I want to show that for a $U(1)$ bundle $P$ over a connected smooth 4-manifold $X$, the moduli space of Yang-Mills connection over $P$ is the torus $H^1(X,\mathbb{R})/H^1(X,\mathbb{Z})$.
Now I reduce the question above as following: suppose $s$ a map $X\to U(1)$(since $U(1)$ is commutative, the $G-$ equivalent property holds automatically), fix a connection $A$ on $P$, consider the gauge transformation $s^*A=s^{-1}ds+s^{-1}As=s^{-1}ds+A$. All I need is to show that for all map $s$, all differential from in the type $s^{-1}ds$ together represent the cohomology group $H^1(X,\mathbb{Z})$. But I have no idea to prove this.
Maybe I get into a wrong way since the gauge transformation formula is a local calculation and the cohomology is a global topological invariant. However since $s^{-1}ds=d{} \mathrm{ln}s$ seems I can get the desired integer cohomology class by "integral" the form $d{} \mathrm{ln}s$. I think I may lack of some fundamental knowledge so that I cannot step forward.
Could anyone help me? Thanks in advance.