Topological transport of vorticity on curved magnetic membranes

In this work, we study the transport of vorticity on curved dynamical two-dimensional magnetic membranes. We find that topological transport can be controlled by geometrically reducing symmetries, enabling processes absent from flat magnetic systems. To this end, the vorticity 3-current is constructed, which obeys a continuity equation immune to local disturbances of the magnetic texture and spatiotemporal fluctuations of the membrane. We show how electric current can manipulate vortex transport in geometrically nontrivial magnetic systems. As an illustrative example, we propose a minimal setup that realizes an experimentally feasible energy storage device and discuss its thermodynamic efficiency in terms of a vortexoelectric counterpart of the thermoelectric figure of merit $ZT$.