Gauge/Liouville Triality
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.
Cite
@article{arxiv.1309.1687,
title = {Gauge/Liouville Triality},
author = {Mina Aganagic and Nathan Haouzi and Can Kozcaz and Shamil Shakirov},
journal= {arXiv preprint arXiv:1309.1687},
year = {2013}
}
Comments
50 pages, 3 figures. v2: Typos corrected, aspects of 3d gauge theory clarified, references added