A direct proof of AGT conjecture at beta = 1

The AGT conjecture claims an equivalence of conformal blocks in 2d CFT and sums of Nekrasov functions (instantonic sums in 4d SUSY gauge theory). The conformal blocks can be presented as Dotsenko-Fateev beta-ensembles, hence, the AGT conjecture implies the equality between Dotsenko-Fateev beta-ensembles and the Nekrasov functions. In this paper, we prove it in a particular case of beta=1 (which corresponds to c = 1 at the conformal side and to epsilon_1 + epsilon_2 = 0 at the gauge theory side) in a very direct way. The central role is played by representation of the Nekrasov functions through correlators of characters (Schur polynomials) in the Selberg matrix models. We mostly concentrate on the case of SU(2) with 4 fundamentals, the extension to other cases being straightforward. The most obscure part is extending to an arbitrary beta: for beta \neq 1, the Selberg integrals that we use do not reproduce single Nekrasov functions, but only sums of them.


Introduction
One of the most recent instructive discoveries in string theory, the AGT conjecture [1] (see [2]- [21] for later progress) states an equivalence between conformal blocks in two-dimensional conformal field theory (with W N symmetry) on one side [22], and the LMNS instanton partition functions [23] in four-dimensional supersymmetric (SU (N )) gauge theory on the other. This relation is important for several reasons. Basically, it provides a very explicit (and rigorously formulated) realization of the string theory idea for a similarity between 4d supersymmetric and 2d conformal field theories, much more concrete than the standard AdS/CFT duality. Serving as a bridge between two different fields of research, the AGT relation stimulates progress in the both of them (say, activates the once abandoned studies of conformal blocks of W N -algebras). It also provides [19] an advanced version [6] of the well-known correspondence [24] between 4d effective low-energy actions (Seiberg-Witten prepotentials [27]) and integrable systems (often formulated in terms of 2d bosons and fermions).
Remarkably, apart from the two initial branches of physics connected by the AGT relation, there is still another, third field of research, which gets naturally involved: the theory of matrix models [7,12,11,15,13]. This was, of course, expected from the very beginning that matrix models belong to the same level of complexity as Seiberg-Witten prepotentials, their partition functions are long known to provide solutions to classical integrable hierarchies [28], etc (see [29] for an exact correspondence between matrix models and Seiberg-Witten theory). Nowadays these expectations turned into a very clearly formulated statement: that matrix models provide explicit integral representations for the conformal blocks. To be more precise, these are integral representations of conformal blocks B(q) in the non-trivially interacting 2d CFT in terms of correlators with screening charge insertions in the free field 2d CFT a-la Dotsenko and Fateev Figure 1: Feynman-like diagram for the 4-point conformal block. The external legs represent primary fields in 2d CFT; the structure of the graph shows the order of contractions in the operator product expansion procedure. [30]. A representative example is the four-point spherical conformal block (related to SU (2) Nekrasov function with four fundamental matter hypermultiplets) [13]: : e bφ(z) : dz (1) whereα i = α i /2b and β = b 2 . The second line is the correlator of normally ordered chiral vertex operators, corresponding to the initial four external fields and additional N 1 + N 2 screening charges, inserted in positions z 1 , . . . , z N 1 +N 2 and integrated with peculiar choices of integration contours. Such correlators are free field (Gaussian) averages, straightforwardly evaluated with help of the Wick theorem : eα 1 φ(z 1 ) : . . . : eα mφ(zm) : = 1≤i<j≤m (z j − z i ) 2α iαj (2) and finally put into the form of multiple integral (1) similar to matrix model eigenvalue integrals. For a generic β = 1, determined by the value of screening charge b and related to the central charge via c = 1 − 6(b − 1/b) 2 , the integral is not, strictly speaking, an ordinary matrix model, it is rather a generalization known as β-ensemble [31,20] or "conformal" matrix model [32,15]. The difference, however, is not too drastic: it is well-known that matrix model theory is easily generalizable from β = 1 to arbitrary values of β, see [20] for a recent summary. There are many different conformal blocks classified by the three main characteristics: a) conformal diagram, i.e. a graph with external legs, which shows the order of their OPE contraction; b) genus of underlying Riemann surface and c) rank N of the symmetry, which is N = 2 for the usual Virasoro conformal blocks, and higher N for conformal blocks of W N algebras. For all of them, the Dotsenko-Fateev integrals can be straightforwardly written: extra internal dimensions are described by adding screening operators with different integration contours [12,13]; higher genera surfaces are described by substitution of free field Green functions by appropriate theta-functions [7,17]; higher rank symmetries are described by making α's and b's vector-valued [32]. Because of this, and also because of their natural simplicity, it is convenient to use the Dotsenko-Fateev integrals to represent the whole variety of conformal blocks in the left hand (conformal) side of the AGT conjecture. This is exactly what we do in the present paper: we use for the conformal blocks the matrix model Dotsenko-Fateev representation [7,12,11,15,13].
On the other (gauge theory) side of the AGT conjecture, there are Nekrasov functions, the ultimate outcome of evaluation of integrals over the instanton moduli spaces in N = 2 SUSY Yang-Mills theories [33]. Since integrals over instanton moduli spaces typically diverge, they need to be regularized, and this, as usual, can be done in many different ways. One of the most popular ways to regularize these integrals [23] relies on introduction of the so-called Ωbackground and associated deformation parameters ǫ 1 , ǫ 2 . The integrals over moduli spaces, regularized in this way, were evaluated in [33] and finally represented as series in instanton parameters, with all terms explicit.  (2) Nekrasov functions with four fundamentals. The external boxes represent the matter hypermultiplets, the central circle represents the gauge group, and the structure of the graph shows transformation properties of the matter hypermultiplets under the gauge group action.
There are many different types of Nekrasov functions, classified according to quiver diagrams [34,1], i.e. graphical representations of the field content of a given theory with detailed indication of gauge groups and transformation properties of the matter multiplets. According to the AGT conjecture [1], each of these types of Nekrasov functions corresponds to a conformal block: the conformal diagram can be simply read off from the quiver diagram, with genus corresponding to the number of loops and with the symmetry (Virasoro or, generally, W N ) fixed by rank of the gauge group (SU (2) or, generally, SU (N )). Such a "dictionary" between 2d and 4d theories extends the one, originally suggested in [24,25,26], and represents one of the most explicit manifestations of the gaugestring duality over the last decades.
It is natural that, apart from generalizations and possible applications, more and more attention is getting attracted to the questions of understanding and proof of the AGT conjecture. The understanding of the otherwise mysterious connection between 2d and 4d theories is generally believed to be based upon existence of a certain unique 6d theory, which is in charge (through compactification) of the AGT relation. However, due to technical complications this direction remains largely philosophical, and has been unable to produce a proof yet.
Since the AGT relation is essentially the equality between the Nekrasov functions and Dotsenko-Fateev integrals, a more concrete approach could be to make use of the well-developed methods of matrix models for the proof. Several suggestions have been proposed on how to deal with the Nekrasov functions within the matrix model framework [7,12,11,15,13].
In [20] in order to proof the AGT conjecture in a more concrete way, we suggested to use that the Nekrasov functions are ǫ 1 , ǫ 2 -deformations of the celebrated Seiberg-Witten prepotentials, and the corresponding Seiberg-Witten theory coincides with the Seiberg-Witten theory of the planar limit of the Dotsenko-Fateev matrix model [7,12,11]. Then, one may restore the ǫ 1 , ǫ 2deformations of the both Seiberg-Witten theories by the topological recursion [35], so that they still would coincide, with the Seiberg-Witten differential in the recursion being given by the exact 1-point resolvent of the matrix model (or, more precisely, of the β-ensemble). Another possibility is to use the Harer-Zagier recursion [36,20]. However, at the moment too little is known about matrix model representation of the Nekrasov functions, thus, this program remains to be accomplished. Development of the Harer-Zagier technique may play an important role here.
In this paper, we suggest to look at Nekrasov functions literally: as explicitly known sums over partitions (Young diagrams, see Figure 3).Such series are indeed available for various Nekrasov functions in the literature [33,37]. For example, in the case of SU (2) with four fundamental hypermultiplets (related to the 4-point spherical conformal block (1)) the Nekrasov function can be written as a sum , with the coefficients N A,B being rational functions of the masses µ 1 , µ 2 , µ 3 , µ 4 , of the Coulomb parameter a and of the deformation parameters ǫ 1,2 . Explicitly, several first coefficients N AB have the form and so on (omitting the trivial N [][] = 1). For an explicit formula for the generic N AB , see (46). This "sum-over-partitions" point of view allows one to make a direct contact with matrix models, where these sums appear after character expansion [38]: decomposition of the integrand in a proper basis of symmetric polynomials. At β = 1, the proper basis is realized by the ordinary Schur polynomials, i.e. by the GL(∞) characters, which are labeled by partitions: etc., where p k = z k i are the power sums. For β = 1, the proper deformation of the Schur polynomials is the Jack polynomials (aka β-characters), which depend on p k and on a single additional parameter β. Further deformations, to the McDonald and, generally, Askey-Wilson polynomials, depend on more additional parameters and are relevant for description of 5d [9] and, perhaps, 6d gauge theories. Figure 4: The core idea of the proof. This is a typical duality, when one unifying structure To interpret series (3) as a character expansion, one needs to express the Nekrasov coefficients N A,B through the Schur polynomials. In this paper, we describe a solution to this problem for β = 1, which corresponds to the case of ǫ 1 + ǫ 2 = 0 for the Nekrasov function (the minus in the argument of the Schur function corresponds to the transposed Young diagram, see (77)): where the brackets + , − denote averaging over two Selberg ensembles with After one substitutes (13) into (3) and takes the sum over A, B, the characters re-combine in the way that precisely reproduces the Dotsenko-Fateev integral (1), where the above two Selberg ensembles correspond to the two groups of variables z i with i ≤ N 1 and i > N 1 , respectively (N ± ≡ N 1,2 ). Thus, the AGT relation for β = 1 is derived through the character expansion of the Dotsenko-Fateev integral, and can be interpreted as duality (as illustrated in Figure 4).
It is tempting to generalize identity (13) to β = 1, i.e. to ǫ 1 + ǫ 2 = 0. Naively, one just has to substitute the Schur polynomials by the Jack polynomials 20) and change the power β of the Van-der-Monde determinant in the definition of the Selberg averages. However, this naive β-deformation fails to reproduce the Nekrasov coefficients N AB ǫ 1 , ǫ 2 . In our opinion, the basic reason for the discrepancy is that N AB (if considered as a rational function of a) has a very special structure of poles, which accidentally coincides with that of Selberg integrals at β = 1, but for generic β is not captured by the Selberg integrals. To reproduce N AB ǫ 1 , ǫ 2 for generic ǫ 1 , ǫ 2 , some clever deformation of the r.h.s. of (13) is required. Clarifying this point would complete the direct proof of the AGT conjecture.
This paper is organized as follows.
Section 2 is devoted to the simple case of the AGT relation for pure SU (2). Consideration of this simple case helps to elucidate some of the important details of the story. We describe the conformal block (as the Dotsenko-Fateev integral), the Nekrasov functions (as explicit sums over Young diagrams) and state the AGT relation between them. Then, using a pure gauge version of the pair-correlator identity (13), we derive the ǫ 1 + ǫ 2 = 0 Nekrasov function from the β = 1 Dotsenko-Fateev integral.
Section 3 similarly deals with the AGT relation for SU (2) with four fundamental matter hypermultiplets. We describe, with the help of the pair-correlator identity (13), the analytical proof of equality between the ǫ 1 + ǫ 2 = 0 Nekrasov function and the β = 1 Dotsenko-Fateev integral, for arbitrary values of masses.
Section 4 is devoted to analysis of the problems, which arise when one attempts to generalize our construction to generic β.

Section 5 is the Conclusion.
The Appendix is a list of various known factorizable 1-character (Jack) and 2-character (Jack) averages in the Selberg and BGW matrix model (β-ensemble) theories, for β = 1 and β = 1, which can also play a role in the future investigations.
2 The case of pure SU (2)

Nekrasov function
The Nekrasov sum over partitions in this case has the form where g A,B denote the contributions of gauge fields into the Nekrasov function (23) which has a characteristic form of a product over all the cells of the Young diagram. For the arbitrary Young diagram Y , the symbols Arm Y (i, j) and Leg Y (i, j) denote the arm-length and leglength of the cell (i, j) in the diagram Y . Algebraically, these lengths are given by the expressions Figure 5: For the cell (i, j) = (2, 2), the armand leg-length are shown in black and grey, respectively. Note that the cell can lie beyond the diagram.
where Y ′ stands for the transposed Young diagram. This algebraic definition is not quite transparent: more enlightening may be the graphical meaning of these quantities, which is shown at Figure 5. Several first Nekrasov coefficients for pure SU (2) have the form Comparing with eqs.(4)- (9), one can see that the case of pure SU (2) can be obtained from the more general case of SU (2) with four fundamental matter hypermultiplets by a particular pure gauge limit (PGL) (note that various ǫ 1,2 -dependent factors emerging in the Nekrasov functions in PGL are completely determined by the way one takes this limit): As one can see, in this limit the Nekrasov functions get simplified. The conformal block in this limit is also simplified [4,18] and coincides with the PGL of the 1-point toric block [17].

Dotsenko-Fateev integral
As explained in [18], the relevant Dotsenko-Fateev integral can be obtained by taking the PGL of the initial integral (1). The result is somewhat non-trivial [18]: where the averaging goes over two independent β-ensembles (labeled with the symbols +, −) with the β-deformed Brezin-Gross-Witten (BGW) partition function [39] in the role of integrand and with δ = (β − 1)/β. It is checked in the same paper [18] that Dotsenko-Fateev integral (32) reproduces correctly the first terms of the Λ-expansion of the conformal block.

The AGT conjecture
The AGT conjecture states that under the following identification of parameters: Let us prove this statement in the case of β = 1.

Proof of (36) at β = 1
We start from rewriting the determinant in eq.(32) in the exponential form: Therefore, the Dotsenko-Fateev integral takes the form To expand this expression in characters, one can use the standard Cauchy-Stanley identity where the sum is taken over all Young diagrams R and j R are the normalized Jack polynomials, j R = J R /||J R || (see the Appendix for the details) which at β = 1 coincide with the ordinary Schur polynomials: The exponent in (39) contains −2β instead of +β; thus (40) is not directly applicable, instead one can use a trick: rewrite it in the following form: At β = 1, the r.h.s. is precisely the Nekrasov function, due to the pair-correlator identity (83): which is the pure gauge limit of (13) and is considered in more detail in the Appendix, see eq.(83). Substituting this into (42), one obtains and this completes the proof. It may even seem that the only non-trivial part of this calculation is the pair-correlator identity (43). However, the identity itself is nothing but a technical detail. Really important is a duality: the existence of the quadrilinear character expansion (42). Eq. (42) contains both a sum over A, B diagrams and an average over " + ", " − " ensembles, and reduces either to the Nekrasov function (43) or to the Dotsenko-Fateev integral (32) if one evaluates either the double average or the double sum, respectively. This is a typical duality, only realized at a very simple algebraic level with the help of characters. Let us now include masses into our consideration.
3 The case of SU (2) with 4 fundamentals

Nekrasov function
The Nekrasov function for this case has the form with coefficients where in addition to contributions (23) of gauge fields, one now has matter contributions: A few first Nekrasov coefficients N A,B are written in eqs.(4)-(8).

Dotsenko-Fateev integral
The Dotsenko-Fateev integral for this case has the form (1), but as was noticed a while ago [15], for the purposes of q-expansion it is more convenient to rewrite this integral (of course, omitting the U (1) prefactors, which are irrelevant for comparison with the Nekrasov functions) as a double average: where the averaging goes over two independent ensembles (labeled with symbols + and − ) of variables x 1 , . . . , x N + and y 1 , . . . , y N − ("eigenvalues" in matrix model terms) as follows: with the normalization constants needed to satisfy 1

The AGT conjecture
The AGT conjecture states that under the following identification of parameters: Let us prove this statement in the case of β = 1.

Proof at β = 1
The proof goes completely similar to the BGW case. Likewise, we start from rewriting the Dotsenko-Fateev integrand in an exponential form, and then use the Cauchy-Stanley identity to perform an expansion in the basis of Schur/Jack symmetric polynomials: where p k = i x k i and p k = i y k i . Therefore, the Dotsenko-Fateev integral takes the form At β = 1, the correlators at the r.h.s. precisely reproduce the Nekrasov function (see (79)): Substituting this into (57), one obtains which completes the proof.
The basic puzzle of the AGT relation for β = 1 is a different structure of poles at the two sides of the equality. The conformal block has poles at zeroes of the Kac determinant, i.e. at z = mǫ 1 + nǫ 2 with mn > 0, while the poles of the particular Nekrasov functions N AB (z) (here z = 2a) occur also at mn ≤ 0. Transition from the conformal blocks to the Selberg or BGW pair correlators of characters, exploited in the present paper, does not help: their poles are still at mn > 0, just as for the conformal blocks.
In this section, Figures 6-11   Because of the problem of poles, it is unclear if it is at all possible to extend the relations like (13) and (43) to β = 1. What happens at β = 1 is that only the difference m − n matters, and all the poles can be projected from the plane to a single line z = (m − n)ǫ 1 , and the difference between the sets with mn > 0 and mn ≤ 0 disappears. This phenomenon at ǫ 1 + ǫ 2 = 0 is illustrated in Figure 8.
Of course, the extra poles of the particular Nekrasov coefficients N AB (z) drop away from their sum, i.e. from the LMNS partition function, which is AGT-related to the conformal block.
Thus the real puzzle is, what at all is the real role of the individual N AB , i.e. why does the linear basis with the nicely factorizable coefficients (as functions of µ's and ǫ's) include extra poles in the z-variable. Anyhow, if N AB (z) are relevant, their Selberg or BGW interpretation is still missed when β = 1. In what follows we illustrate the problem at the first two levels of the Young diagram expansion. For this, in addition to explicit formulas for the Nekrasov functions in (4)- (9) and (25)-(30), one also needs explicit formulas for the pair correlators of the β-characters (i.e. the normalized Jack polynomials, see the Appendix). These are listed in Tables 2-4 and 7-9 for the BGW case. Actually we need only the entries of Table 3, the other two are added to illustrate factorizability properties and to provide some data for the future study of alternatives to eq.(13): clearly, instead of one could also use another types of correlators: or or any linear combination of the three. Tables 2 and 7 are devoted to correlators S AB , tables 3 and 8 to R AB , tables 4 and 9 to Q AB . Figure 10: Poles of N [2][] , N [11][] , N [1] [1] , N [] [2] and N [] [11] , respectively. Figure 9: Poles of the sum N [1][] + N [] [1] . The "extra" poles vanish.
Formulas for the Selberg correlators are more lengthy, but their properties are essentially the same, see Tables 5 and 6. Actual examples below are given for the simpler BGW case, i.e. relevant for the AGT relation in the pure gauge limit. [1]

Level 1. At level one, the relation looks like
The auxiliary poles, which are present at the particular Nekrasov coefficients, but disappear from the whole sum (see Figure 10) in this case are represented by a single pole at z = 0. Moral: the individual N 10 (z) and N 01 (z) can not be expressed through R 10 (±z) and R 01 (±z), but their sum can, as shown by relation (63). At ǫ 1 = −ǫ 2 , however, eq. (63) gets simplified: the l.h.s. sum (N + N ) and the r.h.s. sum (RR + RR) become equal term by term! This is illustrated by  Figure 11: Poles of the sum N [2][] + N [11][] + N [1] [1] + N [] [2] + N [] [11] . The "extra" poles vanish.
in the abbreviated notation (m, n) = (z − mǫ 1 − nǫ 2 ). The same phenomenon of transformation of a complicated equality between the whole sums at generic ǫ 1 + ǫ 2 into the very simple equality between individual terms at ǫ 1 + ǫ 2 = 0 persists at higher levels.
Higher levels. At higher levels, things get even more sophisticated. A new feature, which appears at this level of consideration, is that only the correlators S AB remain factorized, while the correlators R AB and Q AB at β = 1 contain non-factorizable expressions in numerators. Thus, it becomes impossible to illustrate the phenomenon by using the shorthand notation (n, m). However, the phenomenon itself does not change: at ǫ 1 + ǫ 2 = 0, the relation between the Nekrasov side and the Selberg side becomes termwise. Figure 12: The picture of Nekrasov functions/conformal block duality expressed by the Hubbard-Stratonovich type formula (67). The symbol z here denotes integration with the Selberg measure over variables z i , and the symbol A denotes summation over all Young diagrams A.

Conclusion
In this paper we succeeded in interpreting the AGT relation as the standard duality relation of the Hubbard-Stratonovich type, see Fig.5: The role of X a i is played by the GL(∞) characters χ A (p). This provides a very direct and conceptually clear proof of the AGT relation, very different from both the various formal proofs, suggested in [8,14,10,16,21], and more transcendental projects like [7,20] etc. Moreover, as a byproduct we found a new representation for the particular Nekrasov functions N AB through the pair correlators of characters in relevant matrix models (like the Selberg or BGW ones).
Unfortunately, all this works so nicely only in the particular case of β = 1. The extra poles puzzle at β = 1, which we described above in s.4, remains unresolved and generalization of the duality interpretation to β = 1 is still missed. No representation of the individual Nekrasov functions N AB (z) in terms of pair correlators of characters is found for β = 1: they simply possess more poles than the known correlators.
At the same time, generalizations in other directions: from 4-point to generic conformal blocks (at least, spherical and elliptic) and from the U (2)/Virasoro symmetry to U (N )/W N seem straightforward. In both cases polylinear, rather than bilinear combinations of pair correlators are going to arise.
The technical base of our consideration is the further generalization of the Selberg/Kadell formulas from single to pair correlators of characters in the Selberg and BGW models, given by eqs. (79) and (83) respectively. We did not describe a proof of these formulas, it is straightforward done within the standard approaches (e.g., from the singularity analysis to the Ward identities). What deserves to be mentioned, these correlators are different from another important set of correlators recently considered in [21] (see also the end part of the Appendix). The two most important differences are: (i) ours have poles, while the non-trivial part of those in [21] have only zeroes; (ii) ours remain non-trivial in the pure gauge limit (the Selberg correlators turn into the BGW ones), while the non-trivial part of those in [21] becomes trivial. An advantage of the correlators in [21] could be that they remain factorized for β = 1, just like our R AB (z), unfortunately, they are also not sufficient to describe the individual Nekrasov functions N AB (z) for β = 1.
To summarize, the AGT relation is now clearly understood in two limits: for c = ∞ [8], when conformal blocks become ordinary hypergeometric series, and for c = 1 when they possess the free fermion representation and, as we explained in the present paper, are related to the Nekrasov functions by the most naive duality transformation a la (67). An interpolation between these two extreme cases still remains to be found.

Appendix. Averages of Jack polynomials in β-ensembles
The present section lists various known averages of the Jack polynomials in various β-ensembles, which possess remarkably simple factorization properties. In this paper, we demonstrated that, in some cases, averages of the Jack polynomials are directly related to the Nekrasov functions. Further progress in understanding of these averages may lead to a complete reformulation of the Nekrasov functions in terms of matrix model (β-ensemble) theory.

β-ensembles
Here we consider just two β-ensembles. The Selberg averaging is defined as an integral The BGW averaging is most simply defined as PGL [18] of the Selberg averaging: or, equivalently, as the unitary integral average (35). Let us describe various averages of the Jack polynomials in these ensembles.

Selberg model
The average of single Jack polynomial in the Selberg model has the form where the following notation is used: This Kadell formula is proved in [40].

BGW model
The average of single Jack polynomial in the BGW model has the form This formula directly follows from the PGL of the Kadell formula.

Selberg model
The average of product of two Jack polynomials in the Selberg model is known in the form u, v, N) [ with the A, B-independent normalization constant Norm β (u, v, N ) determined from 1 = 1, and with the conventional Pochhammer symbol (x) β defined as As usual, A i in (74) denotes the height of i-th coloumn in the diagram A. The shift w = (v + 1 − β)/β is essential for the correlator to factorize. This formula is proved (at least for zero shift, w = 0) in [41]. Note that eq.(74) contains only the heights of Young diagrams, A i and B j , while the Nekrasov functions contain also the heights of transposed diagrams like A ′ . The transposed diagrams can be obtained by the following identity: which for β = 1 turns into β = 1 : and is used below in eq.(79).

BGW model
In the PGL only the denominator in the last factor survives in (74). The result can be written as where z = βn, and L A , L B are the maximal row lengths of the diagrams A and B: Case of β = 1 Formulas (74) and (78) can look similar to the Nekrasov ones, however, there is also an important difference: the heights of diagrams in denominator enter not in combinations like A i − j, which would correspond to Arm-and Leg-lengths, but rather in combinations like A i − βj: very much different from the Nekrasov side. Clearly, this difference disappears when β = 1, and above eq.(74) can be reduced to where the both sides are taken at β = 1, and the function G has the form Recalling that the gauge contribution to the Nekrasov functions has the form one can see that eq.(79) then directly implies that which is the main identity we use in section 3. Note that we wrote eq.(79), and its PGL eq. (83) below, in terms of χ A (−p) instead of χ A (p), because this is what we need to establish the relation with the DF integral. Eqs. (74) and (78) involve χ A (p), but because of eq.(77), there is essentially no difference at β = 1.
The PGL of the β = 1 case From above eq.(79) it follows that where the both sides are taken at β = 1. Eq.(81) then implies a BGW counterpart of (82): which is the main identity we use in section 2. BGW multiplication on Young diagrams A puzzling observation about these poles is that they always form a new Young diagram, in particular, there are no multiplicities. Zeroes lie over this newly emerging diagram (denoted A ⊙ B in what follows) but form a rather strange configuration. This issue will be discussed elsewhere, here we just illustrate it with the two particular examples: where z-independent normalization prefactors are omitted. Poles (white cells with dots) and zeroes (grey cells) of these correlators are shown in Figure 13. Note that the poles form a new Young diagram, a property not shared by the Nekrasov functions, and the same happens for all choices of A, B. Thus, the BGW averaging allows one to define a new amusing commutative "multiplication" A, B → A ⊙ B on Young diagrams. Moreover, this multiplication seems to be associative! It does not, however, preserve the size-grading: |A ⊙ B| ≥ |A| + |B|.

Alternative 2-Jack average
Selberg model A somewhat similar, still different 2-Jack correlator was recently considered in [21]. The main difference is the inverse powers p −k = z −k i in the second argument: Note that G AB functions appear here not in the denominator, but in the numerator, thus, the r.h.s. of (87) has no poles. The proportionality coefficient in (87) depends only on A and B independently: it is equal to In result, eq.(87) is of less direct use for the purpose of AGT proof than our eq.(79), however, ref. [21] suggests a more involved project with the use of this formula.

BGW model
The r.h.s. of (87) depends not on the BGW variable z = βn = u + v + 2βN + 1 − β, which is the only combination left finite in this limit. Thus, the r.h.s. of (87) becomes trivial in the PGL.