Proof of An AGT conjecture at β = 1

Abstract

AGT conjecture reveals a connection between 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauge theory and 2D conformal field theory. Though some special instances have been proven, others remain elusive and the attempts on its full proof never stop. When the Ω background parameters satisfy −ϵ1/ϵ2 ≡ β = 1, the story can be simplified a bit. A proof of the correspondence in the case of A1 gauge group was given in 2010 by Mironov et al., while the An extension is verified by Matsuo and Zhang in 2011, with an assumption on the Selberg integral of n + 1 Schur polynomials. Then in 2020, Albion et al. obtained the rigorous result of this formula. In this paper, we show that the conjecture on the Selberg integral of Schur polynomials is formally equivalent to their result, after applying a more complicated complex contour, thus leading to the proof of the An case at β = 1. To perform a double check, we also directly start from this formula, and manage to show the identification between the two sides of AGT correspondence.