  Broken symmetries in topological condensed matter systems have implications
for the spectrum of Fermionic excitations confined on surfaces or topological
defects. The Fermionic spectrum of confined (quasi-2D) $^3$He-A consists of
branches of chiral edge states. The negative energy states are related to the
ground-state angular momentum, $L_z = (N/2) \hbar$, for $N/2$ Cooper pairs. The
power law suppression of the angular momentum, $L_z(T) \simeq (N/2)\,\hbar\,[1
- \frac{2}{3}(\pi T/\Delta)^2 ]$ for $0 \le T \ll T_c$, in the fully gapped 2D
chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We
discuss the effects of wave function overlap, and hybridization between edge
states confined near opposing edge boundaries on the edge currents,
ground-state angular momentum and ground-state order parameter of superfluid
$^3$He thin films. Under strong lateral confinement, the chiral A phase
undergoes a sequence of phase transitions, first to a pair density wave (PDW)
phase with broken translational symmetry at $D_{c2} \sim 16 \xi_0$. The PDW
phase is described by a periodic array of chiral domains with alternating
chirality, separated by domain walls. The period of PDW phase diverges as the
confinement length $D\rightarrow D_{c_2}$. The PDW phase breaks time-reversal
symmetry, translation invariance, but is invariant under the combination of
time-reversal and translation by a one-half period of the PDW. The mass current
distribution of the PDW phase reflects this combined symmetry, and originates
from the spectra of edge Fermions and the chiral branches bound to the domain
walls. Under sufficiently strong confinement a second-order transition occurs
to the non-chiral ``polar phase'' at $D_{c1} \sim 9\xi_0$, in which a single
p-wave orbital state of Cooper pairs is aligned along the channel.
