The electroweak monopole supports a single fermion zero mode; spectral mirror symmetry then implies the monopole has fermion number exactly 1/2.
Fermion number 1/2 of sphalerons and spectral mirror sym- metry
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Fermion Zero Modes and Fermion number $1/2$ of the Electroweak Monopole
The electroweak monopole supports a single fermion zero mode; spectral mirror symmetry then implies the monopole has fermion number exactly 1/2.