TITLE: What are the implications of torsion in H^2 for geometric quantization?
QUESTION [11 upvotes]: Given a real manifold $M$ with symplectic $2$-form $\omega$,
one can ask whether the cohomology class $[\omega] \in H^2(M;{\mathbb R})$ lies in the image of 
$H^2(M;{\mathbb Z})$. If so, one can ask for a line bundle ${\mathcal L}$
with $c_1({\mathcal L}) = [\omega]$ (or even better, 
a connection $\alpha$ on $\mathcal L$ whose curvature $curv(\alpha)$ is $\omega$).
In the weakest definition of "geometric quantization", one puts a compatible
almost complex structure on $M$ and uses it to define the pushforward
of $\mathcal L$ to a point in $K$-theory. Call this $Q(M)$.
Are there examples worked out somewhere in which $H^2(M;{\mathbb Z})$
has torsion, so that $\mathcal L$ is not uniquely determined by $\omega$?
Can $Q(M)$ depend on the choice of $\mathcal L$?
I don't have any very good reason for asking this, other than I've felt
it to be a hole in my understanding of geometric quantization. The spaces I care about quantizing never seem to have torsion in $H^2$.

REPLY [4 votes]: It is the group of periods of a closed 2-form $\omega$ which plays a role on the different quantizations. Every closed 2-form $\omega$ on a manifold $M$ (more generally on a diffeological space) is the curvature of a connexion on an integration bundle, a principal bundle with group the torus of periods $T_\omega = {\bf R}/P_\omega$, where $P_\omega$ is the group of periods
$$
P_\omega = \{ \int_\sigma \omega \mid \sigma \in H_2(M,{\bf Z})\} \subset {\bf R}.
$$
The different integration structures (bundle + connexion) are classified by $H^1(M,T_\omega)$, the different integration bundles only are classified by ${\rm Ext}(H_1(M,{\bf Z}),P_\omega)$. The torsion plays a role at the level of $H_1(M,{\bf Z})$. For example, if this group has no torsion and $M$ is compact then the integration bundle is unique, up to an equivalence.
The torus of periods $T_\omega$ is a priori equipped with the quotient diffeology of $\bf R$. Of course $T_\omega$ is a manifold only if the group of periods is generated by one number (we say that $\omega$ is integer), that is, $P_\omega = a{\bf Z}$ one says then that $\omega$ is quantizable if $a$ is a multiple of $\hbar$, $a = k\hbar$, $k \in {\bf Z}$ (it's just a normalization). But note that the construction of the integration structure doesn't need $\omega$ to be integer.
Actually this proposition is true in general if the group of periods is a strict subgroup of $\bf R$, which is always the case for a second countable manifold, but it may be not happen if $M$ is just a general diffeological space, or some very special manifold.

References

La trilogie du moment, Annales de l'Institut Fourier, t. 45, n°3 (1995)
Diffeology, in Mathematical Surveys and Monographs, 185, AMS Providence RI, (2013).