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.
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. Abstract After a very brief recollection of how my scientific collaboration with Ugo started, in this talk I will present some recent results obtained with localization: the deformed gauge theory partition function Z ( τ → | q ) and the expectation value of circular Wilson loops W on a squashed four-sphere will be computed. The partition function is deformed by turning on τ J tr Φ J interactions with Φ the N = 2 superfield. For the N = 4 theory SUSY gauge theory exact formulae for Z and W in terms of an underlying U ( N ) interacting matrix model can be derived thus replacing the free Gaussian model describing the undeformed N = 4 theory. These results will be then compared with those obtained with the dual CFT according to the AGT correspondence. The interactions introduced previously are in fact related to the insertions of commuting integrals of motion in the four-point CFT correlator and the chiral correlators are expressed as τ -derivatives of the gauge theory partition function on a finite Ω -background.