Proving the AGT relation for N_f = 0,1,2 antifundamentals

Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of N=2 superconformal SU(2) quiver gauge theories with N_f = 0,1,2 antifundamental hypermultiplets

Yet another generalization has been recently found in [16].
An explanation of the AGT relation was given by Dijkgraaf and Vafa [17]. The idea was to relate both sides of the correspondence to a certain class of matrix models. These relations were further analyzed in a number of papers [18][19][20][21][22][23][24]. Another M-theory explanation was presented in [25,26].
In spite of all these developments only in the case of N = 2 SU(2) SYM with a single adjoint matter multiplet an analytic proof of the AGT relation is known. The main idea of the proof given by Fateev and Litvinov [52] is to show that the corresponding Nekrasov function and the 1-point conformal block on the torus satisfy exactly the same recursion relations.
These relations were first conjectured by Poghossian [38] and then proven on the CFT side in [41] and on the N = 2 SU(2) SYM side in [52]. The aim of the present paper is to extend this proof to the case of N f = 0, 1, 2 antifundamental hypermultiplets.
In Section 2 we use the results of Marshakov, Mironov and Morozov [10] and Poghossian [38] to derive the recursive relations for all irregular blocks by analyzing appropriate decoupling limits of the Zamolodchikov elliptic recursive relation for the 4-point conformal block on the sphere [53][54][55]. This allows in particular to prove the relation between two representations of the irregular block with two µ parameters conjectured in [9]. As a side topic we clarify the relations of polynomials appearing in the recursive relations to the fusion polynomials and to the null states in the degenerate Verma modules. These new, intriguing result deserves further investigations.
In Section 3 we follow the method of Fateev and Litvinov [52] to analyze singularities of the Nekrasov functions and the factorization of the residues in the case of an arbitrary number of antifundamentals. The complete derivation of recursive formulae along this line requires the large p = a asymptotic of the Nekrasov functions. This is simple in the cases N f = 0, 1. We were also able to calculate the asymptotic in the N f = 2 case. In all three cases the recursions obtained are identical on both sides of the correspondence. Calculating the asymptotics in the cases N f = 3, 4 turned out to be difficult and is still a challenging open problem.

Recursive relations for irregular blocks
The Gaiotto states [9] can be defined by the conditions An explicit form of these states was found in [10] ∆, of the Verma module V c,∆ of the central charge c and the highest weight ∆.
The irregular blocks [9] are defined as scalar products of the Gaiotto states: or in terms of the 3-point conformal block 4 : where the conformal weights are related to µ i parameters by It was shown in [10] that all irregular blocks above can be obtained by appropriate decoupling limits of the 4-point conformal block on the sphere : then [10]: As it was demonstrated in [38] 4 The present notation for 3-point conformal block is related to that used in [41,56] by ξ ′ | V∆(1) |ξ ′′ = ρ(ξ ′ , ν∆, ξ ′′ ). The normalization condition takes the form ∆ ′ | V∆(1) |∆ ′′ = 1.
We close this section by some remarks on Y rs polynomials. They show up in the derivation of the Zamolodchikov recursive relation in the factorization formula for the fusion polynomial where as before the relation between parameters is given by (4) and O rs denotes the combination of the Virasoro algebra generators creating the singular state of level rs out of the degenerate vacuum |∆ rs normalized by the condition that the coefficient in front of L rs −1 |∆ rs is equal 1. Let us note that factorization formula (21) is a direct consequence of the null vector decoupling theorem [57].
Another interpretation of Y rs can be obtained by the derivation of the recursive formulae for irregular blocks directly from their expressions in terms of the inverse Gram matrix. Let us consider the block In the generic case the only singularities of and by an arbitrary basis in the orthogonal complement of Span{L −K O rs |∆ }. Due to the singular behavior of the Gram matrix in the limit ∆ → ∆ rs [41,56] one has at the residue where For our normalization of O rs |∆ rs the exact form (15) of the coefficient A rs was first proposed by Al. Zamolodchikov in [53] and then justified in [58]. In formula (23) only the coefficients corresponding to the states generated from |∆ rs by the operators L −2 , L −1 are present. Let us define such coefficients for the singular state: Then for an arbitrary q, the sum over p in (23) is given in terms of c (rs) p and µ: This yields the factorization formula for the residues R µ rs,n in (22): Expansion (22) and the formula above yield the recursive relation: ∆, µ, n ∆, n = δ n 0 + 1 rs n A rs X rs (µ) ∆ − ∆ rs ∆ rs + rs, µ, n − rs ∆ rs + rs, n − rs .
Comparing with (19) one gets X rs (µ) = Y rs (µ) which provides the factorization formula for the X rs polynomial: and implies an unexpected (from the point of view of the original definition of X rs ) relation with the fusion polynomial (21):

Recursive relations for the Nekrasov partition functions
We shall discuss the instanton contribution to the Nekrasov partition function of the N = 2 supersymmetric, U(2) gauge theory with N f hypermultiplets in the antifundamental representation [27]. It can be written as a sum over pairs of Young diagrams, where 6 p α = a α , α = 1, 2 are the vev-s of the scalar component of the N = 2 gauge supermultiplet, b = ǫ 1 , b −1 = ǫ 2 are the parameters of the Ω background, Q = b + b −1 , 6 Our notation is close to the one in [52]. The contribution to the partition function parameterized by a specific pair of Young diagrams is of the form [59] The N −box diagram Y can be described by an ordered sequence of natural numbers k 1 ≥ while the horizontal distance reads The contribution to the partition functions from instantons of topological charge N can be expressed as a contour integral [27] Z N f Let us calculate the residue of the pole at p 12 = −rb − sb −1 . One has to take into account only those contributions in (28) for which rs out of N integrals are evaluated by calculating the residues at p 1 , All the other contributions are finite in the limit p 12 = −rb − sb −1 . By a suitable re-labeling of indices (which yields a combinatorial factor N rs ) we may denote by φ mn the variable of integration in the integral evaluated by taking a residue at p 1 + (m − 1)b + (n − 1)b −1 . If we declare the indices i, j and k, l to satisfy 1 ≤ i, j ≤ rs, rs < k, l ≤ N, .
is a contribution to the partition function corresponding to the pair Y rs such that Y 1 is a r×s rectangle and Y 2 = ∅, while x mn = p 1 +(m−1)b+(n−1)b −1 .
After some simple algebra we get For p 12 = −rb − sb −1 the factor in the last line of (29) is equal to 1 and with It leads to the relation For the pair of diagrams Y rs = (r × s, ∅) one has and The formula (27)  Let us now assume which corresponds to SU(2) rather than U(2) gauge group. The pole at p 12 = −rb − sb −1 then corresponds to For the pair of diagrams Y rs = (r × s, ∅) we thus have 2 α=1 m,n ∈Yα where Y rs are defined by (16).
the residues at the poles at p 12 = −rb − sb −1 and p 21 = −rb − sb −1 differ only by a sign.
Equations (32) and (33) then yield for 4p 2 → rb + sb −1 2 where A rs is given by formula (15). For p → ∞ relation (27) implies Together with (34) this gives the recursion relations: Regarding the partition function as a function of the conformal dimension ∆ = 1 4 Q 2 − p 2 rather than the function of p, one gets the recursion formulae of the form: where ∆ rs is given by relation (14). Comparing with (20) and (19) one gets the AGT relation for N f = 0, 1: Calculating the large p asymptotic in the case N f = 2 is more involved. Let us first note that for α = β and p = p α = −p β → ∞ : In order to calculate the sum we shall use the integral representation The relation (38) can be obtained using the results of [61] and [62] (formulae 6.4-6.12 in [61]).
Applying the reasoning presented in Appendix to the integral in (38) one can show that the only singularities of Z N (b) result from the collision of the poles at φ k = (r − 1)b + (s − 1)b −1 (above the real axis) with the pole at φ k = −Q (below the real axis), i.e. for rb + sb −1 = 0.
Since r and s are positive integers the only singularities of Z N (b) can occur for purely imagi- is strictly positive. Thus the residue at the would-be pole at rb + sb −1 = 0 actually vanishes unless H Y ( m, n ) = 0, i.e. Y consists solely of the first column and therefore The Liouville's boundedness theorem then implies that Z N (b) is a b independent constant 7 : One thus gets which along with (34) implies the recursive relation Comparing with recursive relations (17) and (18) one gets the AGT correspondence for N f = 2

Appendix. Singularities of the partition function
In this appendix we shall repeat (a slightly elaborated version of) a proof of the following lemma due to Fateev and Litvinov [52]: Lemma: Let f (φ 1 , . . . , φ n ) be a holomorphic function of all its arguments which grows for φ i → ∞ slow enough to ensure the convergence of the integral is a holomorphic function of all a α . Proof. Let us perform the integral over φ N by closing the contour in the upper half-plane.
With this choice the poles at φ N = a α and φ N = φ j + ǫ a are inside the contour, while the poles at φ N = φ j − ǫ a stay outside. When we vary a α , some of these poles move and, when the positions of two poles coincide, a higher order singularity of the integrand may occur.
If this happens for the poles on the same side of the integration contour, then we can move the contour away from such a colliding pair and the integral stays finite. On the other hand, if the colliding poles "squeeze" the integration contour in between then we can deform the contour of integration away only at a price of evaluating a residue (which can become singular in the limit) at one of the poles.
More precisely: take a pole at φ N = φ j − ǫ b and deform the integration contour to the sum of C and −C k,b , where C encloses all the poles of φ N in the upper half-plane and the pole at By the argument above a singularity of the integral over φ N may appear only from the integral over C j,b . Adding such contributions for all j < N and all b (and neglecting for the moment all the factors which do not depend on φ N ) we get: There are indeed several new poles on the right hand side of this equation. The first type is at φ j = a α + ǫ b above the real axis. In order to expose the others let us note that (A.2) is symmetric under (a, i) ↔ (b, j) so that symmetrizing its right hand side we get It follows from the expression above that I N has poles above the real axis at φ i = φ j +ǫ a +ǫ b and poles below the real axis at φ i = φ j − ǫ a − ǫ b . It seems that there is also a pole at φ i = φ j + ǫ a − ǫ b . However, the residue of such a "would be" pole contains a factor ij − (ǫ a + ǫ b ) 2 and I N is actually finite for φ i → φ j + ǫ a − ǫ b .
After performing the integral over φ N we arrive at the expression of the same structure as with {e A } = {0, ǫ a , ǫ a + ǫ b , . . .} and some holomorphic functionf . The thesis of the lemma follows.
Let us now suppose that the function f appearing in (A.1) contains a factor ν N i=1 1 φ i − ξ ν + i0 + , with simple poles below the real axis. Adapting the reasoning given in the proof above we see that F f N is no longer holomorphic. Its singularities may only come from the collision of poles at φ i = a α + e A and φ i = ξ ν with the φ i integration contour squeezed in between and are thus located at In the situation of interest in Section 3 we have α = 1, 2, ξ 1 = a 1 − ǫ and ξ 2 = a 2 − ǫ. The poles in the variables φ i can appear only at φ i = a α + (r − 1)ǫ 1 + (s − 1)ǫ 2 , r, s ∈ Z + and the only singularities of the partition function as a function of a α are the (simple in the generic case) poles at a 1 − a 2 = ∓(mǫ 1 + nǫ 2 ).