I'm getting quite twisted around trying to figure out what all is quantized exactly in IQH looking at it from the Chern number perspective.
Let's suppose quantum hall on a torus -- I can apply a large gauge transform $e^{iqf} = \exp\left(iq\left(\frac{2 \pi N_1 x^1}{qL_1} + \frac{2 \pi N_2 x^2}{qL_2} \right)\right)$ and see that something is being changed by $N_1$ and something else is being changed by $N_2$. Geometrically these seem to just correspond to the $\mathbb{Z} \times \mathbb{Z}$ space corresponding to the number of times you can wrap non-contractable loops around the torus.
Relating this back to the canonical momentum $\Pi_i$, I observe that this large gauge change relates to a shift in momentum $\Pi_i \to \Pi_i + \partial_{x_i}f=\Pi_i+\frac{2 \pi N_i}{L_i}$
It seems that only after introducing a magnetic field and writing down magnetic translation operators that we see some holonomy introduced by moving around closed loops in the $x^1, x^2$ parameter space. Writing the holonomy measured by encircling a little patch $\text{d}A$ and then integrating this over the whole torus leads to the quantization of flux and the Chern number $c_1$.
At this point I'm completely twisted around (haha.), and am struggling to figure out what exactly it is that $N_1$, $N_2$, and $c_1$ measure and if they're at all interrelated, or separate things.
All these numbers appear to be related to different closed loops in $x^1,x^2$ space. In one I integrate in a straight line vertically that loops back to where it started. In another, I integrate a horizontal line that loops back to where it started. In the third, I integrate a bunch of little patches until I've covered the whole torus. What information am I extracting? This was all spurred by the, perhaps misguided, question "What loop extracts the Chern number?"
Side question: If I apply Stokes theorem and stitch together all the $\text{d}A$s, this would seem to be one grand loop in $x^1,x^2$ space on the edges of the square that have been identified! It would therefore seem that I've taken a null loop. Is this what people (Berneveig?) means by "the Chern number measures how Stokes theorem fails globally."
