In defining a nonzero Chern number as the integral of Berry curvature over the parameter manifold: $$n=\frac{1}{2\pi}\int_{S}{\mathcal{F}}{dS}$$ does the integral exist in a general Riemann sense, or is the integration carried out in some limited sense, e.g. as a principal value? My physical intuition suggests that the Dirac string should cancel the rest of the flux if it is included in the integration, so that $n=0$ always. On the other hand if the integral is taken with initial exclusion of a small circle and then shrinking the circle radius to $0$, then the flux of Dirac string wil be left out and $n$ can take nonzero values.
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