Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization? The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
 A: Indeed, the quantization of angular momentum and of spin is an example of the integrality condition in the geometric quantiation of the 2-sphere, regaded as a symplectic phase space with its canonical volume form taken as the symplectic form. This is spelled out in a good bit of detail in the nLab entry geometric quantization of the 2-sphere.
Generally, the integrality condition in geometric quantization affects other "internal" degrees of freedom in quantum physics similarly. For instance it plays a central role in the geometric quantization of phase spaces which are coadjoint orbits of some Lie group $G$. Here the Lie group acts by quantum operators on the resulting Hilbert space and hence geometric quantization here produces (irreducible) complex Lie group representations,  a process known as Kirillov's orbit method.
Physically, the representations appearing this way are precisely the quantization of the Wilson line 1d field theory inside Chern-Simons theory, a phenomenon maybe first hinted at on p. 22 of Witten's famous Jones polynomial article.
What is maybe noteworthy is that in the modern cohomological formulation of geometric quantization as push-forward in K-theory (see at quantum state space as the indexof a Spin^c Dirac operator) what matters is not primarily that the prequantum line bundle provides a lift of the symplectic form of phase space to integral cohomology (really to ordinary differential cohomology), but that it provides a lift to K-theory. Because one finds that what is really going on in geometric quantization in the presence of a complex (Kähler) polarization is that it amounts to computing the index of the înduced $Spin^c$-Dirac operator coupled to the prequantum line bundle, hence the push-forward of the prequantum line bundle regarded now as a representative of a K-theory class. (Doing this in equivariant K-theory produces the qauntum observables acting on the Hilbert space of states, as above in the examples of the orbit method).
So one may wonder if maybe at a deeper leverl the cohomological condition that plays a role - for instance in more general situations not covered by traditional geometric quantization, such as the quantization of Poisson manifolds -- is not really one in ordinary integral cohomology, but one in K-theory. Indeed, this turns out to be the case. For some observations along these lines see the examples-section in


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*Joost Nuiten, Cohomological quantization of local prequantum boundary field theory
A: For a closed 2-form $\omega$ on a manifold $M$, the integrality of the closed 2-form, that is,
$$
\int_\sigma \omega \in a{\bf Z}, \quad \mbox{for all} \quad \sigma \in H_2(M,{\bf Z}),
$$
for some real number $a$, ensures the existence of a principal circle-bundle $Y$ (and its associated line bundle $L$) over $M$ and a connexion $\lambda$ with curvature $\omega$. Then, it is possible to lift some groups of automorphisms of $\omega$ (subgroups of ${\rm Diff}(M,\omega)$) as groups of automorphisms of $(Y,\lambda)$. This procedure is called prequantization because it is the first step of an answer to the Dirac program of quantization consisting in representing symmetries in classical mechanics by unitary transformations in some Hilbert space (that is supposed to have a physical meaning). I would not want to develop why one needs this bundle, at the first place, to answer Dirac's program, and are not contented just with the automorphisms of $\omega$, because it will lead us too far. If you are happy with this answer I'm fine, else I'll try to say a more few words(*).
P.S. The fact that the number $a$ is required to be a multiple of $\hbar$ comes just from physics consideration.
(*) Edited: I added a few words here
