Abstract
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.
Highlights
Defined in terms of the parameter n
We study the AGT correspondence between four-dimensional supersymmetric gauge field theory and two-dimensional conformal field theories in the context of WN minimal models
In this paper we propose AGT-like combinatorial representation for the conformal block functions in WN minimal models
Summary
As an example of the AGT representation for non-degenerate representations of Virasoro algebra W2 we consider a 4-point conformal block on the sphere. The consideration of general k-point correlation functions contains the same ingredients. The 4-point conformal block is a holomorphic contribution of the conformal family [Φ∆0] of the primary field Φ∆0 in the correlation function Φ∆1(x)Φ∆2(0)Φ∆3(1)Φ∆4(∞). The AGT correspondence gives the following power series expansion for the 4-point conformal block [2]. The summation on the right hand side runs over pairs of Young diagrams λ = (λ1, λ2) and the norm |λ| denotes the total number of cells. [9] one can find an orthogonal basis |P, λ numerated by pairs of Young diagrams in AN module that reproduces Nekrasov decomposition (2.2)–(2.3). In the correlation function Φ∆1(x)Φ∆2(0)Φ∆3(1)Φ∆4(∞) between each two of the primary fields
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