NUPHB14397S0550-3213(18)30193-710.1016/j.nuclphysb.2018.07.007The AuthorsHigh Energy Physics – TheoryQuantum periods for N=2 SU(2) SQCD around the superconformal pointKatsushiItoTakafumiOkubo⁎t.okubo@th.phys.titech.ac.jpDepartment of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, JapanDepartment of PhysicsTokyo Institute of TechnologyTokyo152-8551Japan⁎Corresponding author.Editor: Leonardo RastelliAbstractWe study the Argyres–Douglas theories realized at the superconformal point in the Coulomb moduli space of N=2 supersymmetric SU(2) QCD with Nf=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.1IntroductionA 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 [1–3]. The dynamics of AD theories is an interesting subject of recent studies from the viewpoint of M5-branes compactified on a punctured Riemann surface [4–6] and its relation to two-dimensional conformal field theories [7–10].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 four-dimensional spacetime by the torus action with two parameters (ϵ1,ϵ2) [11,12]. The partition function is 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 ϵ1=−ϵ2, 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 ϵ2 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 ħ corresponds to the remaining deformation parameter ϵ1. 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 Nf≤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 [23,25–28].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 Nf=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 ħ.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 Nf=3 AD theory.2Seiberg–Witten curve at the superconformal pointIn 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 Nf=1,2,3 hypermultiplets. We begin with the Seiberg–Witten curve for the N=2 SU(2) gauge theory with Nf(=1,2,3) hypermultiplets which is given by(2.1)C(p)−ΛNf2−Nf22(z+G(p)z)=0, where ΛNf is the QCD scale parameter. C(p) and G(p) are defined by(2.2)C(p)={p2−u,Nf=1,p2−u+Λ228,Nf=2,p2−u+Λ34(p+m1+m2+m32),Nf=3,(2.3)G(p)=∏i=1Nf(p+mi), where u is the Coulomb moduli parameter and m1,…,mNf are the mass parameters of the hypermultiplets. The SW differential is defined by(2.4)λSW=p(dlogG(p)−2dlogz). The SW periods Π(0):=(a(0),aD(0)) are(2.5)a(0)(u)=∮αλSW,aD(0)(u)=∮βλSW 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](2.6)y2=C(p)2−ΛNf4−NfG(p) by introducing(2.7)y=ΛNf2−Nf2z−C(p). The SW differential (2.4) is expressed as(2.8)λSW=pdlog(C(p)−yC(p)+y). The u-derivative of the SW differential becomes the holomorphic differential:(2.9)∂λSW∂u=2∂uzzdp+d(⁎)=2dpy+d(⁎) where ∂u:=∂∂u. Differentiating the SW period Π(0) with respect to u, one obtains the periods for the curve:(2.10)∂ua(0)(u)=∮α2∂uzzdp=∮α2ydp,∂uaD(0)(u)=∮β2∂uzzdp=∮β2ydp. The period ∂uΠ(0) is evaluated as the elliptic integral. For the curve of the form y2=∏i=14(x−ei), it is convenient to introduce the variables(2.11)D=∑i<jei2ej2−6∏i=14ei−∑i<j<k(ei2ejek+eiej2ek+eiejek2),(2.12)w=−27Δ4D3, where Δ is of the discriminant(2.13)Δ=∏i<j(ei−ej)2, and w is inverse of the modular J-function of the curve [30]. Then it is shown that the integral F=(−D)14∫dxy obeys the hypergeometric differential equation(2.14)w(1−w)d2Fdw2+(γ−(α+β+1)w)dFdw−αβF=0 with α=112, β=512 and γ=1. For the SW curve (2.6) this leads to the Picard–Fuchs equation for Π(0) [24,31–33] 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 Δ (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 Nf 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 Nf=1,2,3 hypermultiplets, the AD points are given as follows: For Nf=1, the Coulomb moduli and the mass are chosen as(2.15)u=34Λ12,m1=34Λ1. The SW curve (2.6) becomes(2.16)y2=(p−23Λ1)(p+12Λ1)3. For Nf=2, we have(2.17)u=38Λ22,m1=m2=Λ22, so that the SW curve (2.6) becomes(2.18)y2=(p−32Λ2)(p+Λ22)3. For Nf=3, the superconformal point is given by(2.19)u=132Λ32,m1=m2=m3=Λ38, where the SW curve (2.6) becomes(2.20)y2=(p−78Λ3)(p+Λ38)3.Let us study the SW curve and the SW differential around the superconformal point. By taking the scaling limit, we identify the operators and couplings which deform the superconformal point. Their scaling dimensions are determined by the SW curve and the fact that the SW differential has the scaling dimension one. We first consider in the Nf=1 theory. The branch point p=−Λ12 of the curve (2.16) corresponds to z=±Λ1122. We expand the curve (2.1) around z=−Λ1122 by introducing(2.21)p=ϵp˜−Λ12,z=i212ϵ32Λ1z˜−ϵ2M˜Λ112−ϵp˜Λ112−Λ1122,u=ϵ3u˜+ϵ2M˜Λ1+34Λ12,m1=ϵ2M˜+34Λ1, and consider the scaling limit ϵ→0 with fixed u˜ and M˜. At the leading order in ϵ we obtain the curve for the AD theory of (A1,A2)-type:(2.22)z˜2=p˜3−M˜Λ1p˜−Λ12u˜.Substituting (2.21) into the SW differential (2.4) and expanding around ϵ=0, the SW differential becomes(2.23)λSW=iϵ52212Λ112λ˜SW+…,(2.24)λ˜SW:=−8Λ1z˜dp˜. We read off the scaling dimension of u˜ and M˜ as 65 and 45, respectively, from the curve (2.22). Here u˜ is the operator and M˜ is the corresponding coupling parameter.For Nf=2, defining the new variables as(2.25)p=ϵp˜−ϵM˜3−Λ22,z=i212ϵ32Λ212z˜−ϵp˜−2ϵM˜3,u=ϵ2u˜−(ϵM˜)23+Λ2ϵM˜+3Λ228,m1=Λ22+ϵM˜+ϵ32a˜,m2=Λ22+ϵM˜−ϵ32a˜, and expanding the curve around ϵ=0, we find that the curve (2.1) become(2.26)z˜2=p˜3−u˜p˜−23M˜u˜+827M˜3−C˜2Λ24. Here u˜ is the operator, M˜ is the coupling and C˜2:=2a˜2 is the Casimir invariant of the U(2) flavor symmetry. The corresponding AD theory is of (A1,A3)-type.Substituting (2.25) into (2.4), the SW differential around the superconformal point is(2.27)λSW=iϵ32212Λ212λ˜SW+⋯ up to the total derivatives where(2.28)λ˜SW=−4z˜dlog(p˜+23M˜). The scaling dimension of u˜, M˜ and C˜2 are 43, 23 and 2, respectively.For Nf=3, we define the scaling variables as(2.29)p=ϵ2p˜−ϵM˜+4((ϵM˜)2+ϵ3u˜)3Λ3+16(ϵM˜)39Λ32−Λ38,z=ϵ3iz˜−4(ϵM˜)33Λ332−2(ϵM˜)(ϵ2p˜)Λ312−ϵ3u˜Λ312,u=ϵ3u˜−4(ϵM˜)33Λ3+(ϵM˜)2+3Λ3ϵM˜8+Λ3232,m1=Λ38+ϵM˜+ϵ2c˜1,m2=Λ38+ϵM˜+ϵ2c˜2,m3=Λ38+ϵM˜−ϵ2(c˜1+c˜2), and then consider the limit ϵ→0 limit with keeping u˜, M˜, c˜1 and c˜2 finite. Rescaling the curve (2.1) we obtain the curve of the AD theory of (A1,D4) type:(2.30)z˜2=p˜3−p˜(C2˜2+4M˜u˜Λ3)−u˜2Λ3−8M˜3u˜3Λ32+16M˜627Λ33−2C2˜M˜23Λ3+C3˜3 where(2.31)C2˜:=2(c˜12+c˜1c˜2+c˜22),C3˜:=−3(c˜12c˜2+c˜1c˜22). Here u˜ is the operator and M˜ is the coupling. C˜2 and C˜3 are the Casimir invariants associated with the U(3) flavor symmetry. Then the SW differential (2.4) at the superconformal point becomes(2.32)λSW=iϵ2Λ312λ˜SW+⋯ up to the total derivatives where(2.33)λ˜SW=iΛ312{2p˜dlog(iz˜−2M˜p˜Λ312−4M˜33Λ332−u˜Λ312)−∑i=13p˜dlog(p˜+m˜i)}. m˜i (i=1,…,3) are defined by(2.34)m˜1=4M˜23Λ3+c˜1,m˜2=4M˜23Λ3+c˜2,m˜3=4M˜23Λ3−(c˜1+c˜2). These parameters are interpreted as the mass parameters at the superconformal point. We see that the scaling dimensions of u˜, M˜, C˜2, C˜3 are 32, 12, 2 and 3, respectively.We now study the SW periods for the AD theories associated with SU(2) theory with Nf hypermultiplets. We write the SW curves in the form of(2.35)z˜2=p˜3−ρNfp˜−σNf for the Nf AD theory. Here ρNf and σNf are read off from (2.22), (2.26) and (2.30). We have normalized the SW differential λ˜SW (2.24), (2.28) and (2.33) such that(2.36)∂∂u˜λ˜SW=2dp˜z˜. The SW periods are defined by(2.37)Π˜(0)=(a˜(0),a˜D(0))=(∫α˜λ˜SW,∫β˜λ˜SW), where α˜ and β˜ are canonical 1-cycles on the curve (2.35). Differentiating the SW periods with respect to u˜, we have the period integral ∫dp˜z˜ of the holomorphic differential dp˜z˜:(2.38)ω=∫α˜dp˜z˜,ωD=∫β˜dp˜z˜. As in the case of SU(2) SQCD, the period integral is expressed in terms of the hypergeometric functions of the argument:(2.39)w˜Nf:=−27Δ˜Nf4D˜Nf3=1−27σNf24ρNf3. Here Δ˜Nf and D˜Nf correspond to Δ in (2.13) and D in (2.11), respectively, which are defined by(2.40)Δ˜Nf=4ρNf3−27σNf2,(2.41)D˜Nf=−3ρNf.For example, we will evaluate the integrals (2.38) around the point w˜Nf=0, where the α˜-cycle is chosen as a vanishing cycle. Using the quadratic and cubic transformation [34,35], the periods are given by(2.42)ω0(w˜,D˜)=2π(−D˜)−14F(112,512;1;w˜),(2.43)ωD0(w˜,D˜)=−2iπ(−D˜)−14(3log122πF(112,512;1;w˜)−12πF⁎(112,512;1;w˜)), where F(α,β;γ;z) is the hypergeometric function. F⁎(α,β;1;z) is defined by(2.44)F⁎(α,β;1;z)=F(α,β;1;z)logz+F1(α,β;1;z) and(2.45)F1(α,β;1;z)=∑n=0∞(α)n(β)n(n!)2∑r=0n−1(1α+r+1β+r−21+r)zn. We have omitted the subscript Nf of w˜ and D˜ for brevity. Since the dual period has logarithmic divergence around w˜=0, it does not represent the expansion around the superconformal point, where u˜ and M˜ have fractional scaling dimensions.We will perform the analytic continuation of the solutions around w˜=0 to those of w˜=∞ by using the connection formula [34](2.46)F(α,β;γ;z)=Γ(γ)Γ(β−α)Γ(β)Γ(γ−α)(1−z)−αF(α,γ−β;α−β+1;11−z)+Γ(γ)Γ(α−β)Γ(α)Γ(γ−β)(1−z)−βF(β,γ−α;−α+β+1;11−z), where |arg(1−z)|<π. We then find that the periods (2.42) and (2.43) become(2.47)ω∞(w˜,D˜)=2π(−D˜)−14(Γ(13)Γ(512)Γ(1112)(1−w˜)−112F(112,712;23;11−w˜)+Γ(−13)Γ(112)Γ(712)(1−w˜)−512F(512,1112;43;11−w˜)),(2.48)ωD∞(w˜,D˜)=2iπ(−D˜)−14((−1)56Γ(13)Γ(512)Γ(1112)(1−w˜)−112F(112,712;23;11−w˜)+(−1)16Γ(−13)Γ(112)Γ(712)(1−w˜)−512F(512,1112;43;11−w˜)), respectively. Similarly we can perform the analytic continuation to the solutions around w˜=1. By using the connection formula(2.49)F(α,β;γ;z)=(1−z)−α−β+γΓ(γ)Γ(α+β−γ)Γ(α)Γ(β)×F(γ−α,γ−β;−α−β+γ+1;1−z)+Γ(γ)Γ(−α−β+γ)Γ(γ−α)Γ(γ−β)F(α,β;α+β−γ+1;1−z), we obtain expansion around w˜=1:(2.50)ω1(w˜,D˜)=π−12(−D˜)−14(6Γ(512)Γ(1312)F(112,512;12;1−w˜)−(1−w˜)12Γ(712)Γ(1112)F(712,1112;32;1−w˜)),(2.51)ωD1(w˜,D˜)=−iπ−12(−D˜)−14(6Γ(512)Γ(1312)F(112,512;12;1−w˜)+(1−w˜)12Γ(712)Γ(1112)F(712,1112;32;1−w˜)).Based on these formulas, we discuss the SW periods for the AD theories. For the Nf=1 theory, w˜1 and D˜1 are given by(2.52)w˜1=1−27u˜216Λ1M˜3,(2.53)D˜1=−3Λ1M˜. The superconformal point corresponds to w˜1′:=11−w˜1=0. Therefore eqs. (2.47) and (2.48) give the expansion around the superconformal point:(2.54)∂a˜∂u˜=2ω∞(w˜1,D˜1),∂a˜D∂u˜=2ωD∞(w˜1,D˜1). By integrating them over u˜, we obtain the SW periods(2.55)a˜(0)=312Λ132212⋅5π12(u˜Λ12)56(283Γ(16)Γ(13)F(−512,112;23;w˜1′)+15w˜1′13Γ(−16)Γ(53)F(−112,512;43;w˜1′)),(2.56)a˜D(0)=312Λ132212⋅5π12(u˜Λ12)56(−283(−1)13Γ(16)Γ(13)F(−512,112;23;w˜1′)+15(−1)23w˜1′13Γ(−16)Γ(53)F(−112,512;43;w˜1′)). We note that the SW periods Π˜(0) satisfy the Picard–Fuchs equation [36](2.57)(1−w˜1′)w˜1′∂2∂w˜1′2Π˜(0)+23(1−w˜1′)∂∂w˜1′Π˜(0)+5144Π˜(0)=0. From (2.55) and (2.56) we see that the SW periods scale as u˜56. Since the SW periods a(0) and aD(0) have the scaling dimension one, the scaling dimension of u˜ and M˜ is given by 65 and 45, respectively [2]. The expansion of the coupling constant τ(0):=∂u˜a˜D(0)∂u˜a˜(0) in w˜1′ does not contain logarithmic terms, which implies that the theory is around the superconformal point. The SW periods (2.55) and (2.56) represent the expansions in the coupling M˜ with fixed u˜ in the scaling limit. We note that the present expansions for Nf theories are different from the results in the previous literatures [15,35], where the coupling and the Casimir invariants are chosen to be zero, u˜ is small without taking the scaling limit. In [37] the expansion of the SW periods without taking the scaling limit has been presented.For the Nf=2 theory, we have(2.58)w˜2=1−(272C˜2Λ2−16M˜3+36M˜u˜)2432u˜3,(2.59)D˜2=−3u˜. The superconformal point corresponds to w˜2=1 or w˜2′:=1−w˜2=0. Eqs. (2.50) and (2.51) provide the expansion around the superconformal point:(2.60)∂a˜∂u˜=2ω1(w˜2,D˜2),∂a˜D∂u˜=2ωD1(w˜2,D˜2). Expanding them around w˜2′=0, where M˜2u˜≪1 and C˜2Λ2u˜32≪1, and integrating over u˜, one obtains the SW periods, which are given by(2.61)a˜(0)=Λ232(u˜Λ22)34(24Γ(512)Γ(1312)314π12−23⋅314Γ(712)Γ(1112)π12(M˜2u˜)12−312Γ(712)Γ(1112)π12(C˜22Λ22u˜3)12+⋯),(2.62)a˜D(0)=Λ232(u˜Λ22)34(−24iΓ(512)Γ(1312)314π12−23⋅314iΓ(712)Γ(1112)π12(M˜2u˜)12−312iΓ(712)Γ(1112)π12(C˜22Λ22u˜3)12+⋯). We see again that the scaling dimensions of u˜, M˜ and C˜2 are 43, 23 and 2, respectively. The expansions of the periods (2.61) and (2.62) have no logarithmic behavior.For the Nf=3 theory, we have(2.63)w˜3=1−(−9C˜3Λ33+18C˜2Λ32M˜2−16M˜6+72Λ3M˜3u˜+27Λ32u˜2)2108Λ36(C˜22+4M˜u˜Λ3)3,(2.64)D˜3=−3(C˜22+4M˜u˜Λ3). The superconformal point corresponds to w˜3=∞ or w˜3′:=11−w˜3=0. Then (2.47) and (2.48) provides the periods around the superconformal point:(2.65)∂a˜∂u˜=2ω∞(w˜3,D˜3),∂a˜D∂u˜=2ωD∞(w˜3,D˜3). Expanding these in w˜3′, where M˜3u˜Λ3≪1, C˜23Λ32u˜4≪1 and C˜3Λ3u˜2≪1, and integrating (2.65) over u˜, we obtain the SW periods:(2.66)a˜(0)=Λ332(−1)56(u˜Λ32)23(5Γ(−56)Γ(13)2⋅312π12−23Γ(−13)Γ(56)312π12(M˜3u˜Λ3)13+Γ(−13)Γ(56)2⋅π12(C˜23Λ32u˜4)13+Γ(16)Γ(13)22⋅332π12(C˜3Λ3u˜2)+⋯),(2.67)a˜D(0)=Λ332(−1)16(u˜Λ32)23(5Γ(−56)Γ(13)2⋅312π12−23Γ(−13)Γ(56)312π12(M˜3u˜Λ3)13+iΓ(−13)Γ(56)2⋅π12(C˜23Λ32u˜4)13+Γ(16)Γ(13)22⋅332π12(C˜3Λ3u˜2)+⋯). It turns out that the scaling dimensions of u˜, M˜, C˜2 and C˜3 are 32, 12, 2 and 3, respectively. As in the case of Nf=1 and 2 theories, the expansion of the SW periods has no logarithmic term.Although the SW curves for Nf theories become a common cubic form, their SW differentials take different forms due to the flavor symmetry. This means that we need to introduce different quantization conditions for each Nf as we will discuss in the next section.3Quantum Seiberg–Witten curves and periodsIn 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 Nf(=1,2,3) hypermultiplets. We take the Nekrasov–Shatashvili (NS) limit such that one of the two deformation parameters (ϵ1,ϵ2) of the Ω background is going to be zero. The other parameter plays a role of the Planck constant ħ. 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]:(3.1)∮λSW=inħ,(n∈Z). 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 dλSW. 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 OˆkΠ(0) for some differential operator Oˆk with respect to the moduli parameters. In the following we will construct Oˆ2 and Oˆ4 explicitly and compute the second and fourth order corrections to the SW periods in ħ around the superconformal point.3.1Nf=1 theoryWe start with the Nf=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 coordinate z˜ by the differential operator:(3.2)z˜=−iħ∂∂p˜. Then the SW curve becomes the Schrödinger type equation:(3.3)(−ħ2∂2∂p˜2+Q(p˜))Ψ(p˜)=0, where(3.4)Q(p˜)=−(p˜3−M˜Λ1p˜−Λ12u˜). We study the WKB solution to the equation (3.3):(3.5)Ψ(p˜)=exp(iħ∫p˜Φ(y)dy), where(3.6)Φ(y)=∑n=0∞ħnϕn(y). Substituting the expansion (3.6) into (3.3), one obtains the recursion relations for ϕn(p˜)'s. Note that ϕn(p˜) for odd n becomes a total derivative and only ϕn(p˜) for even n contributes to the period integrals. The first three ϕ2n's are given by(3.7)ϕ0(p˜)=iQ12,(3.8)ϕ2(p˜)=i48∂p˜2QQ32,(3.9)ϕ4(p˜)=−7i1356(∂p˜2Q)2Q72+i768∂p˜4QQ52, up to total derivatives where ∂p˜:=∂∂p˜. We define the quantum SW periods(3.10)Π˜=(a˜,a˜D)=(∮α˜Φ(p˜)dp˜,∫β˜Φ(p˜)dp˜) along the canonical 1-cycles α˜ and β˜. The periods are expanded in ħ as(3.11)Π˜=Π˜(0)+ħ2Π˜(2)+ħ4Π˜(4)+⋯ where Π˜(2n):=∮ϕ2n(p˜)dp˜. Π˜(0) is the classical SW period. Similarly, we define a˜(2n) and a˜D(2n) by(3.12)a˜=a˜(0)+ħ2a˜(2)+ħ4a˜(4)+⋯,(3.13)a˜D=a˜D(0)+ħ2a˜D(2)+ħ4a˜D(4)+⋯. Substituting (3.4) into (3.8) and (3.9), one finds that(3.14)ϕ2(p˜)=1Λ12∂∂M˜∂∂u˜ϕ0(p˜),ϕ4(p˜)=710Λ14∂2∂M˜2∂2∂u˜2ϕ0(p˜). The classical SW periods Π˜(0) satisfy the Picard–Fuchs equation (2.57). It is also found to satisfy the differential equation with respect to M˜ and u˜:(3.15)∂2∂M˜∂u˜Π˜(0)=−3u˜2M˜∂2∂u˜2Π˜(0)−14M˜∂∂u˜Π˜(0). From (3.14), the second and fourth order terms satisfy(3.16)Π˜(2)=1Λ12∂∂M˜∂∂u˜Π˜(0),(3.17)Π˜(4)=710Λ14∂2∂M˜2∂2∂u˜2Π˜(0).We note that the higher order corrections can be calculated by taking the scaling limit of those of the Nf=1 SU(2) theory. The second and fourth order corrections to the SW periods for the Nf=1 theory are given as [24]. We can show that the formulas in [24] reduces to (3.16) and (3.17) in the scaling limit (2.21). The quantization conditions for the AD theories become different although they take the same form for the SQCDs. Therefore it is nontrivial to check that the scaling limit of the quantum SW periods of the SQCDs gives those of the AD theories. In Section 4, we will calculate the deformed SW periods around the superconformal point by using the relations (3.16) and (3.17) up to fourth order.3.2Nf=2 theoryNext we discuss the quantum SW curve for the Nf=2 theory. We introduce a new variable ξ by(3.18)p˜=eξ−23M˜, so that the SW differential (2.27) becomes a canonical form(3.19)λ˜SW=z˜dξ. The SW curve (2.26) takes the form:(3.20)z˜2−(e3ξ−2M˜e2ξ+eξ(4M˜23−u˜)−Λ2C˜24)=0. Replacing z˜ by the differential operator(3.21)z˜=−iħ∂∂ξ, we obtain the quantum SW curve:(3.22)(−ħ2∂2∂ξ2+Q(ξ))Ψ(ξ)=0 where(3.23)Q(ξ)=−(e3ξ−2M˜e2ξ+eξ(4M˜23−u˜)−Λ2C˜24). We consider the WKB solution to the wave function Ψ(ξ) which is defined by (3.5). The leading term ϕ0(ξ) in the expansion (3.6) in ħ is given by ϕ0(ξ)=z˜(ξ), which gives the classical SW periods Π˜(0)=∫ϕ0(ξ)dξ. One can show that (−D˜2)14∂u˜Π˜(0) satisfies the Picard–Fuchs equation (2.14). Π˜(0) also satisfies the differential equation(3.24)∂2∂M˜∂u˜Π˜(0)=L2(4u˜∂2∂u˜2Π˜(0)+∂∂u˜Π˜(0)) where(3.25)L2:=4(4M˜2−3u˜)27Λ2C˜2+24M˜u˜−32M˜3. From (3.8) and (3.9), we find that the second and fourth order corrections are related to the classical SW period as(3.26)Π˜(2)=(14∂∂M˜∂∂u˜+M˜3∂2∂u˜2)Π˜(0),(3.27)Π˜(4)=(7M˜290∂4∂u˜4+120∂3∂u˜3+7160∂2∂u˜2∂2∂M˜2+7M˜60∂3∂u˜3∂∂M˜)Π˜(0). Note that (3.26) and (3.27) are defined up to the Picard–Fuchs equations. We also note that one can derive these relations from those of Nf=2 SU(2) theory, which are given by [24]. We find that the second and fourth order formulas of the Nf=2 theory [24] lead to (3.26) and (3.27) after taking the scaling limit (2.25).3.3Nf=3 theoryFinally we study the quantum SW curve for the Nf=3 theory. We introduce a new coordinate ξ by(3.28)z˜=−i(eξ+2M˜p˜Λ312+4M˜33Λ332+u˜Λ312), so that the SW differential (2.32) becomes the canonical form(3.29)λ˜SW=iΛ3(p˜dξ˜+∑i=13p˜dlog(p˜+m˜i)). Then the SW curve (2.30) can be written as(3.30)e2ξ+(f0p˜+f1)eξ+g(p˜)=0, where(3.31)f0=4M˜Λ312,f1=8M˜33Λ332+2u˜Λ312,g(p˜)=p˜3−ρ3p˜−σ3+(2M˜p˜Λ312+4M˜33Λ332+u˜Λ312)2. Replacing the coordinate ξ by the differential operator(3.32)ξ=−iħ∂∂p˜, 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(3.33)tp˜e−iħ∂p˜Ψ(p˜)+e−iħ∂p˜((1−t)p˜Ψ(p˜))=(p˜−i(1−t)ħ)e−iħ∂p˜Ψ(p˜), parametrized by t (0≤t≤1). We will use the t=12 prescription as in [23]. Then the quantum SW curve (3.30) takes the form(3.34)(exp(−2iħ∂p˜)+(12f0p˜+f1)exp(−iħ∂p˜)+exp(−iħ∂p˜)12f0p˜+g(p˜))Ψ(p˜)=0. We consider the WKB solution (3.5) to the quantum curve. The leading term is given by ϕ0(p˜):=ξ(p˜). To discuss the higher order terms in ħ, we rewrite the quantum curve by introducingJ(α):=exp(−iħ∫p˜Φ(y)dy)exp(−iħα∂p˜)exp(iħ∫p˜Φ(y)dy). The quantum SW curve (3.34) is written as(3.35)J(2)+(f0(p˜−i2ħ)+f1)J(1)+g(x)=0. Substituting (3.6) into (3.35), we can determine ϕn(p˜) in a recursive way. ϕ0(p˜) is expressed as(3.36)ϕ0(p˜)=log(12(−f0p˜−f1+2y˜)) which is equal to ξ˜(p˜). Here y˜ is defined by(3.37)y˜2=14(f0p˜+f1)2−g(p˜). ϕ1(p˜) is shown to be the total derivative:(3.38)ϕ1(p˜)=∂∂p˜(i2ϕ0(p˜)+i4log4y˜). We can show that ϕ3(p˜) is also a total derivative. ϕ2 and ϕ4 are found to be(3.39)ϕ2(p˜)=(−f0p˜−f1)g″(p˜)96y˜3+f02(f0p˜+f1)192y˜3,(3.40)ϕ4(p˜)=g(4)(p˜)((f0p˜+f1)g(p˜)1536y˜5+−f0p˜−f15760y˜3)+g(3)(p˜)(f0g(p˜)480y˜5+f0720y˜3)+g″(p˜)(−7f02(f0p˜+f1)g(p˜)3072y˜7−7f02(f0p˜+f1)7680y˜5)+g″(p˜)2(7(f0p˜+f1)g(p˜)3072y˜7+7(f0p˜+f1)7680y˜5)+7f04(f0p˜+f1)g(p˜)12288y˜7+7f04(f0p˜+f1)30720y˜5, up to the total derivative.For the classical SW periods Π˜(0), (−D˜3)14∂u˜Π˜(0) satisfies the Picard–Fuchs equation (2.14). Π˜(0) also satisfies the differential equation with respect to M˜ and u˜:(3.41)∂2∂M˜∂u˜Π˜(0)=b3∂2∂u˜2Π˜(0)+c3∂∂u˜Π˜(0) where(3.42)b3=4M˜(3Λ3M˜u˜+4M˜4−3Λ32ρ3)ρ3+27Λ32u˜σ33Λ3(9Λ3M˜σ3−4M˜3ρ3−3Λ3u˜ρ3),(3.43)c3=(4M˜3+3Λ3u˜)2−12Λ32M˜2ρ33Λ3(9Λ3M˜σ3−4M˜3ρ3−3Λ3u˜ρ3). ρ3 and σ3 are read off from (2.30). Substituting (3.31) into (3.39) and (3.40) we find that formulas for the second and fourth order corrections in ħ:(3.44)Π˜(2)=(−M˜212∂2∂u˜2−Λ316∂∂u˜∂∂M˜)Π˜(0),(3.45)Π˜(4)=(7M˜41440∂4∂u˜4+Λ3M˜192∂3∂u˜3+7Λ322560∂2∂u˜2∂2∂M˜2+7Λ3M˜2960∂3∂u˜3∂∂M˜)Π˜(0). These formulas can be also obtained by taking scaling limit (2.29) of those in Nf=3 SU(2) SQCD [24].In the next section we will calculate the quantum corrections to the SW periods as expansions in coupling constant and the mass parameters.4Quantum SW periods around the superconformal pointIn the previous section we have constructed the quantum SW curves and the quantum SW periods of the AD theory, 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 ħ. We will consider the expansion in the coupling constant and the mass parameters of the AD theory.4.1Nf=1 theoryWe first discuss the Nf=1 theory around the superconformal point. Substituting (2.55) and (2.56) into (3.16) and changing the variables (u˜,M˜) to (u˜,w˜1′), the second order corrections to the SW periods are expressed in terms of hypergeometric function as(4.1)a˜(2)=1252⋅332π12Λ172(u˜Λ12)−56(F1(2)(w˜1′)−F2(2)(w˜1′,u˜)),(4.2)a˜D(2)=1252⋅332π12Λ172(u˜Λ12)−56((−1)23F1(2)(w˜1′)+(−1)13F(2)2(w˜1′)), where(4.3)F1(2)(w˜1′)=273⋅3Γ(23)Γ(56)(F(512,1112;43;w˜1′)−5F(1112,1712;43;w˜1′)),(4.4)F2(2)(w˜1′)=−7w˜1′23Γ(16)Γ(13)F(1312,1912;53;w˜1′). Similarly, substituting (2.55) and (2.56) into (3.17) and changing the variables (u˜,M˜) to (u˜,w˜1′), we find that the fourth order corrections to the SW periods (3.13) become(4.5)a˜(4)=−72436⋅352⋅5π12Λ1172w˜1′13(w˜1′−1)(u˜Λ12)−52(−F1(4)(w˜1′)+F2(4)(w˜1′)),(4.6)a˜D(4)=−72436⋅352⋅5π12Λ1172w˜1′13(w˜1′−1)(u˜Λ12)−52((−1)13F1(4)(w˜1′)+(−1)23F(4)2(w˜1′)), where(4.7)F1(4)(w˜1′)=23⋅7⋅13Γ(13)Γ(76)((11w˜1′+13)F(1912,2512;53;w˜1′)−5F(1312,1912;53;w˜1′)),(4.8)F2(4)(w˜1′)=213⋅5⋅11⋅17w˜1′13Γ(23)Γ(56)((7w˜1′+17)F(2312,2912;73;w˜1′)−F(1712,2312;73;w˜1′)). Expanding in w˜1′ around w˜1′=0, the quantum SW periods become(4.9)a˜=Λ132(u˜Λ12)56(−272Γ(−56)Γ(13)312π12−7Γ(−76)Γ(23)612π12w˜1′13+⋯)+ħ2Λ172(u˜Λ12)−56(7Γ(−76)Γ(23)216⋅3192π12+⋯)+ħ4Λ1172(u˜Λ12)−52(72⋅13Γ(−56)Γ(13)2196⋅392π12w˜1′13+⋯)+⋯,(4.10)a˜D=Λ132(u˜Λ12)56(272(−1)13Γ(−56)Γ(13)312π12−7(−1)23Γ(−76)Γ(23)612π12w˜1′13+⋯)+ħ2Λ172(u˜Λ12)−56(7(−1)23Γ(−76)Γ(23)216⋅3192π12+⋯)+ħ4Λ1172(u˜Λ12)−52(−72⋅13(−1)13Γ(−56)Γ(13)2196⋅392π12w˜1′13+⋯)+⋯. We define the effective coupling constant11Note that the present definition of the effective coupling constant is inverse of the one in [1]. τ˜ of the deformed theory by(4.11)τ˜:=∂u˜a˜D∂u˜a˜, which is expanded in ħ as(4.12)τ˜=τ˜(0)+ħ2τ˜(2)+ħ4τ˜(4)+⋯. Substituting (4.9) and (4.10) into (4.11) and expanding in ħ, we find(4.13)τ˜=(−(−1)13+312⋅7iπ12Γ(−76)10Γ(−56)Γ(16)w˜1′13+⋯)+ħ2Λ15(−243⋅312iπ12Γ(56)Γ(16)Γ(−56)(u˜Λ12)−53+⋯)+ħ4Λ110(−2⋅332iπ12Γ(56)2Γ(53)Γ(−56)2Γ(16)Γ(13)(u˜Λ12)−103+⋯)+⋯. We can express τ˜ as a function of a˜ by solving (4.9). Then integrating it over a˜ twice, we obtain the free energy. We find that the free energy at M˜=0 agrees with the one obtained from the E-string theory [38]. We note that the present expansions for Nf theories in the coupling parameter are different from those in the self-dual Ω-background [15], where the expansions in the operator have been done with the zero coupling and without taking the scaling limit.4.2Nf=2 theoryWe next compute the quantum corrections to the SW periods for the Nf=2 theory. From (3.26) and (2.60) we find that the second order corrections are given by(4.14)a˜(2)=−124⋅3154π12Λ232(u˜Λ22)−34(F1(2)(w˜2′)−F2(2)(w˜2′)),(4.15)a˜D(2)=i24⋅3154π12Λ232(u˜Λ22)−34(F1(2)(w˜2′)+F2(2)(w˜2′)), where we have defined w˜2′=1−w˜2. Here F1(2)(w˜2′) and F2(2)(w˜2′) are defined by(4.16)F1(2)(w˜2′)=32Γ(112)Γ(512)(22⋅312(M˜2u˜)12F(512,1312;12;w˜2′)−5w˜2′12F(1312,1712;32;w˜2′)),(4.17)F2(2)(w˜2′)=62Γ(712)Γ(1112)(3F(712,1112;32;w˜2′)+7X(2)F(1112,1912;32;w˜2′)), where(4.18)X(2)=−3+2⋅312w˜2′12(M˜2u˜)12. Expanding the second order terms in w˜2′ around w˜2′=0, where M˜2u˜≪1 and C˜2Λ2u˜32≪1, we obtain(4.19)a˜(2)=1Λ232(u˜Λ22)−34(−314Γ(712)Γ(1112)2π12+Γ(112)Γ(512)24⋅354π12(M˜2u˜)12+⋯),(4.20)a˜D(2)=1Λ232(u˜Λ22)−34(−314iΓ(712)Γ(1112)2π12−iΓ(112)Γ(512)24⋅354π12(M˜2u˜)12+⋯).The fourth order corrections can be obtained in a similar manner. We find that(4.21)a˜(4)=129⋅3114⋅5π12Λ2921w˜2′12(w˜2′−1)2(u˜Λ22)−94(F1(4)(w˜2′)−F2(4)(w˜2′)),(4.22)a˜D(4)=−i29⋅3114⋅5π12Λ2921w˜2′12(w˜2′−1)2(u˜Λ22)−94(F1(4)(w˜2′)+F2(4)(w˜2′)), where(4.23)F1(4)(w˜2′)=Γ(112)Γ(512)(−14X1(4)F(112,512;12;w˜2′)+X2(4)F(512,1312;12;w˜2′)),(4.24)F2(4)(w˜2′)=14w˜2′12Γ(712)Γ(1112)(−2X1(4)F(712,1112;32;w˜2′)+X2(4)F(1112,1912;32;w˜2′)). Here the coefficients X1 and X2 are defined by(4.25)X1(4)=−22⋅332w˜2′12(10w˜2′+11)+3(M˜2u˜)12(377w˜2′+127)−23⋅312(M˜2u˜)w˜2′12(13w˜2′+113)+28(M˜2u˜)32(13w˜2′+11),(4.26)X2(4)=−332w˜2′12(1345w˜2′+671)+6(M˜2u˜)12(520w˜2′2+4639w˜2′+889)−22⋅332(M˜2u˜)w˜2′12(593w˜2′+1423)+56(M˜2u˜)32(211w˜2′+77). Expanding the fourth order corrections to the SW periods in w˜2′ around w˜2′=0, where M˜2u˜≪1 and C˜2Λ2u˜32≪1, we get(4.27)a˜(4)=1Λ292(u˜Λ22)−94(−11Γ(112)Γ(512)29⋅354π12−314⋅5⋅7Γ(712)Γ(1112)28π12(M˜2u˜)12+⋯),(4.28)a˜D(4)=1Λ292(u˜Λ22)−94(11iΓ(112)Γ(512)29⋅354π12−314⋅5⋅7iΓ(712)Γ(1112)28π12(M˜2u˜)12+⋯). The effective coupling constant τ˜ is expanded in ħ as(4.29)τ˜=(−i−iΓ(712)Γ(1112)312Γ(512)Γ(1312)(M˜2u˜)12+i312Γ(712)Γ(1112)23Γ(512)Γ(1312)(C˜22Λ22u˜3)12+⋯)+ħ2Λ23(u˜Λ22)−32(312iΓ(712)Γ(1112)24Γ(512)Γ(1312)+32iΓ(712)2Γ(1112)2Γ(112)2Γ(512)2(M˜2u˜)12+⋯)+ħ4Λ26(u˜Λ22)−3(−3iΓ(712)2Γ(1112)229Γ(512)2Γ(1312)2−312i(3Γ(712)3Γ(1112)3+19π2Γ(512)Γ(1312))210Γ(512)3Γ(1312)3(M˜2u˜)12+⋯)+⋯. It would be interesting to compare the free energy with that of the E-string theory, which is left for future work.4.3Nf=3 theoryWe now discuss the Nf=3 case. Using (3.44) and (2.65) we find that the second order corrections to the SW periods are given by(4.30)a˜(2)=12103⋅372π12w˜3′Λ33(u˜Λ32)(−σ3)−56(1+43M˜3u˜Λ3)(F1(2)(w˜3′)+F2(2)(w˜3′)),(4.31)a˜D(2)=i2103⋅372π12w˜3′Λ33(u˜Λ32)(−σ3)−56(1+43M˜3u˜Λ3)×((−1)56F1(2)(w˜3′)+(−1)16F2(2)(w˜3′)), where w˜3′:=11−w˜3. F1(2)(w˜3′) and F2(2)(w˜3′) are defined by(4.32)F1(2)(w˜3′)=18Γ(16)Γ(13)(F(112,712;23;w˜3′)−X(2)F(712,1312;23;w˜3′)),(4.33)F2(2)(w˜3′)=−3w˜3′223Γ(−16)Γ(−13)(F(512,1112;43;w˜3′)−5X(2)F(1112,1712;43;w˜3′)). Here X(2) is given by(4.34)X(2)=1+223⋅3M˜Λ3(3u˜Λ3+4M˜3)(−σ3)13w˜3′23. Expanding the second order corrections to the SW periods in w˜3′, where M˜3u˜Λ3≪1, C˜23Λ32u˜4≪1 and C˜3Λ3u˜2≪1, we obtain(4.35)a˜(2)=1Λ312(u˜Λ32)−23×(−(−1)16Γ(23)Γ(56)2⋅312π12+(1+19(−1)13)Γ(16)Γ(13)34π12(M˜3u˜Λ3)23+⋯),(4.36)a˜D(2)=1Λ312(u˜Λ32)−23×(−(−1)56Γ(23)Γ(56)2⋅312π12+(−1)23(1+19(−1)13)Γ(16)Γ(13)34π12(M˜3u˜Λ3)23+⋯). The effective coupling constant is found to be(4.37)τ˜=(−(−1)13−24iπ2Γ(16)2Γ(13)2(M˜3u˜Λ3)13−2iπ2Γ(16)2Γ(13)2(C˜23Λ32u˜4)13+⋯)+ħ2Λ32(−1)56(u˜Λ32)−43×(−312Γ(23)Γ(56)5Γ(−56)Γ(13)−2103⋅312Γ(−16)Γ(56)252Γ(−56)2Γ(16)(M˜3u˜Λ3)13+⋯)+⋯. We can calculate the ħ4-order correction to the effective coupling constant in a similar way. The result is(4.38)τ˜(4)=(−1)16Λ34(u˜Λ32)−83×(243⋅312πΓ(56)252Γ(−56)2Γ(16)2+23⋅332π12Γ(−16)Γ(56)353Γ(−56)3Γ(16)2(M˜3u˜Λ3)13+⋯).In summary, we have explicitly calculated the quantum corrections to the SW periods in terms of the hypergeometric functions up to the fourth orders in ħ for the AD theories of the (A1,A2), (A1,A3) and (A1,D4)-types.5Conclusions and discussionsIn this paper we studied the quantum SW periods around the superconformal point of N=2 SU(2) SQCD with Nf=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 Nf. 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 ħ, 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 ħ. In particular the resurgence method helps us to understand non-perturbative structure of the ħ-corrections [39–42]. The SW curve for Nf theory at the superconformal point are given by the scaling limit of the SW curve for the original SQCD. The SU(2) theory with the Nf hypermultiplets are obtained by the decoupling limit for the SU(2) theory with Nf=4 hypermultiplets. It is interesting to see the SW curve at the various superconformal points by combining both the scaling limit and the decoupling limit of the SW curve of the SU(2) theory with Nf=4 hypermultiplets.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 ħ-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.It is known that the four-dimensional theories in the NS limit are described by certain quantum integrable systems. The quantum corrections to the periods provide some data of the integrable systems. For Nf=1 case, the curve describes the same AD theory as SU(3) N=2 super Yang–Mills theory [1], whose quantum curve is the Schrödinger equation with cubic polynomial potential. In [44], using the ODE/IM correspondence (for a review see [45]), it is shown that 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].AcknowledgementsWe would like to thank S. Kanno, H. Shu and K. Sakai for useful discussion. 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