Abstract
Let \( {\mathrm{\mathcal{B}}}_{N,n}^{p,p\prime, \mathrm{\mathscr{H}}} \) be a conformal block, with n consecutive channels χ ι , ι = 1, ⋯ , n, in the conformal field theory \( {\mathrm{\mathcal{M}}}_N^{p,p\prime}\times {\mathrm{\mathcal{M}}}^{\mathrm{\mathscr{H}}} \), where \( {\mathrm{\mathcal{M}}}_N^{p,p\prime } \) is a \( {\mathcal{W}}_N \) minimal model, generated by chiral spin-2, ⋯ , spin-N currents, and labeled by two co-prime integers p and p′, 1 < p < p′, while \( {\mathrm{\mathcal{M}}}^{\mathrm{\mathscr{H}}} \) is a free boson conformal field theory. \( {\mathrm{\mathcal{B}}}_{N,n}^{p,p\prime, \mathrm{\mathscr{H}}} \) 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 \( {\overrightarrow{\omega}}_1,\cdots, {\overrightarrow{\omega}}_{N-1} \). We restrict our attention to conformal blocks with vertex operators whose charge vectors point along \( {\overrightarrow{\omega}}_1 \). The charge vectors that label the initial and final states can point in any direction.
Following the \( {\mathcal{W}}_N \) AGT correspondence, and using Nekrasov’s instanton partition functions without modification to compute \( {\mathrm{\mathcal{B}}}_{N,n}^{p,p\prime, \mathrm{\mathscr{H}}} \), 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 \( {\mathrm{\mathcal{B}}}_{N,n}^{p,p\prime, \mathrm{\mathscr{H}}} \). 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 \( {\mathcal{W}}_3 \) irreducible highest weight representations.
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Belavin, V., Foda, O. & Santachiara, R. AGT, N-Burge partitions and \( {\mathcal{W}}_N \) minimal models. J. High Energ. Phys. 2015, 73 (2015). https://doi.org/10.1007/JHEP10(2015)073
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DOI: https://doi.org/10.1007/JHEP10(2015)073