  We consider the ground state of an electron gas embedded in a quantum
gyrotropic cavity. We show that the light-matter interaction leads to a
nontrivial topology of the many-body electron-photon wave function
characterized by a nonzero Berry curvature. Physically, the latter manifests as
the anomalous Hall effect, appearance of equilibrium edge/surface currents and
orbital magnetization induced by vacuum fluctuations. Remarkably, closed
analytical expressions for the anomalous Hall conductivity and macroscopic
magnetization are obtained for the interacting many-body case.
