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.
If one can extend the analytic continuation scheme used in @cite_48 to the full instanton partition functions of the most general conformal blocks, and obtain the same result as in the present work, then this would amount to a proof that the proposed modified AGT expression for @math in equation ) is indeed the required minimal model conformal block up to a Heisenberg factor.
@cite_48
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We identify Moore-Read wave functions, describing non-Abelian statistics in fractional quantum Hall systems, with the instanton partition of N = 2 superconformal quiver gauge theories at suitable values of masses and Ω-background parameters. This is obtained by extending to rational conformal field theories the SU(2) gauge quiver Liouville field theory duality recently found by Alday-Gaiotto-Tachikawa. A direct link between the Moore-Read Hall n-body wave functions and ℤ n -equivariant Donaldson polynomials is pointed out.