  We discuss a recent experiment in which the resistance of a superconducting
film has been measured in magnetic field. A strong decrease of the
superconducting film resistance has been observed when a metallic gate is
placed above the film. We study how the magnetic coupling between vortices in a
thin superconducting film and electrons in a remote unbiased gate suppresses
the tunneling rate of the vortices. We examine two general approaches to
analyze tunneling in the presence of slow low-energy degrees of freedom: the
functional-integral and scattering formalisms. In the first one, the response
of the electrons inside the metallic gate to a change in the vortex position is
described by the "tunneling with dissipation". We consider the Eddy current
induced in the gate by the magnetic flux of the vortex as a result of
tunneling. In the second approach, the response is given in terms of scattering
of the electrons by the magnetic flux of the vortex in a way similar to the
Aharonov-Bohm scattering. A sudden change in the vortex position leads to the
Orthogonality Catastrophe that opposes the vortex tunneling. We show that the
magnetic coupling between the vortices and the electrons inside the gate can
lead to a dramatic suppression of the vortex tunneling, restoring the
superconducting property in accord with the experiment.
