On Combinatorial Expansions of Conformal Blocks

Abstract

In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter \epsilon=\epsilon_1+\epsilon_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6\epsilon^2/\epsilon_1\epsilon_2. If reversed, this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with non-degenerate Virasoro representations, possesses a non-trivial decomposition into sum over sets of the Young diagrams, different from the natural decomposition studied in conformal field theory. We provide some details about this intriguing new development in the simplest case of the four-point correlation functions.