On combinatorial expansion of the conformal blocks arising from AGT conjecture

TL;DR

This paper derives a new basis in conformal field theory that simplifies the computation of conformal blocks, providing a combinatorial expansion linked to AGT conjecture and instanton counting.

Contribution

It introduces a special orthogonal basis for the algebra al, leading to a simplified, factorized form of conformal blocks that confirms the combinatorial expansion from AGT conjecture.

Findings

The basis diagonalizes commuting integrals of motion.

Matrix elements match the bifundamental Nekrasov function.

Simplifies the calculation of conformal blocks.

Abstract

In their recent paper \cite{Alday:2009aq} Alday, Gaiotto and Tachikawa proposed a relation between $\mathcal{N}=2$ four-dimensional 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 $\mathcal{A}=\text{\sf Vir}\otimes\mathcal{H}$ 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 $\mathcal{A}$. The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called $Z_{\text{\sf{bif}}}$ 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 \cite{Alday:2009aq}. We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin-Ono integrable hierarchy.