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?



Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?
Indeed, the quantization of angular momentum and of spin is an example of the integrality condition in the [geometric quantiation](http://ncatlab.org/nlab/show/geometric+quantization) 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](http://ncatlab.org/nlab/show/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](http://ncatlab.org/nlab/show/orbit%20method).
Physically, the representations appearing this way are precisely the quantization of [the Wilson line 1d field theory](http://ncatlab.org/nlab/show/Chern-Simons+theory#WithWilsonLineObservables) inside Chern-Simons theory, a phenomenon maybe first hinted at on p. 22 of Witten's famous [Jones polynomial article](http://ncatlab.org/nlab/show/Chern-Simons+theory#Witten).


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](http://ncatlab.org/nlab/show/geometric%20quantization#AsIndexOfSpinCDiracOperator)*) 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


* Joost Nuiten, *[Cohomological quantization of local prequantum boundary field theory](http://ncatlab.org/schreiber/show/master+thesis+Nuiten)*


