The paper by Steven S. Gubser explores the spontaneous breaking of Abelian gauge symmetry near the horizon of electrically charged black holes in anti-de Sitter (AdS) space. The author argues that coupling the Abelian Higgs model to gravity plus a negative cosmological constant leads to black holes that spontaneously break the gauge invariance via a charged scalar condensate slightly outside their horizon. This suggests that black holes can exhibit superconductivity. The study focuses on a simple Lagrangian that includes an Einstein-Hilbert term, a negative cosmological constant, and an Abelian Higgs term. The key finding is that for certain parameter ranges, the effective mass of the scalar field becomes negative near the horizon, leading to an unstable mode. This instability can result in a charged scalar condensate, which is interpreted as a superconducting layer forming just outside the black hole horizon. The paper also discusses the scaling symmetries of the black hole background and the properties of marginally stable modes. It shows that these modes can exist in asymptotically AdS4 geometries, providing evidence for spontaneous symmetry breaking. The author concludes by discussing the implications of these findings for the understanding of superconductivity in the context of black holes and the potential for further theoretical exploration.The paper by Steven S. Gubser explores the spontaneous breaking of Abelian gauge symmetry near the horizon of electrically charged black holes in anti-de Sitter (AdS) space. The author argues that coupling the Abelian Higgs model to gravity plus a negative cosmological constant leads to black holes that spontaneously break the gauge invariance via a charged scalar condensate slightly outside their horizon. This suggests that black holes can exhibit superconductivity. The study focuses on a simple Lagrangian that includes an Einstein-Hilbert term, a negative cosmological constant, and an Abelian Higgs term. The key finding is that for certain parameter ranges, the effective mass of the scalar field becomes negative near the horizon, leading to an unstable mode. This instability can result in a charged scalar condensate, which is interpreted as a superconducting layer forming just outside the black hole horizon. The paper also discusses the scaling symmetries of the black hole background and the properties of marginally stable modes. It shows that these modes can exist in asymptotically AdS4 geometries, providing evidence for spontaneous symmetry breaking. The author concludes by discussing the implications of these findings for the understanding of superconductivity in the context of black holes and the potential for further theoretical exploration.