AGT, Burge pairs and minimal models

We consider the AGT correspondence in the context of the conformal field theory ℳp,p' ⊗ ℳℋ, where ℳp,p' is the minimal model based on the Virasoro algebra V p,p' labeled by two co-prime integers {p, p'}, 1 < p < p', and ℳℋ is the free boson theory based on the Heisenberg algebra ℋ. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in ℳp,p' ⊗ ℳ ℋ leads to ill-defined or incorrect expressions. Let B np,p',ℋ be a conformal block in ℳp,p' ⊗ ℳℋ, with n consecutive channels χι, ι = 1, ⋯, n, and let χι carry states from Hrι,sιp, p' ⊗ ℱ, where Hrι,sιp, p' is an irreducible highest- weight Vp, p'- representation, labeled by two integers {rι, sι }, 0 < rι < p, 0 < sι < p', and ℱ is the Fock space of ℋ. We show that restricting the states that flow in χι, ι = 1, ⋯, n, to states labeled by partition pairs {Y1ι Y2ι that satisfy Y2,σι,T - Y 1, σ +rι-1ι, T ≥ 1 sι and Y1,σι, T - Y 2, σ + p - r ι - 1ι, T ≥ 1 - p' + sι, where Yi, σι, T is the σ-column of Yiι, i ∈ {1, 2}, we obtain a well-defined expression that we identify with ℬnp,p',H. We check the correctness of this expression for 1. Any 1-point ℬ1p, p', H on the torus, when the operator insertion is the identity, and 2. The 6-point ℬ33, 4, H on the sphere that involves six Ising magnetic operators. © 2014 The Author(s).