Exact partition functions for the Ω-deformed \( \mathcal{N}={2}^{\ast } \) SU(2) gauge theory

Journal of High Energy Physics, Jul 2016

We study the low energy effective action of the Ω-deformed \( \mathcal{N}={2}^{\ast } \) SU(2) gauge theory. It depends on the deformation parameters ϵ 1, ϵ 2, the scalar field expectation value a, and the hypermultiplet mass m. We explore the plane \( \left(\frac{m}{\upepsilon_1},\frac{\upepsilon_2}{\upepsilon_1}\right) \) looking for special features in the multi-instanton contributions to the prepotential, motivated by what happens in the Nekrasov-Shatashvili limit ϵ 2 → 0. We propose a simple condition on the structure of poles of the k-instanton prepotential and show that it is admissible at a finite set of points in the above plane. At these special points, the prepotential has poles at fixed positions independent on the instanton number. Besides and remarkably, both the instanton partition function and the full prepotential, including the perturbative contribution, may be given in closed form as functions of the scalar expectation value a and the modular parameter q appearing in special combinations of Eisenstein series and Dedekind η function. As a byproduct, the modular anomaly equation can be tested at all orders at these points. We discuss these special features from the point of view of the AGT correspondence and provide explicit toroidal 1-blocks in non-trivial closed form. The full list of solutions with 1, 2, 3, and 4 poles is determined and described in details.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP07%282016%29066.pdf

Exact partition functions for the Ω-deformed \( \mathcal{N}={2}^{\ast } \) SU(2) gauge theory

HJE SU(2) gauge theory Matteo Beccaria 0 Guido Macorini 0 0 Via Arnesano , 73100 Lecce , Italy We study the low energy e ective action of the gauge theory. It depends on the deformation parameters 1; 2, the scalar eld expectation value a, and the hypermultiplet mass m. We explore the plane ( m ; 2 ) looking for special features in the multi-instanton contributions to the prepotential, motivated by what happens in the Nekrasov-Shatashvili limit 2 ! 0. We propose a simple condition on the structure of poles of the k-instanton prepotential and show that it is admissible at a nite set of points in the above plane. At these special points, the prepotential has poles at xed positions independent on the instanton number. Besides and remarkably, both the instanton partition function and the full prepotential, including the perturbative contribution, may be given in closed form as functions of the scalar expectation value a and the modular parameter q appearing in special combinations of Eisenstein series and Dedekind function. As a byproduct, the modular anomaly equation can be tested at all orders at these points. We discuss these special features from the point of view of the AGT correspondence and provide explicit toroidal 1-blocks in non-trivial closed form. The full list of solutions with 1, 2, 3, and 4 poles is determined and described in details. ArXiv ePrint: 1606.00179 Extended Supersymmetry; Nonperturbative E ects; Supersymmetric E ec- - 2 INFN, Via Arnesano, 73100 Lecce, Italy Dipartimento di Matematica e Fisica \Ennio De Giorgi", Universita del Salento, -deformed N = 2 SU(2) 1 tive Theories 1 Introduction and results 2 Looking for simplicity beyond the Nekrasov-Shatashvili limit 2.1 Back to the instanton partition functions 3 AGT interpretation 3.1 Explicit CFT computations 4 Perturbative part of the prepotential at the points Xi 5 Full prepotential and generalization to N -poles points 5.1 Constraints from the modular anomaly equation 5.2 A worked out 3-pole example 6 Predictions for the torus 1-block A 3- and 4-poles Nekrasov functions data B On the structure of higher instanton Nekrasov functions 1 4 5 -deformed N = 2 SU(2) gauge theory in four dimensions and present novel closed expressions for its low energy e ective action at special values of the deformation parameters. On general grounds, before deformation, the e ective action of N = 2 theories is computed by the Seiberg-Witten (SW) curve [1, 2]. It is the sum of a 1-loop perturbative correction and an in nite series of non-perturbative instantonic contributions that are weighted by the instanton counting parameter q = ei where is the complexi ed gauge coupling constant at low energy. Due to N = 2 supersymmetry, the full e ective action may be expressed in terms of the analytic prepotential F (a; m) depending on the vacuum expectation value a of the scalar in the adjoint gauge multiplet and on the mass m of the adjoint hypermultiplet [3]. Instead of applying the SW machinery, one may compute the e ective action by topological twisting the theory and exploiting localization on the many-instanton moduli spaces [4{6]. Technically, this is made feasible by introducing the so-called -deformation of the theory, i.e. a modi cation breaking 4d Poincare invariance and depending on two parameters 1; 2 such that the initial theory is recovered when 1; 2 ! 0. The role of { 1 { the -deformation is that of a complete regulator for the instanton moduli space integration [7{14]. In this approach, it is natural to introduce a well de ned partition function Zinst( 1; 2; a; m) and its associated non-perturbative -deformed prepotential by means of F inst( 1; 2; a; m) = 1 2 log Zinst( 1; 2; a; m): (1.1) It is well established that the quantity in (1.1) is interesting at nite values of the deformation parameters 1; 2, i.e. taking seriously the deformed theory. This is because the amplitudes appearing in the expansion F pert + F inst = Pn1;g=0 F (n;g) ( 1 + 2)2n ( 1 2)g are related to the genus g partition function of the N = 2 topological string [15{21] and satisfy a powerful holomorphic anomaly equation [22{25]. Actually, understanding the exact dependence on the deformation parameters is an interesting topic if one wants to resum the above expansion in higher genus amplitudes. Clearly, this issue is closely related to the Alday-Gaiotto-Tachikawa (AGT) correspondence [26] mapping deformed N = 2 instanton partition functions to conformal blocks of a suitable CFT with assigned worldsheet genus and operator insertions. AGT correspondence may be checked by working order by order in the number of instantons [27{29]. For the N = 2 -deformed SU(2) gauge theory the relevant CFT quantity is the one-point conformal block on the torus, a deceptively simple object of great interest [27, 28, 30{37]. The AGT interpretation emphasizes the importance of modular properties in the deformed gauge theory. Indeed, it is known that SW methods can be extended to the case of non-vanishing deformation parameters 1; 2 [38, 39] and modular properties have been clari ed in the undeformed case [40, 41] as well as in presence of the deformation [42, 43]. The major outcome of these studies are explicit resummations of the instanton expansion order by order in the large a regime. The coe cients of the 1=a powers are expressed in terms of quasi-modular functions of the torus nome q. This approach can be pursued in the gauge theory [44{50], in CFT language by AGT correspondence [34{36, 51], and also in the framework of the semiclassical WKB analysis [52, 52{57]. An important simpler setup where these problems may be addressed is the so-called Nekrasov-Shatashvili (NS) limit [58] where one of the two parameters vanishes. In this case, the deformed theory has an unbroken two dimensional N = 2 super-Poincare invariance and its supersymmetric vacua are related to the eigenstates of a quantum integrable system. Under this Bethe/gauge map, the non-zero deformation parameter plays the role of ~ in the quantization of a classically integrable system. Saddle point methods allow to derive a deformed SW curve [59, 60] that can also be analyzed by matrix model methods [33, 52, 53, 61, 62]. In the speci c case of the N = 2 theory, the relevant integrable system is the elliptic Calogero-Moser system [58] and the associated spectral problem reduces to the study of the celebrated Lame equation. Besides, if the hypermultiplet mass m is taken to be proportional to with de nite special ratios m = n + 12 , where n 2 N, the k spectral problem is n-gap. Remarkable simpli cations occur in the k-instanton prepotential contributions F inst [63] that may be obtained by expanding the eigenvalues of a Lame equation in terms of its Floquet exponent. As a byproduct of this approach, it is possible to clarify the meaning of the poles that appear in the k-instanton prepotential at special { 2 { values a = O( ) of the vacuum expectation value a. Indeed, the pole singularities turn out to be an artifact of the instanton expansion. In this paper, we inquire into similar problems when both the deformation parameters are switched on, i.e. by going beyond the Nekrasov-Shatashvili limit. In particular, we explore the ( ; ) plane where ; are real parameters entering the scaling relation m = 1 ; 2 = 1 : (1.2) In other words, we keep the hypermultiplet mass to be proportional to one deformation parameter with ratio , but 1; 2 are generic ( is just a convenient replacement of 2). By dimensional scaling, the prepotential is a function Fe( ; )( ) of the combination at the xed point ( ; ).1 The dependence on q is not written explicitly. After this stage = 2 a= 1 preparation, the claim of this paper is the following There exists a nite set of N -poles points ( ; ) such that the k-instanton prepotential is a rational function of with poles at a xed set of positions 2 f 1; : : : ; N g independent on k. This claim is motivated by our previous analysis in the restricted Nekrasov-Shatashvili limit [63] and is far from obvious. Most important, it has far reaching consequences. At the special N -poles points, we show that the instanton partition function and the perturbative part of the prepotential take the exact form hm log with coe cients depending on ; , and hm 2 N. The total prepotential is thus remarkably simple and reads Fe( ; )( ) = hm log N n=1 log 1 + X conformal dimension hm | the perturbative part providing interesting special instances of the 3-point DOZZ Liouville correlation function. These results are derived and tested by giving a complete list of all the N 4 poles points. These turns out to be 4, 7, 12, and 11 (1.3) (1.4) at N = 1; 2; 3; 4 respectively. of the variable . 1We shall systematically add a tilde to quantities that are considered under (1.2) and expressed in terms { 3 { Our results complement a similar analysis that has been performed in [64] on a special class of N = 2 gauge theories on S4 with fundamental matter. In this reference, it has been shown that, for a speci c choice of the masses, the partition function localizes around a nite set of critical points where it can be explicitly evaluated and written in terms of generalized hypergeometric functions. From the AGT perspective, these special points may be viewed as a four point correlator involving the insertion of a degenerated eld. The plan of the paper is the following. In section 2 we determine the 1-pole points by a direct inspection of the instanton prepotential contributions. In section 2.1 we discuss the special features of the instanton partition function at the 1-pole points. The AGT interpretation is analyzed in section 3 where we also provide various explicit CFT tests of the proposed partition functions. In section 4 we discuss the perturbative part of the prepotential at the 1-pole points. In section 5 the analysis is extended to N -poles points and the cases N = 2; 3 are fully classi ed. Finally, section 6 presents a list of special toroidal blocks. Various appendices are devoted to additional comments. 2 Looking for simplicity beyond the Nekrasov-Shatashvili limit As discussed in the Introduction, we are interested in the scaling limit (1.2). The instanton partition function is Zinst = Zinst( 1; 2; a; m) and it is convenient to introduce Ze(ins;t )( ) = Zinst 1 ; 1 ; 1 ; 2 1 = Zinst 1; ; ; ; 2 where we used dimensional scaling independence to remove 1. Similarly, we de ne F inst = 1 2 log Zinst; with a polynomial Pk( ) and a single pole 1 0 in the variable j j. The Ansatz (2.3) is a non-trivial requirement. It is motivated by the analysis in [63], but its admissibility is actually one of the results of our investigation. To explore the constraints that (2.3) imposes, we begin by looking at the simple one-instanton case.2 For k = 1 we have the explicit expression e1 F inst( ) = (2 2The functions Fekinst may be computed by the beautiful Nekrasov formula [5]. Alternatively, for a gauge algebra g 2 fAr; Br; Cr; Drg one can also apply the methods described in [4, 5, 8, 11, 65{67]. { 4 { (2.1) (2.2) (2.3) and there is a simple pole 1 = j + 1j. At the two-instanton level, k = 2, the denominator of Fe2( ) turns out to vanish at j j = + 1(order 2); Special cases occur when one of the poles coincides with those at 1 = j +1j. This happens for 2 3 only non-trivial cases consistent with (2.3)3 is not in the set (2.6), one checks that Fe2 takes the form (2.3) if Pushing the calculation up to 12 instantons, we con rm that the points in (2.7) and (2.8) agree with the Ansatz (2.3). Thus, the 1-pole condition (2.3) selects the following distinct 4 special points 2.1 Back to the instanton partition functions We could analyze further the structure of the prepotential in (2.3) at the special points Xi in (2.9) by looking for regularities in the polynomials Pk( ). However, it is much more convenient to go back to the instanton partition function. To see why, let us consider X1 as a rst illustration. We nd indeed the simple expansion 2 After some educated trial and error, we recognize that (2.10) is the expansion of the 5 2 + 4 4 14 ZeXin1st( ) = 1 following expression where ZeXin1st( ) = 2 E2(q) 1 q 31 ( )4; 1 k=1 3Here, trivial means a constant Fek( ). and E2 is an Eisenstein series.4 Similar expressions are found at the other three special points. The detailed formulas are The associated all-instanton Nekrasov functions are ZeXin2st( ) = ZeXin3st( ) = ZeXin4st( ) = FeXin1st( ) = 8 log[q 112 ( )] + 2 log FeXin2st( ) = 3 log[q 112 ( )] + FeXin3st( ) = 4 3 1 log[q 12 ( )] + FeXin4st( ) = 2 log[q 112 ( )] + 3 2 2 3 1 2 log log log 2 2 Equations (2.13) and (2.14) are already remarkable because they are non-trivial closed expressions for the instanton partition function, or prepotential, at all instanton numbers. It is clear that it would be nice to provide some clarifying interpretation for this features at the special points Xi. In the next section, we shall examine the clues coming from AGT correspondence. 3 AGT interpretation According to the AGT correspondence, the instanton partition function of N = 2 SU(2) gauge theory is [26{28] Zinst(q; a; m) = q2k) " 1 Y(1 k=1 # 1+2 hm Fhhm (q); where F m(q) is the 1-point toroidal block of the Virasoro algebra of central charge c = 1 + 6 Q2 on a torus whose modulus is q, with one operator of dimension hm inserted and a primary of dimension h in the intermediate channel. The precise dictionary in terms of the deformation parameters is b = p 2= 1; hm = Q2 4 m2 { 6 { Assuming the scaling relations (1.2), the expressions in (3.2) read In particular, at the four points Xi we obtain the following values for (c; hm) Of course, points appear in pairs with the same central charge and values related by ! 1 . More remarkably, the associated values of the parameter is always such that hm is a positive integer. The toroidal block has a universal prefactor q 12 = ( ) that is its value at h ! 1. Comparing (3.1) with (2.13) we can write the general form for all four Xi points as log 1 1 2 2 1 = j + 1j; ( ) 1 + c 1 24 h (E2(q) 1) ; (c; hm) = (0; 2) or ( 2; 3): (3.7) We remark that the above (c; hm) may well be pathological for a physical CFT. Nevertheless, the toroidal block is de ned by the Virasoro algebra for abritrary values of c; hm, and h. Eq. (3.7) must be taken in this sense. We checked (3.7) against Zamolodchikov recursive determination of the toroidal block [69{71] with perfect agreement. Of course, by AGT, this is same as Nekrasov calculation. The remarkably simple form (3.7) is clearly consistent with general results for the torus block. For instance, at leading and next-to-leading order and generic operator dimensions we have [35] Fhhm (c; q) = 1 + F1(h; hm; c) q2 + F2(h; hm; c) q4 + : : : ; where F1(h; hm; c) = 1 + hm (hm 2 h 1) ; F2(h; hm; c) = [4 h (2 c h + c + 16 h2 10 h)] 1 { 7 { Thus, in full agreement with (3.7) for c = 0; 2. Notice also that (3.7) may be written 1 ( ) 1 + c 1 ( ) = 1 X k=0 1 + 1 h c k Pk q2k; (3.11) where Pk are the coe cients of the expansion of Qk1=1(1 unrestricted partitions of k (nm means m copies of n) q2k) 1, i.e. the number of P1 = #f(1)g = 1; P2 = #f(2); (12)g = 2; P3 = #f(3); (2; 1); (13)g = 3; P4 = #f(4); (3; 1); (22); (2; 12); (14)g = 5; P5 = #f(5); (4; 1); (3; 2); (3; 12); (22; 1); (2; 13); (15)g = 7; and so on. 3.1 Explicit CFT computations The (c; hm) = (0; 2) conformal block. It is instructive to derive the result (3.7) at c = 0 from a direct CFT calculation.5 In other words, we want to show that h Fhm=2(q; c = 0) = 1 ( ) 1 1 24 h (E2(q) 1) = 1 + k Pk q2k; 1 X k=0 1 h where we used (3.11). The toroidal block is obtained as Fhhm (q; c) = q h+ 1c2 Trh qL0 1c2 'hm (1) ; where the trace is over the descendants of 'h. The starting point is thus conformal descendant decomposition of the diagonal part of the OPE 'hm (x) 'h(0) = X x hm+jY j Y L Y 'h(0) Y = x hm (1 + x (1) L 1 + x2 ( (2) L 2 + (1;1) L2 1) + : : : ) 'h(0); 5CFT at c = 0 is obviously quite special since the 2-point function of the stress energy tensor is then zero, so strictly speaking, the theory is not conformal any more (since the stress-tensor vanishes identically). However, as we remarked, we are considering the toroidal block as a well-de ned function of (c; hm; h) that may be regarded as the solution to the Zamolodchikov recursion relations. It would be interesting to revisit our calculation in the language of logarithmic CFT, see for instance [72, 73]. { 8 { (n) = 1 h ; (1) = Computing them at (c; hm) = (0; 2) we see that indeed Hence, if we apply (3.15) to the vacuum, we get6 (1) = (2) = where Y denotes a unrestricted partition of jY j and Ln are Virasoro generators they are all equal The coe cients in (3.15) are determined by conformal symmetry and are functions of h; hm; c. As a consequence of unbroken conformal symmetry (c = 0) and of the fact that hm = 2 is the same dimension as that of the energy momentum tensor, one nds that the coe cients are those associated with simple L n descendants. Besides Using we obtain with one index Now, to get the torus block, we need to evaluate the diagonal matrix elements of '2(x). [Ln; '2(x)] = xn (x @ + 2 (n + 1)) '2(x); 1 h ; 1 h 1 X x n=0 { 9 { (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) HJEP07(216) '2(x) L k jhi = [L k; '2(x)] jhi + L k '2(x) jhi = = h 1 X1 (2k n=0 + 1 x2 n) xn k 2 L njhi + n 2 L kL n jhi 1 h 1 X x n=0 1 + k h L kjhi + : : : ; where we have shown only the diagonal entry. Adding one index each time, a similar calculation shows that for any number of indices '2(x) L Y jhi = + 1 x2 1 + jY j h L Y jhi + : : : : Thus the diagonal matrix element of '2(x) associated with the Y descendent depends only on jY j. The number of Y with xed jY j is the number PjY j of unrestricted partitions of jY j. Summing over Y with jY j = k we prove (3.13). 6Notice that hm = 2 is not enough to achieve such simpli cation. A vanishing central charge is also needed to remove descendants with multiple applications of Virasoro operators. 3 2h ; (2) = 12 8h + 1 ; for certain coe cients MY functions of h. At level 1, we have only '3(x) L Y jhi = 1 + x3 MY L Y jhi + : : : ; hm = 3 is apparently quite less trivial. The main reason is that the coe cients in the conformal decomposition (3.15) do not trivialize in this case. This complication forbids us to prove the wanted result in general. Nevertheless, we provide an explicit check at level 4. Of course, one could simply use the recursive de nition of the toroidal block, but our brute force calculation is perhaps more transparent. Besides, it emphasizes the di erence compared with the previous (c; hm) = (0; 2) case. The starting point is again the OPE (3.15) that now takes the following form up to level 4 descendants with the simple but non trivial coe cients (1) = (2;1) = (3;1) = (2;1;1) = As in (3.23), we write At level 2, At level 3, M4 = M2;2 = MY = 5 + 60 h X jY j=4 2. It would be nice to prove the agreement at all levels, possibly working in a de nite c = 2 CFT like the triplet model considered 4 Perturbative part of the prepotential at the points Xi The prepotential has also a perturbative part, related by AGT to the DOZZ 3-point function in the Liouville theory [70, 76, 77]. The general expression for the perturbative part of the prepotential is ( me = m + 1+2 2 ) [4, 5] F pert = 1 2 1; 2 (2 a) + 1; 2 ( 2 a) 1; 2 (2 a + me) 1; 2 ( 2 a + me) ; 1; 2 (x) = s Z 1 dt (s) 0 ts e t x t (e 1 t 1) (e 2 t 1) and is a renormalization scale. Evaluating Fpert by expanding at small 1 and resumming, we nd that at all Xi points it is possible to write d ds 4 F pert = Again, this appears to be a special feature of the Xi points because it is not possible to give such a simple expression for F pert at generic 1; 2 from (4.1). With a rede nition of the UV cuto , this may be written in the following suggestive form that we shall generalize later e F pert( ) = Full prepotential and generalization to N -poles points If we combine the perturbative (4.4) and instanton (3.6) parts of the prepotential, we obtain the remarkably simple expression Fe = Fepert + Feinst = 4 a2 : (4.1) (4.2) (4.3) (4.4) (5.1) with a certain coe cient . This suggests that it is convenient to organize the total prepotential in the form Fe = where M2n(q) is a polynomial in E2;4;6 of (quasi-) modular degree 2n. We emphasize again that the Ansatz (5.2) is non trivial because Fetot is a combination of the perturbative and instanton contributions. Our claim is that (5.2) can be truncated at maximum degree n = N for a special set of points ( ; ). At such points, the instanton partition function takes the special form, see (1.3) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) 4 = 2;2 = 1 23040 1 To see how this works in practice, let us parametrize Replacing these combinations in (5.3) and comparing with the Nekrasov function at 5 instanton we obtain explicit expressions for the coe cients in terms of ; . The rst coe cients are The non trivial solutions of 4 = 2;2 = 0 (as always, up to the choice > 0) are that is precisely the four points Xi. The coe cient 2 is also an output of the calculation and its general expression is 2 = ( 2 + 1)( 2 + Finally, comparing (5.2) and (5.3) we determine the perturbative part of the prepotential to be, see (1.3) that is a nice generalization of (4.4). In all cases, we have con rmed that this is in full agreement with the explicit evaluation of the general expression (4.1). We remark once again that it is remarkable that at the special n-poles points, it is possible to get such a simple expression. Performing the above analysis for 2- and 3- poles Nekrasov functions, we nd the following complete results. We have four 1- and seven 2-poles Nekrasov functions that are fully characterized by the following tables where we have separated by a horizontal line points with di erents (c; hm) 5 2 5 4 7 4 7 6 2 1 2 Looking for 3-poles points, we identify 12 cases whose full data is collected in the table in (A.1). With 4-poles, we found 11 solutions collected in the two tables (A.2) and (A.3). Notice that in all presented cases, the parameter is always rational negative. This implies that central charge takes the form of extended minimal models with coprime integers p, q. Here, the extension is due to the fact that the minimum value of p; q is one instead of 2, see also gure 1. 5.1 Constraints from the modular anomaly equation An important test of the expression (5.2) is the validity of the modular anomaly equation expressing S-duality [44{49]. This is a non-trivial constraint capturing the dependence on the quasi-modular series E2 and reads [45] An alternative form is obtained by expanding at large and identifying the coe cients h` in Then, (5.11) may be written in the equivalent form Fe = h0 log 1 X `=1 h` 21 ` ` 2` 1 ; h0 = hm: n=0 X hn h` 1 n + ` (2 ` 1) 12 h` 1: HJEP07(216) (5.10) (5.11) (5.12) (5.13) - 20 - 15 - 10 - 5 8 6 4 2 0 c 2;2 2 2;2;2 2;2 2;4 4 Just to give a simple example, let us consider a 1-pole function and write the -dependent part of Fe as F ( ) = h0 log e Imposing (5.11) or (5.13), we recover the expression (5.7) for 2. Besides, we also get the following constraint between and (excluding trivial solutions with constant prepotential) have presented. ratio 2;2= 2: The condition (5.15) is indeed satis ed by the values in the rst table of (5.9). However, there are in nite other pairs ( ; ) that make (5.14) a solution of (5.11) that is not realized in the gauge theory. Of course, this is because (5.14) predicts all the higher order terms in the large only for a expansion and this is correct in comparison with the actual Nekrasov formulas nite set of values of , and . Actually, the admissibility of the Ansatz (5.3) is de nitely non trivial. Anyway, we checked the validity of (5.11) for all the solutions we As a nal comment, we notice that using the explicit expressions for 2;2 and 2 it turns out that the constraint (5.15), up to an multiplicative factor, can be expressed as the 2;2 = 0 is of course a necessary condition in the Ansatz (5.3) to truncate the sum at N = 1. Looking at higher values of N , one gets similar constraints for any N . In fact, since the modular anomaly equation controls the dependence of Fe on E2, the consistency of our Ansatz with (5.11) imposes relations between all the coe cients of the form 2;X with X . For example, in the 2-poles case we have that (5.14) (5.15) HJEP07(216) Similarly for 3-poles we get 2;2;2;2 2;2;2 2;2;4 2;4 and 2 2;6 6 and the constraint for ensuring that 2;2;2 = 2;4 = 0 is To appreciate the result of our analysis, let us consider in some details the rst line of (A.1). The instanton partition function for X = ( ; ) = ( 11 ; 4) is up to 12 instantons 2 This is far more involved than (2.10) and a brute force guess would not be possible. Nevertheless, it is a straightforward calculation to check that this is the expansion of ZeXinst( ) = [q 112 ( )]12 6 14 E2 4 + 1430 E22 + 73 E4 ( 2 9) ( 2 2 4) ( 2 in agreement with the data in (A.1). The perturbative part of the prepotential is computed from (4.1) by expanding at large a. This gives, in the variable eX F pert( ) = 28 log 56 2 196 4 3176 and this is indeed the large expansion of eX F pert( ) = 28 log + 4 log 1 1 2 4 2 1 9 2 ; (5.23) where 28 = hm = ( 4) 7. Hence, the full quantum prepotential is in this case Fe = 28 log + 4 log 1 14 2 E2 + 7 3 4 (20 E22 + E4) 9 6 (140 E23 + 21 E2 E4 + E6) ; (5.24) and one can check that (5.11) is satis ed. 1 2 (5.20) +: : : : ; (5.21) Predictions for the torus 1-block We have already seen that AGT implies explicit expressions for special torus blocks that we write stripping o the large h dominant term From the 1-pole partition functions, we have obtained From the 2-poles partition functions, we get similar expressions Notice that there are multiple entries in the tables (5.9) and (A.1) with the same value of (c; hm), but di erent ( ; ). Nevertheless, the associated block is consistently the same as soon as is expressed in terms of h. From the 3-poles partition functions, we get H5h(q; 7) = 1 + H4h q; 22 5 = 1 + H4h(q; 1) = 1 + H5h(q; 0) = 1 + 1 24h E2 ; 15E22 + E4 25E22 + 2E4 E 2 2 3 5E22 H3h(q; 2) = 1 + 1 8h E2 : 80E2(3 h + 1) + 16(15 h + 4) 144 h(4 h + 1) 30E2(40 h + 9) + 3(400 h + 81) 960 h(5 h + 1) E4 48E2 h + 48 h 48 h(4 h 1) ; 2E4 10E2(24 h + 1) + 240 h + 1 ; 5 E23 + 3 E4 E2 + 2 E6 45 E2(8 h + 1)2 + 9(5 E22 30 E2(48 h + 1)2 + 3(60 E22 17 E4)(48 h + 1) 3312 h + 1 ; 126 E2(16 h + 9)2 + 21(20 E22 + E4)(16 h + 9) ; 2(140 E23 + 21 E4 E2 + E6) + 9(3584 h2 + 3248 h + 729) 32256 h(7 h + 2)(7 h + 3) 23040 h(5 h 2)(5 h + 1) While the structure is extremely similar, the selection rule is no longer true here. At the point X1, the Laurent expansion of FeXin1s;tk( ) around = 1 has the generic form FeXin1s;tk( ) = ( d(k) 0 1)k + ( d(k) where all the coe cients are non vanishing. The same is true for all the other three 1-pole cases X2; X3; X4. As in [63], we can rewrite the k-instanton functions in terms of the d(pk) in the exact form FeXin1s;tk( ) = ck + independent part (i.e. the term proportional to the logarithms of the Dedekind function. The coe cients d(pk) can be obtained from the expansion of the exact expressions of FeXinis;tk( ). For X1 they read 1) 1 (B.5) Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and con nement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE]. [2] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE]. [3] E. D'Hoker and D.H. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, in Theoretical physics at the end of the twentieth century. Proceedings, Summer School, Ban Canada June 27{July 10 1999, pg. 1 [hep-th/9912271] [INSPIRE]. [4] N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE]. [7] R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coe cients [8] U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE]. [9] R. Flume, F. Fucito, J.F. Morales and R. Poghossian, Matone's relation in the presence of gravitational couplings, JHEP 04 (2004) 008 [hep-th/0403057] [INSPIRE]. [hep-th/0404225] [INSPIRE]. [11] M. Marin~o and N. Wyllard, A note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP 05 (2004) 021 [hep-th/0404125] [INSPIRE]. [12] M. Billo, L. Ferro, M. Frau, L. Gallot, A. Lerda and I. Pesando, Exotic instanton counting and heterotic/type-I' duality, JHEP 07 (2009) 092 [arXiv:0905.4586] [INSPIRE]. [13] F. Fucito, J.F. Morales and R. Poghossian, Exotic prepotentials from D( 1)D7 dynamics, JHEP 10 (2009) 041 [arXiv:0906.3802] [INSPIRE]. [14] M. Billo, M. Frau, F. Fucito, A. Lerda, J.F. Morales and R. Poghossian, Stringy instanton corrections to N = 2 gauge couplings, JHEP 05 (2010) 107 [arXiv:1002.4322] [INSPIRE]. [15] I. Antoniadis, E. Gava, K.S. Narain and T.R. Taylor, Topological amplitudes in string theory, Nucl. Phys. B 413 (1994) 162 [hep-th/9307158] [INSPIRE]. [16] I. Antoniadis, S. Hohenegger, K.S. Narain and T.R. Taylor, Deformed topological partition function and Nekrasov backgrounds, Nucl. Phys. B 838 (2010) 253 [arXiv:1003.2832] 95 (2011) 67 [arXiv:1007.0263] [INSPIRE]. [17] D. Kre and J. Walcher, Extended holomorphic anomaly in gauge theory, Lett. Math. Phys. [18] M.-X. Huang and A. Klemm, Direct integration for general backgrounds, Adv. Theor. Math. Phys. 16 (2012) 805 [arXiv:1009.1126] [INSPIRE]. [19] I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Non-perturbative Nekrasov partition function from string theory, Nucl. Phys. B 880 (2014) 87 [arXiv:1309.6688] [INSPIRE]. [20] I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Worldsheet realization of the re ned topological string, Nucl. Phys. B 875 (2013) 101 [arXiv:1302.6993] [21] I. Florakis and A. Zein Assi, N = 2? from topological amplitudes in string theory, Nucl. Phys. B 909 (2016) 480 [arXiv:1511.02887] [INSPIRE]. [22] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological eld theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE]. [INSPIRE]. exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [INSPIRE]. [25] M.-X. Huang and A. Klemm, Holomorphicity and modularity in Seiberg-Witten theories with matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [INSPIRE]. [26] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [27] R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [28] V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [29] V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [30] V.A. Fateev, A.V. Litvinov, A. Neveu and E. Onofri, Di erential equation for four-point correlation function in Liouville eld theory and elliptic four-point conformal blocks, J. Phys. A 42 (2009) 304011 [arXiv:0902.1331] [INSPIRE]. [31] L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE]. [32] P. Menotti, Riemann-Hilbert treatment of Liouville theory on the torus, J. Phys. A 44 (2011) [33] A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys. 61 (2011) 1203 [34] A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 [35] M. Piatek, Classical torus conformal block, N = 2? twisted superpotential and the accessory parameter of Lame equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE]. [36] A.-K. Kashani-Poor and J. Troost, Quantum geometry from the toroidal block, JHEP 08 (2014) 117 [arXiv:1404.7378] [INSPIRE]. [37] K.B. Alkalaev and V.A. Belavin, Holographic interpretation of 1-point toroidal block in the semiclassical limit, JHEP 06 (2016) 183 [arXiv:1603.08440] [INSPIRE]. [38] D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE]. [39] A. Mironov, A. Morozov and S. Shakirov, Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [INSPIRE]. [40] J.A. Minahan, D. Nemeschansky and N.P. Warner, Instanton expansions for mass deformed N = 4 super Yang-Mills theories, Nucl. Phys. B 528 (1998) 109 [hep-th/9710146] [INSPIRE]. [41] M. Billo, M. Frau, L. Gallot and A. Lerda, The exact 8d chiral ring from 4d recursion relations, JHEP 11 (2011) 077 [arXiv:1107.3691] [INSPIRE]. deformed B-model for rigid N = 2 theories, Annales Henri Poincare 14 (2013) 425 [arXiv:1109.5728] [INSPIRE]. [43] M.-X. Huang, On gauge theory and topological string in Nekrasov-Shatashvili limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE]. recursion relations and S-duality, JHEP 04 (2013) 039 [arXiv:1302.0686] [INSPIRE]. kernel and S-duality in N = 2 theories, JHEP 11 (2013) 123 [arXiv:1307.6648] [INSPIRE]. [46] M. Billo et al., Modular anomaly equations in N = 2? theories and their large-N limit, JHEP N = 2? theories (II): the non-simply laced algebras, JHEP 11 (2015) 026 theories with arbitrary gauge groups, arXiv:1602.00273 [INSPIRE]. [50] S.K. Ashok, E. Dell'Aquila, A. Lerda and M. Raman, S-duality, triangle groups and modular anomalies in N = 2 SQCD, JHEP 04 (2016) 118 [arXiv:1601.01827] [INSPIRE]. [51] A.-K. Kashani-Poor and J. Troost, Transformations of spherical blocks, JHEP 10 (2013) 009 [arXiv:1305.7408] [INSPIRE]. [52] A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: the case of SU(N ), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [INSPIRE]. JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE]. [53] A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, [54] W. He and Y.-G. Miao, Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory, Phys. Rev. D 82 (2010) 025020 [arXiv:1006.1214] [INSPIRE]. [55] W. He and Y.-G. Miao, Mathieu equation and elliptic curve, Commun. Theor. Phys. 58 (2012) 827 [arXiv:1006.5185] [INSPIRE]. [56] A.V. Popolitov, Relation between Nekrasov functions and Bohr-Sommerfeld periods in the pure SU(N ) case, Theor. Math. Phys. 178 (2014) 239 [Teor. Mat. Fiz. 178 (2014) 274] [57] W. He, Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations, Annals Phys. 353 (2015) 150 [arXiv:1401.4135] [INSPIRE]. [58] N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), (2009) [arXiv:0908.4052] [INSPIRE]. [59] R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE]. [60] F. Fucito, J.F. Morales, D.R. Paci ci and R. Poghossian, Gauge theories on -backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] JHEP 08 (2012) 046 [arXiv:1206.1696] [INSPIRE]. JHEP 05 (2013) 047 [arXiv:1212.4972] [INSPIRE]. theories, JHEP 10 (2013) 178 [arXiv:1307.6612] [INSPIRE]. HJEP07(216) for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE]. in Liouville eld theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE]. cells can tell us about LCFT, J. Phys. A 46 (2013) 494007 [arXiv:1307.5844] [INSPIRE]. [arXiv:1502.07742] [INSPIRE]. arXiv:1605.03959 [INSPIRE]. [hep-th/0509075] [INSPIRE]. [10] N. Nekrasov and S. Shadchin , ABCD of instantons, Commun. Math. Phys. 252 ( 2004 ) 359 [44] M. Billo , M. Frau , L. Gallot , A. Lerda and I. Pesando , Deformed N = 2 theories, generalized [45] M. Billo , M. Frau , L. Gallot , A. Lerda and I. Pesando , Modular anomaly equation, heat [61] J.-E. Bourgine , Large-N limit of beta-ensembles and deformed Seiberg-Witten relations , [62] J.-E. Bourgine , Large-N techniques for Nekrasov partition functions and AGT conjecture , [63] M. Beccaria , On the large -deformations in the Nekrasov-Shatashvili limit of N = 2? SYM , [64] F. Fucito , J.F. Morales , R. Poghossian and D. Ricci Paci ci , Exact results in N = 2 gauge [65] F. Fucito , J.F. Morales and R. Poghossian , Multi instanton calculus on ALE spaces , Nucl. [66] S. Shadchin , Saddle point equations in Seiberg-Witten theory , JHEP 10 ( 2004 ) 033 [67] M. Billo et al., Non-perturbative gauge/gravity correspondence in N = 2 theories , JHEP 08 [68] N.I. Koblitz , Introduction to elliptic curves and modular forms , Grad. Texts Math. 97 , Springer Science & Business Media, Germany ( 2012 ). [71] E. Perlmutter , Virasoro conformal blocks in closed form , JHEP 08 ( 2015 ) 088 [75] M.R. Gaberdiel and I. Runkel , The logarithmic triplet theory with boundary , J. Phys. A 39


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP07%282016%29066.pdf

Matteo Beccaria, Guido Macorini. Exact partition functions for the Ω-deformed \( \mathcal{N}={2}^{\ast } \) SU(2) gauge theory, Journal of High Energy Physics, 2016, 1-24, DOI: 10.1007/JHEP07(2016)066