On Combinatorial Expansions of Conformal Blocks

Understand

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.