On Combinatorial Expansions of Conformal Blocks

TL;DR

This paper explores the combinatorial structure of conformal blocks in Liouville theory, revealing a novel decomposition linked to Nekrasov partition functions and Young diagrams, with implications for understanding 2D conformal field theories.

Contribution

It introduces a new combinatorial expansion of conformal blocks related to Nekrasov functions, extending the understanding of their structure beyond traditional conformal field theory decompositions.

Findings

Conformal blocks can be decomposed into sums over Young diagrams.

The decomposition relates to Nekrasov partition functions.

Provides detailed analysis for four-point correlation functions.

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.