From Horizon Microstates to the Black Hole Membrane

The membrane paradigm represents a black-hole horizon by a fictitious conducting surface. We derive a microscopic electromagnetic membrane from black-hole matrix quantum mechanics. The fuzzy-sphere horizon carries a Berry monopole, placing its fundamental fermionic partons in lowest-Landau-level states with Ohmic and Hall responses. Off-diagonal bifundamental modes connecting the horizon and exterior matrix blocks become tachyonic near the horizon and condense, dynamically coupling the horizon gauge field to the exterior Maxwell field. In the low-frequency regime $\omega R\ll1$, the fixed-parton transport description predicts frequency- and helicity-dependent reflectivity. At larger frequency, real parton excitations require a black-hole $S$-matrix. Ohm's law then fixes the inclusive absorption probability; unitarity bounds the local absorption cross section by the horizon area, with the classical conductivity $1/4\pi$ saturating this maximal-absorption bound.