Effective theory of two-dimensional chiral superfluids: gauge duality and Newton-Cartan formulation

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 $c\to \mathrm{\infty }$. Using Newton-Cartan geometry we construct the covariant formulation of the effective theory and calculate the energy current.