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

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. Thereforeor 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.