  We apply a recently developed effective string theory for vortex lines to the
case of two-dimensional trapped superfluids. We do not assume a perturbative
microscopic description for the superfluid, but only a gradient expansion for
the long-distance hydrodynamical description and for the trapping potential.
For any regular trapping potential, we compute the spatial dependence of the
superfluid density and the orbital frequency and trajectory of an off-center
vortex. Our results are fully relativistic, and in the non-relativistic limit
reduce to known results based on the Gross-Pitaevskii model. In our formalism,
the leading effect in the non-relativistic limit arises from two simple Feynman
diagrams in which the vortex interacts with the trapping potential through the
exchange of hydrodynamical modes.
