Proving AGT conjecture as HS duality: extension to five dimensions

TL;DR

This work extends the AGT correspondence to five dimensions by a direct $q$-deformation of the 4d Hubbard-Stratonovich framework, replacing Jack with MacDonald polynomials and Selberg integrals with Jackson sums. The construction yields a 5d Dotsenko-Fateev block $B^{5D}(\Lambda)$ and a 5d Nekrasov partition function, clarifying how the AGT relation operates under $q$-deformation and identifying the pivotal role of the deformation parameter $\beta$ (with $\beta=1$ restoring the HS duality). For $\beta\ne 1$, the correspondence no longer factors into Nekrasov expressions; instead, a non-Nekrasov bi-Selberg decomposition emerges, which remains pole-free but does not exhibit straightforward factorization. The $\beta=1$ case, where MacDonald polynomials reduce to Schur functions and Jackson integrals to their classical counterparts, explicitly recovers the 5d Nekrasov function, providing a concrete higher-dimensional generalization of the AGT framework and a foundation for exploring further extensions (e.g., to 6d).

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 β= 1, i.e. for q=t in MacDonald's notation. For β\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.

Figures (2)

• Figure 1: Feynman diagram for the 4-point conformal block.
• Figure 2: Picture of the Nekrasov functions/conformal block duality expressed by the Hubbard-Stratonovich type formula (\ref{['HS']}). The symbol $\int_z$ here denotes integration with the Selberg measure over variables $z_i$, and the symbol $\sum_A$ denotes summation over all Young diagrams $A$.