Hey,


For personal exercising purposes I try to give a proof, that a U(1)-principal-bundle has curvature $\alpha$ iff the cohomology class of $\alpha$ is integral:


By the Cech-deRham-isomorphism a $[\alpha] \in H^2\_{DR}(M, \mathbb{Z})$ iff $[\alpha] \in H^2\_{Cech}(M, \mathbb{Z})$. Then we can use the isomorphism $H^2\_{Cech}(M, \mathbb{Z}) \tilde= H^1\_{Cech}(M, U(1))$ to construct an U(1)-principal bundle P up to equivalence by defining its transition functions.
To define a connection 1-Form $\varphi$ use the local trivialization $h\_j: U\_j \times U(1) \mapsto \pi^{-1}(U\_j)$ and set
\begin{equation}
h\_j^\* \varphi = \pi^\* \beta\_j + d\vartheta,
\end{equation}
where $d \beta\_j = \alpha$ on $U\_j$ and $\vartheta$ is the coordinate in U(1). This defines a connection with curvature $d \varphi = \alpha$.


I think this should it be. Any remarks/corrections are welcome!


But I read, that the various choices of such an principal bundle with given curvature are parametrized by $H^1(M,U(1))$, which in view of my proof makes absolutely no sense! Where lies the error?


Thank you for your help! 


P.s. - a second question: How does the curvature of the principal bundle defines the curvature of an associated (line)bundle? In the special case of a U(1)-bundle with curvature $\alpha$ and the representation $x \mapsto exp(-i k x)$, is the curvature of the assoziated line bundle given by $k \cdot \alpha$??



Principal Bundle with given Curvature
EDIT: I'm not very happy the the exposition below. But I hope you get the idea.


Your proof is correct. Given a principal bundle with connection having the given curvature, and a second connection on a possibly different with the same curvature, the difference between the two is a bundle with a flat connection (note that the category of $U(1)$-bundles with connection is a symmetric 2-group). Thus one gets an ambiguity in the choice of bundle and connection. This information is reflected in the short exact sequence


$$
0 \to H^1(M,U(1)) \to \check{H}^1(M) \to \Omega^2\_\mathbb{Z}(M) \to 0
$$


where the left cohomology group has *discrete* coefficients and classifies bundles with flat connections, and the group on the right is that of *integral* differential 2-forms on $M$. The central group is the [ordinary differential cohomology](http://ncatlab.org/nlab/show/ordinary+differential+cohomology) of $M$. One way it may be thought of is as a moduli space (which has a group structure) of $U(1)$-bundles with connection. The right map is the map that sends a bundle with connection to its curvature form.


There is another short exact sequence dealing with the characteristic class of the bundle, see Proposition 1 at [this nLab page](http://ncatlab.org/nlab/show/ordinary+differential+cohomology). It describes the kernel of the (surjective) homomorphism $\check{H}^1(M) \to H^2(M,\mathbb{Z})$, sending a bundle with connection to its first Chern class (which one can think of as being a Cech class).


