Surface defects in gauge theory and KZ equation

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