Brezin-Gross-Witten model as"pure gauge"limit of Selberg integrals

The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (beta-ensemble) representations the latter being polylinear combinations of Selberg integrals. The"pure gauge"limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.

PGL in the matrix model formulation. Fortunately, matrix model (i.e. the Dotsenko-Fateevlike β-ensemble) representations are already known both for B (0) [12,51] and for B (1) [46]. The first one is represented [36] as an AMM decomposition [59,61] into two spherical Selberg integrals (2) while the second one is the single, but elliptic (toric) Selberg integral Then, eqs.(1), (2) and (3) imply that (1) * (A|Λ) (4) and this is the straightforward matrix model formulation of eq.(1). The partition functions Z (0) * ± and Z (1) * denote here what we call the PGL of the corresponding Selberg integrals. To describe this PGL, explicit expressions for the Selberg integrals are needed. Note that the second equality in (4) automatically provides an AMM decomposition formula for Z (1) * into a pair of the Z (0) * models, even before explicit expressions are discussed.
Parameters of Selberg integrals. To study the PGL in terms of the Selberg integrals, one needs to describe clearly the values of their parameters. In the spherical case [12,51], they are given by and, in the toric case [46], they are given by where the α-parameters are related to the initial ∆-parameters (conformal dimensions) via and is the central charge. Here it is convenient to write all these formulas in terms of the Nekrasov parameters ǫ 1 , ǫ 2 , which are in one-to-one correspondence with the matrix model parameters β (the power of the Vandermonde determinants) and g s = g (the "string" coupling constant, aka the genus expansion parameter): or vice versa This completes the list of relations between the parameters, and allows one to look at the PGL of the Selberg integrals. This limit is simple in terms of the external dimensions: α ext −→ ∞, α 4 ext q = fixed = Λ 4 while in terms of the matrix model parameters it gets more sophisticated. We will now describe the limit in terms of N, u, v (for the spherical Selberg models) and of N, A (for the elliptic model).
PGL of Selberg integrals. Relations (7)- (8) imply that, in the PGL, the parameters u, v, N of the spherical Selberg integrals all tend to infinity. However, the same relations indicate that a particular combination of parameters, that is, u + v + 2βN remains finite in the PGL, since it does not depend on the external dimensions. We find it most convenient to parametrize this combination by a single variable n which is equal to for our "+" and "-" Selberg models. Consequently, the PGL for the spherical Selberg model looks like a non-trivial double scaling limit, where the parameters u, v, N tend to infinity as where distinguished combinations (uN + βN 2 ) + and (uN + βN 2 ) − in the PGL are proportional to µ 1 µ 2 and µ 3 µ 4 , respectively (µ's are mass parameters of the Nekrasov function, linearly related to α's). This particular rescaling of the time-variables t k → t k (uN + βN 2 ) −k is necessary to suppress a growth of correlators in the model: only with variables defined in this way, the partition function has a finite PGL. In particular, only with such a rescaling of variables the decomposition formula (2) remains non-trivial in the PGL (14) and, moreover, turns into formula (4), i.e.
Similarly, in the toric case, relations (9) imply that the PGL for the toric Selberg integral is where, since no time-variables are introduced, no additional rescalings are required. As one can see, the PGL's of the Selberg models are quite sophisticated: it is by no means transparent that eqs. (17) or (19) do at all have a finite limit. However, as we shall see below, they do, and the main problem is to give some constructive description of this limit. This paper is devoted to finding a (at least, partial) solution to this problem.
PGL of spherical Selberg: BGW model. As the first (simplest) part of solution to this problem, in this paper we demonstrate that Z (0) * , the PGL of the Selberg partition function Z S is actually the partition function of the (β-deformed) celebrated BGW model [62,63,61] of size n and in the character phase [63]: where the integral over U is the β-deformed unitary integral, and Vol β (n) is the β-deformed volume of the unitary group. As usual for the BGW model, the time-variables are identified with traces of the external field powers t k = tr Ψ k /k, and this brings us directly to the topic of β-ensembles with external fields, which is somewhat underinvestigated and not exhaustively covered in the existing (physical) literature. The point is that the β-deformations are usually defined for integrals of eigenvalues only. Whereas the notion of trace (of determinant, etc.) in (20) remains well-defined as combinations of the eigenvalues, the treatment of the external field term tr U Ψ in (20) deserves some comments. Actually one needs only the U -integrals (averages) of such quantities, they will be defined in s.2 with the help of a β-ensemble version of the Harish-Chandra-Itzykson-Zuber integral.
In this paper we prove eq. (20) in two independent, but complementary ways. After the βunitary BGW model is defined in s.2, in s.3 we demonstrate that its Jack expansion (the β-ensemble counterpart of the character expansion) coincides with the PGL of the Jack expansion for the spherical Selberg model. This is just an algebraic exercise, which still may be not too much transparent. A more conceptual way may be the method of s.4, where one instead takes the PGL of the Virasoro constraints (Ward identities) for the spherical Selberg model, and then shows that they coincide with the known Virasoro constraints [63,61] for the character phase of the BGW β-ensemble.
Formula (20) fully defines the middle part of formula (4). It still remains to explain how the time derivatives can be taken in the external field BGW model: the simplest possibility is provided by a Fourier-like transform in the β-character calculus, which is reminded in s.2.
PGL of elliptic Selberg: double BGW model. The second, harder part of the solution, taking the PGL of the elliptic Selberg model is not completely finalized in the present paper. The corresponding partition function Z (1) * (A|Λ) should be given by some β-ensemble with the partition function Note that the problem of finding the PGL of the elliptic Selberg model is just the same as finding the PGL B * (∆|Λ) of the conformal block/Nekrasov function. Having in mind the general context of the problem and lessons from the (successful) solution in the spherical case, one can go further in several directions.
One way to solve this problem is to attack it directly from the elliptic side, by trying to take the Inozemtsev limit [64] of the Ward identities for the elliptic Selberg model Z (1) * . This remains to be done.
Another way is to make a direct educated guess for what the β-ensemble in question should be. Such an attempt has actually been made long ago in [65]. The conjecture was that Z (1) * is again a BGW model, but this time in another phase: the double-cut DV phase, i.e.
This suggestion can be checked, for example, by a derivation of AMM decomposition of Z DV 2 BGW and comparison with the second equality in (4). This also remains to be done.
The third possibility is to apply the character calculus of [66], either to the elliptic Selberg integral itself or to the decomposition formula (4). This is what we do in the present paper, in s.5 below. Applying the character calculus to the decomposition formula (4), one obtains a double β-unitary-ensemble where ks k = tr U k , ks k = trŨ k and m ± = n ± +(β−1)/β. Thus, Z BGW 's in the integrand are actually functions of U andŨ , while their conjugates U † andŨ † enter through the mixing (intertwining) determinant. Note that the β-unitary integrals in (23) have sizes n ± = ±2a/ǫ 1 , while the BGW models in the integrand have sizes m ± = ±2α/ǫ 1 . As explained in s.5, this shift of sizes can be seen as a natural property of the Fourier transformation for the β-ensembles. Eq. (23) or, at least, the first terms of its Λ-expansion (21) can be checked in practice by substituting (20) into (23) and taking the remaining averages  (24) which is valid when f (U ) is an invariant function (depends only on eigenvalues or, what is the same, on traces of powers of U ). The averages, necessary to reproduce the first two orders in Λ, are Using them, one easily verifies that (23) reproduces (21), so that operationally the model is welldefined and gives correct results. However, there are many conceptual questions left, in particular, the relation of the model (23) to the Inozemtsev limit of the elliptic Selberg model and, most importantly, to the double-cut BGW integrals. If the DV conjecture (22) is true, the model (23) should probably be interpreted as an integral representation of Z DV 2 BGW . We do not consider this topic in the present paper, this will be done elsewhere.

Comments on the definition of β-deformed BGW model
Generalization of unitary integrals to β = 1 deserves comments. The problem is to give some concrete definition for the β-deformed unitary integral of the form which for β = 1 is the well-defined integral over the compact Lie group U (n) with the Haar invariant measure [dU ] β=1 . We are unaware of any similar group theory definition for generic β (occasionally, such a definition exists for β = 1/2 and β = 2 in terms of the groups O(n) and Sp(n), respectively). Instead, various other, more or less natural definitions can be suggested. The simplest case is when f (U ) is an invariant function, i.e. it depends only on the traces of powers of U or, equivalently, only on the eigenvalues of U . In this case, the most natural definition of β-generalization, motivated by consideration of the above-mentioned three group integrals (U (n) for β = 1, O(n) for β = 2 and Sp(n) for β = 1/2) is the following eigenvalue integral: i.e. the role of the β-deformation is just to have the power 2β of the Van-der-monde determinant. It is this definition which is most commonly recalled when the words "β-ensemble" are mentioned.
However, for the purpose of present paper this definition is not enough. The case when the integrand f (U ) depends only on the eigenvalues, does not cover all the eigenvalue models [69,54]. In particular, the main object of the present paper, which we use to describe the PGL of the Selberg integrals, has an integrand which is not a function of the eigenvalues of U only, it involves an "external field" matrix Ψ in the following way so that definition (29) is not applicable. Note that for β = 1 the integral Z BGW (Ψ) obviously depends only on the eigenvalues of the matrix Ψ. We can quite naturally assume the same property to hold for all β. After that, the two options still remain: to consider the integral as a function of traces of positive powers kt k = tr Ψ k or of negative powers kτ k = tr Ψ −k . As is well-known in the theory of the ordinary (β = 1) BGW model, these two choices lead to different results, commonly known as the character phase Z BGW c (t) and the Kontsevich phase Z BGW k (τ ), respectively [63]. In this paper we are interested in the definition of the first one, Z BGW c (t).
Our definition refers to the Harish-Chandra-Itzykson-Zuber (HCIZ) integral over the unitary To see this at β = 1, it is enough to diagonalize the matrices, is indeed a function of the eigenvalues {Ψ i } and {U j } only, and thus of t and s (the symmetry under permutation of the eigenvalues is obvious). Once Z IZ is defined, one can write which is now a function of the positive time-variables t. Moreover, the integrand also depends only on the eigenvalues of the integration matrix-variable U , so that definition (29) is applicable. Thus, to define Z BGW c (t), it suffices to define the Itzykson-Zuber integral (31) in some independent way. Such a possibility is provided by the character calculus [66]: for β = 1, the IZ integral can be defined as an expansion where χ R are the characters, and d R , D R are their values at particular points d R = χ R (t k = δ k,1 ), D R (n) = χ R (t k = n/k). The quantities d R and D R have an important representation theory meaning: they are dimensions of representations of the symmetric-and GL(n)-groups, respectively (labeled by the Young diagram R). Thus, for arbitrary β, one can naturally define where j R are the well-known β-characters (actually, the properly normalized Jack polynomials [67], reviewed in detail in Appendix 1 of the present paper) and d R , D R are their values at particular points d R = j R (t k = δ k,1 ), D R (n) = j R (t k = n/k). For arbitrary β, these quantities do not have any straightforward representation theory meaning, they can be only thought of as β-deformations of the dimensions of symmetric and general linear groups. Note that the n-dependence in formula (34) emerges only due to the n-dependent quantity D R (n). Eqs. (32) and (34) provide our constructive definition of the BGW partition function in the character phase. It should be emphasized, that in this way the β-deformation of any unitary eigenvalue model with no more than one external field in character phase can be defined. This is enough for our purposes here, but not enough in principle: to study the β-deformations of the Kontsevich phases or the β-deformations of the models with multiple external fields (known as non-eigenvalue models), some other definitions have to be invented. Hopefully, there exists a unifying framework for the β-deformations of any matrix model, not necessarily of the eigenvalue type. This framework remains to be discovered.
Given such a definition of Z BGW c β-ensemble, it is possible to study if it indeed coincides with the PGL Z (0) * of the spherical Selberg β-ensemble. The next two sections are devoted to two ways of proving this. The first method makes a direct use of properties of the β-characters j R , while the second method relies upon Ward identities for the β-ensembles.

Eq. (20) via Jack expansion
To check the equality Z (0) * = Z BGW c , we expand the both quantities in the basis of Jack polynomials.
The right hand side. For Z BGW c , eqs. (32) and (34) imply that To deal with the r.h.s. integral, one needs just two properties of the β-characters: the completeness and the Haar orthogonality where n is the size of U . The first identity (with kt k = tr (U † ) k , kt k = δ k,1 ) implies that and the second property allows one to perform the integration over U : The expansion obtained is of course a β-deformation of the well-known character expansion [66] Z BGW c (t) = but not quite a naive one (because of the non-trivial δ-shift in the denominator). Note that the matrix sizes enter the character expansions only through the explicit factors D R . Let us now derive a similar expansion for the Selberg partition function.
The left hand side. The Jack expansion for the Selberg β-ensemble using the same completeness condition (36) can be written as where the coefficients j R are averages of the Jack polynomials of the z-variables, Fortunately, the averages of Jack polynomials in the Selberg model are well-known to be simple quantities [70]. They factorize nicely, and can be generally expressed by the Kadell formula where notation [...] R (see Appendix 1) stands for Using this Kadell formula, it is immediate to take the PGL (15): As one can see, the correlators grow like (uN + βN 2 ) |R| in the PGL. This is actually the reason to introduce the corresponding rescaling of time-variables in the PGL of the partition function: Substituting here eq.(48), one finds what precisely coincides with (41). In this way, the PGL limit of the Kadell formula reproduces the BGW model: we conclude that Z (0) * = Z BGW c , since their Jack expansions are just the same. Let us now pass to the second method of proving this statement.

Eq. (20) via Virasoro constraints
Another way to see that Z (0) * = Z BGW is to study the PGL of the Virasoro constraints for the Selberg model. Just as any matrix model (or the β-ensemble), the Selberg model can be characterized by certain linear differential equations, which arise as a consequence of the reprarametrization invariance of the multiple integral (i.e. as the Ward identities). In the case of the Selberg model, these linear differential equations have the form Their derivation is given in Appendix 2. These equations completely determine the partition function of the Selberg model, therefore, one can take the PGL in these equations, not in the integral. In the PGL, all the three parameters u, v, N of the Selberg models tend to infinity in such a way that their combination u + v + 2βN is held finite. In this limit, the Virasoro constraints get simplified (many terms can be thrown out) and turn into for any k > 0, and these are precisely the Ward identities for integral (32). They can be found in [63,61], of course, only for the most popular case of β = 1. Despite there are no doubts that eqs. (52) hold for arbitrary β, in principle, it would be nice to derive them directly from the definition of the BGW β-ensemble (in the present paper, the role of such a definition is played by the character calculus). This remains to be done.

Eq. (23) via Fourier transform
Definition (41) can be directly used to convert the action of the intertwining operator in (4) into a kind of a 2-matrix BGW model. This can be done via the Fourier transform, widely used in the character calculus [66]. The basic and most important Fourier relation has the form where ks k = tr U k . This relation is a direct consequence of the completeness condition (36) and the Haar orthogonality (37): one substitutes (36) into the r.h.s. and calculates integral with the help of (37). Contracting both sides of this relation with the coefficients of the BGW expansion, one finds 54) and the sum in the r.h.s. is the BGW partition function again, only of a different size: This relation can be understood as a Fourier transform for the BGW β-ensemble. It is important for two reasons. First, it allows one to write t-derivatives of the partition function (correlators) as integrals: say, [dU ] β Vol β (n) Z BGW c n + δ s βtr U k 1 . . . βtr U km (56) whereas without the Fourier transform one could only write which is largely a symbolical notion: the integrand in the r.h.s. does not depend on the t-variables, only the integral does. Thus, the Fourier transform appears to be a convenient tool in this case. Second, the Fourier transform allows one to convert the decomposition operator into an integral. Indeed, we have Substituting each of the BGW partition functions by its Fourier transform, one then finds where m ± = n ± + (β − 1)/β are the shifted sizes. Such a shift looks like a natural property of the Fourier transforms for the β-ensembles. For β = 1, the shift vanishes.

Conclusion
In this paper, we have obtained two BGW-model-representations of the Selberg models (β-ensembles), elementary constitutients of the conformal blocks, in the pure gauge limit. The spherical Selberg model after the PGL turned out to become the character phase of the BGW model, while the elliptic Selberg model after the PGL is converted into some other model X. In this paper we succeeded in rewriting X as a double BGW model, but we still expect some simpler representations for this model. A possible candidate on the role of such a simpler representation is the double-cut BGW model, suggested long ago by Dijkgraaf and Vafa [65]. If this is correct, then the double BGW model of the present paper is yet another integral representation for this double-cut model.

Appendix 1. Properties of Jack polynomials
The Jack polynomials form an important class of symmetric polynomials, which are often useful in calculations with arbitrary β-ensembles, not only of the Dotsenko-Fateev type. For convenience, in this Appendix we list several basic formulas related to the Jack polynomials, which are well-known but scattered in the literature.

Jack polynomials
The Jack polynomials J Y are the polynomial eigenfunctionŝ of theŴ -like operator kms k+m ∂ 2 ∂s k ∂s m + β(k + m)s k s m ∂ ∂s k+m (61) normalized with a condition J Y = m Y + . . . (i.e. the coefficient in front of m Y is equal to unity). Explicitly, several first Jack polynomials with this normalization convention have the form Normalized Jack polynomials: β-characters Jack polynomials are orthogonal with respect to the following scalar product: which is often called intersection product. The expression for the norm is rather complicated: where P Y and Q Y are the two "Young normalization factors" which have a form of products over the cells (i, j) of the Young diagrams. In applications of the Jack polynomials to matrix models, it is often convenient to normalize them with respect to their norm: Such normalized polynomials j Y satisfy and, in general, formulas in terms of j Y are simpler than formulas in terms of J Y . However, the polynomials j Y themselves are more complicated and depend on β irrationally. Explicitly, several first Jack polynomials with this normalization convention have the form j 111 (s k ) = 6β 3 (2β + 1)(β + 1) Particular values of Jack polynomials d R and D R Frequently encountered quantities are values of the Jack polynomials at particular points s k = δ k,1 = (1, 0, 0, . . .) k and s k = n (equivalently, t k = δ k,1 and t k = n/k). The Jack polynomials have the following values at these points: where [...] Y is the so-called "Young-Pochhammer symbol" where Y is any Young diagram and Y ′ stands for its transposed diagram. Note that the Young-Pochhammer symbol is a generalization of the classical Pochhammer falling and rising factorials which simply correspond to the row and coloumn diagrams Y , respectively. For the normalized Jack polynomials j Y , the values under consideration are usually denoted through d Y and D Y , respectively: This implies, that which is useful to estimate the large-n behaviour of D Y (n).
The Jack polynomials are also orthogonal with respect to the integral product which is the β-deformed unitary integral. The norm is where Vol β (n) = [dU ] β , and J Y [U ] denote the Jack polynomials of power sums s k = a U k a of eigenvalues U a = e iφa . In terms of the normalized Jack polynomials, one has where δ = β − 1 β is an often encountered shift.
A useful formula is the Cauchy-Stanley identity, sometimes also called the completeness condition: Just as many other formulas, this expansion looks simpler in terms of j-polynomials.
The character expansion of the β-deformed IZ integral has a form The only origin of n-dependence lies in the explicit factors D R (n) (equivalently, [βn] R ).
The Carlsson-Okounkov identity has a form The r.h.s. of this identity has a typical structure of denominators of Nekrasov functions. 5d deformations.
All formulas in this Appendix possess direct generalization to the McDonald polynomials, which depend on two parameters. Generalizations of the IZ and BGW integrals can also be defined in this way. The McDonald case complements β by the "quantum-group" q-deformation and, therefore, is relevant for the AGT relation between 5d SYM theories and the conformal blocks of the "quantum" Virasoro algebra. For some recent work in this direction, see [16]. In principle, even further deformations and generalizations may exist, for example, to the Askey-Wilson polynomials [71]. The role of Askey-Wilson polynomials in the context of AGT relation remains to be understood.

Appendix 2. Derivation of Ward identities for the Selberg model
The Ward identities can be most simply derived by requiring the vanishing of the integral of full derivative: where L is an auxiliary parameter introduced for the purpose of derivation of the Ward identities (we put L = 1 in the end). Whenever possible, we substitute the zero indices of the correlators with appropriate powers of N , in order not to work with the t 0 variable. Differentiating the expression in the brackets, and using for k m > 0 the relations the same relations can be rewritten as differential equations known as generalized Virasoro constraints: This completes the derivation of the Virasoro constraints for the Selberg model. The trick (82) with insertion of a new dimensional parameter L does not work for the elliptic Selberg integral, as it does not work in the original Dotsenko-Fateev integrals: dimensionless parameters are present in the both cases. Still analogues of the Virasoro constraints in these both cases exist, they will be considered and analyzed elsewhere.