Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory

We study the partition function of the matrix model of finite size that realizes the irregular conformal block for the case of the ${\cal N}=2$ supersymmetric $SU(2)$ gauge theory with $N_f =2$. This model has been obtained in [arXiv:1008.1861 [hep-th]] as the massive scaling limit of the $\beta$ deformed matrix model representing the conformal block. We point out that the model for the case of $\beta =1$ can be recast into a unitary matrix model with log potential and show that it is exhibited as a discrete Painlev\'{e} system by the method of orthogonal polynomials. We derive the Painlev\'{e} II equation, taking the double scaling limit in the vicinity of the critical point which is the Argyres-Douglas type point of the corresponding spectral curve. By the $0$d-$4$d dictionary, we obtain the time variable and the parameter of the double scaled theory respectively from the sum and the difference of the two mass parameters scaled to their critical values.

Study of correlation functions in lower dimensional quantum field theory and statistical system has sometimes led us to a surprising occurrence of nonlinear differential/difference equations that they obey. In two dimensional physical systems, these equations are typically Painlevé equations that have attracted interest of both physicists and mathematicians, and that govern the scaling behavior of the systems. They first appeared in the study of Ising two point correlation functions, and take the form of Painlevé III [1,2,3,4,5].
The second such development was made in the context of two dimensional quantum gravity (2d gravity for short) and c < 1 non-critical strings [6]. For a review, see, for example, [7]. Equilateral triangulation of a two dimensional random surface generates the double infinite sum for its partition function by the number of triangles and by the number of holes of the discretized surface. Through dual Feynman diagrams, the partition function is recast into multiple integrals of a hermitean matrix of finite size N and hence is the finite N hermitean matrix model having a bare cosmological constant parameter. The method of orthogonal polynomials permits us to relate the partition function with a set of recursion relations. A specific set of recursion relations forms a system of difference equations called string equations, and a phenomenon of genus enhancement/resummation takes place in a limit commonly referred to as the double scaling limit where a difference operator is replaced by the corresponding derivative. Painlevé I equation has been exhibited this way as a universal equation governing the nonperturbative scaling function for the partition function containing all genus contributions.
In recent years, a certain class of β-deformed ensembles for matrix models containing log potentials have been serving as integral representations [8,9,10,11,12,13,14] of 2d conformal and irregular 2d conformal block [15,16]. They in fact generate directly [13,14] the expansion of the block in the form of the instanton expansion in accordance with the AGT correspondence [17]. The matrix model free energy F is thus equal to the instanton part of the Seiberg-Witten prepotential F augmented by the higher genus contributions [18]: F = F . For a review, see, for example [19]. In [20], Painlevé VI equation has been derived for a Fourier transform of the c = 1 conformal block with respect to the intermediate momentum. See [21,22,23,24,25,26] for subsequent analyses.
In this letter, we will study the simplest prototypical case of the irregular block, namely the case of the N = 2 supersymmetric SU(2) gauge theory with the number of hypermultiplets N f = 2 in the form of the matrix model integral representation derived in [14]. Unlike [20], our procedure is closer in spirit to that 2d gravity in its unitary counterpart. We see that the finite N system formulated by the orthogonal polynomials which we devise is already regarded as a discretized Painlevé system. We are able to take the double scaling limit of this system to its critical point to derive the Painlevé II equation for the scaling function.
The "time" variable t is obtained from the limit of the sum of the two hypermultiplet masses of the gauge theory to its critical value by the 0d-4d dictionary while the parameter M in the equation from the limit of the difference of the two masses. Details of the derivation to our findings will be given elsewhere.
The partition functions of the β-deformed matrix models which directly generate [13,14] the instanton expansion of the four-dimensional N = 2 SU(2) gauge theories with N f fundamental matters can be generically presented as and C (N f ) I certain integration contours below.
For N f = 4, namely, the case of 2d conformal block, the potential W (4) (w) is given by the three-Penner (logarithmic) potential Let N = N L + N R . The N L contours C The β-deformed matrix model for N f = 4 contains seven parameters α 1 , α 2 , α 3 , α 4 , β, N L , N R undergoing one constraint (the momentum conservation) These are transcribed into six unconstrained 4d parameters of N f = 4 SU(2) gauge theory by the 0d-4d dictionary [13] 1 : The omega background parameters ǫ 1,2 are related to β as ǫ 1 = √ β g s and ǫ 2 = −g s / √ β.
The momentum conservation (7) becomes Now α 3+4 = (2m 1 +ǫ)/g s and q 02 = Λ 2 /(2 g s ). The potential of the resultant N f = 2 irregular matrix model takes the following form: 1 Here in comparison to [13], we have renamed the mass parameters as m We are interested in the large N behavior of these irregular matrix models, in particular, the double scaling limit of them. In the limit, the leading part of the partition function does not depend on the details of the contours. Moreover, it has been argued [27,28,29] that two-cut hermitean matrix model and the unitary matrix model share the critical properties.
Therefore, the planar scaling or the double scaling limit of our irregular models would lie in the same universality class which the unitary matrix model belongs to.
Let us consider the following unitary matrix model where the potential is given by W U (w) = W (w) + N log w. In particular, we study the N f = 2 case for simplicity: where M ≡ α 3+4 + N = (m 1 − m 2 )/g s . In order for the contour integrals to be well-defined, we assume that M is an integer. In [14], the original integration path on the real axis of the N f = 4 model was deformed into a contour in the complex plane by an analytic continuation to avoid the singularity which is induced by the N f = 3, 2 potential in the limit. In N f = 2 case, the contour derived is the one wrapping the positive real axis from the origin to infinity.
When M is an integer, this contour becomes a closed circle around the origin. Our N f = 2 model is in fact equivalent to the above unitary matrix model. The unitary matrix model can be solved [30,31,32] by the method of orthogonal polynomials [33,34]. Let us use the monic orthogonal polynomials [31,32] In [31,32], M = 0 case was considered withp n (w) = p n (w) (B . Through explicit computation, we have found that the moments of this model are given by the modified Bessel functions up to phase factors. Let where I ν (z) is the modified Bessel function of the first kind and g s ≡ 1/(2 q 02 ) = g s /Λ 2 . The  (11) can be written in terms of these objects: The string equations lead to the following recursion relations for A n and B n : With the initial conditions A 0 = B 0 = 1, and the remaining constants A n and B n are completely characterized by the recursion relations (17), (18).
Let A n = R n D n and B n = R n /D n . The partition function (16) becomes Eliminating D n from (17) and (18), we obtain the recursion relation for R 2 n : This is equivalent to where ξ n ≡ R 2 n , η n ≡ n g s , ζ ≡ M g s . In the planar limit (ξ n , η n , ζ) → (ξ, η, ζ), the second line of (22) is ignored and the three roots out of four in the resulting quartic equation in ξ become degenerate to zero at η = ±1, ζ = 0, where we take the continuum limit. In fact, setting ξ = a 2 u, η = ± 1 − (1/2) a 2 t, ζ = ± a 3 M, we obtain at O(a 6 ) Eq.(22) also becomes the defining relation of an algebraic variety. With the introduction of the homogeneous coordinates (X : Y : Z : W) = (ξ : η : ζ : 1) of the three-dimensional complex projective space P 3 , this algebraic variety is the union of the hyperplane Y = 0 (with multiplicity two) and the singular K3 surface The singular loci of this surface consist of three spheres whose intersections are represented by the A 3 Dynkin diagram.
In order to present the critical behavior at the planar and the double scaling limit better, let us rewrite the potential (12) as Here, S ≡ Ng s = g s N/Λ 2 = −(m 1 + m 2 )/Λ 2 is the parameter we fine tune to ±1, and is the counterpart of the bare cosmological constant in 2d gravity. Also note that ζ = (m 1 − m 2 )/( SΛ 2 ) = O(a 3 ) and the two masses are fine tuned to be equal in this limit. It is easy to see what this critical point corresponds to in the Seiberg-Witten curve [35,36] (quartic one), which is the spectral curve obtained from the planar loop equation/Virasoro constraints [37,38,39]. Omitting the standard procedure of this derivation, the curve (y(z), z), where the resolvent ω(z) = lim N →∞ 1/(z − w I ) lies, is given by Here, we have used (9) and the residue relation of the resolvent at z = ∞. We have parametrized the coefficient of z 2 by the coordinate of the moduli space of the curve. Clearly, at our critical point m 1 /Λ 2 = m 2 /Λ 2 = ∓1/2, this genus one curve shrinks to a point at u/Λ 2 2 = 3/8. Our limit is, therefore, the limit to the Argyres-Douglas point [40,41,42].
Here, we have taken the upper sign without losing generality. With these scaling ansatze, the double scaling limit is defined as the N → ∞ (a → 0) limit while simultaneously sending S to its critical value 1 by (28). The original 't Hooft expansion parameter 1/N gets dressed by the combination which is kept finite in this limit: with γ being the susceptibility of the system. This last point can be checked from the free energy F computation from (20): In the double scaling limit, the string equation (22) turns into the Painlevé II equation It is noteworthy that the parameter M in the original model survives the limit. We can convert (32) into standard form as follows. By using p u ≡ −u ′ /u, this equation (32) can be written as a Hamilton system with the Hamiltonian H II (u, p u ; t) = − 1 2 p 2 u u +