Quantum periods for $\mathcal{N}=2$ $SU(2)$ SQCD around the superconformal point

We study the Argyres-Douglas theories realized at the superconformal point in the Coulomb moduli space of $\mathcal{N}=2$ supersymmetric $SU(2)$ QCD with $N_f=1,2,3$ hypermultiplets in the Nekrasov-Shatashvili limit of the Omega-background. The Seiberg-Witten curve of the theory is quantized in this limit and the periods receive the quantum corrections. By applying the WKB method for the quantum Seiberg-Witten curve, we calculate the quantum corrections to the Seiberg-Witten periods around the superconformal point up to the fourth order in the parameter of the Omega background.


Introduction
A large class of N = 2 supersymmetric gauge theories has a superconformal fixed point at strong coupling in the Coulomb moduli space, where mutually non-local BPS states become massless. This theory becomes an interacting N = 2 superconformal field theory, which is called the Argyres-Douglas (AD) theory [1,2]. The BPS spectrum of the AD theory can be studied by the Seiberg-Witten (SW) curve, which are obtained from degeneration of the curve of N = 2 gauge theories [1,2,3]. The dynamics of AD theories is an interesting subject of recent studies from the viewpoint of M5-branes compactified on a punctured Riemann surface [4,5,6] and its relation to two-dimensional conformal field theories [7,8,9,10].
In the weak coupling region, one can compute the partition function of N = 2 gauge theories based on the microscopic Lagrangian in the Ω-background, which deforms fourdimensional spacetime by the torus action with two parameters (ǫ 1 , ǫ 2 ) [11,12]. The partition function are related to conformal blocks of two-dimensional conformal field theories [13,14], the partition functions of topological strings [15,16], and the solutions of the Painléve equations [17], where the Ω-deformation parameters enter into the formulas of the central charges and the string coupling. It would be interesting to study the effects of the Ω-deformations in the strong coupling region. However in the strong coupling region such as the superconformal point, we have no appropriate microscopic Lagrangian. In the case of the self dual Ω-background with ǫ 1 = −ǫ 2 , the Argyres-Douglas theories have been studied by using the holomorphic anomaly equation [15,18] and the E-strings [19].
The purpose of this paper is to study the Argyres-Douglas theories in the Ω-background realized at the superconformal point of N = 2 supersymmetric gauge theories. In particular, we consider the Nekrasov-Shatashvili (NS) limit [20] of the Ω background where one of the deformation parameters ǫ 2 is set to be zero. In this limit the SW curve becomes a differential equation which is obtained by the canonical quantization procedure of the symplectic structure induced by the SW differential. The Planck constant corresponds to the remaining deformation parameter ǫ 1 . The WKB solution of the differential equation gives the Ω-deformation of the SW periods which is the main subject of this paper.
The quantum SW curve has been studied for N = 2 theories in the weak coupling regions. A simple example is SU(2) pure Yang-Mills theory where the quantum SW curve becomes the Schrödinger equation with the sine-Gordon potential [21] and the WKB solution is shown to agree with that obtained from the NS limit of the Nekrasov function.
The expansion of the periods around the massless monopole point in the Coulomb moduli space has been studied in [22]. For N = 2 SU(2) SQCD with N f ≤ 4 hypermultiplets, the WKB solutions of the quantum SW curves have been studied in [23] in the weak coupling region, while in the strong coupling region the solutions around the massless monopole point have been studied in [24]. Generalization to other N = 2 theories and their relations to the Nekrasov partition functions have been studied extensively [25,23,26,27,28].
In this paper we will study the quantum SW periods around the superconformal point of the moduli space of N = 2 SU(2) SQCD with N f = 1, 2, 3 hypermultiplets. The SW curve degenerates into a simpler curve which represents the SW curve of the Argyres-Douglas theory. We will calculate the WKB solution of the quantum SW curve of the AD theory and compute the quantum corrections up to the fourth order in . This paper is organized as follows: In Section 2, we review the SW curve and the SW differential near the superconformal point of the N = 2 SU(2) SQCD. In Section 3, we quantize the SW curve of the AD theories and derive the differential equations satisfied by quantum periods. In Section 4, we calculate the quantum corrections to the SW periods near the superconformal point, which are expressed in terms of the hypergeometric function. Section 5 is devoted to conclusions and discussion. In the Appendix, we present detailed analysis of the fourth order terms in the quantum SW periods for the N f = 3 AD theory.

Seiberg-Witten curve at the superconformal point
In this section we study the Argyres-Douglas theory which appears at the superconformal point in the moduli space of N = 2 SU(2) SQCD with N f = 1, 2, 3 hypermultiplets. We begin with the Seiberg-Witten curve for the N = 2 SU(2) gauge theory with N f (= 1, 2, 3) hypermultiplets which is given by 1) where Λ N f is the QCD scale parameter. C(p) and G(p) are defined by where u is the Coulomb moduli parameter and m 1 , . . . , m N f are the mass parameters of the hypermultiplets. The SW differential is defined by where α and β are the canonical one-cycles on the curve. Here the superscript (0) refers the "undeformed" (or classical) period. The SW curve (2.1) can be written into the standard form [29] The SW differential (2.4) is expressed as The u-derivative of the SW differential becomes the holomorphic differential: where ∂ u := ∂ ∂u . Differentiating the SW period Π (0) with respect to u, one obtains the periods for the curve: 2∂ u z z dp = β 2 y dp. (2.10) The period ∂ u Π (0) is evaluated as the elliptic integral. For the curve of the form where ∆ is of the discriminant and w is inverse of the modular J-function of the curve [30]. Then it is shown that the integral F = (−D) 1 4 dx y obeys the hypergeometric differential equation with α = 1 12 , β = 5 12 and γ = 1. For the SW curve (2.6) this leads to the Picard-Fuchs equation for Π (0) [31,32,33,24] as the third order differential equation with respect to u.
There are singularities on the u-plane where some BPS particles become massless and the discriminant ∆ (2.13) becomes zero. We consider the superconformal or Argyres-Douglas (AD) point on the u-plane where mutually nonlocal BPS particles become massless [1,2]. For the SU(2) theory with N f hypermultiplets, the squark and monopole/dyon are both massless at the AD point, where the SW curve degenerates and has higher order zero. For the SU(2) theories with N f = 1, 2, 3 hypermultiplets, the AD points are given as follows: For N f = 1, the Coulomb moduli and the mass are chosen as The SW curve (2.6) becomes (2.16) For N f = 2, we have so that the SW curve (2.6) becomes (2.18) For N f = 3, the superconformal point is given by where the SW curve (2.6) becomes up to the total derivatives wherẽ These parameters are interpreted as the mass parameters at the AD point. We see that the scaling dimensions ofũ,M ,C 2 ,C 3 are 3 2 , 1 2 , 2 and 3, respectively. We now study the SW periods for the AD theories associated with SU(2) theory with N f hypermultiplets. We write the SW curves in the form of for the N f AD theory. Here ρ N f and σ N f are read off from ( The SW periods are defined bỹ whereα andβ are canonical 1-cycles on the curve (2.35). Differentiating the SW periods with respect toũ, we have the period integral dp z of the holomorphic differential dp z : ω = α dp z , ω D = β dp z . (2.38) As in the case of SU(2) SQCD, the period integral is expressed in terms of the hypergeometric functions of the argument: Here∆ N f andD N f correspond to ∆ in (2.13) and D in (2.11) , respectively, which are defined by∆ For example, we will evaluate the integrals (2.38) around the pointw N f = 0, where theα-cycle is chosen as a vanishing cycle. Using the quadratic and cubic transformation [34,35], the periods are given by where F (α, β; γ; z) is the hypergeometric function. F * (α, β; 1; z) is defined by and We have omitted the subscript N f ofw andD for brevity. Since the dual period has logarithmic divergence aroundw = 0, it does not represent the expansion around the superconformal point, whereũ andM have fractional scaling dimensions.
We will perform the analytic continuation of the solutions aroundw = 0 to those of w = ∞ by using the connection formula [34] F (α, β; where | arg(1 − z)| < π. We then find that the periods (2.42) and (2.43) become respectively. Similarly we can perform the analytic continuation to the solutions around w = 1. By using the connection formula we obtain expansion aroundw = 1: Based on these formulas, we discuss the SW periods for the AD theories. For the N f = 1 theory,w 1 andD 1 are given bỹ The superconformal point corresponds tow ′ 1 := 1 1−w 1 = 0. Therefore eqs. (2.47) and (2.48) give the expansion around the superconformal point: By integrating them overũ, we obtain the SW periods We note that the SW periodsΠ (0) satisfy the Picard-Fuchs equation [36] ( From (2.55) and (2.56) we see that the SW periods scale asũ 5 6 . Since the SW periods a (0) and a (0) D have the scaling dimension one, the scaling dimension ofũ is 6 5 [2]. Since we have taken the scaling limit ǫ → 0, the SW periods (2.55) and (2.56) give different expansions from those in [35,37], which include full ǫ corrections.

Quantum Seiberg-Witten curves and periods
In this section we study the deformation of the SW periods in the Ω-background at the superconformal point for the SU (2) gauge theory with N f (= 1, 2, 3) hypermultiplets.
We take the the Nekrasov-Shatashvili (NS) limit such that one of the two deformation parameters (ǫ 1 , ǫ 2 ) of the Ω background is going to be zero. The other parameter plays a role of the Planck constant . From the analysis of the Ω-deformed low-energy effective action, the deformed periods in the NS limit are shown to satisfy the Bohr-Sommerfeld quantization condition [20]: This condition also follows from the quantization of the SW curve, which is introduced by the canonical quantization of the holomorphic symplectic structure defined by dλ SW .
The quantum SW curve becomes the ordinary differential equation. Its WKB solution gives the quantum correction to the SW periods, which can be represented in the form O k Π (0) for some differential operatorÔ k with respect to the moduli parameters. In the following we will constructÔ 2 andÔ 4 explicitly and compute the second and fourth order corrections to the SW periods in around the superconformal point.

N f = 1 theory
We start with the N f = 1 theory. The SW differential (2.24) defines a symplectic form dλ SW = dz ∧ dp on the (z,p) space. We quantize the system by replacing the coordinatẽ z by the differential operator:z Then the SW curve becomes the Schrödinger type equation: where We study the WKB solution to the equation (3.3): Note that φ n (p) for odd n becomes a total derivative and only φ n (p) for even n contributes to the period integrals. The first three φ 2n 's are given by up to total derivatives where ∂p := ∂ ∂p . We define the quantum SW periods along the canonical 1-cyclesα andβ. The periods are expanded in as Π =Π (0) + 2Π(2) + 4Π(4) + · · · (3.11) whereΠ (2n) := φ 2n (p)dp.Π (0) is the classical SW period. Similarly, we defineã (2n) and a (2n) D byã =ã (0) + 2ã(2) + 4ã(4) + · · · , (3.12) Substituting (3.4) into (3.8) and (3.9), one finds that (3.14) The classical SW periodsΠ (0) satisfy the Picard-Fuchs equation (2.57). It is also found to satisfy the differential equation with respect toM andũ: From (3.14), the second and fourth order terms satisfỹ We note that the higher order corrections can be calculated by taking the scaling limit of those of the N f = 1 SU(2) theory. The second and fourth order corrections to the SW periods for the N f = 1 theory are given as [24] Π (2) = 1 12 2u

N f = 2 theory
Next we discuss the quantum SW curve for the N f = 2 theory. We introduce a new variable ξ byp so that the SW differential (2.27) becomes a canonical form The SW curve (2.26) takes the form: Replacingz by the differential operator we obtain the quantum SW curve: We consider the WKB solution to the wave function Ψ(ξ) which is defined by (3.5). The leading term φ 0 (ξ) in the expansion (3.6) in is given by φ 0 (ξ) =z(ξ), which gives the classical SW periodsΠ (0) = φ 0 (ξ)dξ. One can show that (−D 2 ) 1 4 ∂ũΠ (0) satisfies the Picard-Fuchs equation (2.14).Π (0) also satisfies the differential equation where the classical SW period as Note that (3.30) and (3.31) are defined up to the Picard-Fuchs equations. We also note that one can derive these relations from those of N f = 2 SU(2) theory, which are given by [24] Π Taking the scaling limit (2.25), We find that the quantum SW periods Π (2) and Π (4) become Then the SW curve (2.30) can be written as (3.39) Replacing the coordinate ξ by the differential operator one obtains the quantum SW curve. But we need to consider the ordering of the operators.
In general we can define the ordering of the operators by (3.41) parametrized by t (0 ≤ t ≤ 1). We will use the t = 1 2 prescription as in [23]. Then the quantum SW curve (3.38 ) takes the form We consider the WKB solution (3.5) to the quantum curve. The leading term is given by

The quantum SW curve (3.42) is written as
Substituting (3.6) into (3.43), we can determine φ n (p) in a recursive way. φ 0 (p) is expressed as which is equal toξ(p). Hereỹ is defined bỹ is shown to be the total derivative: We can show that φ 3 (p) is also a total derivative. φ 2 and φ 4 are found to be up to the total derivative.

Quantum SW periods around the superconformal point
In the previous section we have constructed the quantum SW curves and the quantum SW periods which are obtained by acting the differential operators on the classical SW periods. In this section we will calculate an explicit form of the quantum SW periods around the superconformal point up to the fourth order in .

N f = 2 theory
We next compute the quantum corrections to the SW periods for the N f = 2 theory.
From (3.30) and (2.60) we find that the second order corrections are given bỹ The fourth order corrections can be obtained in a similar manner. We find that a (4) = 1 (4.29) It would be interesting to compare the free energy with that of the E-string theory, which is left for future work.

N f = 3 theory
We now discuss the N f = 3 case. Using (3.52) and (2.65) we find that the second order corrections to the SW periods are given bỹ Expanding the second order corrections to the SW periods inw ′ 3 , whereM In a similar way, we can calculate the fourth order corrections. Since they are rather cumbersome, we will show them in the Appendix.

Conclusions and Discussions
In this paper we studied the quantum SW periods around the superconformal point of N = 2 SU(2) SQCD with N f = 1, 2, 3 hypermultiplets, which is deformed in the Nekrasov-Shatashvili limit of the Ω-background. The scaling limit around the superconformal point gives the SW curves of the corresponding Argyres-Douglas theories. The SW curves take the form of cubic elliptic curve for all N f . But the SW differentials take the different form, which introduce the different quantization condition. We have computed the quantum corrections to the SW periods up to the fourth order in , which are obtained from the classical periods by acting the differential operators with respect to the moduli parameters. They are shown to agree with the scaling limit of the SW periods of the original SQCD. We wrote down the explicit form of the quantum corrections in terms of hypergeometric functions. It is interesting to explore the higher order corrections in . In particular the resurgence method helps us to understand non-perturbative structure of the -corrections [39,40,41,42].
So far we have studied the AD theories around the superconformal fixed point, where the SW periods and the effective coupling constant are expanded in the Coulomb moduli parameter with fractional power. It would be interesting to study the -corrections to the beta functions around the conformal point [43]. Note that the moduli space of these AD theories contains the point, where one of the periods shows the logarithmic behavior around the point. It would be interesting to describe the theory around the point by the Nekrasov partition function.
For N f = 1 case, it gives the same AD theory as SU(3) N = 2 super Yang-Mills theory [1], which gives the Schrödinger equation with cubic polynomial potential. In [44], using the ODE/IM correspondence (for a review see [45]), it is shown that the the exponential of the quantum period can be regarded as the Y-function of the quantum integrable model associated with the Yang-Lee edge singularity. It is interesting to study this relation further by computing further higher order corrections by using the ODE/IM correspondence. It is also interesting to generalize the quantum SW curve for the AD theories associated with higher rank gauge theories [37].