Holography and defect conformal field theories

We develop both the gravity and field theory sides of the Karch-Randall conjecture that the near-horizon description of a certain D5-D3 brane configuration in string theory, realized as ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ bisected by an ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$ ``brane,'' is dual to $\mathcal{N}=4$ super Yang-Mills theory in ${\mathbf{R}}^{4}$ coupled to an ${\mathbf{R}}^{3}$ defect. We propose a complete Lagrangian for the field theory dual, a novel ``defect superconformal field theory'' wherein a subset of the fields of $\mathcal{N}=4$ SYM theory interacts with a $d=3 \mathrm{SU}(N)$ fundamental hypermultiplet on the defect preserving conformal invariance and 8 supercharges. The Kaluza-Klein reduction of wrapped D5 modes on ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$ leads to towers of short representations of $OSp(4|4),$ and we construct the map to a set of dual gauge-invariant defect operators ${\mathcal{O}}_{3}$ possessing integer conformal dimensions. Gravity calculations of $〈{\mathcal{O}}_{4}〉$ and $〈{\mathcal{O}}_{4}{\mathcal{O}}_{3}〉$ are given. Spacetime and N dependence matches expectations from DCFT, while the behavior as functions of $\ensuremath{\lambda}{=g}^{2}N$ at strong and weak coupling is generically different. We comment on a class of correlators for which a nonrenormalization theorem may still exist. Partial evidence for the conformality of the quantum theory is given, including a complete argument for the special case of a $U(1)$ gauge group. Some weak coupling arguments which illuminate the duality are presented.