Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_{f}=4$ flavours

We consider the $\Omega$-deformed $\mathcal{N}=2$ $SU(2)$ gauge theory in four dimensions with $N_{f}=4$ massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters $\varepsilon_{1}, \varepsilon_{2}$, the scalar field expectation value $a$, and the hypermultiplet masses ${\bf m}=(m_{1}, m_{2}, m_{3}, m_{4})$. Motivated by recent findings in the $\mathcal{N}=2^{*}$ theory, we explore the theories that are characterized by special fixed ratios $\varepsilon_{2}/\varepsilon_{1}$ and ${\bf m}/\varepsilon_{1}$ and propose a simple condition on the structure of the multi-instanton contributions to the prepotential determining the effective action. This condition determines a finite set $\Pi_{N}$ of special points such that the prepotential has $N$ poles at fixed positions independent on the instanton number. In analogy with what happens in the $\mathcal{N}=2^{*}$ gauge theory, the full prepotential of the $\Pi_{N}$ theories may be given in closed form as an explicit function of $a$ and the modular parameter $q$ appearing in special combinations of Eisenstein series and Jacobi theta functions with well defined modular properties. The resulting finite pole partition functions are related by AGT correspondence to special 4-point spherical conformal blocks of the Virasoro algebra. We examine in full details special cases where the closed expression of the block is known and confirms our Ansatz. We systematically study the special features of Zamolodchikov's recursion for the $\Pi_{N}$ conformal blocks. As a result, we provide a novel effective recursion relation that can be exactly solved and allows to prove the conjectured closed expressions analytically in the case of the $\Pi_{1}$ and $\Pi_{2}$ conformal blocks.

ABSTRACT: We consider the Ω-deformed N = 2 SU(2) gauge theory in four dimensions with N f = 4 massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters ε 1 , ε 2 , the scalar field expectation value a, and the hypermultiplet masses m = (m 1 , m 2 , m 3 , m 4 ). Motivated by recent findings in the N = 2 * theory, we explore the theories that are characterized by special fixed ratios ε 2 /ε 1 and m/ε 1 and propose a simple condition on the structure of the multi-instanton contributions to the prepotential determining the effective action. This condition determines a finite set Π N of special points such that the prepotential has N poles at fixed positions independent on the instanton number. In analogy with what happens in the N = 2 * gauge theory, the full prepotential of the Π N theories may be given in closed form as an explicit function of a and the modular parameter q appearing in special combinations of Eisenstein series and Jacobi theta functions with well defined modular properties. The resulting finite pole partition functions are related by AGT correspondence to special 4-point spherical conformal blocks of the Virasoro algebra. We examine in full details special cases where the closed expression of the block is known and confirms our Ansatz. We systematically study the special features of Zamolodchikov's recursion for the Π N conformal blocks. As a result, we provide a novel effective recursion relation that can be exactly solved and allows to prove the conjectured closed expressions analytically in the case of the Π 1 and Π 2 conformal blocks.

Introduction and results
In this paper, we consider the four dimensional N = 2 SU(2) theory with N f flavour hypermultiplets in the fundamental representation of the gauge group. At the special value N f = 4 the one loop β-function vanishes and conformal invariance is broken by the flavour masses m = (m 1 , . . . , m 4 ). Due to N = 2 supersymmetry, the full effective action may be expressed in terms of the analytic prepotential F (a, m), where a is the vacuum expectation value of the scalar in the adjoint gauge multiplet [1]. Non-perturbative effects are present due to instantons and they are captured by the exact Seiberg-Witten (SW) solution [2,3]. The instanton expansion is weighted by the instanton counting parameter x which is a cross-ratio of the four roots of the original SW curve. In the following, it will be convenient to trade x by the modular parameter q defined by, see App. (A), where K is the complete elliptic integral of the first kind. Indeed, the quantity τ appearing in the parametrization q = e i π τ receives only corrections proportional to the hypermultiplet masses. It is the analogue of the complexified gauge coupling constant of the related N = 2 * theory whose matter content is a massive adjoint hypermultiplet (see [4] for a detailed recent discussion).
At finite values of ε 1 , ε 2 , the deformed partition function is a central quantity in the Alday-Gaiotto-Tachikawa (AGT) correspondence [31]. 2 This is a map between deformed N = 2 instanton partition functions and conformal blocks of a suitable CFT with certain worldsheet genus and operator insertions. AGT correspondence may be checked perturbatively in the number of instantons [34][35][36] and it has been partially proved in [37][38][39][40][41][42], see also [43]. For the N = 2 * ε-deformed SU(2) gauge theory the relevant CFT quantity is the one-point conformal block on the torus. In the case of the SU (2) theory with N f = 4 the generalized prepotential is related to the logarithm of the conformal blocks of four Liouville operators on a sphere with the bare gauge coupling constant being associated with the cross-ratio of the punctures where the four operators are inserted [44,34,45,35,[46][47][48][49][50][51]. 1 For the string interpretation of the Ω-background and its BPS excitations see [16][17][18][19]. The approach in these papers provides a geometric interpretation for the localization (in terms of a dilaton potential), and gives a clear understanding of all the possible deformation parameters (ε and mass terms). It also gives a way to write down an effective action that contains exactly the same information as the partition function. 2 A simpler setup is the Nekrasov-Shatashvili (NS) limit [32] where one of the two ε parameters vanishes.
The supersymmetric vacua of the deformed theory are related to the eigenstates of a quantum integrable system. In the case of the N = 2 * theory, the relevant integrable system is the elliptic Calogero-Moser system [32] and the associated spectral problem reduces to the study of the celebrated Lamé equation. If the hypermultiplet mass m is taken to be proportional to ε with definite special ratios m ε = n + 1 2 , where n ∈ N, the spectral problem is n-gap and major simplifications occur in the k-instanton prepotential contributions [33].
The AGT perspective suggests to inspect the modular properties that have been studied in the undeformed case [52,53] as well as for ε 1,2 = 0 [54,55]. It is possible to sum in closed form the instanton expansion order by order at large vacuum expectation value a → ∞. In more details, for the N = 2 * theory, the coefficients of the various (1/a) n terms are written in terms of quasi-modular Eisenstein series depending on q. Similar results are available for the N f = 4 theory by adding contributions involving the Jacobi theta functions ϑ 2 , ϑ 4 . This approach has been successful in the gauge theory [53,[56][57][58][59][60][61][62], in CFT language by AGT correspondence [48,49,63,50], and by a semiclassical WKB analysis [64][65][66][67][68][69].
In a recent paper [70], the Ω-deformed partition function of the N = 2 * theory is analyzed in the (α, β) plane where α, β are the ratios α = m/ε 1 (m being the hypermultiplet mass) and β = ε 2 /ε 1 . At each instanton number, the partition function is a rational function of the ratio ν = 2 a/ε 1 . Special points (α, β) do exist with the property that the k-instanton prepotential has poles at a fixed set of positions ν ∈ {ν 1 , . . . , ν N } independent on k. At these special N-poles points, the instanton partition function and the perturbative part of the prepotential take an exact closed form to be reviewed in Sec. (2). These explicit expressions satisfy the modular anomaly equation expressing S-duality formulated in [56][57][58][59][60][61]. Besides, by applying the AGT correspondence, it is possible to obtain particular toroidal blocks in closed form at specific values of the central charge c and of the inserted operator conformal dimension. Similarly, the perturbative part of the prepotential can also be given in closed form and provides interesting special cases of the 3-point DOZZ Liouville correlation function [71][72][73]. In [70], the complete list of all the N ≤ 4 poles points has been given, i.e. a set of 4, 7, 12, and 11 solutions at N = 1, 2, 3, 4 respectively. Recently, these exact partition functions have been analyzed from the perspective of the Zamolodchikov's recursion relation for toroidal and spherical conformal blocks in [74]. This approach reveals once more the non-triviality of the solutions. In particular, the all order resummation of the instanton expansion in terms of quasi-modular forms remains an open question from the purely CFT point of view.
Building on these results, in this paper we extend the analysis to the N f theory with four hypermultiplet in the fundamental representation. 3 Indeed, as we discuss in Sec. (3), the explicit results obtained in [56] and our previous experience in the N = 2 * theory suggest the possibility of obtaining again closed partition functions in terms of modular Eisenstein series plus additional contributions involving two Jacobi theta functions. To 3 A complementary analysis has been performed in [75] on a class of N = 2 gauge theories on S 4 with gauge group U(N), and N fundamental plus N anti-fundamental hypermultiplets. It has been shown that for a specific values of the masses (being linear combinations of ε i and a), the partition function is non trivial only at certain values of a so that the full partition function da|Z(a)| 2 is given in terms of a finite number of residues and can be explicitly evaluated by generalized hypergeometric functions. From the AGT perspective, these configurations are associated with four point correlators involving the insertion of a degenerated field. Our setup is quite different. Our hyper masses are proportional to linear combinations of the ε 1,2 parameters only and the instanton partition functions will be a non-trivial function of a. Using AGT, the associated conformal blocks will have a generic intermediate conformal dimension and well defined modular properties.
-3 -illustrate our results, we begin the discussion by presenting in great details two special one-pole and one two-poles partition functions. The first example has central charge c = 1 (in the dual AGT conformal theory) and it is discussed in Sec. (4). The mass and ε parameters are constrained by the relations An educated guess based on explicit high instanton number computation reads 4 This expression may be read as an all-order truncation of the results of [56] at the special point (1.3). To prove it, we can exploit AGT correspondence. As explained later, this amounts to the calculation of Virasoro conformal block in closed form with external conformal dimensions ∆ i = ( 1 16 , 9 16 , 9 16 , 1 16 ), and internal generic dimension ∆ = ν 2 4 . This block can be computed in the Ramond sector of a massless scalar field and has been fully worked out by Zamolodchikov and Apikyan in [76], generalizing the celebrated determination of the block of primary fields of dimension ∆ i = ( 1 16 , 1 16 , 1 16 , 1 16 ) and generic internal dimension ∆ in [77]. The result confirms perfectly the guessed expression (1.4).
We remark that the calculation in [76] is standard but somewhat laborious, in particular if one wants to extend it to other cases. It would be quite nice if it were possible to obtain results like (1.4) by directly solving the Zamolodchikov recursion for conformal blocks at special points like (1.3). This happens to be quite non-trivial, according to the analysis of [74] that is plagued by important technical problems. In particular, the special combination of quasi-modular forms in (1.4) is a fact that remains a conjecture within this approach -although it is true according to the monodromy approach of [76]. We solve these difficulties and provide a new effective recursion that allows to compute the block in closed form. Again, this may appear as a mere confirmation of the result in [76], but the advantage of the effective recursion is that it may be applied to other situations and prove more complicated expressions extending (1.4) to two-poles partition functions, as explained later. 5 We also emphasize that arbitrary conformal blocks at c = 1 are computed in principle by generic Painlevé VI tau functions [87], see also [88][89][90][91][92]. Special Painlevé solutions (of Riccati, Picard, Chazy and algebraic type) are in correspondence with specific blocks. For instance, the ( 1 16 , 1 16 , 1 16 , 1 16 ) conformal block is obtained from the Picard solution of [93] and it would be interesting to discuss the block associated with (1.4) in this perspective.
In the special deformed point (1.3), we show that it is possible to give also the perturbative part of the prepotential in closed form. Thus, we obtain the exact quantum prepotential as F(ε, a, m) = F pert (ε, a, m) + F inst (ε, a, m), (1.5) where the perturbative and instanton parts are respectively (Λ is a perturbative ultraviolet scale) The second example that we illustrate in details has central charge c = −2 and is discussed in Sec. (5). The mass and ε parameters are constrained by the relations Our guess is in this case and it matches several terms of the small instanton expansion. Now, the associated conformal block has external dimensions ∆ i = (− 3 32 , 5 32 , 5 32 , 21 32 ) and generic internal dimension ∆ = 1 8 4ν 2 − 1 . It may be realized in the Z 4 orbifold sector of the (η, ξ) ghost logarithmic CFT considered in [94]. Many 4-point functions are available for such theories, see for instance [95][96][97][98][99][100][101]. However, our case in (1.8), and with generic internal dimension ∆, is new. Nevertheless, we are able to compute it by applying the effective recursion mentioned above. The result is in full agreement with the Ansatz (1.8). As in the previous c = 1 case, the perturbative part can be computed exactly, and we are able to give a closed formula for the full prepotential (1.9) In the same Sec. (5) we also discuss a more complex example, where the scaling relations among the ε and the m lead to a two-poles Nekrasov partition function. While it is slightly more complex from the technical point of view, we can again provide an exact form for the partition function and the complete prepotential In Sec. (6.1), the previous examples are generalized and we provide a full list of onepole Nekrasov partition functions with their associated exact conformal blocks. They are -5 -characterized by special values of the 5 ratios α = m/ε 1 and β = ε 2 /ε 1 . The general form of the one-pole instanton partition function is (1.11) where with certain explicit coefficients c 1,2,3 (α, β). The relevant 1-pole sets Π 1 = {(α, β)} contain 29 elements, up to additional discrete transformations to be discussed later. In each case, the perturbative prepotential can be given in closed form and the full quantum prepotential turns out to be given by the following compact expression (this is (6.11) copied here for the sake of presentation) The prepotential (1.13) may be checked to pass a non trivial test, i.e. the S-duality modular anomaly equation [56][57][58][59][60][61]. All these solutions reduce to the massless N f = 4 theory in the undeformed limit ε → 0. Nevertheless, in the deformed theory, they are non-trivial exact special partition functions.
Finally, in Sec. (6.2), we present similar results for 2-pole solutions. In this case we find a set Π 2 with 74 elements. The instanton partition function takes again the form (1.11), but with a function H 2-poles ν (q) fully discussed in (6.13) and involving higher degree modular forms. Again, the perturbative prepotential can be computed in closed form and nicely combine with the instanton contribution.
To conclude, the expression (1.13) and the analogous ones for the two-pole solutions, see Sec. (6.2), together with the full data characterizing the sets Π 1 and Π 2 are the main result of our paper. Once again, we remark that they can be proved by applying the novel tool of the effective recursion relation discussed in Sec. (4.3).

Exact partition functions in the deformed N = * theory
In this section we briefly summarize the results of [70] for the Ω-deformed N = 2 * SU(2) gauge theory. Exact partition functions are found at particular values of the ratios where m is the mass of the hypermultiplet transforming in the adjoint representation. We can parametrize the scalar field expectation value by setting a = ν 2 ε 1 . The partition where we exploited the dimensional scaling indipendence to remove ε 1 and we use a tilde to emphasize the new variables. We also define which is the deformed prepotential that can be decomposed in its classical, perturbative, and instanton contributions The main claim of [70] is that there exists a finite set of N-poles points (α, β) such that the k-instanton prepotential is a rational function of ν with poles at a fixed set of positions This claim is definitely non-trivial and it has important consequences. At the special N-poles points, it has been shown that the instanton partition function and the perturbative part of the prepotential read, see App.
where M 2n is a polynomial in the Eisenstein series E 2 , E 4 , E 6 with modular degree 2n and coefficients depending on α, β, and ∆ m ∈ N. The quantity Λ is the Seiberg-Witten UV scale. The total prepotential is therefore remarkably simple and reads The properties of this result have been discussed in the introduction. Here, we just add that according to AGT duality, the instanton partition function is related to the 1-point Here, the conformal dimension of the internal operator of the block is 6 We remind that the torus conformal block F ∆ ∆ m (q) is the trace of the external primary O ∆ m over the descendants of O ∆ with a suitable q-dependent weight.
-7 -and gives the dependence on the vacuum expectation value a appearing in ν, see (2.2). Finally, the central charge is The four 1-pole solutions found in [70] are associated with the CFT data (c = 0, ∆ m = 2) and (c = −2, ∆ m = 3). The prediction for the conformal block is then with similar results for the 2-and 3-pole solutions. In the rest of the paper, we shall generalize such kind of results to the N f = 4 theory. Here, AGT predicts exact expressions for spherical conformal blocks with four external operators and an exchanged internal operator with generic conformal dimension. 7

Partition function and prepotential in the deformed N f = 4 theory
After the brief review of the N = 2 * case, we now move to the Ω-deformed N = 2 SYM theory with gauge group SU(2) and four flavour hypermultiplets in the fundamental representation. For non-vanishing masses of the hypermultiplets, the classical prepotential of the theory receives both perturbative (1-loop) and non-perturbative (instanton) quantum corrections where the instanton contribution is related to the Nekrasov partition function by the well known relation (1.2) that we repeat here for the reader's convenience The perturbative part instead can be expressed in terms of the generalized Barnes Γ functions [5,6]. The precise definition is the following. Given the function the perturbative part of the prepotential is Of course, in a definite CFT the internal operator is constrained by the fusion algebra, but the conformal block may be defined off-shell and as such it is the object appearing in AGT correspondence.
-8 -where we used the shorthand notation ε + = ε 1 + ε 2 8 . As anticipated in the introduction, our exact results for the Nekrasov partition functions (and consequently for the prepotential) will involve simple combinations of modular and Jacobi theta functions. Similar structures appear in the large a expansion of the prepotential analysed in [56]. We recall here explicitly some of the results of [56], since they will be useful in the following sections for a direct comparison. The complete quantum prepotential can be arranged in a series expansion in inverse powers of a of the form The logarithmic term comes from the perturbative part, and both the perturbative and instanton parts contribute to the coefficients h k . Rewriting the expansion (3.6) in terms of q instead of x, the authors of [56] managed to reconstruct the complete q dependence of the first coefficients h k , i.e. their results encode the complete all-instanton contribution. The result is that the functions h k can be expressed as combinations of modular forms E 2 , E 4 , E 6 and Jacobi theta functions ϑ 2 , ϑ 4 , with ε i and m f dependent polynomial coefficients. Explicitly, the first cases are where R, T 1 , T 2 and N are given by the following SO(8) invariant combinations of the masses m 8 We warn the reader that there are different conventions in the literature on the parametrization of the hypermultiplets masses, differing by a sign choice for m f and/or shift of the masses by (ε 1 + ε 2 )/2. Our conventions are the same, for example, of [56].
Starting from h 3 , the expressions involve also the modular form E 6 . Note that the combination of masses defined in (3.8) transform non trivially under the action of the modular group [3]. Taking into account the modular properties of the Jacobi theta functions, these transformations ensure that each h k transforms as a modular form of degree 2k, up to the anomalous term coming from E 2 , see [56] for details. Compared to the previous expansion, the meaning of the result (1.11) is that it is possible to choose special ratios m i /ε 1 and ε 2 /ε 1 in such a way that all the higher degree modular forms and powers of Jacobi theta functions in the prepotential are generated by the expansion of the logarithm of a finite number of terms in the partition function.

Instanton partition function
The instanton partition function in the Ω-deformed N = 2 SYM theory can be expanded order by order in the number of instantons according to As we discussed in the Introduction, it is convenient to trade x by the modular parameter q by means of the inverse of the relation (1.1) . (3.10) We will refer to the Z k in expansion (3.9) as the Nekrasov functions. At fixed instanton number k, the Nekrasov function can be calculated as a sum over pairs of U(k) Young diagrams (Y 1 , Y 2 ), such that the total number of boxes |Y 1 | + |Y 2 | = k [31]. To give a glimpse of the results and fix the notation, the simplest function Z 1 is given by the sum of the two terms Increasing k, the expressions for the Nekrasov functions become quickly cumbersome, but the structure of the Z k is always a rational function of the parameters, i.e., where P k (ε, a, m) and Q k (ε, a, m) are polynomials in all the variables with increasing degree as k grows.

Predictions from AGT correspondence
Before discussing AGT in this context, let us briefly summarize our notation for the conformal blocks and recall Zamolodchikov's recursion. Given a four point correlation function of primaries with dimensions ∆ i we can use a conformal transformation to set where C ∆ i ,∆ j ,∆ k are the 3-point structure constants. The function B is the conformal block and depends on the external dimensions ∆ i , the internal dimension ∆ and the central charge c of the CFT (we omit this obvious dependence). 9 It may be represented by the diagram and admits the following elliptic representation [102] (3.15) where the relation between x and q has been given in (3.10) and the function H obeys the Zamolodchikov's recursion relation [102]. Using the following parametrization for the conformal dimensions in terms of the λ i variables and defining the quantities (∆ m,n are the entries in the Kac table) 9 It is convenient to strip off the explicit power of x as in (3.14) so that the small x expansion of the conformal block start at one -11 -the recursion formula for the function H reads The coefficients R m,n appearing in (3.18) are defined in terms of the λ i , λ r,s introduced in (3.16) and (3.17) as follows Equation (3.18) is understood as a sum over the residues of the conformal block in the poles ∆ m,n . The products appearing in P m,n are exactly those needed to cancel the contribution of null-states at the level L = nm of the Verma module.

The AGT dictionary for the N f = 4 theory
The AGT correspondence states the equivalence between the Nekrasov partition function and a four point conformal block, identified through the following map between the gauge and CFT side parameters [31,34] 20) More precisely, the AGT statement is that the two functions Z inst (ε, a, m; x) and B(∆, ∆ i , c; x) are the same, using the above map, up to an overall U(1) factor (3.22) Following [34], it is convenient to rewrite the Nekrasov partition function in a form that mimics the elliptic form of the conformal block (3.15). For compactness, we introduce the definition of the three exponents (3.23) With these definitions, one can write [34], see (3.15), . (3.24) In the following sections we are going to analyze particular cases where the Nekrasov functions have a very simple pole structure. In this context, the recursion (3.18) has the clear advantage to make transparent the pole structure from the beginning.
The AGT dictionary is completed with the correspondence between the perturbative part of the prepotential and the DOZZ formula for the 3-point function in the Liouville theory, see section 4.1 of [103].

The Ramond sector of scalar CFT and a special point with c = 1
In this section we discuss a first example of a special point and we show how the partition function can be obtained in closed form. This is the point specified by the relations (1.3) that we repeat here for convenience Given the AGT map, the special point corresponds to the following set of external dimensions and central charge Alternative equivalent choices are possible because of the global SO(8) flavour symmetry of the gauge theory for N f = 4 and this symmetry is non trivial on the CFT side [104]. The choice in (4.2) is particularly convenient for the following analysis.

Perturbative contribution in closed form
Given the relations (4.1) the calculation of the perturbative part is straighforward. In fact for this special point, the perturbative part reduces to In particular, this result means that all the coefficients h k with k > 0 in (3.6) are completely determined by the instanton contribution only.

Instanton partition function at all orders
A direct, brute force calculation of the Nekrasov partition function leads to the following result, see Sec. (3.1) where the polynomials P k (ν) have degree 2k + 2. The first polynomials are Going to the function H according to (3.24), we see that the dependence on ν in the polynomials P k (ν) is fully removed by the first factor in the r.h.s. of (3.24), i.e. the power of the ratio x/(16 q) with a ν-dependent exponent. Including all the other factors appearing in the r.h.s. of (3.24), we obtain From this series one may guess the following exact q dependence in terms of E 2 , ϑ 2 , and Upon the substitution ∆ = ν 2 /4, the exact form for H in (4.7) turns into a prediction for the conformal block with conformal data as in (4.2). In other words, our conjecture and AGT imply the remarkable result In fact, as we mentioned in the Introduction, the relation (4.8) has been obtained by Zamolodchikov and Apikyan in [76] (see also [105]). They considered correlation functions in the Ramond sector of a free scalar field, which contains an infinite set of fields σ k -14 -with dimensions ∆ k = (2k + 1) 2 /16. From the analysis of the operator product expansion they derived a set of operatorial relations for the σ k fields, and using these relations they were able to obtain, algorithmically, exact expressions for the conformal blocks. They also pointed out the simplicity of the pole structure of the conformal blocks. Among the explicit result presented in the paper, the formula (4.23) of [76] It is straightforward to use the properties of Jacobi functions in App. (A) to show that (4.8) and (4.9) are equivalent.

The CFT side: a new method to derive exact conformal blocks
Exact expressions for the conformal blocks are rarely known. In this respect, the example (4.8) considered in this section is a notable exception. In the rest of the paper we will extend the list of exact predictions from AGT, and in all the cases we will find similar structures in terms of the functions E 2 , ϑ 2 , and ϑ 4 . On the other hand, the method adopted in [76] is in principle applicable to other cases, but rather laborious. It is then natural to investigate whether this particular example -where everything is well under control both on the AGT and CFT side -is characterised by some hidden structure that allows for an exact calculation by means of standard CFT techniques.
There are various approaches that allow the perturbative computation of the expansion of the conformal block in powers of the cross-ratio x or of the elliptic parameter q. To give an example, the coefficient B k of x k in the expansion of the conformal block can be derived using the so called hypergeometric recursion in the following form [106] where (a) n is the Pochhammer symbol, and the coefficients χ are functions of the central charge, external and internal dimensions. The χ coefficients can be computed recursively and quite efficiently as nested sums as explained in [106] (see [107] for a recent application based on this formalism). 11 The first non trivial cases, at the CFT data in (4.1), are , (4.11) Of course, using these χ , one can reproduce the correct expansion of the complete block, see ( Computationally the method is highly efficient, and one can push the above expansion to high order. On the other hand, it is hard to see any regularity or structure that may help in the search of the exact solution. From the previous discussion, these difficulties are somehow expected. Using the relations collected in appendix (A), one can rewrite the exact expression of the block using only the x variable as follows This form makes clear that the dependence on x of the block is rather intricate.

Effective elliptic recursion
A more promising strategy is provided by the elliptic recursion. Following the results presented in [70], this approach to the problem was investigated in [74]. The solution of the recursion is in principle straightforward. Starting from equation (3.18) and expanding the function H in series of q as one gets a recursive definition for the coefficients H k (∆)  The usual strategy, given the external dimensions, is to compute the R m,n leaving c generic, to avoid poles in (3.19), solve the recursion and finally take the limit c → 1. Following this -16 -route one recovers the correct series expansion, but the nested structure of the recursive solution (4.18) is still not very transparent. On the other hand, one knows that at the point under consideration the block has the quite simple dependence on ∆ as in If one starts at c = 1 from the beginning in order to possibly truncate the recursion using (4.16) in (4.15) the problem is that of the reducibility of Verma module. In other words, the equation ∆ m,n + m n = ∆ m ,n , (4.17) has integer solutions. This means that a naive application of (4.15) leads to undetermined ratios 0/0 preventing truncation, see [74].
Here, we propose another strategy that overcomes these problems and is able to provide the function H in (4.16) in closed form with little effort. To illustrate the basic idea let us consider the case of the R 2,2 term. As a function of c, in the special point one gets and clearly lim c→1 R 2,2 (c) = 0. Instead, working with c = 1, the arguments of the products P m,n and A m,n become 1 16 (r − s + 1) 2 (r − s + 2) 2 and k − l respectively. Taking into account the range of the indices, for R 2,2 we have a "0 2 " in the numerator from the pair (r, s) = (−1, 1), but also two factors vanishing in the denominator for (k, l) = (−1, −1) and (k, l) = (1, 1). The final 0 2 /0 2 expression is finite. This order of limits ambiguity has been clearly pointed out in [74] 12 . To evaluate practically the products with c = 1, we need some regularization. We propose the simplest possible prescription, introducing a shift δ in one of the indices of the products and then taking the δ → 0 limit at the end. Our effective definition of the R m,n coefficients is then based on the definitions with (in our special c = 1 point) At the end, we remove the regularization and compute the effective coefficients R eff m,n = lim δ→0 P m,n A m,n .
(4.21) 12 A similar problem is discussed in appendix B of the dissertation at the following link. A theoretical framework for this kind of problems has been recently formulated in [109].
-17 - The first advantage of this approach is that the coefficients R eff m,n can be easily computed in closed form. They read R eff m,n = − 2 16 m 2 (−1) m m 2 δ m,n , (4.22) and, as expected, they are non vanishing only for ∆ m,n = 0. Plugging the Ansatz (4.16) into (4.15) and using R → R eff , we get the effective recursion relation We can solve (4.23) for H as the ratio of two series The denominator is nothing but ϑ 4 , while the numerator is its q ∂ q . In this way one recovers our result in the form of eq. (4.9). Later, we shall show that this approach is general and powerful providing similar simple computation of the special exact conformal blocks appearing in our analysis.

Closed form of the full prepotential
Putting together the results for the perturbative and instanton parts we can now write the full prepotential F pert (ε, a, m) = 2 ε 2 1 log a Λ − a 2 log 16, (4.25) Expanding for a → ∞, and discarding all the a independent terms, we can compare directly with the the large a expansion (3.6). The term ( x 16 q ) −a 2 /ε 2 1 in the instanton part combines with the corresponding term proportional to a 2 in the perturbative part and gives the renormalization of the gauge coupling constant [56]. The check for the first coefficient h 1 is trivial. Moreover, equation (4.25) implies that all the coefficients h k with k > 1 must simply be powers of h 1 , up to the numerical coefficient arising from the expansion of the logarithm. This is indeed what happens. The second coefficient for a −4 arising from the expansion of (4.25) is which is in agreement with (3.6), evaluating h 2 at the special point and using the identity E 4 = ϑ 8 2 + ϑ 4 4 ϑ 4 2 + ϑ 8 4 . The same check can be performed with h 3 , and confirms again the pattern.

Logarithmic CFT and examples with c = −2
In this section we present other two examples of special points where the Nekrasov partition function has a finite number of poles and can be worked out exactly. They are characterized by the scaling relations, see (1.7) and respectively. For the first one, the Nekrasov functions have again a single pole, for the second choice two poles. Through AGT they are related to a logarithmic CFT with c = −2, as discussed in more detail below.
Before moving to the calculation of the instanton partition functions, we give here the result for the perturbative part of the prepotential. In both cases the calculation can be easily done exactly, and the result is with κ = 1 for the first special point, and κ = 2 for the second one. We remark that the perturbative part of the prepotential may be given in clean form in the undeformed massive limit, see for instance (2.16) of [59]. The simplicity of (5.3) holding for non zero deformation is remarkable and similar to the recent findings in [70].

A one-pole point
Given the scaling relations in (5.1), the AGT map leads to the corresponding set of external dimensions , (5.4) and the central charge of the CFT is c = −2. It is easy to recognize that the values in (5.4) appear in the extended Kac table of the logarithmic minimal model for critical dense polymers, usually denoted by L (1,2). This is the first (and best studied) example of a family of integrable logarithmic minimal model L(p, p ), with 1 < p < p and p, p coprimes, describing non-intersecting loops in a lattice. The CFT description corresponds to the Z 4 sector of the (η, ξ) ghost logarithmic CFT [94], usually referred as symplectic fermions [97]. The dimension − 3 32 corresponds to the ground state of the Z 4 sector. The states with dimensions 21 32 and 5 32 are obtained by acting on the ground state with a single creation mode of the η and ξ fields, respectively. To discuss our results, we follow the same steps of the previous example. The calculation of the Nekrasov partition function gives suggesting the following exact form To prove (5.9) by a direct CFT calculation we use the effective recursion presented in Sec. (4.3.1). Given the external dimensions and c = −2, the arguments of the products P m,n and A m,n are found to be The coefficients R m,n are calculated with the same regularization prescription discussed in Sec. .

Complete prepotential
Our results for the perturbative and instanton contribution to the prepotential are Summing them, we have again a closed form for F = F pert (ε, a, m) + F inst (ε, a, m; x). In this case the perturbative part contributes to all the h k coefficient, and combining the logarithms one gets the very compact expression As a further check, we have successfully tested (5.19) by comparing it with the general large a expansions, as we did in the previous example.
Removing as usual the prefactor, the resulting H has the expansion Despite the structure of the expansion is clearly more involved than in the previous cases, one can guess again a closed form where h 1 and h 2 are the coefficients in equation (3.7) evaluated at the special point. Explicitly they are: where the two functions H 1 and H 2 are given by the following combinations of E 2 and Jacobi theta functions: The strategy to prove our guess is again to use the effective recursion. In this case we have 28) and the effective R eff m,n coefficients are given by: Using the expressions for the R eff m,n and separating the contributions coming from the two poles, one gets, after some simple algebra, an equation for the two functions H 1 and H 2 . As in the previous cases this allows to write the H i as a combination of series. The result is slightly more involved than in the one pole cases and reads where the S j are the following series We do not attempt to put (5.30) in the form (5.27), but this equivalence may be checked at very high order in q by exploiting the explicit sums in (5.33). This proves in a very non-trivial way the power of the effective recursion approach.

Complete prepotential
As we did in the previous cases we can collect our results and give the complete prepotential in closed form. 6 Systematical study of one and two poles partition functions

The set Π 1 of one-pole partition functions
In the previous sections, we provided special one-pole partition functions showing how to extend to the N f = 4 theory the results derived in [70] for the N = 2 * theory. In general, one has a set Π 1 of special one-pole points (α, β) where α and β are the following fixed ratios In terms of the dual CFT, the ratio β determines the central charge, while α determine the conformal dimensions ∆ 1,2,3,4 of the relevant 4-point function, according to (3.20). Due to the symmetry of the partition function under a permutation of m, it is obvious that a simple permutation of the α i will lead to the same result for Z inst (but not to the same set -24 -of ∆ i in the dual CFT, see below). Moreover, one can check that if a set α gives a one-pole partition function, then any arbitrary sign choice for the α i still gives a one-pole Z inst that may be different from the initial one. The position of the pole is uniquely determined by value of the ratio β. For instance, with these definitions, the c = 1 case discussed in Sec. (4) belongs to the group of special points while the one-pole example in Sec. (5) is part of the group As stated above, if one is interested in the precise correspondence between a special point and its dual conformal block, the order of the α i is important, even if Z inst is invariant for permutations. Indeed, the external dimensions for the conformal block in the dual CFT, written as functions of α and β are, see again (3.20), and are not invariant for permutations of the α i . Looking again at the Π 1 point in (6.2), we see that external dimensions associated to the α = ( 1 2 , 1 2 , 1, 1) are ∆ = (0, 1 4 , 1, 0). Instead, the exchange α 2 ↔ α 3 gives ∆ = ( 1 16 , 9 16 , 9 16 , 1 16 ) that are the values we used in (4.2) to exploit the Ramond scalar CFT realization. The symmetry of Z inst ensures that the two conformal blocks have the same analytic form, as one can explicitly check order by order in the instanton expansion.
The examples (6.2) and (6.3) are part of a complete list that may be found in various ways, for instance by a brute force analysis of the Nekrasov functions or, alternatively, by the method suggested in [74]. We exploited these approaches, but we also devised a more interesting search algorithm that is based on the special simple form of F pert at the points Π N . As we observed in the illustrative examples and in the analysis of [70], the perturbative prepotential is essentially the sum of log(1 − ν 2 k /ν 2 ) over all pole positions ν k . This is a peculiar property of the points Π N and may be exploited to propose candidate solutions (α, β). This method is mandatory beyond the one-pole case and is described in App. (B). Each solution in Π 1 has input data (α, β) from which one can write the instanton partition function in the general form  (6.6) and the function H ν (q) can be written (the combination inside the square brackets can be identified with h 1 , in the large a expansion (3.6)) where ν 1 is the position of the pole, and with and The quantities in (6.6) and (6.9) are invariant under permutations. Different choices of signs of the four α i modify only X 1 and S 1,1,1,1 , i.e. the coefficient c 2 .
Before listing all the solutions, we complete the collection of formulas writing also the closed form for the perturbative part, which is given by Putting together the above perturbative part with the logarithm of the instanton partition function (6.5), one gets the complete prepotential. Note that the arguments of the logarithms nicely combine, and the resulting formula (discarding as usual all the a independent terms) is very compact In Tab. (1) and (2), we give the complete list of the groups of one-pole solutions. In particular Tab. (1) collects all the special points such that the single pole is already present in the one-instanton Nekrasov function Z 1 . For the special points in Tab. (2), Z 1 has no poles, and the single pole is present starting from the two-instanton partition function -26 -Z 2 . All these solutions have been further cross-checked with a high instanton explicit calculation. According to [74], modular transformation properties imply that the number of poles of a spherical finite pole block is given by the nice expression Consistently with our remarks, this expression depends on α through the combination S 2 and is thus independent on permutations and change of signs of α. One checks that the expression in (6.12) is indeed equal to 1 for the points Π 1 listed in Tab. (1) and (2). We also remark that the analysis of [74] shows that finite pole conformal blocks may only be found at minimal model values of the central charge, c = 1 − 6 (n−m) 2 nm , with n, m being positive coprime integers. This implies that β must be a negative rational number. Again, this statement is true in all the solutions we found.
As a final comment, we remind that after equation (3.8), we remarked that the h k combinations are quasi-modular forms, taking into account the transformation properties of the SO(8) invariant combinations of masses. Once the masses are fixed, as in the special points, this is of course no longer true. Nevertheless, from the explicit form of c 2 , one can note that there are only two possible outcomes for the prepotential within each group of special points. In fact, two distinct outcomes arise only for the special points in Tab. (1). For the special points in Tab. (2) the prepotential is unique for each group, since one of the α i is zero. It turns out that the T generator of the modular transformations maps the two outcomes into each other for each group in Tab. (1), while the single prepotentials for any group in Tab. (2) are T -invariant.

The set Π 2 of two-poles partition functions
Using the algorithm in Sec. (B), we can extend the analysis presented in the previous section and consider the complete list of special points Π 2 = {(α, β)}, leading to two poles solutions. Again, we are able to write a closed form for the instanton partition function for all the Π 2 special points. Of course, the complete Z inst is still of the form (6.5), so we write here only the function H ν (q), that for the Π 2 cases is given by (6.13) In the previous formula ν 1,2 are the positions of the two poles, H 1 is the same combination c 1 (α, β) E 2 + c 2 (α, β) ϑ 4 2 + c 3 (α, β) ϑ 4 4 defined in the previous section, see (6.7) and (6.8), and finally H 2 is given by where we introduced the additional quantities (extending the list in (6.9)) 16) In all cases, the perturbative prepotential takes the following natural closed form As in the case of 1-pole points, the two logarithms coming from F pert and F inst = −ε 1 ε 2 log Z inst combine nicely in order to give the following simplified form of the full prepotential The number of solutions Π 2 is 74, significantly larger than for Π 1 , so the result are collected in two tables, Tab. (3) and Tab. (4). All the Π 2 points have been tested successfully by comparing them with a direct calculation of Z inst at high instanton number. One finally checks that the quantity defined in (6.12) is equal to 2 in all cases.

A Special functions and useful identities
In this paper the relation between the modular parameters τ and q is q = e i π τ . Accordingly, the Dedekind η function reads The definition of the Jacobi ϑ 2,3,4 (q) functions as series expansions around q = 0 is  Table 4. Second part of the complete list of the two-poles special points. We list the groups Π 2 = {(α, β)}, the corresponding value of the CFT central charge and the position of the two poles.
If we define, see (1.1), , (A.5) B An efficient algorithm to find the sets Π N Let us illustrate an efficient algorithm that is able to locate special N-pole solutions with moderate computational effort. We examine in details the case of 1-pole solutions. Our starting remark is that the perturbative prepotential takes the special form at all special points. This is an extremely useful Ansatz to locate special points that can be ultimately cross checked by a dedicated high-instanton ( ≥ 20 ) computation. As a first step, we match the large a expansion of the perturbative contribution (3.4) with the Ansatz in (B.1). The comparison allows to express the sums Q n = ∑ 4 i=1 α 2n i with n ≥ 2 in terms of the three unknown quantities Q 1 , ν 2 1 , and β. For instance, we find and so on. On the other hand, we can exploit the Viète-Newton theorem [111] to build a polynomial P (x) with roots x = α 2 1,2,3,4 compatible with the constraints on Q n≥2 , see for instance (B.2). In particular, the coefficients of the terms 1, x, x 2 , x 3 , x 4 are fixed by Q 2,3,4,5 . Instead, the values of Q n≥6 must be compatible with the claim that P (x) has degree 4. These conditions impose algebraic constraints between the three quantities Q 1 , ν 1 , and β. For example, the condition that P (x) has no x 5 term reads with similar conditions from the vanishing of the x 6 and x 7 terms. The solution of this reduced problem is affordable and for each solution (Q 1 , ν 1 , β) we obtain a specific polynomial P (x) whose roots can be identified with α 2 i .