Surface Defects from Fractional Branes -- I

We show that the Gukov-Witten monodromy defects of supersymmetric Yang-Mills theory can be realized in perturbative string theory by considering an orbifold background of the Kanno-Tachikawa type and placing stacks of fractional D3-branes whose world-volume partially extends along the orbifold directions. In particular, we show that turning on a constant background value for some scalar fields in the closed string twisted sectors induces a non-trivial profile for the gauge field and one of the complex scalars of the world-volume theory, and that this profile exactly matches the singular behavior that one expects for a Gukov-Witten surface defect in the $\mathcal{N}=4$ super Yang-Mills theory. To keep the presentation as simple as possible, in this work we restrict our analysis to surface defects corresponding to a $\mathbb{Z}_2$ orbifold and defer the study of the most general case to a companion paper.