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This is why people start to abandon calling the zero modes in vortices "Majorana fermions" because they are NOT fermions. $\gamma$ is a Majorana zero mode, which means it always has to pair up with another Majorana zero mode to form a 2-dimensional Hilbert space. Exchanging vortices generates a nontrivial unitary transformation on the degenerate space.


if by "gauge transformation in the vielbein" you mean the local lorentz transformation that acts on one of the indices of the vielbein, then the answer is no. Because this local symmetry is in addition to the general coordinate transformation, and not part of it. In other words all veilbeins that are related by a gauged lorentz transformation correspond to ...

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