Let @math be a conformal block, with @math consecutive channels @math , @math , in the conformal field theory @math , where @math is a @math minimal model, generated by chiral fields of spin @math , and labeled by two co-prime integers @math and @math , @math , while @math is a free boson conformal field theory. @math 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 @math , spanned by weight vectors @math . We restrict our attention to conformal blocks with vertex operators whose charge vectors point along @math . The charge vectors that label the initial and final states can point in any direction. Following the @math AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute @math , leads to ill-defined expressions. We show that restricting the states that flow in the channels @math , @math , to states labeled by @math partitions that satisfy conditions that we call @math -Burge partitions, leads to well-defined expressions that we identify with @math . We check our identification by showing that a specific non-trivial conformal block that we compute, using the @math -Burge conditions satisfies the expected differential equation.
Recently an orthogonal basis ofWN -algebra (AFLT basis) labeled by N-tuple Young diagrams was found in the context of 4D 2D duality. Recursion relations among the basis are summarized in the form of an algebra SH c which is universal for any N. We show that it has an S3 automorphism which is referred to as triality. We study the level- rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by N or M Young diagrams with the rows of boxes appropriately shued. The reshuing of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH c . Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the N-Burge condition in the Hilbert space. We construct a representation of the affine W-algebra of ( g l _ r ) on the equivariant homology space of the moduli space of U r -instantons, and we identify the corresponding module. As a corollary, we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). 