On Combinatorial Expansions of Conformal Blocks

TL;DR

This work provides an explicit, hands-on check of the AGT conjecture by mapping the universal part of the four-point Virasoro conformal block to Nekrasov’s instanton partition functions for a $U(2)$ gauge theory with four flavors. The authors derive the block from the operator expansion, compute the Shapovalov form and triple-vertex factors for non-degenerate Verma modules, and demonstrate that the low-order coefficients match Nekrasov expansions when the Liouville dimensions and gauge-theory parameters are related by $\Delta_\ ext{alpha}=\alpha(\epsilon-\alpha)/(\epsilon_1\epsilon_2)$ and $c=1+6\epsilon^2/(\epsilon_1\epsilon_2)$. They reveal eight equivalent parameter mappings corresponding to reflections and leg permutations, and illustrate the free-field case where the block factorizes into a bilinear Young-diagram sum that coincides with a known U(1)/Schur-function decomposition. The work concludes with a systematic demonstration at low order and discusses poles, zeroes, and symmetries, highlighting the robustness and potential generalizations of the AGT framework to higher points and other chiral algebras.

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 ε=ε_1+ε_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6ε^2/ε_1ε_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.