I'm currently reading the paper "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase" by Barry Simon.

Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is a $U(1)$ vector bundle. Parallel transport gives rise to the holonomy group, which assigns to each curve $C$ a number $e^{i\gamma(C)}$ that indicates how a vector is "rotated" when transporting it along the curve. In turns out that the phase change $\gamma(C)$ can be expressed as an integral of the curvature form over any surface $S$ that delimits the curve, $C = \partial S$,

$$ \gamma(C) = \int_{S} F^{\nabla} .$$

I am interested in the integral of the curvature form over the whole manifold, which turns out to be an integer multiple of $2\pi$,

$$ \int_{M} F^{\nabla} = 2\pi k, k\in\mathbb{Z}$$

Simon notes that this "standard fact" is a consistency condition on the holonomy group. I can understand that: integrating over the whole manifold is like taking the holonomy of the constant path, which must be the identity.

What I would like to understand is the generalization to higher Chern classes. For instance,

Why is the integral of the second Chern form an integer multiple of $4\pi^2$?

$$ \int_{M} F^{\nabla}\wedge F^{\nabla} = 4\pi^2 k, k\in\mathbb{Z}$$

I have a pedestrian proof for special cases, but I would like to understand a general reason behind this phenomenon. Is there a "higher holonomy" at work here?

Obviously, my knowledge of vector bundles and characteristic classes is rather limited. I can find my way around the book "From Calculus to Cohomology", but have by no means absorbed all the material. Basically, my question is why the Chern classes defined via connections are normalized with a factor of $1/(2\pi)^k$.