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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 the same bundle a possibly different with the same curvature, since the space of connections is affine we get a 1-form on the base which measures the 'difference' difference between the two connectionsis 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 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. 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).

2 added 108 characters in body

Your proof is correct. Given a principal bundle with connection having the given curvature, we can add and a flat second connection to on the given connection, and end up same bundle with the same curvature, since the space of connections is affine we get a 1-form on the base which measures the 'difference' between the two connections. Thus one gets an ambiguity in the choice of 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 the group on the right is that of integral differential 2-forms on $M$. The central group is the 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. 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).

1

Your proof is correct. Given a principal bundle with connection having the given curvature, we can add a flat connection to the given connection, and end up with the same curvature. Thus one gets an ambiguity in the choice of 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 the group on the right is that of integral differential 2-forms on $M$. The central group is the 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. 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).