Superconductors are bosonic at low energy yet carry a gravito-magnetic anomaly from fermion parity gauging that forbids trivial massive phases in 3D and 4D.
Braiding statistics and link invariants of bosonic/fermionic topological quantum matter in 2+1 and 3+1 dimensions,
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Gauging in superconductors and other electronic systems
Superconductors are bosonic at low energy yet carry a gravito-magnetic anomaly from fermion parity gauging that forbids trivial massive phases in 3D and 4D.