Surface defects in gauge theory and KZ equation

TL;DR

This paper demonstrates that the vacuum expectation value of a surface defect in Omega-deformed SU(N) gauge theory satisfies the Knizhnik-Zamolodchikov equation, establishing a link between four-dimensional gauge theories and two-dimensional conformal field theories.

Contribution

It proves the vacuum expectation value of a regular surface defect obeys the KZ equation, elucidating the BPS/CFT correspondence in this context.

Findings

VEV satisfies KZ equation for 4-point conformal block

Parameters of Omega-background determine level and vertex operators

Branching rule is parametrized by Coulomb moduli

Abstract

We study the regular surface defect in the Omega-deformed four-dimensional supersymmetric gauge theory with gauge group SU(N) with 2N hypermultiplets in fundamental representation. We prove its vacuum expectation value obeys the Knizhnik-Zamolodchikov equation for the 4-point conformal block of current algebra of a two-dimensional conformal field theory. The level and the vertex operators are determined by the parameters of the Omega-background and the masses of the hypermultiplets; the cross-ratio of the 4 points is determined by the complexified gauge coupling. We clarify that in a somewhat subtle way the branching rule is parametrized by the Coulomb moduli. This is an example of the BPS/CFT relation.