Each of the $M$ vortices contributes one bulk <Majorana zero mode>. A finite fermionic system must have an even total number of Majorana zero modes. The boundary <Chiral Majorana edge mode> has no zero momentum in the antiperiodic sector but has one $k=0$ Majorana mode in the periodic sector. Therefore
$$
c(x+L)=
\begin{cases}
-c(x),&M\text{ even},\\
+c(x),&M\text{ odd},
\end{cases}
$$
or compactly $c(x+L)=(-1)^{M+1}c(x)$. This <boundary condition of a chiral Majorana edge mode> supplies the extra boundary zero mode exactly when the vortex count is odd.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 342 / 3 / e / Solution

On a closed boundary, a chiral Majorana field is antiperiodic in the sector with even enclosed vortex parity and periodic in the sector with odd enclosed vortex parity. Only the periodic sector contains a zero-momentum Majorana mode.


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