Holomorphic D-brane embeddings in D-brane backgrounds

We describe families of probe D$q$-brane embeddings in the extremal black D$p$-brane backgrounds of type IIA and type IIB supergravity, specified by an arbitrary holomorphic function of a complex coordinate on the worldvolume of the D$q$-branes. These embeddings preserve one-quarter of the supersymmetry of the D$p$-brane background, or sometimes one-half of the supersymmetry when $p = q$. We discuss the holography of two example families of holomorphic probe branes in the near-horizon limit of the D3-brane background. The first is probe D5-branes, dual to defect hypermultiplets with a holomorphic mass, which in the infrared flow to Wilson lines located at the zeros of the mass. The second is probe D3-branes, holographically dual to states in the presence of Gukov--Witten surface defects in the dual $\mathcal{N}=4$ supersymmetric Yang--Mills theory.