TITLE: Intuitive explanation for the Atiyah-Singer index theorem
QUESTION [77 upvotes]: My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem.
I'm trying to learn the Atiyah-Singer index theorem from standard and less-standard sources, and what I really want now is some soft, heuristic, not-necessarily-rigourous intuitive explanation of why it should be true. I am really just looking for a mental picture, analogous somehow to the mental picture I have of Gauss-Bonnet: "increasing Gaussian curvature tears holes in a surface".
The Atiyah-Singer theorem reads $$\mathrm{Ind}(D)=\int_{T^\ast M}\mathrm{ch}([\sigma_m(D)])\smile \mathrm{Td}(T^\ast M \otimes \mathbb{C})$$
What I want to understand is what the Chern character cup Todd class is actually measuring (heuristically- it doesn't have to be precisely true), and why, integrated over the cotangent bundle, this should give rise to the index of a Fredholm operator. I'm not so much interested in exact formulae at this point as in gleaning some sort of intuition for what is going on "under the hood". The Chern character is beautifully interpreted in this answer by Tyler Lawson, which, however, doesn't tell me what it means to cup it with the Todd class (I can guess that it's some sort of exponent of the logarithm of a formal group law, but this might be rubbish, and it's still not clear what that should be supposed to be measuring). Peter Teichner gives another, to my mind perhaps even more compelling answer, relating the Chern character with looping-delooping (going up and down the n-category ladder? ), but again, I'm missing a picture of what role the Todd class plays in this picture, and why it should have anything to do with the genus of an elliptic operator. I'm also missing a "big picture" explanation of Fei Han's work, even after having read his thesis (can someone familiar with this paper summarize the conceptual idea without the technical details?). Similarly, Jose Figueroa-O'Farrill's answer looks intriguing, but what I'm missing in that picture is intuitive understanding of why at zero temperature, the Witten index should have anything at all to do with Chern characters and Todd classes.
I know (at least in principle) that on both sides of the equation the manifold can be replaced with a point, where the index theorem holds true trivially; but that looks to me like an argument to convince somebody of the fact that it is true, and not an argument which gives any insight as to why it's true.
Let me add background about the Todd class, explained to me by Nigel Higson: "The Todd class is the correction factor that you need to make the Thom homomorphism commute with the Chern character." (I wish I could draw commutative diagrams on MathOverflow!) So for a vector bundle $V\longrightarrow E\longrightarrow X$, you have a Thom homomorphism in the top row $K(X)\rightarrow K_c(E)$, one in the bottom row $H^\ast_c(X;\mathbb{Q})\rightarrow H_c^\ast(E;\mathbb{Q})$, and Chern characters going from the top row to the bottom row. This diagram doesn't commute in general, but it commutes modulo the action of $\mathrm{TD}(E)$. I don't think I understand why any of this is relevant.
In summary, my question is

Do you have a soft not-necessarily-rigourous intuitive explanation of what each term in the Atiyah-Singer index theorem is trying to measure, and of why, in these terms, the Atiyah-Singer index theorem might be expected to hold true.

REPLY [22 votes]: I am going to give a topologically biased answer, which will proceed by restating what the Index Theorem says so that its plausibility (though not its truth) is more immediate.  [Oops - I see that Paul Siegel said much of this in the comments following his answer - oh well.]

Philosophy: It is a good thing to take a homology and cohomology theory and find geometric/ analytic models for them as well as the linear and Poincare-Lefshetz duality between them, when the spaces in questions are manifolds.

The most basic example is of course ordinary homology and cohomology, which are quite familiar, but worth revisiting since there are actually a few different models of this philosophy. The standard geometric model for homology is chains, and then cohomology can be obtained as a "formal linear dual."  But in an oriented manifold, one can show that differential forms provide functionals on chains and go on to establish the de Rham theorem.  Or one can take proper submanifolds as partially-defined cochains and get an intersection-theoretic interpretation of cohomology.  Or one can decide that de Rham theory is so wonderful that it should be the basic theory, and then passing to linear duals from there leads to the theory of currents to represent homology.
Another example which works very easily but is not as well known is bordism and cobordism.  By definition bordism groups are defined by maps from manifolds.  Cobordism is typically defined by maps to Thom spectra, but standard transversality shows that such maps are represented by proper submanifolds.  The linear/Poincare/Lefshetz duality between bordism and cobordism is given by intersection theory.
The Index Theorem manifests this philosophy for K-theory.  K-theory cohomology classes are of course represented by formal sums (or complexes) of vector bundles.  K-homology, as developed in Higson and Roe's book, is represented by Fredholm operators.  The pairing between them is roughly "counting the dimensions of spaces of solutions of an operator on a vector bundle."  (On some planet with very advanced topologists but relatively weak other flavors of math, one could envision the topologists "inventing" this kind of analysis just so they could have more fun with K-theory.)  While the analysis (pseudodifferential operators and the like) which shows that Fredholm operators "pair" with vector bundles is standard enough, showing that there is such a homology theory is of course involved, and takes up much of the Higson-Roe book.
Thus, the Index Theorem follows immediately from the stronger statement that one can define K-homology theory using differential operators and that the pairing between that and K-theory is given by the Index. In order to get the cohomology statement, as Paul Siegel mentions above, one uses the Chern character to translate from K-theory to cohomology but then must multiply by the Todd genus because the Chern character does not preserve Thom classes.
I realize that this stronger statement while perhaps feeling plausible is more of a formal answer than the geometric answer (like for Gauss-Bonnet) that you sought.  But when I studied the Index Theorem, mostly on the topological side, years ago I found that this formalism did help me develop some "pictures" for some of the arguments (for example the Fredholm theoretic proof Bott periodicity etc).
By the way, while we're mentioning this philosophy, there is one case which is very much a hot/interesting topic, namely for elliptic cohomology where objects like conformal nets, higher categories and "stringy" topology of manifolds all need to be further developed to tell the story.