In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F\_\nabla$, and then we get Chern classes of $E$ from the curvature form. A priori it looks like these live in $H^\*(X;\mathbb{C})$, but by an argument that I don't really understand they're actually in the image of $H^\*(X;\mathbb{Z})$ (which is where they're usually considered to live). Meanwhile, I've heard people say that whenever I see a arbitrary real constants that end up having to be integers I should wonder whether the Atiyah-Singer index theorem is lurking in there somewhere. Is there anything to this wild guess?



Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?
It's true that in some sense the Atiyah-Singer Index Theorem has led to some integrality results. The theorem states that for an elliptic operator on a compact manifold two numbers are equal. One of them, the "analytic index", is obviously an integer. The other one, the "topological index", which depends on the symbol of the operator, is obviously a rational number but not so obviously an integer. 


On the other hand, you can argue that the integrality phenomena revealed in this way don't really have anything to do with the analysis. 


There are two equivalent ways to define the topological index. There is the cohomological formula in terms of characteristic classes that live in rational cohomology. There is also the conceptually more fundamental $K$-theory definition of the topological index, in which you let the symbol of the operator give you an element of the compactly supported $K^0$ of the cotangent manifold and then use Bott periodicity and the fact that $K$-theory is a cohomology theory to get from here to an element of $K^0(point)\cong\mathbb Z$. The fact that the integer thus defined coincides with the rational number obtained from characteristic classes is a calculation involving the relationship between $K$-theory and ordinary cohomology, and does not rely on the fact that there was a differential operator involved.


