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 β=1β=1, i.e. for q=tq=t in MacDonaldʼs notation. For β≠1β≠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.