Integrable Conformal Defects in N=4 SYM

In this paper we classify integrable conformal defects in N=4 SYM theory for which the scalar fields pick up a non-trivial vacuum expectation value. Defects of this form correspond to Dirichlet boundary conditions that have a pole at the defect. These set-ups typically appear on the field theory side of probe brane set-ups in the AdS/CFT correspondence. We show that such defects, for any codimension, are related to fuzzy spheres. We discuss the properties of the different possible fuzzy spheres that can appear and present the corresponding Matrix Product States. We furthermore set-up the quantum field theoretic framework by computing the mass matrix and finding the propagators.