Proving AGT conjecture as HS duality: extension to five dimensions

TL;DR

This paper extends the proof of the AGT conjecture as a Hubbard-Stratonovich duality to five-dimensional gauge theories by incorporating q-deformation, showing the extension is straightforward with certain limitations.

Contribution

It demonstrates that the AGT relation as HS duality can be extended to 5d theories through simple substitutions, highlighting the case where   t and addressing the pole issue.

Findings

Extension is straightforward via q-deformation substitutions.

Proof works explicitly for  = 1, i.e., q = t.

For   1, conformal blocks relate to a non-Nekrasov decomposition.

Abstract

We extend the proof from arXiv:1012.3137, 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 trivial: 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\ne 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.