It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is depended on level quantization of Chern-Simon theory.
$$ \frac{k}{4\pi}\int_W AdA $$ where $W$ is our three manifold. As I saw in literatures, for example at page 151, https://arxiv.org/abs/1606.06687, they look at the partition function of the theory at the finite temperature and take the spatial directions as a sphere $S^2$. So the total $2+1$ space-time is $W=S^2*S^1$ and it is easy to see that the level $k$ must be an integer for consistency of the quantum theory if we have a unit flux threading through the sphere.
$$ \frac{1}{2\pi}\int_{S^2} F=\frac{h}{e} $$
my question is that, in real situation, our sample is not a sphere and usually is a rectangular and (I think) experiments are performed at zero temperature. How Chern-Simon level have to be quantized in $W=\mathbb{R^2}*\mathbb{R}$ background.