AGT, N-Burge partitions and W_N minimal models

Abstract: Let (Formula Presented) be a conformal block, with n consecutive channels χ_ι, ι = 1, ⋯ , n, in the conformal field theory (Formula Presented) where (Formula Presented) is a W_N$$ {\mathcal{W}}_N $$ minimal model, generated by chiral spin-2, ⋯ , spin-N currents, and labeled by two co-prime integers p and p^′, 1 < p < p^′, while ℳ^ℋ is a free boson conformal field theory. (Formula Presented) is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra A_{N − 1}, spanned by weight vectors (Formula Presented). We restrict our attention to conformal blocks with vertex operators whose charge vectors point along (Formula Presented). The charge vectors that label the initial and final states can point in any direction. Following the W_N AGT correspondence, and using Nekrasov’s instanton partition functions without modification to compute (Formula Presented), leads to ill-defined expressions. We show that restricting the states that flow in the channels χ_ι, ι = 1, ⋯ , n, to states labeled by N partitions that we call N-Burge partitions, that satisfy conditions that we call N-Burge conditions, leads to well-defined expressions that we propose to identify with (Formula Presented). We check our identification by showing that a non-trivial conformal block that we compute, using the N-Burge conditions satisfies the expected differential equation. Further, we check that the generating functions of triples of Young diagrams that obey 3-Burge conditions coincide with characters of degenerate W₃ irreducible highest weight representations.