TITLE: Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?
QUESTION [18 upvotes]: 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?

REPLY [5 votes]: My first remark about this question is a little bit pithy - the standard cohomological index formula works only for even dimensional manifolds while Chern-Weil theory is of course more general.  
For a more substantive answer, let's look at the index formula itself:
$Index(D) = \int_{T^*M} ch(\sigma_D) Todd(TM \otimes \mathbb{C})$
Here $ch(\sigma_D)$ refers to the chern character of the "symbol class" in K-theory of the elliptic operator $D$.  My first observation is that this formula places special emphasis on the Todd class of the tangent bundle, and it seems to me that one would have to place similar emphasis on the Todd class in Chern - Weil theory in order to answer your question in the affirmative.  I certainly do not claim that this is impossible - the Todd polynomial seems to take a privileged position in many places where it does not seem to belong - but I do not know how to do it.
Additionally, we are forced to ask what characteristic classes can be written in the form $ch(\sigma_D)$.  As Tom Goodwillie points out, we will get an integrality result if we plug in any K-theory class $x$ rather than a K-theory class of the form $\sigma_D$ (although it may be that all classes in $K(T^*M)$ are symbol classes), but there are still constraints on which characteristic classes can appear.  For example, we will only get classes in cohomology of even degree.
So there are a number of hints that the right hand side of the index formula is too specialized for Chern-Weil theory.  But the left hand side also helps answer your question.  Here are two general principles in index theory that will help me explain:

The index theorem for any operator can be deduced from the index theorem for a special class of operators on a special class of manifolds - the "spin$^c$ Dirac operators".  These operators are completely determined by the geometry of the manifolds on which they live.
The index depends continuously on the operator in an appropriate sense.  In particular, if $D_t$ is a continuous family of operators then $Index(D_t)$ is constant in $t$.

Combining these two facts, we see that the integers which we can obtain from the index theorem depend only on homotopy theoretic information about the manifold and the coefficient bundle (if applicable).  This is not true of all integers that can be obtained via Chern-Weil theory, and it demonstrates that index theoretic invariants are much too crude for ordinary cohomology theory.