I'm looking for a reference for the following.
Fix a connected, closed differentiable manifold $M$. Let $d\theta$ be the angle form $S^1$.
On one hand, given a differentiable map $f : M \to S^1$, we get a closed $1$-form $f^*(d\theta)$; this $1$-form has the property that it integrates to an integer on any closed path.
On the other hand, given a closed $1$-form $\omega$ on $M$ with the property that it integrates to an integer on any closed path, one can get a differentiable map $f : M \to S^1$ as follows: pick $x \in M$ arbitrarily, and for all $y \in M$, pick any path $p : [0,1] \to M$ from $x$ to $y$ and let $$ f(y) = \left(\int_0^1 \omega_{p(t)}(p'(t)) dt\right) \mod \mathbb{Z}, $$ where $\mod \mathbb{Z}$ refers to the fact that I am defining $S^1 = \mathbb{R}/\mathbb{Z}$.
This defines a bijection between the set of closed $1$-forms of $M$ that integrate to an integer on any closed path, on one hand, and the set of differentiable maps $M \to S^1$ modulo rotations of $S^1$, on the other hand.