On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture
Abstract
In their recent paper, Alday et al. (Lett Math Phys 91:167197, 2010) proposed a relation between {mathcal{N}=2} fourdimensional supersymmetric gauge theories and twodimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the 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}={Vir} ⊗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_{{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 Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to BenjaminOno integrable hierarchy.
 Publication:

Letters in Mathematical Physics
 Pub Date:
 October 2011
 DOI:
 10.1007/s110050110503z
 arXiv:
 arXiv:1012.1312
 Bibcode:
 2011LMaPh..98...33A
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 References added, misprints corrected, the exposition has been substantially improved