Abelian flat connection maps to zero

Question

In an abelian gauge theory, a flat connection is an $A_\mu$ such that $F_{\mu\nu}=0$. I have to prove that these connections are equivalent to $0$, i.e. there is a gauge transformation that maps the flat connection to zero.

Here's my attempt:

$$A_{\mu}~~|~~F_{\mu\nu}=0$$

$$0=A'_\mu=A_\mu+\partial_\mu\alpha~~\Rightarrow~~A_{\mu}=-\partial_\mu\alpha,$$ so I have to prove that $\exists~\alpha~|~A_{\mu}=-\partial_\mu\alpha$.

In three dimensions, if a vector field satisfies $\vec{\nabla}\times\vec{V}=0$, then it can be written as the gradient of a scalar function $\vec{V}=\vec{\nabla}\phi$. I would generalize this to $d$ dimensions this way:

$$\epsilon^{\mu_1\mu_2\cdot\cdot\cdot\mu_d}\partial_{\mu_1}A_{\mu_2}=0~~\Rightarrow~~A_\mu=-\partial_\mu\alpha$$

But $F_{\mu\nu}=0$ implies $\epsilon^{\mu_1\mu_2\cdot\cdot\cdot\mu_d}\partial_{\mu_1}A_{\mu_2}=0$ so it is proven.

My question is if the generalization to $d$ dimensions is right, and how could I prove it.

Answer 1

Hint: Given an abelian flat connection $F=\mathrm{d}A=0$, from Poincare Lemma we know that there exists a locally$^1$ defined $0$-form gauge transformation $\alpha$, so that 1-form gauge potential $A=\mathrm{d}\alpha$.

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$^1$ The corresponding global problem can have topological obstructions.