Proving AGT conjecture as HS duality: Extension to five dimensions

We extend the proof from Mironov et al. (2011) [25]. which interprets the AGT relation as the Hubbard Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be straightforward: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for beta = 1, i.e. for q = t in MacDonald's notation. For beta not equal 1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams. (C) 2011 Elsevier B.V. All rights reserved.