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 \({{\mathcal {N}}}=2\) $\mathcal{N}=2$ supersymmetry, in the \(\Omega \) $\mathrm{\Omega }$ -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 \({{\mathcal {N}}}=1\) $\mathcal{N}=1$ gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.