I am trying to understand Wen and Zee's article on topological quantum numbers of Hall fluid on curved space: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.953(adsbygoogle = window.adsbygoogle || []).push({});

They passingly mentiond the fact that a spinning particle with orbital angular momentum $s$ moving on a manifold with Christoffel connection $\omega$ will acquire a Aharanov-Bohm like phase of $s \oint \omega$.

I can sort of see why the Christoffel connection will give rise to such a phase, since by analogy with the magnetic vector potential, the Christoffel connection will enter into the covariant derivative the same way as a magnetic vector potential. However, I do not understand why it would couple to the orbital angular momentum. I would really appreciate if someone could show me a derivation or point me to some references.

Thanks.

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# A Orbital Angular momentum couples to Christoffel connection

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