  We present a theory of Galilean-invariant conventional and chiral $p_x \pm
ip_y$ fermionic superfluids at zero temperature in two spatial dimensions in
terms of a dual gauge theory. Our formulation is general coordinate invariant.
The parity-violating effects are encoded in the Wen-Zee term that gives rise to
the Hall viscosity and edge current. We show that the relativistic superfluid
with the Euler current reduces to the chiral superfluid in the limit
$c\to\infty$. Using Newton-Cartan geometry we construct the covariant
formulation of the effective theory and calculate the energy current.
