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In the Hall effect, the edge modes that possess an anomaly are connected to the bulk in such a way that the total system is gauge invariant and and has a conserved current. The Bardeen Zumino consistancy conditions arise from considering the current $J_{\mu {\rm consistent}}$ as the functional derivative with respect to $A_\mu$ of the edge effective action ...


In 1969 Adler [https://inspirehep.net/record/55000] and, Bell & Jackiw [https://inspirehep.net/record/54998] showed that in the UV divergent triangle Feynman diagram made up of one axial and two vector currents, only the vector current is conserved, whereas the conservation law of the axial current is broken. Hence the names ABJ or chiral anomaly. ...

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