Let $A$ be a *flat* connection on a principal $G$-bundle $G\hookrightarrow P\to M$.
Consider an *homotopically trivial* loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the boundary of an smoothly embedded disk.
I was trying to prove that the holonomy of $A$ along $\gamma$ is trivial (must be by flatness).
The holonomy of $A$ along $\gamma$ is given by
$$\exp^{-\int\_\gamma A} \quad ( \ = 1 \text{ by flatness}) $$
Therefore applying Stokes, we get ($F\_A=0$)
$${\int\_\gamma A}=\int\_D dA = \int\_D (F\_A - \frac 1 2 [A\wedge A])= \int\_D - \frac 1 2 [A\wedge A]$$



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> Supposing that $G$ is not abelian, then I would like to understand why $\int\_D [A\wedge A]$ lies in the kernel of $\exp$.
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I know other proofs of this fact, e.g flatness implies the horizontal distribution is trivial hence we can use charts where $A$ is identically zero. But I hope there is a simpler explanation for this, i.e. we can prove that trivial loops have trivial holonomy just using that $F\_A=0$ instead of the deeper/equivalent integrability of the horizontal distribution.



Flat connections, curvature and holonomy
The Stokes theorem must be modified first to deal with the nonabelian case.


See <http://arxiv.org/abs/0802.0663>,
Section 3.2, Theorem 3.4 and the displayed formula on top of page 48
for an appropriate formulation.


