AGT, N-Burge partitions and WN minimal models

Abstract

Abstract: Let (Formula Presented) be a conformal block, with n consecutive channels χι, ι = 1, ⋯ , n, in the conformal field theory (Formula Presented) where (Formula Presented) is a WN$$ {\mathcal{W}}_N $$ minimal model, generated by chiral spin-2, ⋯ , spin-N currents, and labeled by two co-prime integers p and p′, 1 < p < p′, while ℳℋ 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 AN − 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 WN 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 χι, 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.