The light asymptotic limit of conformal blocks in Toda field theory

We compute the light asymptotic limit of An−1 Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of A2 Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions.


Introduction
Semiclassical limits play important role since they link quantum physics to the Lagrangian approach. In the Liouville and Toda field theories there are three semiclassical limits: mini-superspace [1][2][3][4], the light and heavy [4][5][6]. All three asymptotics are the large central charge limits. The difference comes in the treatment of the primary fields. In the minisuperspace limit one considers a limit where only the zero mode dynamics survives. In this limit the Liouville and Toda field theories reduce to the corresponding quantum mechanical problems [1,2,4]. In the light asymptotic limit one keeps the conformal dimensions fixed. Then the correlation functions are given by the finite dimensional path integral over solutions of the equations of motion with a vanishing energy-momentum tensor. And finally in the heavy asymptotic limit the conformal dimensions blow up, scaling as the classical action and correlation functions are given by the exponential of the action evaluated over the singular solutions.
To be more specific recall that primary fields in the Liouville and Toda field theories are related to the vertex operators V α = e iαφ . The spectrum is given by α = Q 2 + iP . In the light asymptotic limit we set α = η l b and keep η l fixed for b → 0, whereas in the heavy asymptotic limit we take α = η h b and hold η h fixed again for b → 0. In the minisuperpsace limit one should take for some of the vertex operators α = η m b and for some P = η m b.
These semiclassical limits were used in [3,4,6] to relate the quantum three-point functions in the Liouville and Toda theories with the corresponding classical actions. In papers [7][8][9] these techniques were generalized to the boundary Liouville and Toda theories.

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The heavy and light asymptotic limits were reconsidered in [10] also for complex solutions of the analytically continued Liouville theory. In [11] both limits were used in the Liouville field theory with defects.
Discovery of AGT correspondence [12][13][14][15] relating 2d CFT conformal blocks to the Nekrasov partition function [16,17] in N = 2 supersymmetric gauge theory provides powerful tools to investigate CFT correlators using gauge theory methods or alternativly to apply advanced CFT methods in gauge theory (see e.g. [18,19]. The essential point here is the fact that there are explicit combinatorial formulas for the Nekrasov partition function [20,21], which now can be successfully applied in 2d CFT. The heavy asymptotic limit in the AGT context was considered in [22][23][24]. In [25][26][27] the light asymptotic limit was used to test the AGT correspondence. Investigation of the light asymptotic limit of the Nekrasov functions for U(2) N = 2 gauge theory in [25,27] revealed a very interesting fact, that in this limit only the Young diagrams consisting of a single row contribute. This could be anticipated since in the light asymptotic limit the infinite Virasoro symmetry reduces to SL(2) algebra whose representations are classified with one row Young tableaux.
The sum over these one-row Young tableaux yields the Gauss Hypergeometric function, which is the well known light asymptotic limit of the Virasoro conformal blocks [28].
In [26] the AGT correspondence was tested in the light asymptotic limit between conformal blocks of A 2 Toda field theory and Nekrasov functions of U(3) N = 2 gauge theory. Namely some conformal blocks for A 2 Toda field theory were computed in this limit and the result was checked against the corresponding limit of the Nekrasov functions up to the 5-th order terms. But such an elegant picture as it was found for the Liouville field theory was missing.
In this paper we consider the light asymptotic limit of the U(n) Nekrasov partition functions for an arbitrary n. We find that for the certain choice of fields the Nekrasov partition functions in the light asymptotic limit are simplified drastically and given by the sum over Young diagrams having at most n − 1 rows. We compute the corresponding W 3 conformal block using the light asymptotic integral representation and found perfect agreement with the two-row Nekrasov partition functions. Note that in the light asymptotic limit the W n symmetry reduces to SL(n) group [26,29] and this already hints on the existence of the limiting procedure where survive only Young diagrams corresponding to the SL(n) representations.
The paper is organized as follows. In section 2 we compute the light asymptotic limit of the Nekrasov partition functions. In subsection 2.1 we review the necessary facts on the Nekrasov partition functions. In subsection 2.2 we review Toda conformal field theory and the AGT relation. In subsection 2.3 we explain the details on the light asymptotic limit and show that choosing the data as it is specified in eq. (2.18) and (2.19) truncates the Nekrasov functions in the light asymptotic limit to the sum over Young tableaux containing at most n − 1 rows. In subsection 2.4 we compute the Nekrasov partition function in the light asymptotic limit. The formulas (2.34) and (2.35) are our main result. In section 3 we compute the corresponding conformal block in A 2 Toda field theory using that in the JHEP05(2016)087 light asymptotic limit conformal blocks admit an integral representations. In appendix A the details of the integral calculation in section 3 are delivered. Consider N = 2 SYM theory with gauge group U(n) and 2n fundamental (more precisely n fundamental plus n anti-fundamental) hypermultiplets in Ω-background. The instanton part of the partition of this theory can be represented as where Y is an array of n Young diagrams, | Y | is the total number of boxes and z is the instanton counting parameter related to the gauge coupling in a standard manner. The coefficients F Y are given by Here A λ (s) and L λ (s) are correspondingly the arm-length and leg-length of the square s towards the Young tableau λ, defined as oriented vertical and horizontal distances of the square s to outer boundary of the Young tableau λ (see figure 1). Let us clarify our conventions on gauge theory parameters a (0,1,2) u , u = 1, 2, . . . , n. The parameters a (1) u are expectation values of the scalar field in vector multiplet. Without loss of generality we'll assume that the "center of mass" of these expectation values is zerō

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In fact this is not a loss of generality since a nonzero center of mass can be absorbed by shifting hypermultiplet masses. Furthermore a (0) u (a (2) u ) are the masses of fundamental (anti-fundamental) hypers. Finally the ǫ 1 , ǫ 2 are the Ω-background parameters. Sometimes we will use the notation ǫ = ǫ 1 + ǫ 2 .
Due to AGT duality, this partition function is directly related to specific four point conformal block in 2d A n−1 Toda field theory. Before describing this relation let us briefly recall few facts about Toda theory.

Preliminaries on A n−1 Toda CFT and AGT relation
These are 2d CFT theories which besides the spin 2 holomorphic energy momentum W (2) (z) ≡ T (z) are endowed with additional higher spin s = 3, 4, . . . , n currents W (3) , . . . , W (n) with Virasoro central charge conventionally parameterised as where the vector "background charge" with ρ being the Weyl vector of the algebra A n−1 and b is the dimensionless coupling constant of Toda theory. In what follows it would be convenient to represent the roots, weights and Cartan elements of A n−1 as n-component vectors with the usual Kronecker scalar product, subject to the condition that sum of components is zero. Of course this is equivalent to more conventional representation of these quantities as diagonal traceless n×n matrices with the pairing given by trace. In this representation the Weyl vector is given by and for the central charge we'll get where for the later use we have introduced the parameter For further reference let us quote here explicit expressions for the highest weight ω 1 of the first fundamental representation and for its complete set of weights The primary fields V α (here we concentrate only on left moving holomorphic parts) are parameterized by vectors α with vanishing center of mass. Their conformal weights are given by On the left: the quiver diagram for the conformal U(n) gauge theory. On the right: the diagram of the conformal block for the dual Toda field theory.
In what follows a special role is played by the fields V λω 1 with the dimensions: A four point block: (2.9) where α specifies the W-family running in s-channel, is closely related to the gauge partition function Z inst see (2.15) (AGT relation). First of all, the instanton counting parameter z gets identified with the cross ratio of insertion points in CFT block as it was already anticipated in (2.9) and the Toda parameter b is related to Ω-background parameters via (2.10) The map between the gauge parameters in (2.1) and conformal block parameters in (2.9) should be established from the following rules (see figure 2). To formulate them we define the rescaled gauge parameters • The differences between the "centers of mases" of the successive rescaled gauge parameters (2.11) give the charges of the "vertical" entries of the conformal block: (2.12) • The rescaled gauge parameters with the subtracted centers of masses give the momenta of the "horizontal" entries of the conformal block:

Light asymptotic limit
In this paper we are interested in so called "light" asymptotic limit i.e. the central charge is sent to infinity (i.e. b → 0) while keeping the dimensions finite. It follows from (2.7) that to reach this limit one can simply put keeping all the parameters η finite. As for the parameters λ of the special fields V λω 1 , there are two inequivalent alternatives: Though in both cases the conformal dimension takes the same value (see eq. (2.8)) these fields are not identical, which can be seen e.g. from the fact that the zero mode eigenvalues of odd W-currents for these fields have the same absolute values but opposite signs. In fact the fields V bηω 1 and V (nq−bη)ω 1 can be considered as conjugate to each other in the usual sense, since their two point function is non-zero. It is easy to check that V (nq−bη)ω 1 is equivalent to V bηω n−1 (ω n−1 is the highest weight of the anti-fundamental representation) since the corresponding momentum parameters Q − bηω 1 and Q − (nq − bη)ω n−1 are related by a Weyl reflection. In this paper we will investigate in great detail the case when V λ (3) ω 1 is a light field of type (i) while V λ (2) ω 1 is of type (ii). In other words we set nq − λ (2) = bη (2) .  For such choice we will see below, that the corresponding instanton sum simplifies drastically and leads to a simple explicit expression for the conformal block. Note that this choice is very convenient since the prefactor in front of conformal block in (2.15) now goes to 1 in the light asymptotic limit. The opposite case when two special fields are of the same type, has been investigated in [26] in particular case of A 2 Toda. In the case considered in [26] the above mentioned prefactor survives.
Coming back to our case of interest using (2.18), (2.16) we can rewrite the AGT map (2.14) as In view of (2.10) the small b limit is equivalent to ǫ 1 → 0. Hence we are interested in the ǫ 1 → 0 limit of (2.2). We will see that the degree of ǫ 1 (denote it by N ) is non-negative for arbitrary array of Young diagrams Y v and that the degree N = 0 (hence a finite non-zero limit exists) if and only if each Young diagram Y v (v = 1, 2, . . . , n) has at most v − 1 rows. From (2.2) we see that N = n 1 + n 2 − n 3 (2.20) with n 1 , n 2 being the ǫ 1 degrees of the first and second factors in the numerator of (2.2) respectively and n 3 is the ǫ 1 degree of its denominator. Let us derive n 1 , using A factor in (2.21) contributes to the degree of ǫ 1 if its part proportional to ǫ 2 vanishes. Evidently this happens when A Yv (s) = v − u. Since the box s ∈ Y v , A Yv (s) ≥ 0, we see that when v = 1 the only admissible value for u is u = 1. It is obvious from figure 3 that Obviously the degree n 1 is nothing but the sum of all entries of this table.
With almost identical arguments it is possible to show that n 2 = n 1 . Finally, an analogous consideration for the degree n 3 gives Each term here is non-negative and in order to get a vanishing total degree N = 0, the array of Young diagrams should satisfy the conditions Y 1,1 = Y 2,2 = · · · Y n,n = 0, which means that each Young diagram Y u consists of at most u − 1 rows.

Nekrasov partition function of N = 2 SYM theory in the light asymptotic limit
Now our purpose is to derive F Y explicitly in the light asymptotic limit. To do this let us study the first factor in the numerator of (2.2) which, according to (2.3) and (2.19), is given by

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Let Y (1) v be the set of such boxes s of the Young diagram Y v (with at most v − 1 rows) that the coefficient of ǫ 2 vanishes in the respective factor of (2.25)), i.e. v − u − A Yv (s) = 0 . (

2.26)
This can happen only when i ≡ v−u ≥ 0. Thus for the part of (2.25) under discussion we get We have already seen in previous section that there are exactly Y v,i+1 boxes, satisfying (2.26). These boxes are distributed in Y v in such a way that there is a single box on j-th column (denote it by s j ) for each j = 1, . . . , Y v,i+1 (see figure 3). Taking into account that L ∅ (s j ) = −j, we can rewrite (2.27) as Now let's look on the alternative case of the set Y (2) v of those boxes which do not satisfy (2.26) so that in the related factors we can safely set ǫ 1 = 0. Again from (2.21) we'll get

Combining (2.28) with (2.30) we obtain
Similar arguments for the second factor in the denominator of (2.2) lead to the expression The derivation of the denominator of (2.2) though somewhat lengthier but still is quite straightforward and leads to where the products on the second (third) line comes from the terms u < v (u > v) and the last line results in from diagonal u = v terms. Notice that, as we have already proved earlier, the order in ǫ 1 of the numerator and the denominator coincide safely providing a finite ǫ 1 → 0 limit. Also dependence of the ratio in ǫ 2 disappears (as it should from scaling arguments). Inserting (2.31), (2.32) and (2.33) in (2.2) for F Y in the light asymptotic limit we finally get where Y v,i is the number of boxes in the i'th row of diagram Y v . As we have mentioned already, with the prescription (2.18) of the field, in the light asymptotic limit the prefactor in (2.15) becomes 1, and hence, remembering also that the field V (nq−bη (2) )ω 1 is equivalent to V bη (2) ω n−1 , we can write The sum is taken over all Young diagrams Y u , u = 2, . . . , n, with at most u − 1 rows, i.e. over all allowed row lengths Y u,1 ≥ Y u,2 ≥ · · · ≥ Y u,u−1 ≥ 0. Let us consider the particular cases when n = 2 (Liouville) and n = 3 separately.

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When n = 2 we have a single sum a, b; c; x) is the Gauss hyper-geometric function, and we set in (2.36) η (4) ≡ η 1 and η ≡ η 1 . This is a well known result in Liouville theory [25][26][27][28]. When n = 3 we get This formula completes the result of [26] where the light four-point function of W 3 -theory has been computed in the case when both the second and the third insertions were light primaries of the same sort: whereas (2.37) is obtained with the choice specified in (2.18). In the next section we present an alternative calculation of (2.37) based on the integral representation of the conformal blocks in the light asymptotic limit used in [26].

Light asymptotic limit for the four point block in W 3
It has been shown in [26] that the multi-point conformal blocks of the W 3 theory in the light asymptotic limit can be constructed in the terms of sl(3) three-point invariant functions. For the details we refer the reader to the original paper. Here we'll introduce the necessary notations and state the relevant results. It is well known that the sl(3) generators can be represented as operators acting on the triple of the isospin variables Z = (w, x, y). To construct a multi-point block one should multiply several three-point functions then identify pairs of isospin variables corresponding to the internal states and integrate them out with an appropriate measure. At the end one specializes the external leg variables putting

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where z is the insertion point. In particular the light asymptotic limit of the four-point block of the same type as in (2.37) can be represented as: where E 1 and E 2 are the appropriate three point invariants given by 1 the quantities j = (r, s), j * = (2 − r, 2 − s), j ω = (s, r) (see [26]) specify the primary fields and are related to the charge vectors η u introduced in section (2.3) as and, finally, 3 ; 3 ; s 2 = η (2) ; r 2 = 0 ; As usual, using projective invariance we can specify the insertion points as (z 4 , z 3 , z 2 , z 1 ) → (∞, 1, x, 0), see figure 2. Under this specification, after dropping out an unimportant constant (infinite ) factor, E 2 gets simplified . (3.8) Putting

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as instructed in (3.1) and dropping out the usual factor z hα s −h α (1) −h λ (2) ω 1 = z rs+ss−(r 1 +s 1 )−s 2 , up to an unimportant constant multiplier we get the integral After the change of the variables we'll get Though this expression looks different from (2.37) below we argue that in fact they coincide. First we will prove this analytically up to the second order in z. It is convenient to rewrite (3.12) and (2.37) in terms of parameters A 1 , A 2 , B 1 , B 2 defined as (3.14) For (3.12) we will get (3.15) and (2.37) is given

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It is easy to see from (3.15) that the term proportional to z is and for (3.16) it is Combining the first two terms in (3.17) we will get (3.18). The details of the second order calculations can be found in appendix B.
Using Mathematica code we have checked up to the 8th order in z, that (2.37) agrees with (3.12).
Note that the expression (2.37), besides the physical poles at r s ∈ Z ≤0 , s s ∈ Z ≤0 , r s + s s − 1 ∈ Z ≤0 , has apparent poles at the positive integer values of s s . In fact explicit calculations ensure that these apparent poles get cancelled in the final expressions.

Discussion
The methods developed in this paper can be extended to other cousins of the Liouville and Toda field theories, like super Liouville and super Toda. Also it is an important problem to compute the conformal blocks in the light asymptotic limit for an arbitrary n via the integral representation, as it was done for A 2 Toda field theory in the last section.
Let us also recall that in the light asymptotic limit the W n symmetry reduces to SL(n) group. This means that the function (2.34) in fact should have a group theoretical meaning. It would be interesting to discover it.
It remains to use (A.5) to arrive at (3.12).