How to sit Maxwell and Higgs on the boundary of AdS

Within the holographic correspondence, boundary conditions play a fundamental role in determining the nature of the dual field theory. In this talk, I will show how to exploit mixed boundary conditions to obtain dynamical electromagnetism in the boundary theory. This is necessary to apply AdS-CFT to many real-world applications, e.g., magnetohydrodynamics, plasma physics, superconductors, etc. where dynamical gauge fields and Coulomb interactions are fundamental. As a proof of concept, I will show two emblematic cases. First, I will prove that the results from the 4-dimensional Einstein-Maxwell bulk theory with these deformed boundary conditions are in perfect agreement with relativistic magneto-hydrodynamics in 2+1 dimensions. Second, I will discuss the collective excitations of a bona-fide holographic superconductor and prove the existence of the Anderson-Higgs mechanism therein.