  Using boson-vortex duality, we formulate a low-energy effective theory of a
two-dimensional vortex lattice in a bosonic Galilean-invariant compressible
superfluid. The excitation spectrum contains a gapped Kohn mode and an
elliptically polarized Tkachenko mode that has quadratic dispersion relation at
low momenta. External rotation breaks parity and time-reversal symmetries and
gives rise to Hall responses. We extract the particle number current and stress
tensor linear responses and investigate the relations between them that follow
from Galilean symmetry. We argue that elementary particles and vortices do not
couple to the spin connection which suggests that the Hall viscosity at zero
frequency and momentum vanishes in a vortex lattice.
