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I am following David Tong's lecture notes on the quantum Hall effect and am confused by his derivation on page 60, where he shows a connection between the Hall conductivity and the Chern number.

What is the motivation for and/or physical meaning of taking the integral over $\mathcal{F}_{xy}$ over the parameter torus in the expression $\sigma_{xy} = -\frac{e^2}{\hbar} \mathcal{F}_{xy}$? Is it trivial that after this integration, the left hand side of the equation still equals $\sigma_{xy}$?

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As this question has not yet attracted an answer, I will post the answer that I have so far come up with here. Note that this is not yet as strong as I would like, so additions would be very welcome.

The thing to realize is that the derivation in the mentioned lecture notes holds for the lowest Landau level (LLL) only. If the system is not in the LLL, the Berry curvature may depend on momentum: $ \mathcal{F}_{xy} = \mathcal{F}_{xy} (k) $. In this case, the equation $ \sigma_{xy} = \frac{e^2}{\hbar} \mathcal{F}_{xy} $ no longer holds without taking the average over the Brillouin zone.

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