Gauge/Liouville Triality

Conformal blocks of the Virasoro algebra have a Coulomb-gas representation as Dotsenko-Fateev integrals over the positions of screening charges. In q-deformed Virasoro, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in Dotsenko-Fateev representation, as the partition functions of a class of 3d U(N) gauge theories with N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {N}}}=2$$\end{document} supersymmetry, in the Omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Virasoro vertex operator; the two real mass parameters determine the momentum and position of the puncture. The Dotsenko-Fateev integrals can be computed by residues. The result is the instanton sum of a five dimensional N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {N}}}=1$$\end{document} gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.