Gauge/Liouville Triality

TL;DR

This paper explores the deep correspondence between Liouville conformal blocks, their q-deformed versions, and partition functions of 3d and 5d supersymmetric gauge theories, revealing a novel gauge/Liouville triality.

Contribution

It establishes a new triality linking Liouville theory, 3d U(N) gauge theories, and 5d N=1 gauge theories through integral representations and residue calculations.

Findings

Liouville conformal blocks match 3d gauge theory partition functions

Residue calculations yield Nekrasov sums for 5d theories

Coupling to flavors encodes vertex operator insertions

Abstract

Conformal blocks of Liouville theory have a Coulomb-gas representation as Dotsenko-Fateev (DF) integrals over the positions of screening charges. For q-deformed Liouville, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in DF representation, as the partition functions of a class of 3d U(N) gauge theories with N=2 supersymmetry, in the Omega-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Liouville vertex operator; the two real mass parameters determine the momentum and position of the puncture. The DF integrals can be computed by residues. The result is the instanton sum of a five dimensional N=1 gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.