AGT, N-Burge partitions and Wn Nn n n n minimal models

摘要

Let ( {mathrm{mathcal{B}}}_{N,n}^{p,pprime, 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,pprime}times {mathrm{mathcal{M}}}^{mathrm{mathscr{H}}} ), where ( {mathrm{mathcal{M}}}_N^{p,pprime } ) 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,pprime, 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