Edge modes, edge currents, and gauge invariance in $p_x+ip_y$ superfluids and superconductors

The excitation spectrum of a two-dimensional $p_x+ip_y$ fermionic superfluid, such as a thin film of $^3$He-A, includes a gapless Majorana-Weyl fermion which is confined to the boundary by Andreev reflection. There is also a persistent ground-state boundary current which provides a droplet containing $N$ particles with angular momentum $\hbar N/2$. Both of these boundary effects are associated with bulk Chern-Simons effective actions. We show that the gapless edge-mode is required for the gauge invariance of the total effective action, but the same is not true of the boundary current.