Recursion relation for instanton counting for SU(2) N=2 SYM in NS limit of Ω background
PPrepared for submission to JHEP
Recursion relation for instanton counting for
S U (2) N = 2 SYM in NS limit of Ω background Hasmik Poghosyan a,ba
Sezione INFN di Bologna and Dipartimento di Fisica e AstronomiaUniversit`a di Bologna, Via Irnerio 46, 40126 Bologna, Italy b Yerevan Physics InstituteAlikhanian Br. 2, 0036 Yerevan, Armenia
E-mail: [email protected]
Abstract:
In this paper we investigate different ways of deriving the A-cycle periodas a series in instanton counting parameter q for N = 2 SYM with up to four antifun-damental hypermultiplets in NS limit of Ω background. We propose a new method forcalculating the period and demonstrate its efficiency by explicit calculations. The newway of doing instanton counting is more advantageous compared to known standardtechniques and allows to reach substantially higher order terms with less effort. Thisapproach is applied for the pure case as well as for the case with several hypermultiplets.We also investigate a numerical method for deriving the A -cycle period valid forarbitrary values of q . Analyzing large q asymptotic we get convincing agreement withan analytic expression deduced from a conjecture by Alexei Zamolodchikov in a differentcontext. a r X i v : . [ h e p - t h ] O c t ontents A cycle via Floquet-Bloch monodromymatrix 113.3 Explicit demonstration of the numerical approach 12 SU (2) SYM with hypermultiplets 13 N f = 1 15 P N f and Q N f for less then four flavors 18B A-cycle derivation for pure SYM 19 B.1 A-cycle derivation with the SW differential 19B.2 A cycle period for pure SYM via instanton counting 19 C Two instanton expressions for the A cycle period with N f = 4 , , , flavors 20D From the difference equation to the differential equation 22 D.1 From the difference equation to Mathieu equation 22
E The differential equation for N f = 1 – 1 – Introduction
The focus of this paper is the A-cycle period for the N = 2 SYM with gauge group SU (2) in Ω background (see [1, 2] and further developments [3–5]). The backgroundis parameterized by two parameters (cid:15) and (cid:15) which can be interpreted as angularvelocities on two orthogonal planes of the space time. We will be interested in the casewhen one of the parameters say (cid:15) is sent to zero while the other one is kept finite,commonly referred as Nekrasov-Shatashvili (NS) limit [6].According to the AGT relation [7] the instanton partition function in Ω backgroundis closely related to the conformal block of 2 d Liouville CFT. Thus any result relatedto partition function can be reinterpreted in terms of the conformal block and viceversa. The NS limit corresponds to the so called heavy classical limit of the conformalblock [8, 9]. The four point conformal block satisfies a well known recursion relationdiscovered by Alexei Zamolodchikov [10, 11] (for the analog in gauge theory side see[12] and for generalizations of CFT see [13–15]). Surely, Zamolodchikov’s recursionrelation may be explored to investigate the heavy conformal block, by computing firstthe exact block and only afterwords tacking the heavy limit. This procedure appearsto be rather inefficient. Meanwhile constructing a heavy analog of Zamolodchikov’srecursion relation directly is not straightforward, due to arising strong singularities[16–18]. A natural question arises whether a kind of alternative procedure efficientlyworking in heavy limit can be found. The current article provides a positive answer tothis question.The method we suggest is the following: the Nekrasov partition function can berepresented as a sum over pairs of ( N -tuples if the gauge group is U ( N )) Young di-agrams [2, 3]. In NS limit only a single term of this sum contributes dominantly. Amajor role is played by an entire function whose zeros are determined by the columnlength of dominant Young diagrams mentioned above. This function satisfies a differ-ence equation [19], which can be reformulated in such a way that closely resembles theordinary Seiberg-Witten curve equation [20, 21]. We have made use of this differenceequation to obtain a recursion relation in terms of continued fraction.In this paper we also investigate a numerical approach to derive the A-cycle periodagain directly applicable in NS limit. Via Fourier transform from the already men-tioned difference equation a second order ordinary differential equation (ODE) can bederived [22] (for earlier works using different approach see [23–25]). Since the coef-ficients entering in this differential equation are periodic one deduces that it admitsquasi periodic solutions. The index of quasi periodicity or the characteristic exponentcommonly referred as Floquet exponent is just the A-cycle period. In particular whenone considers pure SU (2) SYM the differential equation is just the Mathieu equation– 2 –hich is well studied in mathematical literature (for example see [26]). The fact thatit can serve as a basis for numerical computations is emphasized in [27, 28]. In thiswork we demonstrate how the corresponding differential equations for SU (2) SYM withseveral hypermultiplets can be used for numerical computations in similar manner. Inparticular in the case when one has four hypers, due to the AGT correspondence, thisnumerical approach can be used to investigate the heavy conformal block.There was recent progress in generalizing the SW curve for generic Ω-backgroundtoo [29–31]. Extension of our analyses is an interesting task, though beyond the scopeof current paper.This article is organized as follows: section 2 we review few known things con-nected to instanton counting and the A-cycle period to make clear the notations weused. In section 3 we give our recursion relation for pure SYM. A numerical approachis presented in subsections 3.2, 3.3 which is applied to investigate the A-cycle in SU (2)SYM. This numerical method was previously used in [27] to investigate the Floquet ex-ponent in the context of Ordinary Differential Equation/Integrable Model (ODE/IM)correspondence [32, 33] and also for SU (3) pure SYM [28]. We show that results ob-tained by this numerical method are consistent with our recursion relation. In additionwe numerically checked Alexei Zamolodchikovs conjecture about asymptotic behaviorof Floquet exponent [27, 35]. Section 4 we extend previous results to the case withhypermultiplets. Equation (4.5) expresses our recursion relation for arbitrary numberof hypermultiplets. Using our new recursive method in final section 5 we compute theheavy conformal block as a series in cross ratio of insertion points. In this section we briefly review the combinatorial expression for instanton partitionfunction, the difference relation emerging in NS limit and define the period cycles.Connection between the differences relation and generalized SW curve is explained.We discus some of the similarities and differences between the ordinary SW curve andits generalization for NS limit of Ω background. The A-period computation is performedusing two approaches, first, using instanton counting combined with Matone relationand the second by integrating deformed SW differential.
Consider N = 2 SYM with gauge group SU (2) and four hypermultilets in Ω-backgroundparameterized by (cid:15) and (cid:15) . The instanton part of the partition function [2] of this For a nice review on ODE/IM correspondence see [34]. – 3 – s s Figure 1 : Arm and leg length with respect to the Young diagram whose borders areoutlined by dark black: A ( s ) = − L ( s ) = − A ( s ) = 2, L ( s ) = 3, A ( s ) = − L ( s ) = −
4. The coordinate ( i, j ) of the box s is (3 , Z inst ( a, (cid:15) , (cid:15) , q ) = (cid:88) (cid:126)Y Z f ( (cid:126)Y ) Z g ( (cid:126)Y ) q | (cid:126)Y | , (2.1)where (cid:126)Y is a pair of Young diagrams (cid:126)Y = ( Y , Y ) and | (cid:126)Y | is the total number of boxes.The sum is over all possible pairs of Young diagrams and q is the instanton countingparameter related to the gauge coupling g and the CP violating parameter θ in thestandard manner: q = exp(2 πiτ ), with τ = ig + θ π . We denote the VEV of adjointscalar of N = 2 vector multiplet by a = − a = a . The contribution of antifundamentalhypermultiplets Z f and the gauge multiplet Z g can be represented as [3, 5] Z f ( (cid:126)Y ) = N f (cid:89) (cid:96) =1 2 (cid:89) u =1 (cid:89) ( i,j ) ∈ Y u ( m (cid:96) + a u + ( i − (cid:15) + ( j − (cid:15) ) , (2.2) Z g ( (cid:126)Y ) = (cid:89) u,v =1 (cid:89) s ∈ Y u (cid:0) a u − a v − (cid:15) L µ ( s ) + (cid:15) (1 + A λ ( s )) (cid:1) × (2.3) × (cid:89) s ∈ Y v (cid:0) a u − a v + (cid:15) (1 + L λ ( s )) − (cid:15) A µ ( s ) (cid:1) . Here by m (cid:96) we denote the masses of the hypermultiplets, A λ ( s ) and L λ ( s ) are the arm-length and leg-length of the box s with respect to the Young diagram λ respectively.The arm-length A λ ( s ) (leg-length L λ ( s )) is the number of steps needed to reach fromthe box s to the outer boundary of λ in vertical (horizontal) direction as demonstratedin Fig.1. The coordinates ( i, j ) in (2.3) specify the position of a box (see figure 1).The deformed prepotential in the NS limit is defined as F inst ( a, (cid:15) , q ) = − lim (cid:15) → (cid:15) (cid:15) log Z inst ( a, (cid:15) , (cid:15) , q ) . (2.4)– 4 –rom here on the notation (cid:15) ≡ (cid:15) will be used. We will need also the Matone relation[36] u = (cid:104) tr φ (cid:105) = 2 a + 2 q ∂F inst ∂q = 2 q ∂F∂q , (2.5)which holds also in the presence of Ω background [37]. With the help of this expressionsone can derive the A-cycle period as a power series in q (see appendix B.2 for explicitcalculations). Most of the time instead of the VEV parameter u we will us the parameter p defined as p ≡ u (cid:104) tr φ (cid:105) . (2.6) According to [19] the sum (2.1) in NS limit is dominated by a single term correspondingto a unique pair of Young diagrams (cid:126)Y ( cr ) . By using this fact one defines an entirefunction Y ( z ) whose zeros z u,k are determined by (rescaled) column lengths λ u,k of (cid:126)Y ( cr ) : z u,k = a u + ( k − (cid:15) + λ u,k , u = 1 , . (2.7)For later use we will also need the fact that λ u,k ∼ O ( q k ). It was shown in [19] thatsuch function Y ( z ) , if defined properly satisfies the difference equation Y ( z + (cid:15) ) + (cid:15) − Q N f ( z + (cid:15) ) Y ( z − (cid:15) ) = (cid:15) − P N f ( z + (cid:15) ) Y ( z ) . (2.8)This difference equation leads to a kind of generalization of the Seiberg-Witten curveequation. Introducing the meromorphic function y ( z ) = (cid:15) Y ( z ) Y ( z − (cid:15) ) , (2.9)from (2.8) one immediately gets y ( z ) + Q N f ( z ) y ( z − (cid:15) ) = P N f ( z ) . (2.10)In the case N f = 4 one has Q ( z ) = q (cid:89) j =1 ( z + m j − (cid:15) ) ; (2.11) P ( z ) = (1 + q ) z + ( s − (cid:15) ) qz + q (cid:0) s − (cid:15)s + p + (cid:15) (cid:1) − p , (2.12) We adopted a convention (not universally used), where Y ( z ) is dimensionless. – 5 –here s and s are elementary symmetric polynomials of masses s = (cid:88) i =1 m i ; s = (cid:88) ≤ i 1) = z − p . (3.1)Formally one can represent y ( z ) as a continued fraction in two alternative ways, bysubsequently shifting the parameter z either in negative or positive direction. We willsee below that the latter continued fraction is divergent for generic values of z . But,fortunately, at z = a this continued fraction becomes convergent (at least when q issufficiently small), a key fact which eventually leads to our recursion relation.First let us write y ( z ) as a continued fraction with negative shifts. From (3.1) wesee that y ( z ) = z − p − qy ( z − , ... , y ( z − k + 1) = ( z − k + 1) − p − qy ( z − k ) , therefore y ( z ) = z − p − q ( z − − p − q ( z − − p − . . . − qy ( z − k ) . (3.2) Formaly sending k → ∞ we get y ( z ) = z − p − q ( z − − p − q ( z − − p − q ( z − − p − ... . (3.3)As we will see, this continued fraction is convergent for generic values of z .Now let us write y ( z ) as a continued fraction with positive shifts of z . From (3.1) y ( z ) = q ( z +1) − p − y ( z +1) , ... , y ( z + k − 1) = q ( z + k ) − p − y ( z + k ) , – 8 –o that y ( z ) = q ( z + 1) − p − q ( z + 2) − p − . . . − q ( z + k ) − p − y ( z + k ) . (3.4) Again in formal k → ∞ one would obtain y ( z ) = q ( z + 1) − p − q ( z + 2) − p − q ( z + 3) − p − ... . (3.5)In this case however as explained later this continued fraction converges only for veryspecific values of z .Coming back to (3.3) Re ( z ) → −∞ the asymptotic behavior y ( z ) ∼ z is valid,thus truncating the fraction (3.3) at sufficiently large positive integer k , the reminderterm (3.2) qy ( z − k ) ∼ qk → y ( z + k ) (3.4) for generic z diverges at k → ∞ . Luckily at specific values e.g. when z = a the situation is much better. From the definition (2.9) of y ( z ) we see that it is ameromorphic function with zeros, and poles located at a + ( k − 1) + λ ,k , a + k + λ ,k ; k = 1 , , ... (3.6)respectively. So, separating k + 1’th zero and k ’th pole (which are close to each otherat large k ) y ( z ) can be represented as y ( z ) = ˜ y ( z ) z − ( a + k + λ ,k +1 ) z − ( a + k + λ ,k ) , (3.7)where ˜ y has neither zero nor pole at z = a + k . Hence y ( a + k ) = ˜ y ( a + k ) λ ,k +1 λ ,k ∼ q , (3.8)since λ ,k ∼ O ( q k ). This is why truncating (3.4) at z = a on the level k produces onlyan error of order O ( q k +1 ).Now we can use the continued fractions we have built to obtain the recursionrelation. From (3.3) and (3.5) it is straightforward to see that y ( a ) + y ( − a ) − a + p = 0 , (3.9)– 9 –here the equality holds in the sense of power expansion in q .By using (3.9) and (3.3) we will obtain p as a series in q with a dependent coeffi-cients. For instance up to order O ( q ) from (3.3) we get y ( a ) = a − p − q ( a − − p − q ( a − − p + O ( q ) , (3.10) y ( − a ) = a − p − q ( a + 1) − p − q ( a + 2) − p + O ( q ) . (3.11)Representing p as power series in qp = v + v q + v q + O (cid:0) q (cid:1) (3.12)and inserting it in (3.10) and (3.11) from (3.9) we get( a − v ) − q (cid:16) a − − v + a +1) − v + v (cid:17) − (3.13) − q (cid:18) ( a v − av − v v +4 v +1 ) ( a − a − v +4)( a − a − v +1) + ( a v +4 av − v v +4 v +1 ) ( a +2 a − v +1) ( a +4 a − v +4) + v (cid:19) + O ( q ) = 0 . This equality uniquely specifies v , v and v inserting which in (3.12) one obtains p = a + 2 q a − a + 7) q a − 1) (4 a − + O (cid:0) q (cid:1) . (3.14)In fact without much efforts with simple mathematica code we have extended this seriesup to 10 instantons. Of course inverting the series (3.14) on can express a in terms p and q , but this goal can be achieved also directly from the recursion relation. In asimilar manner without having to derive the series (3.14) we will get a as a series in q .Consider p fixed and representing the A-cycle as a series in qa = a + a q + a q + O (cid:0) q (cid:1) . (3.15)Again with the help of (3.9)-(3.10) we find (cid:0) a − p (cid:1) + q (cid:16) − a − − p − a +1) − p + 2 a a (cid:17) + q (cid:18) a a ( a ( a − p +2 ) +( p − p +3) )( − a ( p +1)+ a +( p − ) −− a +1) − p ) (( a +2) − p ) − a − − p )(( a − − p ) + a + 2 a a (cid:17) + O (cid:0) q (cid:1) = 0 , (3.16) which immediately determines a , a and a . The result is a = p + qp (1 − p ) + 5(7 − p ) p − p − p (4 p − q + O (cid:0) q (cid:1) . (3.17)– 10 –he results for the A -cycle and p (3.14) could be derived using at least two othermethods, presented in appendix B.2 and B.1, which are in agreement with our result.Notice that the symmetry a → − a is manifest in equation (3.9). In the cases with extrahypermultiplets this property no longer holds. Nevertheless exploring two inequivalentrepresentations of y ( a ) as continued fractions we will find analogues recursive represen-tation for these cases too. A cycle via Floquet-Bloch monodromymatrix As demonstrated in appendix D.1 from the difference equation one can derive a secondorder ordinary differential equation which, in the pure case, coincides with the Mathieuequation. We will use this differential equation as a basis for numerical computations.This method was explored earlier in [28] for pure SU (3) SYM case. Here we will startwith pure SU (2) and then generalize to the case with one fundamental hypermultiplet(generalization to the cases with more hypermultiplets is straightforward). In ourcontext the Mathieu equation conveniently is presented as (see appendix (D.1)) f (cid:48)(cid:48) ( x ) − (cid:0) cosh x + p (cid:1) f ( x ) = 0 , q = Λ . (3.18)Consider solutions f ( x ), f ( x ) satisfying the standard initial conditions f (0) = 1 , f (cid:48) (0) = 0 , (3.19) f (0) = 0 , f (cid:48) (0) = 1 .f ( x ) and f ( x ) are commonly referred as basic solutions. From the initial conditions we see that the Wronskian W [ f ( x ) , f ( x )] ≡ f ( x ) f (cid:48) ( x ) − f ( x ) f (cid:48) ( x ) = 1is different from zero so that the basic solutions are linearly independent. Hence anarbitrary solution can be expressed as their linear combination. Due to periodicityof the coefficients of the equation (3.18), obviously f ( x + 2 πi ) and f ( x + 2 πi ) aresolutions too. The monodromy matrix M is defined as f n ( x + 2 πi ) = (cid:88) k =1 f k ( x ) M kn . (3.20)Clearly, from (3.19) for the matrix elements we get M k,n = f ( k − n (2 πi ) . From (3.18) we see that the Wronskian from two of its solutions does not depend on x – 11 – .1 0.2 0.3 0.4 Λ - ( Λ ) Figure 3 : The black line is the a cycle derived with the recursion relation until teninstantons for p = 0 . 17, the blue (cyan) line is the real (imaginary) part of the A-cyclederived with (3.23).The Mathieu equation (3.18) has a quasi periodic solution (see (D.1)) obeying f + ( x + 2 πi ) = e πia f + ( x ) . (3.21)Due to the symmetry of the differential equation under x → − x , the function f − ( x ) ≡ f + ( − x ) is another quasi periodic solution satisfying f − ( x + 2 πi ) = e − πia f − ( x ) . (3.22)Representing above quasiperiodic solutions as linear combinations of basic solutionsfrom (3.20) we deduce that e ± πia are the two eigenvalues of monodromy matrix M k,n .So that tr M = 2 cos(2 πa ) . (3.23)Above results enables us to derive the A -cycle numerically. For the particular value p = 0 . 17 we have derived the A-cycle up to ten instantonsand compered it against the non perturbative numerical result obtained by the methodof section 3.2. Fig.3 confirms that these two approaches are consistent provided theinstanton counting parameter is small.According to Alexei Zamolodchikov [27, 35] the A-cycle is connected to the Baxters T function as follows (Zamolodchikov suggested it in the context of ODE/IM corre-spondence its role for the SYM theory was clarified in [38]): T = 2 cos (2 πa ) . (3.24)– 12 – .5 1.0 1.5 2.0 Λ ( π a ) Λ - × - × - × - × ( π a ) Figure 4 : The graphics present the dependence of 2 cos(2 πa ) on Λ. The blue is for p = 0 . 1, cyan for p = 0 . 2, gray for p = 0 . p = 2. The black line isthe asymptotic curve given by (3.25). Notice that in the second picture Λ reaches 14corresponding to q = 38416.Though a is not a single valued function of q nevertheless cos(2 πa ) behaves much bettersince T is an entire function. Fig.4 demonstrates that cos(2 πa ) is quite regular in theinterval 0 ≤ q ≤ a depicted in Fig.3.Using the results of [27, 35] we can deduce the large instanton behavior ( | Λ | (cid:29) | arg Λ | < π :cos (2 πa ) ∼ exp (cid:32) π / ΛΓ (cid:0) (cid:1) (cid:33) cos (cid:32) π / ΛΓ (cid:0) (cid:1) (cid:33) . (3.25)These behavior is in agreement with our numerical results presented in Fig.4. As onecan see from (3.25) large Λ asymptotic behavior is independent of p , nevertheless thesmaller p is, the faster the asymptotic region is reached. SU (2) SYM with hypermultiplets In this section we are going to obtain a recursion relation for the A-cycle in the presenceof several hypermultiplets. We will generalize the numerical approach using the differ-ential equation for one fundamental hypermultiplet and demonstrate that the resultsare in agreement with the recursion relation.From (2.10) we see that y ( a ) = P N f ( a ) − Q Nf ( a ) y ( a − , y ( a − 1) = P N f ( a − − Q Nf ( a − y ( a − , ... . (4.1)– 13 –o that y ( a ) = P N f ( a ) − Q N f ( a ) P N f ( a − − Q N f ( a − P N f ( a − − Q N f ( a − P N f ( a − − ... . (4.2)Again from (2.10) we observe that y ( a ) = Q Nf ( a +1) P Nf ( a +1) − y ( a +1) , y ( a + 1) = Q Nf ( a +2) P Nf ( a +2) − y ( a +2) , ... , (4.3)hence y ( a ) = Q N f ( a + 1) P N f ( a + 1) − Q N f ( a + 2) P N f ( a + 2) − Q N f ( a + 3) P N f ( a + 3) − ... . (4.4)By taking the difference of (4.2) and (4.4) we will obtain our final recursion relationfor the A-cycle (or alternatively p ) for arbitrary number of flavors P N f ( a ) − Q N f ( a ) P N f ( a − − Q N f ( a − P N f ( a − − ... − Q N f ( a + 1) P N f ( a + 1) − Q N f ( a + 2) P N f ( a + 2) − ... = 0 , (4.5) where P N f and Q N f for arbitrary N f can be found in appendix A and like in the purecase the equality holds in the sense of power expansion in q . Below we do some explicitdemonstration of these approach for four hypermultiplets. The recursion (4.5) in oneinstanton order is P ( a ) − Q ( a ) P ( a − − Q ( a + 1) P ( a + 1) + O (cid:0) q (cid:1) = 0 , (4.6)where P and Q are given in (2.12) and (2.11) respectively. After inserting a = a + a q + O (cid:0) q (cid:1) (4.7)one gets two equations by solving which determine a and a uniquely. Here is theresult a = p + − p + s ) + p ( s − s ) + s p − p q + O (cid:0) q (cid:1) . (4.8)Results for two instantons can be found in appendix B. Alternatively we could consid-ered p = v + v + O (cid:0) q (cid:1) (4.9)– 14 –eading to p = a − − a + s ) + a ( s − s ) + s a − q + O (cid:0) q (cid:1) . (4.10)To carry out computations for less number of flavors one should use coefficients (A.2)-(A.6).Notice also that due to the AGT duality the four point conformal block in 2d CFTis related to the instanton partition function with four hypermultiplets. This allows usto derive the heavy conformal block directly from recursions relation, as demonstratedin section 5. N f = 1We shall generalize the method explored in subsection 3.3 for the cases with one hy-permultiplet. Here instead of the Mathieu equation we have ψ (cid:48)(cid:48) ( x ) − Λ (cid:18) e x + m Λ e x + e − x (cid:19) ψ ( x ) − p ψ ( x ) = 0 . (4.11)Once again, we have two solutions such that ψ (0) = 1 , ψ (cid:48) (0) = 0 , (4.12) ψ (0) = 0 , ψ (cid:48) (0) = 1 . Notice that their Wronskian is W [ ψ ( x ) , ψ ( x )] = 1. The monodramy matrix M as inMathieu case is defined by ψ n ( x + 2 πi ) = (cid:88) k =1 ψ k ( x ) M kn . (4.13)Now it is easy to check that W [ ψ (2 πi ) , ψ (2 πi )] = det M kn = µ µ , where µ and µ are the eigenvalues of M kn . So, taking into account that the Wronskian does notdepend on x we conclude that µ µ = 1 . (4.14)It follows from (E.3) that (4.11) admits a quasiperiodic solution ψ + ( x + 2 πi ) = e πia ψ + ( x ) . (4.15)Hence one of the eigenvalues of M is e πia but due to (4.14) the remaining eigenvaluemust be e − πia . Consequentially as in Mathieu casetr M = 2 cos(2 πa ) , (4.16)– 15 – .1 0.2 0.3 0.4 Λ ( Λ ) Figure 5 : The black line is the a cycle derived with the recursion relation until fiveinstanton for p = 0 . m = 0 . 7, the blue line is the real part of the A-cycle and thecyan is its imaginary part derived with (4.16).and we can derive A-cycle numerically. Fig.5 demonstrates that numerical resultsobtained with instanton series is in agreement with this numerical approach. Ourstudy indicates that the same method can be successfully applied also for the caseswith more hypermultiplets as well as in quiver theories and theories with higher rankgauge groups. The differential equations for them can be found in [39, 40]. According to AGT conjecture [7] the instanton partition function with f = 4 antifun-damental hypermultiplets is related to the generic 4-point conformal block B as Z (4) inst ( a, m i , q ) = x ∆ +∆ − ∆ (1 − x ) λ + Q )( λ + Q ) B (∆ , ∆ i , x ) , (5.1)where ∆ i , i = 1 , , , x , 1 and ∞ respectively) and ∆ is the internal dimension parameterized as∆ i = Q − λ i , ∆ = Q − α . (5.2) Q = b + 1 /b is related to the central charge of the Virasoro algebra through c = 1 − Q . (5.3)To define the heavy asymptotic limit let us introduce new parametrs η i and η by λ i = η i b , α = ηb (5.4)– 16 –nd assume that in b → η i and η are kept fixed. In this limit the conformalblock B is conveniently represented as B (∆ , ∆ i , x ) = e b f ( η,η i ,x ) (5.5)where the function f ( η, η i , x ) has finite limit at b → q to the cross ratio x of insertionpoints in CFT block. The background charge parameter b is related to the Ω back-ground parameters by b = (cid:114) (cid:15) (cid:15) , (5.6)the masses ot anti-fundamental hypermultiplet m i are related to CFT parameters as m √ (cid:15) (cid:15) = (cid:18) λ + λ + Q (cid:19) , m √ (cid:15) (cid:15) = (cid:18) λ − λ + Q (cid:19) , (5.7) m √ (cid:15) (cid:15) = (cid:18) λ + λ + Q (cid:19) , m √ (cid:15) (cid:15) = (cid:18) λ − λ + Q (cid:19) (5.8)and finally the expectation value a is related to the internal conformal dimensionthrough a √ (cid:15) (cid:15) = α . (5.9)Thus from (5.9) and (5.6) in heavy limit we get a(cid:15) = η (5.10)and similarly from (5.7) and (5.4) m (cid:15) = (cid:18) η + η + 12 (cid:19) , m (cid:15) = (cid:18) η − η + 12 (cid:19) , (5.11) m (cid:15) = (cid:18) η + η + 12 (cid:19) , m (cid:15) = (cid:18) η − η + 12 (cid:19) . (5.12)From (2.4) and (5.1) we see that F inst (cid:15) = ( η + η − η − ) log x − (cid:0) η + (cid:1) (cid:0) η + (cid:1) log(1 − x ) − f ( η, η i , x ) . (5.13) With the recursion relation (4.5) we can derive u as a series in instanton countingparameter q (see (4.10)) which can be inserted in the formula (2.5) allowing us to Of course the (cid:15) dependence should be recovered. – 17 –btain ∂F inst ∂q . Integrating the result with respect to q (integration constant is fixedfrom condition f inst → q → 0) we can apply (5.13) and restore f as a series incross ratio x . The result is f ( η, η i , x ) = ( η + η − η − ) log x − ( η +4 η − η − )( η +4 η − η − ) η − x + ... , (5.14)which is in agreement with known results in literature (for a recent paper see [41]).The second order calculation can be inferred from (C.14). Acknowledgments I would like to thank Rubik Poghossian for helpful discussions and comments. I amgrateful to Davide Fioravanti for introducing me to the subjects related to ODE/IMcorrespondence. A P N f and Q N f for less then four flavors It is obvious from the expressions of Q ( z ) and P ( z ) above that they are invariant underthe exchange of the masses. From here we can obtain the cases with less flavors byrenormalizing the instanton coupling and sending some of the masses to infinity. To getthe N f = 0 case from (2.11) and (2.12) we must simultaneously m → µ, . . . , m → µ , q → qµ and µ → ∞ P ( z ) = z − p , Q ( z ) = q . (A.1)The procedure is similar for higher flavors: • For N f = 1 m i → µ , i = 1 , , , q → qµ and then µ → ∞ P ( z ) = z − p , (A.2) Q ( z ) = q ( m + z − . (A.3) • For N f = 2 m i → µ , i = 1 , q → qµ and then µ → ∞ P ( z ) = z − p + q , (A.4) Q ( z ) = q ( m + z − 1) ( m + z − . (A.5) • For N f = 3 m → µ q → qµ and then µ → ∞ P ( z ) = z − p + q ( z − 1) + ( m + m + m ) q , (A.6) Q ( z ) = q ( m + z − 1) ( m + z − 1) ( m + z − . (A.7)– 18 – A-cycle derivation for pure SYM B.1 A-cycle derivation with the SW differential We will derive the A-cycle from the formula (2.20) for pure SYM a = (cid:73) C A dz πi z∂ z log y ( z ) , (B.1)where the contour C A contains half of the poles in SW differential to be specified below.Let us write y ( z ) as a series in qy ( z ) = z − p + qy ( z ) + q y ( z ) + O ( q ) (B.2)after inserting this in (3.1) we will get y ( z ) = 1 p − ( z − ; y ( z ) = 1( p − ( z − ) ( p − ( z − ) . (B.3)From here and (B.2) by direct computation we obtain z∂ z log y ( z ) = z z − p + q (cid:16) z ( z − p ) (( z − − p ) + z ( z − z − p )(( z − − p ) (cid:17) + (B.4)+ q (cid:16) z ( p − ( z − ) ( z − p ) + z (1 − z )( p − ( z − ) ( z − p ) − z ( p − ( z − ) ( p − ( z − )( z − p ) ++ z − z ( p − ( z − ) ( p − ( z − )( z − p ) + z − z ( p − ( z − ) ( p − ( z − ) ( z − p ) (cid:17) + O ( q ) . To derive the A-cycle we need to insert the last result in (B.1) and perform integration.From the above expression we observe that the poles of it are located at: q : ± pq : ± p, ± p + 1 q : ± p, ± p + 1 , ± p + 2The contour C A encloses all the poles where p appears with plus sign (the alternativechoice with minus signs would give − a instead). The final result is (3.17). B.2 A cycle period for pure SYM via instanton counting We can derive u by using (2.1) together with (2.3). The result is Z inst = 1 + q(cid:15) (cid:15) ( − a + (cid:15) + (cid:15) )(2 a + (cid:15) + (cid:15) ) + (B.5)+ q ( − a +8 (cid:15) +8 (cid:15) +17 (cid:15) (cid:15) ) (cid:15) (cid:15) ( − a + (cid:15) + (cid:15) )(2 a + (cid:15) + (cid:15) )( − a +2 (cid:15) + (cid:15) )(2( a + (cid:15) )+ (cid:15) )( − a + (cid:15) +2 (cid:15) )(2( a + (cid:15) )+ (cid:15) ) + O ( q ) . – 19 –fter inserting this in (2.4) we will obtain F inst ( a, (cid:15), q ) = 2 q a − (cid:15) + q (20 a + 7 (cid:15) )4 ( a − (cid:15) ) (4 a − (cid:15) ) + O (cid:0) q (cid:1) . (B.6)Thus from (2.5) we will get u = 2 a + 4 q a − (cid:15) + q (20 a + 7 (cid:15) )( a − (cid:15) ) (4 a − (cid:15) ) + O (cid:0) q (cid:1) . (B.7)This coincides with (3.14) after setting (cid:15) = 1. By inverting this series we will arrive at(3.17). C Two instanton expressions for the A cycle period with N f =4 , , , flavors In this section we perform two instanton computations using our recursion relation(4.5). In two instanton approximation we have P N f ( a ) − Q N f ( a ) P N f ( a − − Q N f ( a − P N f ( a − − Q N f ( a + 1) P N f ( a + 1) − Q N f ( a + 2) P N f ( a + 2) + O (cid:0) q (cid:1) . (C.1) Using (2.11), (2.12) (or (A.2)-(A.6) for the cases with less number of flavors), andinserting the expansion a = a + a q + a q + O (cid:0) q (cid:1) (C.2)into (C.1) we’ll find equations, uniquely specifying the coefficients a , a and a . Hereare the results: • For N f = 1 a = p + (1 − m ) q p − p + q ( − m − m ( p − p +2 ) +24 p − p +19 p − ) p ( p − p − + O ( q ) , (C.3) p = a + (1 − m ) q − a + q ( − a + ( a +7 ) ( m − m +11 a +1 ) a − a − + O ( q ) , (C.4) • For N f = 2 a = p + s − ( p + s ) p − p q + A p ( p − p − q + O ( q ) , (C.5) p = a + ( a + s ) − s a − q + p a − a − q + O ( q ) , (C.6)– 20 –here A = 2 s (cid:0) ( p − p + 1) (cid:0) p + 1 (cid:1) p + (cid:0) p − p + 2 (cid:1) s (cid:1) − (C.7) − (cid:0) p + s (cid:1) (cid:0) ( p − p + 1) (cid:0) p + 1 (cid:1) p + (cid:0) p − p + 2 (cid:1) s (cid:1) ++ (cid:0) p − p + 19 p − (cid:1) s ,p = (cid:0) a + 7 (cid:1) s + (cid:0) a − (cid:1) (cid:0) a (cid:0) a − (cid:1) − (cid:0) a + 1 (cid:1) ( s − s (cid:1) + (C.8)+ s (cid:0) a − (cid:0) a + 7 (cid:1) s − a + 5 (cid:1) , where s and s are elementary symmetric polynomials in m and m (i.e. s = m + m and s = m m ). • For N f = 3 a = p + − p s + p + s − s p − p q + A p ( p − p − q + O ( q ) , (C.9) p = a + − a s + a + s − s − a q + p a − a − q + O ( q ) , (C.10)where A = 2 p s (cid:0) ( p − p + 1) (cid:0) p + 1 (cid:1) (cid:0) p + s (cid:1) + (cid:0) − p + 46 p − (cid:1) s (cid:1) −− (cid:0) p − p + 2 (cid:1) s + 2 (cid:0) p − p + 2 (cid:1) s (cid:0) p + s (cid:1) + (cid:0) − p + 22 p + 2 p (cid:1) s −− ( p − p + 1) (cid:0) p + s (cid:1) (cid:0) p + 2 p + p + (cid:0) − p + 18 p − (cid:1) s (cid:1) , (C.11) p = (cid:0) a + 7 (cid:1) s + (cid:0) a − (cid:1) a (cid:0) a + (cid:0) a − (cid:1) ( s − s − a − (cid:1) ++ (cid:0) − a + 11 a + 1 (cid:1) s − (cid:0) a − (cid:1) s (cid:0) a − (cid:0) a + 1 (cid:1) s + 2 a + 1 (cid:1) + (C.12)+ s (cid:0) − a − (cid:0) a + 7 (cid:1) s + 13 a + (cid:0) a − a + 5 (cid:1) s − (cid:1) . In this case s , s and s are the elementary symmetric polynomials in m , m and m . • For N f = 4 a = p + − (cid:0) p + s (cid:1) + p ( s − s ) + s p − p q + A p ( p − 1) (4 p − q + O (cid:0) q (cid:1) , (C.13) p = a − − (cid:0) a + s (cid:1) + a ( s − s ) + s a − q + p a − 1) (4 a − q + O (cid:0) q (cid:1) , (C.14) where A = 2 p s (cid:0)(cid:0) p − (cid:1) p (cid:0) p + (cid:0) p + 1 (cid:1) s + 2 (cid:0) p − (cid:1) s − p + 1 (cid:1) + (cid:0) p − p + 2 (cid:1) s (cid:1) −− (cid:0) p − p + 2 (cid:1) s + ( p − p + 1) (cid:0) − p (cid:0) p + s (cid:1) (cid:0) p + (cid:0) p + 1 (cid:1) s − p + 1 (cid:1) ++2 p s (cid:0) p + (cid:0) p + 1 (cid:1) s − p + 1 (cid:1) + (cid:0) p − p + 1 (cid:1) s (cid:1) − s (cid:0)(cid:0) − p + 35 p − (cid:1) s ++ p (cid:0) p − p + 10 p + (cid:0) p − p + 1 (cid:1) s − (cid:1)(cid:1) + (cid:0) − p + 6 p + p + p (cid:1) s – 21 – = (cid:0) a + 7 (cid:1) s + (cid:0) a − (cid:1) (cid:0) − (cid:0) a + 1 (cid:1) s + a (cid:0) a + s (cid:1) (cid:0) a + (cid:0) a − (cid:1) s − a + 1 (cid:1) ++ s (cid:0) − a + (cid:0) a + 1 (cid:1) s + 13 a − (cid:1)(cid:1) + (cid:0) a − a + a (cid:1) s + s (cid:0)(cid:0) − a + 13 a − (cid:1) s −− (cid:0) a − (cid:1) (cid:0) a − a + (cid:0) a − (cid:1) a s + a + (cid:0) a + 2 a + 1 (cid:1) s (cid:1)(cid:1) ++ s (cid:0) a − a − (cid:0) a + 7 (cid:1) s + 22 a + (cid:0) a − a + 5 (cid:1) s − (cid:1) D From the difference equation to the differential equation In this section we will derive differential equations from the difference equation (2.8)with the help of inverse Fourier transform: f ( x ) = (cid:88) z ∈ Z + a e x ( z +1) Y ( z ) . (D.1) D.1 From the difference equation to Mathieu equation According to (2.8), the difference equation for pure SYM (A.1) is Y ( z + 1) + qY ( z − − (cid:0) ( z + 1) − p (cid:1) Y ( z ) = 0 . (D.2)By means of inverse Fourier transform (D.1) we can derive the following second orderdifferential equation f (cid:48)(cid:48) ( x ) − (cid:0) qe x + e − x + p (cid:1) f ( x ) = 0 . (D.3)Shifting x by √ q one immediately arrives at the Mathieu equation (3.18), where q =Λ = e θ . E The differential equation for N f = 1 The cases with flavors are similar to the pure one. When N f = 1 we have (A.2), sothat (2.8) becomes Y ( z + 1) + q ( m + z ) Y ( z − − (( z + 1) − p ) Y ( z ) = 0 . (E.1)From this and (D.1) we obtain f (cid:48)(cid:48) ( x ) − qe x f (cid:48) ( x ) − (cid:0) qm e x + e − x + p (cid:1) f ( x ) = 0 . (E.2)– 22 –y taking f ( x ) = e qex ψ ( x ) , (E.3)it is straightforward to see that ψ (cid:48)(cid:48) ( x ) − (cid:18) q e x + (cid:18) m − (cid:19) qe x + e − x + p (cid:19) ψ ( x ) = 0 . (E.4)Shifting x → x − ln( q/ 2) one gets ψ (cid:48)(cid:48) ( x ) − (cid:16) q (cid:17) / (cid:18) e x + 2 (cid:18) m − (cid:19) (cid:16) q (cid:17) − / e x + e − x (cid:19) ψ ( x ) − p ψ ( x ) = 0 . (E.5)Using the notations m ≡ (cid:18) m − (cid:19) , Λ ≡ (cid:16) q (cid:17) / , (E.6)we will obtain (4.11). References [1] A. Lossev, N. Nekrasov, and S. L. Shatashvili, Testing Seiberg-Witten solution , in Strings, branes and dualities. Proceedings, NATO Advanced Study Institute, Cargese,France, May 26-June 14, 1997 , 1997. hep-th/9801061 .[2] N. A. Nekrasov, Seiberg-Witten prepotential from instanton counting , Adv. Theor.Math. Phys. (2003), no. 5 831–864, [ hep-th/0206161 ].[3] R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of thecoefficients of the Seiberg-Witten prepotential , Int. J. Mod. Phys. A18 (2003) 2541,[ hep-th/0208176 ].[4] N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions , Prog.Math. (2006) 525–596, [ hep-th/0306238 ].[5] U. Bruzzo, F. Fucito, J. F. Morales, and A. Tanzini, Multiinstanton calculus andequivariant cohomology , JHEP (2003) 054, [ hep-th/0211108 ].[6] N. A. Nekrasov and S. L. Shatashvili, Quantization of Integrable Systems and FourDimensional Gauge Theories , in Proceedings, 16th International Congress onMathematical Physics (ICMP09) , 2009. arXiv:0908.4052 .[7] L. F. Alday, D. Gaiotto, and Y. Tachikawa, Liouville Correlation Functions fromFour-dimensional Gauge Theories , Lett. Math. Phys. (2010) 167–197,[ arXiv:0906.3219 ]. – 23 – 8] N. Seiberg, Notes on quantum Liouville theory and quantum gravity , Prog. Theor.Phys. Suppl. (1990) 319–349.[9] A. B. Zamolodchikov and A. B. Zamolodchikov, Structure constants and conformalbootstrap in Liouville field theory , Nucl. Phys. B477 (1996) 577–605,[ hep-th/9506136 ].[10] A. B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrenceformula for the conformal partial wave amplitude , Commun. Math. Phys. (1984)419–422.[11] A. B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursionrepresentation of conformal block , Theor. Math. Phys. (1987) 1088–1093.[12] R. Poghossian, Recursion relations in CFT and N=2 SYM theory , JHEP (2009)038, [ arXiv:0909.3412 ].[13] R. Poghossian, Recurrence relations for the W conformal blocks and N = 2 SYMpartition functions , JHEP (2017) 053, [ arXiv:1705.00629 ]. [Erratum: JHEP 01,088 (2018)].[14] L. Hadasz, Z. Jaskolski, and P. Suchanek, Recursion representation of theNeveu-Schwarz superconformal block , JHEP (2007) 032, [ hep-th/0611266 ].[15] L. Hadasz, Z. Jaskolski, and P. Suchanek, Elliptic recurrence representation of the N=1superconformal blocks in the Ramond sector , JHEP (2008) 060, [ arXiv:0810.1203 ].[16] S. Alekseev, A. Gorsky, and M. Litvinov, Toward the Pole , JHEP (2020) 157,[ arXiv:1911.01334 ].[17] M. Beccaria, On the large Ω -deformations in the Nekrasov-Shatashvili limit of N = 2 ∗ SYM , JHEP (2016) 055, [ arXiv:1605.00077 ].[18] A. Gorsky, A. Milekhin, and N. Sopenko, Bands and gaps in Nekrasov partitionfunction , JHEP (2018) 133, [ arXiv:1712.02936 ].[19] R. Poghossian, Deforming SW curve , JHEP (2011) 033, [ arXiv:1006.4822 ].[20] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, andconfinement in N=2 supersymmetric Yang-Mills theory , Nucl. Phys. B426 (1994)19–52, [ hep-th/9407087 ]. [Erratum: Nucl. Phys.B430,485(1994)].[21] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2supersymmetric QCD , Nucl. Phys. B431 (1994) 484–550, [ hep-th/9408099 ].[22] F. Fucito, J. F. Morales, D. R. Pacifici, and R. Poghossian, Gauge theories on Ω -backgrounds from non commutative Seiberg-Witten curves , JHEP (2011) 098,[ arXiv:1103.4495 ]. – 24 – 23] A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-ZommerfeldIntegrals , JHEP (2010) 040, [ arXiv:0910.5670 ].[24] A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: The Case ofSU(N) , J. Phys. A43 (2010) 195401, [ arXiv:0911.2396 ].[25] K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System andLiouville Field Theory , Nucl. Phys. B (2010) 388–425, [ arXiv:1006.4505 ].[26] “ NIST Digital Library of Mathematical Functions .” http://dlmf.nist.gov/16Chapter 16, A. B. Olde Daalhuis, R. A. Askey, Generalized Hypergeometric Functionsand Meijer G-Function.[27] A. Zamolodchikov, Generalized Mathieu equation and Liouville TBA, 2000 , inQuantum Field Theories in Two Dimensions, vol. 2, World Scientific, 2012 .[28] D. Fioravanti, H. Poghosyan, and R. Poghossian, T , Q and periods in SU (3) N = 2 SYM , JHEP (2020) 049, [ arXiv:1909.11100 ].[29] N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equationsand qq-characters , arXiv:1512.05388 .[30] G. Poghosyan and R. Poghossian, VEV of Baxter’s Q-operator in N=2 gauge theoryand the BPZ differential equation , JHEP (2016) 058, [ arXiv:1602.02772 ].[31] G. Poghosyan, VEV of Q -operator in U (1) linear quiver 5d gauge theories , arXiv:1801.04303 .[32] P. Dorey and R. Tateo, J. Phys. A , L419-L425 (1999)doi:10.1088/0305-4470/32/38/102 [arXiv:hep-th/9812211 [hep-th]].[33] V. V. Bazhanov, S. L. Lukyanov and A. B. Zamolodchikov, J. Statist. Phys. ,567-576 (2001) doi:10.1023/A:1004838616921 [arXiv:hep-th/9812247 [hep-th]].[34] P. Dorey, C. Dunning and R. Tateo, J. Phys. A , R205 (2007)doi:10.1088/1751-8113/40/32/R01 [arXiv:hep-th/0703066 [hep-th]].[35] A. B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordonmodel , J. Phys. A (2006) 12863–12887, [ hep-th/0005181 ].[36] M. Matone, Instantons and recursion relations in N=2 SUSY gauge theory , Phys. Lett. B357 (1995) 342–348, [ hep-th/9506102 ].[37] R. Flume, F. Fucito, J. F. Morales, and R. Poghossian, Matone’s relation in thepresence of gravitational couplings , JHEP (2004) 008, [ hep-th/0403057 ].[38] D. Fioravanti and D. Gregori, Integrability and cycles of deformed N = 2 gauge theory , arXiv:1908.08030 .[39] R. Poghossian, Deformed SW curve and the null vector decoupling equation in Todafield theory , arXiv:1601.05096 . – 25 – 40] S. K. Ashok, D. P. Jatkar, R. R. John, M. Raman, and J. Troost, Exact WKB analysisof N = 2 gauge theories , JHEP (2016) 115, [ arXiv:1604.05520 ].[41] A. Litvinov, S. Lukyanov, N. Nekrasov, and A. Zamolodchikov, Classical ConformalBlocks and Painleve VI , JHEP (2014) 144, [ arXiv:1309.4700 ].].