On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

Abstract

In their recent paper [1] Alday, Gaiotto and Tachikawa proposed a relation between N = 2 fourdimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired from the exact form of instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra A = Vir ⊗ H which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of A. The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called Z bif appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in [1]. We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin-Ono integrable hierarchy.

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

“…In our approach, it is still an open question how to systematically obtain the whole tower of integral of motion. We have verified that our third order integral of motion coincides with the expression I 4 conjectured in Appendix C of [28].…”

“…These eigenstates are indexed by k Young tableaux and are obtained by inserting into the conformal block a particular u(1)⊗WA k−1 descendant field. We will prove that the descendant states associated to the CS eigenfunctions form an orthogonal basis which, in the case of the u(1) ⊗ V ir(g) algebra, corresponds to the basis introduced in [28] to investigate the AGT conjecture [29], i.e. the expansion of the conformal blocks of Liouville theory (or more generally of CFTs based on Virasoro algebra [30,31]) in terms of Nekrasov instanton functions [32] of SU (2) gauge theories 2 .…”

“…In some sense, our finding is not surprising. The basis of descendant states have been shown [28] to diagonalize a series of commuting integrals of motion which, in the classical limit, correspond to the Benjamin-Ono conserved quantities. The Benjamin-Ono integrable hierarchy is in turn related to the CS system [33].…”

“…The Benjamin-Ono integrable hierarchy is in turn related to the CS system [33]. The advantage of our approach is that the connection between the CS Hamiltonian and the set of commuting integral of motion obtained in [28] is direct and explicit. For general k, the basis of descendant states associated to CS eigenfunctions is expected [28] to play an analogous role in the generalization of the AGT conjecture concerning the SU (k) gauge theories and the WA k−1 theories [34,35].…”

Generalized Jack polynomials and the AGT relations for the SU(3) group

“…Finding the corresponding basis | Y in CFT is an interesting problem. This basis was found explicitly in the case of c = 1 [6] and it was argued to exist in the general case [7]. Concretely, if one performs the bosonization of the Virasoro algebra, the basis vectors are expressed through the generalized Jack polynomials which are defined as the polynomial eigenfunctions of a certain differential operator 1 [8].…”