Surface defects in gauge theory and KZ equation

Nikita Nekrasov, Alexander Tsymbaliuk

Published 2021-03-23Version 1

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 the $\widehat{\mathfrak{sl}}_{N}$-current algebra, originally introduced in the context of 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.