  We formulate a quantum theory of vorticity (hydro)dynamics on a general
two-dimensional bosonic lattice. In the classical limit of a bosonic
condensate, it reduces to conserved plasma-like vortex-antivortex dynamics. The
nonlocal topological character of the vorticity flows is reflected in the
bulk-edge correspondence dictated by the Stokes theorem. This is exploited to
establish physical boundary conditions that realize, in the coarse-grained
thermodynamic limit, an effective chemical-potential bias of vorticity. A Kubo
formula is derived for the vorticity conductivity|which could be measured in a
suggested practical device|in terms of quantum vorticity-flux correlators of
the original lattice model. As an illustrative example, we discuss the
superfluidity of vorticity, exploiting the particle-vortex duality at a bosonic
superfluid-insulator transition.
