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.
2. In @cite_16 , Estienne, Pasquier, Santachiara and Serban study conformal blocks of vertex operators such that @math , and @math , @math , and @math , @math , or @math , @math , @math , and @math , @math . in @math minimal models. From the null-state conditions of these vertex operators, Estienne show that these specific conformal blocks are labeled by @math -partitions that satisfy specific conditions. While the notation used in @cite_16 is different from that in this work, one can check, in simple cases, that their @math -partitions are equivalent to those that appear in this work.
@cite_16
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We study the properties of the conformal blocks of the conformal eld theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate elds, the conformal blocks obey second order dierential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with nontrivial braiding properties. A generalized duality property relates the two types of second order degenerate elds. By studying this duality we found that the excited states of the CalogeroSutherland Hamiltonian are characterized by two partitions, or in the case of WAk 1 theories by k partitions. By extending the conformal eld theories under consideration by a u(1) eld, we nd that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in generalk, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.