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Broken symmetries in topological condensed matter systems have implicationsfor the spectrum of Fermionic excitations confined on surfaces or topologicaldefects. The Fermionic spectrum of confined (quasi-2D) $^3$He-A consists ofbranches of chiral edge states. The negative energy states are related to theground-state angular momentum, $L_z = (N/2) \hbar$, for $N/2$ Cooper pairs. Thepower 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 2Dchiral A-phase reflects the thermal excitation of the chiral edge Fermions. Wediscuss the effects of wave function overlap, and hybridization between edgestates 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 phaseundergoes 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 PDWphase is described by a periodic array of chiral domains with alternatingchirality, separated by domain walls. The period of PDW phase diverges as theconfinement length $D\rightarrow D_{c_2}$. The PDW phase breaks time-reversalsymmetry, translation invariance, but is invariant under the combination oftime-reversal and translation by a one-half period of the PDW. The mass currentdistribution of the PDW phase reflects this combined symmetry, and originatesfrom the spectra of edge Fermions and the chiral branches bound to the domainwalls. Under sufficiently strong confinement a second-order transition occursto the non-chiral ``polar phase'' at $D_{c1} \sim 9\xi_0$, in which a singlep-wave orbital state of Cooper pairs is aligned along the channel.

Weyl Fermions and Broken Symmetry Phases of Laterally Confined $^3$He Films