm{st}}$ being the susceptibility of the system. For later convenience, we write

[TABLE]

Note that

[TABLE]

and

[TABLE]

We can see that

[TABLE]

[TABLE]

Hence, the double scaling limit of $\xi_{n\pm 1}$ is given by

[TABLE]

Using $\eta_{n}\rightarrow 1-(1/2)a^{2}t$, $\zeta\rightarrow a^{3}M$, $\xi_{n}\rightarrow a^{2}u(t)$ and (4.40), the string equation (3.50) turns into

[TABLE]

Note that the value of $\tilde{c}$, introduced in (4.35), is irrelevant to this order.

Therefore, we obtain the Painlevé II equation as the double scaling limit of (3.50):

[TABLE]

This is the PII equation obtained from the Lax pair of Flaschka-Newell [66]. It can be converted into more standard form of PII (4.49) as follows. By using $p_{u}\equiv-u^{\prime}/u$, eq. (4.42) is equivalent to the following system of equations

[TABLE]

This is a Hamilton system

[TABLE]

with the Hamiltonian

[TABLE]

By the canonical transformation $(u,p_{u})\rightarrow(y,p_{y})$ with $u=-p_{y}$ and $p_{u}=y+(M/p_{y}$), this Hamiltonian becomes the following form [67, 68]

[TABLE]

and the system of equations is converted into

[TABLE]

This means that $y=y(t)$ obeys the following form of the Painlevé II equation:

[TABLE]

By appropriate rescaling of $y$ and $t$, this can be converted into the standard form of PII:

[TABLE]

with $\alpha=(1/2)-M$.

When there is no logarithmic potential ($M=0$), the appearance of the Painlevé II equation in the unitary matrix model was shown in [23, 24]. (See also, [69].) In this case, by setting $u=y^{2}$, (4.42) turns into

[TABLE]

Hence in [23, 24], this is interpreted as the zero parameter ($\alpha=0$) case of PII. But it can be also interpreted as the $\alpha=1/2$ case of PII.

4.2.2 Double scaling limit and Argyres-Douglas point

It is easy to see what this critical point corresponds to in the Seiberg-Witten curve (quartic one), which is the spectral curve obtained the planar loop equation/Virasoro constraints. For the $N_{f}=2$ potential $W^{(2)}(z)$ (F.30), let

[TABLE]

The spectral curve (G.11) for this potential $W(z)$ is given by

[TABLE]

This is the “first realization” of the Seiberg-Witten curve in the Gaiotto form for the $N_{f}=2$ theory [70, eq.(10.11)]. The Seiberg-Witten differential is given by $dS_{\mathrm{SW}}=y(z)\,\mathrm{d}z$. The coefficient of $z^{-2}$ in the right-handed side is identified with the Coulomb moduli parameter $\mathfrak{u}=\langle\mathrm{Tr}\,\phi^{2}\rangle$ of the Seiberg-Witten curve. The parameter $\mathfrak{u}$ is related to the (planar) resolvent $\omega(z)$ (G.8) as follows:

[TABLE]

where

[TABLE]

Clearly, at our critical point $m_{1}/\Lambda_{2}=m_{2}/\Lambda_{2}=\mp 1/2$, this genus one curve (4.52) shrinks to a point at $\mathfrak{u}/\Lambda_{2}{}^{2}=3/8$:

[TABLE]

For simplicity, let us consider the limit to the following critical values:

[TABLE]

Recall that

[TABLE]

In the double scaling limit, these mass parameters are given by

[TABLE]

We require that $z$ and $\mathfrak{u}$ approach their critical values respectively by

[TABLE]

Then we can see that

[TABLE]

Hence by setting

[TABLE]

the curve (4.52) in $a\rightarrow 0$ limit turns into the first realization of the Seiberg-Witten curve for the Argyres-Douglas theory of type $H_{1}=(A_{1},A_{3})$ [14]:

[TABLE]

Our limit is, therefore, the limit toward the $H_{1}$ Argyres-Douglas point .

Acknowledgments

We thank Chuan-Tsung Chan, Masafumi Fukuma, Hikaru Kawai, Vladimir Kazakov, Kazunobu Maruyoshi, Shun’ya Mizoguchi, Hajime Nagoya and Kentaroh Yoshida for useful discussions. We also thank the anonymous referee of our previous paper [1] for valuable suggestions.

Appendix A The volume of the unitary group

In this section, we show that the volume of the $U(N)$ group, constructed from the measure (2.2), is given by (2.3).

A unitary matrix $U\in U(N)$ can be expressed by an $N\times N$ Hermitian matrix $H$ as follows [71, 72]

[TABLE]

Let $[\mathrm{d}U]$ be a Haar measure for the unitary matrix defined by the metric

[TABLE]

Also let $[\mathrm{d}H]$ be a measure for the Hermitian matrix defined by the metric $\mathrm{Tr}\,(\mathrm{d}H)^{2}$. Then, (A.1) leads to the following relation between the two measures:

[TABLE]

The unitary matrix $U$ and the Hermitian matrix $H$ can be diagonalized by a unitary matrix $V$:

[TABLE]

and the eigenvalues are related as

[TABLE]

Note that there is an ambiguity in the choice of $V$. For a diagonal matrix $A_{D}\in U(N)$, $A_{D}V$ also diagonalizes $U$ and $H$.

Let us write the measure for the Hermitian matrix as

[TABLE]

Using the Gaussian integral

[TABLE]

and an integral formula

[TABLE]

we can show that

[TABLE]

Here $G_{2}(z)$ is the Barnes function, defined by $G_{2}(z+1)=\Gamma(z)G_{2}(z)$ with $G_{2}(1)=1$.

Using another integral formula

[TABLE]

we can evaluate the volume of the unitary group $U(N)$:

[TABLE]

Hence the volume of the unitary group obtained from the metric (A.2) is given by

[TABLE]

This completes the proof of (2.3). This volume coincides with the one adopted in the $U(N)$ Chern-Simons theory [73].

Using (A.6), we can see that

[TABLE]

where

[TABLE]

Therefore, the unitary Haar measure can be written as

[TABLE]

It holds that

[TABLE]

Appendix B Derivation of string equations

B.1 Derivation of (2.45), (2.46) and (2.47)

The left-handed side of (2.45) can be written as

[TABLE]

Note that

[TABLE]

Thus the first term of (B.1) is evaluated as

[TABLE]

Here we have used (2.14).

Also, by using

[TABLE]

the second term of (B.1) is given by

[TABLE]

By adding (B.3) and (B.5), we obtain (2.45).

The remaining equations (2.46) and (2.47) can be obtained similarly.

B.2 Derivation of (3.25), (3.26) and (3.27)

Eq. (3.25) can be obtained by using the following relations:

[TABLE]

[TABLE]

[TABLE]

Here we have used the orthogonality (2.10) and (2.32).

The remaining equations (3.26) and (3.27) can be derived similarly.

Appendix C alt-dPII

We learned the contents of this appendix from the referee of our previous paper [1].

Let

[TABLE]

Then (3.30) leads to

[TABLE]

By taking the difference of these two equations and shifting $n$ by one, we find

[TABLE]

Hence

[TABLE]

Also, (3.31) leads to

[TABLE]

Hence

[TABLE]

By multiplying $x_{n-1}$ to the first equation of (C.2) and using (C.6), we find

[TABLE]

i.e.,

[TABLE]

From this equation, we have

[TABLE]

By substituting this into (C.4) and with some work, we obtain the following recursion relation for $x_{n}$:

[TABLE]

Similarly, the recursion relation for $y_{n}$ can be obtained:

[TABLE]

These are the alternate discrete Painlevé II equations [57, 58]. By comparing these equations with the alt-dPII equation [58, eq.(1.3)]

[TABLE]

where $\tilde{z}_{n}=\tilde{a}n+\tilde{b}$, we find

[TABLE]

$\tilde{\mu}=2(-M-1)\underline{g}{}_{s}$ for $\tilde{x}_{n}=x_{n}$ and $\tilde{\mu}=2(M-1)\underline{g}{}_{s}$ for $\tilde{x}_{n}=y_{n}$. Note that the value of the parameter $\tilde{\mu}$ for $x_{n}$ and that for $y_{n}$ are different.

We also mention that our solutions

[TABLE]

belong to a class of the Casorati determinant solutions to the alt-dPII considered in [58].

Appendix D $A_{n,k}(M)$

The coefficients $A_{n,k}(M)$ are introduced in (3.64). For $0\leq k\leq 4$, they are given in (3.65). For $5\leq k\leq 11$, they are determined as follows.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Appendix E $\widetilde{J}_{j}(M)$

We give a proof of (3.75). Recall that the infinite sum (3.74) is defined by

[TABLE]

From the definition (3.56), the coefficients $\widetilde{\mathcal{C}}_{k+j}(M)$ obey the following relations:

[TABLE]

Using these recursion relations, we can see that

[TABLE]

Hence

[TABLE]

where

[TABLE]

We can see that

[TABLE]

and

[TABLE]

Therefore

[TABLE]

which completes the proof of (3.75).

Note that

[TABLE]

Then we have

[TABLE]

Appendix F Conformal block and its irregular limit

The Coulomb-gas representation of the conformal block [6] can be interpreted as a matrix model [74, 75, 76, 7, 77]. By considering “colliding limits” of the matrix model, the irregular conformal blocks can also be treated within the framework of the matrix model [41, 5, 78, 79, 80].888 In [81], the $N_{f}=0$ limit of the conformal block is considered and is interpreted as the $\beta$-deformed GWW model. Following [5], we review the matrix models corresponding to the $SU(2)$ gauge theory with $N_{f}=2,3,4$ cases. Here we adopt integration contours slightly different from those found in [5]. With these modifications, it is easy to see the relation between a special case of $N_{f}=2$ model and the unitary matrix model with the logarithmic potential considered in Section 3.

F.1 Conformal block and $SU(2)$ gauge theory with $N_{f}=4$

For simplicity, we assume that the parameter $q_{0}$ is real and $0<q_{0}<1$. For the matrix model (1.1), let us specify the integration contours to the intervals: $C_{L}^{(4)}=[0,q_{0}]$, $C_{R}^{(4)}=[1,\infty]$, the potential $W^{(4)}(z)$ to

[TABLE]

and the normalization constant $\mathcal{C}^{(4)}$ to

[TABLE]

Here the superscript $4$ in parenthesis represents the number of flavors of the corresponding gauge theory. The matrix model (1.1) in this case becomes

[TABLE]

Here $C^{(4)}_{I}=C^{(4)}_{L}$ for $1\leq I\leq N_{L}$ and $C^{(4)}_{I}=C^{(4)}_{R}$ for $N_{L}+1\leq I\leq N$.

It is a Coulomb gas representation of the conformal block $\mathcal{F}$:

[TABLE]

[TABLE]

where the central charge is given by $c=1-6Q_{E}^{2}$ with $Q_{E}=\sqrt{\beta}-1/\sqrt{\beta}$, and the scaling dimensions are

[TABLE]

Here $\alpha_{4}=-2Q_{E}-2\sqrt{\beta}N-\alpha_{1}-\alpha_{2}-\alpha_{3}$ and $\alpha_{I}=\alpha_{1}+\alpha_{2}+2\sqrt{\beta}N_{L}$. The OPE coefficients $C_{21}{}^{I}$ and $\widetilde{C}_{43}{}^{I}$ in the Coulomb gas model are respectively given by the Selberg integrals:

[TABLE]

[TABLE]

This matrix model contains seven parameters

[TABLE]

with one constraint

[TABLE]

These are related to six unconstrained parameters of $4$d $SU(2)$ gauge theory with $N_{f}=4$

[TABLE]

by the [math]d-$4$d dictionary [8]999Here we have renamed the mass parameters such that the ordering of mass to infinity limit is natural.:

[TABLE]

The omega background parameters $\epsilon_{1,2}$ are related to $\beta$ as $\epsilon_{1}=\sqrt{\beta}\,g_{s}$ and $\epsilon_{2}=-g_{s}/\sqrt{\beta}$. Hence $g_{s}^{2}=-\epsilon_{1}\,\epsilon_{2}$ and $\epsilon\equiv\epsilon_{1}+\epsilon_{2}=Q_{E}\,g_{s}$. The cross ratio $q_{0}$ is identified with the exponentiated ultraviolet gauge coupling constant $q_{0}\equiv e^{\mathrm{i}\pi\tau_{0}}$, $\tau_{0}\equiv(\theta_{0}/\pi)+8\,\pi\,\mathrm{i}/g_{0}^{2}$.

F.2 Irregular limit to $N_{f}=3$

In order to obtain well-defined $\beta$-deformed matrix models corresponding to the $N_{f}=3,\,2$ gauge theories, we should take the analytic continuation of (F.1).

Let us consider an integral of the form

[TABLE]

for the integration contour $C^{(4)}_{L}$ or $C^{(4)}_{R}$. Here $c_{n}$ are constants and $\alpha$ a complex parameter which is not an integer. Let $C(r)$ be the contour which is an arc of radius $r$ in the complex $w$-plane, running counterclockwise from $r+\mathrm{i}0$ to $r-\mathrm{i}0$. It avoids the cut of $w^{\alpha}$ along positive $\mathrm{Re}\,w$-axis (See Fig.1).

For $\mathrm{Re}\,\alpha>0$, the integral over the interval $C^{(4)}_{L}=[0,q_{0}]$ can be converted to a complex integral along $C(r)$ with $r=q_{0}$ by

[TABLE]

Similarly, for $\mathrm{Re}\,\alpha<0$, the integral over $C^{(4)}_{R}=[1,\infty]$ becomes the following complex integral along $C(r)$ with $r=1$:

[TABLE]

Using these relations, we assume that (F.1) can be rewritten as

[TABLE]

where $\mathcal{C}_{L}$ and $\mathcal{C}_{R}$ are the normalization factors. Up to phase factors, they are given by

[TABLE]

The contour $C^{\prime(4)}_{I}$ is the complex contour $C_{L}(r_{0})$ (see Fig.2) for $1\leq I\leq N_{L}$ and $C(1)$ for $N_{L}+1\leq I\leq N$. Here $r_{0}$ is an arbitrary real number satisfying $0<q_{0}<r_{0}<1$. The contour $C_{L}(r_{0})$ is a deformation of $C(q_{0})$.

The $N_{f}=3$ limit of the gauge theory is taken by $m_{4}\rightarrow\infty$ with $\Lambda_{3}\equiv 4\,q_{0}\,m_{4}$ fixed. Using above dictionary (F.12), this corresponds to the $q_{0}\rightarrow 0$ limit with $2\,q_{03}\equiv q_{0}\left(-\alpha_{1}+\alpha_{2}\right)$ and $\alpha_{1+2}\equiv\alpha_{1}+\alpha_{2}$ kept finite. The parameter $q_{03}$ is related to the dynamical mass scale $\Lambda_{3}$ of the $N_{f}=3$ theory as $q_{03}=\Lambda_{3}/(4\,g_{s})$.

Note that the out of the seven constraint parameters in (F.9), six ones are kept intact:

[TABLE]

The constraint (F.10) reduces to

[TABLE]

In (F.12), the dictionary for $\alpha_{1}$ and that for $\alpha_{2}$ get replaced with

[TABLE]

We can then define the well-defined limit of the matrix model which corresponds to the $N_{f}=3$ limit of the gauge theory,

[TABLE]

In this limit, the contour $C_{L}(r_{0})$ goes to $C_{0}(r_{0})$ which are shown in Fig.3. The resultant contour $C_{I}^{(3)}$ is $C_{0}(r_{0})$ for $1\leq I\leq N_{L}$ and $C(1)$ for $N_{L}+1\leq I\leq N$. The corresponding potential is given by

[TABLE]

F.3 Irregular limit to $N_{f}=2$

We can take the $N_{f}=2$ limit in the gauge theory subsequently after the $N_{f}=3$ limit by $m_{3}\rightarrow\infty$ with the dynamical scale $\Lambda_{2}\equiv(m_{3}\,\Lambda_{3})^{1/2}$ fixed. In the matrix model, this corresponds to the $q_{03}\rightarrow 0$ limit with $q_{02}{}^{2}\equiv(1/2)q_{03}\left(\alpha_{3}-\alpha_{4}\right)$ and $\alpha_{3+4}\equiv\alpha_{3}+\alpha_{4}$ kept finite. In (F.9), five parameters are kept intact:

[TABLE]

The momentum conservation (F.19) becomes

[TABLE]

Now, the dictionary for $\alpha_{3}$ and that for $\alpha_{4}$ get replaced with

[TABLE]

in (F.12) and $q_{02}$ is related to $\Lambda_{2}$ as $q_{02}=\Lambda_{2}/(2\,g_{s})$.

We rescale the integration variables as $w_{I}\rightarrow(q_{03}/q_{02})w_{I}$, then we get

[TABLE]

where the integration contour $C^{\prime(3)}_{I}$ is $C_{0}(r_{0}\,q_{02}/q_{03})$ for $1\leq I\leq N_{L}$ and $C_{R}(q_{02}/q_{03})$ (see Fig. 4) for $N_{L}+1\leq I\leq N$.

For simplicity, we set $r_{0}=q_{03}/q_{02}$. Then $C_{0}(r_{0}\,q_{02}/q_{03})=C_{0}(1)\equiv C_{0}$. (See Fig. 5). The normalization factor $\mathcal{N}^{(3)}$ is given by

[TABLE]

where

[TABLE]

We define the $N_{f}=2$ limit for the matrix model by

[TABLE]

The resultant integration contours $C_{I}^{(2)}$ are $C_{0}$ for $1\leq I\leq N_{L}$ and $C_{\infty}$ (see Fig.6) for $N_{L}+1\leq I\leq N$. The corresponding potential is given by

[TABLE]

Note that if $\beta$ is a positive integer and $\sqrt{\beta}\alpha_{1+2}\in\mathbb{Z}$, then the cut for the integrand of (F.3) vanishes. In this case, the open paths $C_{0}$ and $C_{\infty}$ can be deformed to the closed circles with unit radius around the origin. Thus, we can represent (F.3) as the multiple integral over the same unit circles. In particular, for $\beta=1$ with $\alpha_{1+2}\in\mathbb{Z}$, we can identify $N_{f}=2$ matrix model (F.3) with an unitary matrix model with a logarithmic potential.

To see the connection with the unitary matrix model explicitly, let us rewrite the partition function (F.3) as follows:

[TABLE]

For $\beta=1$, it holds that $M=\alpha_{1+2}+N$ and $\alpha_{1+2}\in\mathbb{Z}$ is equivalent to $M\in\mathbb{Z}$. Thus for this case with $\beta=1$, (F.31) is indeed a unitary matrix model. Note that in this case, the partition function $Z^{(2)}=Z^{(2)}(N_{L},N_{R})$ depends only on the sum $N=N_{L}+N_{R}$:

[TABLE]

So, their generating function (1.3) becomes

[TABLE]

Hence, in gauge theory parameters, this does not depend on the vev $\mathfrak{a}$ of the Higgs field. In the matrix model/gauge theory correspondence, the $N_{f}=2$ instanton partition function is identified with $Z^{(2)}/(\mathcal{N}^{(2)}Z^{(2)}_{L}Z^{(2)}_{R})$ (see (H.7)). The dependence on $\mathfrak{a}$ comes from $Z^{(2)}_{L}(N_{L})Z^{(2)}_{R}(N_{R})$.

Appendix G Loop equations

In this section, we briefly review the finite $N$ loop equations for the $\beta$-deformed matrix models and their planar limit. See [82, 83, 84] for earlier literature at $\beta=1$. See also [75, 85] for examples at higher rank extensions.

G.1 Finite $N$ loop equations

For the matrix model (1.1) or (1.3), we assume that the potential $W(z)$ has the following property:

[TABLE]

where $F(w)$ is a holomorphic function of $\{w_{J}\}$ in a domain which includes the integration contours.

Let $\omega_{N}(w)$ be a finite $N$ resolvent defined by

[TABLE]

It obeys the finite $N$ loop equation:

[TABLE]

where

[TABLE]

Here $\langle\!\langle\dotsm\rangle\!\rangle$ means the average with respect to $Z(N_{L},N_{R})$ (1.1) or $\underline{Z}(N;\mu_{L},\mu_{R})$ (1.3).

We denote the power sum by

[TABLE]

Then

[TABLE]

G.2 Planar loop equation and Seiberg-Witten curve

In this subsection, we rescale the potential $W(z)$ of the matrix model by $W(z)\rightarrow(1/g_{s})W(z)$, so

[TABLE]

Let us consider the planar limit of the finite $N$ loop equation. It is the limit of $N\rightarrow\infty$ and $g_{s}\rightarrow 0$ with keeping the ’t Hooft coupling $\widetilde{S}:=\sqrt{\beta}\,g_{s}N$ finite.

The (planar) resolvent $\omega(z)$ is defined by

[TABLE]

Here $\langle\!\langle\dotsm\rangle\!\rangle$ is the average with respect to (G.7). Also, let

[TABLE]

In the planar limit, the finite $N$ loop equation (G.3) turns into the planar loop equation

[TABLE]

By introducing $y(z):=w(z)+(1/2)W^{\prime}(z)$, this can be rewritten as

[TABLE]

In the matrix model/gauge theory correspondence, this spectral curve is identified with the Seiberg-Witten curve of the corresponding gauge theory.

Appendix H Instanton expansion for $N_{f}=2$ matrix model

Let us consider the instanton expansion (small $q_{02}$ expansion) of the partition function for the $N_{f}=2$ model (F.3). See [8] and [86, 87, 3, 74, 75, 76, 77, 4, 85]. To consider the $q_{02}$ expansion, it is convenient to divide the $N$ integration variables $w_{I}$ ($I=1,2,\dotsm,N$) into two sets $\{w_{I}\}_{I=1,2\dotsm,N_{L}}$ and $\{w_{N_{L}+J}\}_{J=1,2,\dotsm,N_{R}}$ and rename the latter variables by $u_{J}:=w_{N_{L}+J}$ ($J=1,2,\dotsm,N_{R}$). Here $N=N_{L}+N_{R}$.

Furthermore, we rescale the integration variables as $w_{I}\rightarrow q_{02}\,w_{I}$ ($I=1,2,\dots,N_{L}$) and $u_{J}\rightarrow(q_{02})^{-1}u_{J}$ ($J=1,2,\dotsm,N_{R}$). In this rescaling, the integration contours $C_{0}$ and $C_{\infty}$ are also rescaled, but can be smoothly deformed back to $C_{0}$ and $C_{\infty}$ respectively. Then, (F.3) becomes

[TABLE]

where $\mathcal{N}^{(2)}=(q_{02})^{\sqrt{\beta}(N_{L}-N_{R})M-2\beta N_{L}N_{R}}$,

[TABLE]

with

[TABLE]

Let us define the left and right partition functions by

[TABLE]

[TABLE]

Then

[TABLE]

Here $\langle\!\langle f(w,u)\rangle\!\rangle_{N_{L},N_{R}}$ means the expectation value of $f(w,u)$ with respect to $Z_{L}^{(2)}Z_{R}^{(2)}$.

Now, $Z^{(2)}/(\mathcal{N}^{(2)}Z^{(2)}_{L}Z^{(2)}_{R})$ has the following $q_{02}$ expansion

[TABLE]

which is identified with the instanton part of the $N_{f}=2$ Nekrasov function.

For simplicity, we evaluate the first coefficient $\mathcal{A}_{1}$

[TABLE]

with help of the finite $N$ loop equations. Let us introduce the finite $N$ resolvent for $\{w_{I}\}$ and that for $\{u_{J}\}$ respectively by

[TABLE]

They respectively satisfy the finite $N$ loop equations

[TABLE]

where $\langle\!\langle\dotsm\rangle\!\rangle_{N_{\mathcal{I}}}$ is the average with respect to $Z_{N_{\mathcal{I}}}$, and

[TABLE]

[TABLE]

They are evaluated as

[TABLE]

[TABLE]

By comparing the $\mathcal{O}(z^{-2}),\mathcal{O}(z^{-3})$ terms of the loop equation for the left resolvent $\omega_{N_{L}}(z)$, we find

[TABLE]

In addition, from $O(z^{-1})$ terms, we have

[TABLE]

Similarly, from the loop equation for $\omega_{N_{R}}(w)$, we obtain

[TABLE]

By substituting (H.16) and (H.18) into (H.8), we find

[TABLE]

which is the same result as that in [5] and coincides with the one instanton part of the instanton partition function.

Note that

[TABLE]

[TABLE]