Conformal blocks of W_N Minimal Models and AGT correspondence

We study the AGT correspondence between four-dimensional supersymmetric gauge field theory and two-dimensional conformal field theories in the context of W_N minimal models. The origin of the AGT correspondence is in a special integrable structure which appears in the properly extended conformal theory. One of the basic manifestations of this integrability is the special orthogonal basis which arises in the extended theory. We propose modification of the AGT representation for the W_N conformal blocks in the minimal models. The necessary modification is related to the reduction of the orthogonal basis. This leads to the explicit combinatorial representation for the conformal blocks of minimal models and employs the sum over N-tupels of Young diagrams with additional restrictions.


Introduction
The bootstrap approach to 2d CFTs is based on the requirements of conformal symmetry, associativity of the operator algebra and crossing symmetry for the correlation functions [1]. One of the main ingredients in the conformal bootstrap approach is the conformal block function, which sums up the contributions of conformal descendants of a given primary field. The bootstrap approach allows in principle to define the structure constants of the operator algebra and then to construct arbitrary multi-point correlation functions. Important class of the conformal field theories is the minimal models, where there is a finite number of irreducible representations of the conformal algebra closed with respect to the operator algebra.
The AGT correspondence [2] and its generalizations establish connections between different 2d conformal field theories and instanton moduli spaces in 4d N = 2 supersymmetric gauge quiver theories in the Omega background [3]. In the framework of this correspondence the conformal blocks are represented by the instanton partition functions which are known explicitly [3]. The AGT representation for W N conformal blocks was considered in Refs. [4,5,6]. In Ref. [7] the connection between an integrable structure of the theory with chiral algebra 1) and W N conformal blocks was found. Here H denotes the Heisenberg algebra and the second term gives the standard coset realisation of the W N algebra with the central charge defined in terms of the parameter n. The theory possesses the following remarkable property. The algebra A N is some special limit of the quantum toroidal gl 1 (q 1 , q 2 ) algebra with simple action on the cohomologies of equivariant K theories. In particular, in the module of the toroidal algebra there exists the orthogonal basis enumerated by plane partitions. A N module inherits the structure of the basis of the module of the toroidal algebra [8]. As a result of the limiting procedure the basis is enumerated by N ordinary Young diagrams. In the A N module there exists some special orthogonal basis such that the matrix element of the composite vertex operators in this basis are known explicitly in terms of the simple rational functions of the basic parameters [9,7]. For this reason, using the orthogonal basis automatically leads to the explicit results for arbitrary correlation functions.
It is interesting to study the consequences of the AGT correspondence for the conformal blocks of W N minimal models. The first example of the AGT correspondence for rational CFT models with Virasoro symmetry was considered in Ref. [10]. In [11] the properties of the conformal blocks of some special degenerate fields in the conformal field theories with extended W N symmetry were studied in the context of the AGT correspondence. However, the problem of constructing the general combinatorial representations for conformal blocks in W N minimal models and more generally the problem of constructing the correlation functions remains open. Interesting applications of W N minimal models, where the question of constructing conformal blocks is relevant, can be found within the AdS 3 /CF T 2 higher spin correspondence, see, e.g., Refs. [12,13].
In this paper we propose AGT-like combinatorial representation for the conformal block functions in W N minimal models. We formulate a necessary modification of the AGT combinatorial representation for the Virasoro minimal models. The modification is reduced to additional restrictions on the region of summation over Young diagrams. In particular, we perform some checks for Virasoro conformal blocks comparing with the exact results which follows from the definition of the conformal block. Also, we formulate the conjecture on the form of AGT representation for conformal blocks of W N minimal models. Our conjecture relies upon the properties of the orthogonal basis in A N modules described in Refs. [14,15].

AGT for non-degenerate Virasoro representations
As an example of the AGT representation for non-degenerate representations of Virasoro algebra W 2 we consider a 4-point conformal block on the sphere. The consideration of general k-point correlation functions contains the same ingredients. The general answer including degenerate representations of W N algebras will be given in Section 5.
The 4-point conformal block is a holomorphic contribution of the conformal family A standard Liouville parametrization of the conformal dimensions and central charge is The AGT correspondence gives the following power series expansion for the 4-point conformal block [2] and The summation on the right hand side runs over pairs of Young diagrams λ = (λ 1 , λ 2 ) and the norm | λ| denotes the total number of cells. The explicit form of Z f (µ i , a; λ) and and Functions E λ,µ (x s and φ(x, s) are defined as (2.6) To explain our notation we adjust (λ) i to ith row of the Young diagram λ and denote (λ) T j the length of the jth column, where T stands for a matrix transposition. For a cell s = (i, j) such that i and j label a respective row and a column, the arm-length function a λ (s) and the leg-length function l λ (s) are given by The parameters of Nekrasov partition function are related to the parameters of the conformal block as follows Following Ref. [9] one can find an orthogonal basis |P, λ numerated by pairs of Young diagrams in A N module that reproduces Nekrasov decomposition (2.2)-(2.3). The Nekrasov sum is obtained simply by inserting the following unity decomposition λ |P, λ P, λ| P, λ|P, λ = I (2.9) between each two of the primary fields.
3 The AGT-like representation for the W 2 minimal models Conformal field theories M p,p ′ are characterized by the central charge of the Virasoro algebra There are (p − 1) × (p ′ − 1)/2 primary fields Φ l,k (l = 1, . . . , p − 1 and k = 1, . . . , p ′ − 1) in the model. The conformal dimensions are determined by the Kac formula Note that there is a symmetry ∆ m,n = ∆ p−m,p ′ −n and ∆ m,n = ∆ p+m,p ′ +n . In the Liouville parametrization the values of the parameter P corresponding to the degenerate values are For the Virasoro minimal models the fusion rules which describe conformal families that appear in the operator product expansion of two primary fields are [1] (3.4)

Reduction of the basis in the minimal models
The AGT representation (2.2) is not directly applicable to minimal models. The reason is that the fields of minimal models are degenerate. Indeed, vectors of invariant submodules possess zero norms so that expression (2.2) which contains these norms in the denominator is singular in this case. One comment about relations between general and minimal models conformal block is in order. The 4-point conformal blocks on the sphere in M p,p ′ minimal model can be derived from the expression for the non-degenerate fields by means of the following procedure of the analytic continuation.
Let us consider first CFT for general value of the central charge parameter, and let be the correlation function of four non-degenerate fields. Suppose we have some orthogonal basis in the Verma module [Φ ∆ ] denoted by |N . For the conformal block we have Now, we are interested in the conformal blocks of minimal models. They can be derived from (3.6) in two steps. First, we fix c = c p,p ′ (3.1) and external dimensions . We note that the set of the dimensions should be admissible for the fusion rules of the minimal models. Second, we take a limit ∆ → ∆ mn . More precisely, we use the parametrization ∆ = a(Q − a) and take a → a mn . Among the descendants of the primary field Φ ∆ will appear singular vectors (and their descendants). Let us denote the singular vector creation operator as D mn . One can derive [16] that the norm of the vector D mn Φ ∆(a) has zero of the first order in the limit a → a mn , or, explicitly, One can check that each of the three-point functions in the numerator of (3.6) has also zero of the first order in this limit as it was also shown in Ref. [16]. Hence we get a secondorder zero in the numerator and a first-order zero in the denominator. The same result occurs for all descendants of the basic singular vectors D mn Φ ∆(a) and D p−m,p ′ −n Φ ∆(a) . So we can just drop out the contribution of the vectors in the decomposition (3.6) which fall in the invariant subspace generated by the singular vector D mn Φ ∆(a) . In fact, this procedure can be considered as a definition of the conformal blocks of the minimal models.
On the other hand, the above procedure can be effectively used to re-derive conformal blocks in minimal models from the AGT representation for non-degenerate fields.

Combinatorial representation
Using the idea of the orthogonal basis for minimal models we must drop out all basis elements belonging to invariant submodules. Even though the explicit construction of the vectors is not known we can use (2.2) to find for which elements of the AGT basis the norm vanishes λ| λ = 0 .
The proof is relegated to Appendix A. Next, we consider a minimal model M p,p ′ so that parameter b is given by formula (3.1). It follows that the function Z v can have additional zeros that restrict the basis. One proves 1 is not equal to zero provided the set of Young diagrams belongs to the region where (λ α ) i are lengths of i-th rows of Young diagrams λ α , and (n 1 , m 1 ) = (n, m) and The proof is relegated to Appendix A. From the Proposition 3.2 we derive the following explicit representation for the 4-point conformal block in the Virasoro minimal models where P n i ,m i denote external conformal dimensions, P n,m denote internal one and the summation region R n,m is defined in (3.10). An analogues result on the conformal blocks in the Virasoro minimal models is obtained in Ref. [17].

Testing AGT for Virasoro minimal models
In this section we consider a 4-point conformal block B(∆ 1,n j ; x) with at least one degenerate field Φ 1,2 . This function satisfies the null vector equation which turns out to be the Riemann equation [1] with the parameters Here, the parameter κ is related to the labels (p, p ′ ) of the minimal model as κ = p/p ′ . The equation (4.1) has two linearly independent solutions with power-law behavior at x = 0. They correspond to two conformal blocks in the S-channel In what follows we reproduce the above hypergeometric functions in the form of the diagrammatic decomposition (2.2), (3.11) applied to the Lee-Yang model.

Lee-Yang model
where At level 2, we have ( , ∅) and (∅, ). The corresponding contributions can be easily derived, and one can check that Let us compare these results with the diagrammatic decomposition at the arbitrary level N. Recall that intermediate fields are Φ 1,1 and Φ 1,3 , while all external fields are Φ 1,2 .
Intermediate field Φ 1,3 . Quite analogously, we consider the case of another intermediate dimension. One obtains that ∆ 1,2 corresponds to P 0 = a = P 1,3 = 1 2 (b + 3b −1 ). An equivalent form of a reads a = Q/2 + b −1 or Q/2 = a − b −1 . Also, We conclude here that a decomposition involves pairs of diagrams of the form (λ 1 = ∅, λ 2 = a row of length N). Provided these facts the diagrammatic decomposition yields the final formula F (N ) These ratios are again expansion coefficients of the hypergeometric function in (4.4).

Generalization for W N minimal models
In this section we are interested in unitary W N minimal models M p,p+1 (N) with the central charge The primary fields are labelled by two sl N weights Λ + = N −1 s=1 (m s − 1)ω s and Λ − = where m N and n N are defined by the above formulas. In the Liouville-like parametrization we write Φ P , where the vector P = (P (1) , ..., P (N −1) ), and where ρ is the Weyl vector (a half-sum of positive roots). Unlike the Virasoro case, in a W N theory the conformal blocks are not fixed by conformal and W N invariance [18]. The bootstrap program for k-point correlation functions can be performed only if the charges of k − 2 fields are proportional to the first fundamental weight ω 1 of the Lie algebra sl N [19]. We consider the correlation functions of this kind where parameters in points z 2 , ..., z k−1 correspond to degenerate representations of minimal models described above, while fields in z 1 and z k are general primary fields of the minimal models. It is convenient to change the variables The holomorphic dependence of the correlation functions is encoded in the conformal block functions B(q 1 , ..., q k−3 |P, P 1 , ..., P k−3 ,P |a 2 , ..., a k−1 ) , (5.6) where the momenta P 1 , ..., P k−3 correspond to the fields in the intermediate channels of the conformal block decomposition. Recall that in the minimal models, the weights of all external and intermediate fields are related by fusion rules.
In [14,15] there was proposed the orthogonal basis for modules of the toroidal algebra in some special limits. This basis is labelled by the special sets of N-tuples of Young diagrams. It was shown that the characters of these diagrams coincide with the characters of W N minimal models up to a contribution related to the presence of extra Heisenberg algebra. It follows that the found basis should define AGT basis in the highest weight representations of the H ⊗ W N algebra [7] (see also [20]) restricted for the minimal models thereby giving rise to the AGT representation for conformal blocks in W N minimal models. The basis vectors are enumerated by N-tuples of Young diagrams with some additional restrictions formulated below.
We conjecture the following explicit form of the k-point conformal block in W N minimal models M p,p+1 (p ≥ N − 1). In this case the conformal block (5.6) for non-degenerate parameters is related to the instanton part of the Nekrasov partition function for the quiver gauge theory and can be written explicitly [7,4] and where j = (j 1 , ..., j k−3 ), with the coefficients The norms are N( λ, P ) = F λ λ (0, −P, P ), and the general matrix element is given by [7] F λ λ ′ (a, P, Extending the Proposition 3.2 we conjecture that the summation in formula (5.9) is further restricted by the following region R m,n = λ λ (s) j ≥ λ (s+1) j+ms−1 − n s + 1, where s = 1, ..., N, j ∈ Z >0 , (5.11) where λ (N +1) ≡ λ (1) , while m s and n s are components of extended sl N weight vectors m = (m 1 , ..., m N ) and n = (n 1 , ..., n N ), see (5.2).

Conclusions
In this paper we studied application of the AGT correspondence to minimal models of Virasoro and W N algebras. We used the conjecture (supported by the comparison of the characters of the corresponding representations) that the chiral W N algebra appears in the conformal limit from the toroidal algebra gl 1 (q 1 , q 2 ) [14,15]. This connection reveals a nice integrable structure of degenerate modules of the algebra A N = H ⊗ W N . It appears for the W N central charge corresponding to the minimal models M p,p ′ arising from gl 1 (q 1 , q 2 ) once the parameters are constrained by the following wheel condition q p 1 q p ′ 2 = 1. In particular, A N module inherits some (reduced) orthogonal basis defined naturally in gl 1 (q 1 , q 2 ) modules. Hence, it is natural to use this basis for the evaluation of the W N conformal block functions. We have checked that using this basis for Virasoro minimal models we get the AGT-like representation for the conformal blocks of minimal models once the consequences of emergence of invariant subspaces in the degenerate representations is taken into account.
Our main result is the explicit expression for the conformal blocks in the Virasoro minimal models (3.11) and the conjecture on the form of AGT representation for W N conformal blocks (5.9) -(5.11). The difference from the original AGT expression for nondegenerate representations is encoded in additional restrictions on the summation region over Young diagrams.

A Proof of the propositions
In order to analyse zeros of functions Z f and Z v we use simple method described below. For the minimal models M p,p ′ the Liouville coupling takes particular value (3.1) and therefore equation (A.1) allows for more (infinitely many) solutions, namely, i = L ± αp ′ and j = K + αp. Here arbitrary parameter α ∈ Z because p and p ′ are coprimes. Note that the case α = 0 reproduces zeros described above for general Liouville coupling b. It follows that for the minimal models more zeros appear but sometimes new zeros are "weaker" than those for ∀ b. It is worth noting that actually there are infinitely many new zeros but generally values their coordinates are restricted from below by minimal values that define the form of admissible diagrams.
Proof of the Proposition 3.1. First of all we note that the terms with E λα,λα x s are not equal to zero. The only source of zeros is provided by terms containing E λα,λ β x s with the pair of different Young diagrams.
We start with empty diagram λ 2 and subsequent increase a number of its rows. Another trick is to consider cases ∆ 1,m and ∆ n,1 separately so that the general case of ∆ n,m is a straightforward combination of the previous ones. On the plane (b −1 , b) we have and Consider diagrams (λ 1 , ∅) with any λ 1 , and show that these do not produce zeros of As the next step one considers an arbitrary diagram λ 2 with m + l rows, where l = 1, 2, .... The diagram λ 2 naturally splits in two subdiagrams λ 2 = λ ′ 2 ⊕ λ ′′ 2 , where the first factor is a diagram composed of first m − 1 rows of λ 2 , while the second one is a diagram composed of the remaining rows of λ 2 . Considering equations (A.6) one shows that zeros are absent when λ ′′ 2 ⊆ λ 1 . Equivalently, In the case of λ 2 = the first equation is automatically satisfied for = j = 1, ..., N 1 . The second equation says that zeros appear when the first row of λ 1 is of length less than N 1 − n. This is to say that zeros are absent for pairs of diagrams with lengths subject to In particular, this relation implies that in order to produce a zero the second diagram Y 2 should include a rectangle of length n and height m.
To conclude the proof one notes that zeros are also contained in E λ 2 ,λ 1 −2a s −Q = 0. Equations that define coordinates of a zero are those in (A.5) but with m → m + 1 and n → n + 1. Admissible diagrams are defined by inequality for any α ∈ Z because coordinates are integers, while p and p ′ are coprimes. These equations coincide with those in (A.5) but n → n + αp and m → m + αp ′ . For α ∈ Z + one obtains that resulting restrictions of diagrams are weaker than (3.9). To consider the case of α ∈ Z − one recalls that by definition of minimal models n < p and m < p ′ . Then one notices that n + αp < 0 and m + αp ′ < 0 which is to say that zeros are absent. We conclude that condition (A.4) does not produce new zeros.
New zeros appear due to condition (A.2). In this case coordinates of zeros are where α ∈ Z. Zeros are possible for α ∈ N only. The resulting equations coincide with those in (A.14) provided λ 1 ↔ λ 2 and n ↔ (α + 1)p − n and m ↔ (α + 1)p ′ − m, where now α ∈ Z + . Repeating arguments below formula (A.14) one concludes that equations (A.15) impose the following restrictions