Holomorphic D-brane embeddings in D-brane backgrounds

Abstract

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.

Figures (1)

• Figure 1: The relation between class 1 embeddings and class 1${}^\prime$ embeddings. (a): Cartoon of a class 1 embedding. The thick, horizontal lines represent the D$p$-branes sourcing the background \ref{['eq:Dp_brane_background']}. The curve represent the probe D$q$-branes, which have embedding specified by how $y$ depends on $z$. (b): Cartoon of a class 1$'$ embedding. The thick, vertical lines represent the D$p$-branes, and the curve again represents the probe D$q$-branes, which again have embedding specified by how $y$ depends on $z$. Figures (a) and (b) are the same up to a $\pi/2$ rotation, representing a reparameterisation of the D$q$-branes, and relabelling of variables $y \leftrightarrow z$. Note however that not every class 1$'$ embedding can be thought of as a simple reparameterisation of class 1 embedding, as discussed in the main text.