Edge modes, edge currents, and gauge invariance inpx+ipysuperfluids and superconductors

The excitation spectrum of a two-dimensional $p_x{+ip}_y$ fermionic superfluid, such as a thin film of ${}^3\mathrm{He}-A,$ 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 $ħN/2.\mathrm{}$ 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.