AGT, Burge pairs and minimal models

We consider the AGT correspondence in the context of the conformal field theory, M-p,M-p' circle times M-H, where M-p,M-p' is the minimal model based on the Virasoro algebra V-p,V-p' labeled by two co-prime integers {p, p'}, 1 < p < p', and M-H is the free boson theory based on the Heisenberg algebra H. Using Nekrasov's instanton partition functions without modification to compute conformal blocks in M-p,M-p' circle times M-H leads to ill-defined or incorrect expressions. Let B-n(p,p'H) be a conformal block in M-p,M-p' circle times M-H, with n consecutive channels chi(iota), iota = 1,center dot center dot center dot, n, and let chi(iota) carry states from H-r iota,s iota(p,p') circle times F, where H-r iota,s iota(p,p') is an irreducible highest-weight V-p,V-p' -representation, labeled by two integers {r(iota),s(iota)}, 0 < r(iota) < p, 0 < s(iota) < p',and F is the Fock space of H. We show that restricting the states that flow in chi(iota), iota = 1, center dot center dot center dot , n, to states labeled by partition pairs {Y-1(iota), Y-2(iota)} that satisfy , Y-2,sigma(iota,T) - Y-1,sigma+r iota-1(iota,T) >= 1 - s(iota,) and Y-1,sigma(iota,T) - Y-2 sigma+p-r iota-1(iota,T) >= 1 -p' vertical bar s(iota,) where Y-i,sigma(iota,T) is the sigma-column of Y-i(iota) i is an element of{1,2}, we obtain a well-defined expression that we identify with B-n(p,p',H) . We check the correctness of this expression for 1. Any 1-point B-1(p,p',H) on the torus, when the operator insertion is the identity, and 2. The 6-point B-3(3,4,H) on the sphere involves six Ising magnetic operators.