A universal 4N-step word W_N=(Y^{-1}X^{-1})^N(YX)^N measures mutual statistics in d=p+q+1 and diagnoses mixed Bockstein anomalies.
Generalized Symmetries Phase Transitions with Local Quantum Fields
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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.
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Bockstein braiding statistics
A universal 4N-step word W_N=(Y^{-1}X^{-1})^N(YX)^N measures mutual statistics in d=p+q+1 and diagnoses mixed Bockstein anomalies.
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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.