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Berry phases are, in some sense, the common origin of all anomaly effects.
This is stated, for example in http://www.sciencedirect.com/science/article/pii/0550321388902556?via%3Dihub
https://michaelberryphysics.files.wordpress.com/2013/07/berry178.pdf and https://courses.cit.cornell.edu/ece5390/7_berry_phase.pdf
For a nice introduction, see Quantum Phases And Angles by R. Jackiw