Timeline for Intuitive explanation for the Atiyah-Singer index theorem
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| May 23, 2017 at 15:44 | comment | added | Sebastian Goette | The heat kernel proof surely does not really nicely explain the topological meaning of the $\hat A$-class. It captures some of its geometric meaning, though. Even better in this respect is the proof by Berline-Vergne using the Laplacian on principal bundles, which gives one possible truely geometric explanation of $\hat A$ through the Jacobian on the principal bundle. | |
| Sep 5, 2010 at 12:26 | vote | accept | Daniel Moskovich | ||
| May 5, 2010 at 13:00 | comment | added | Paul Siegel | One last vague comment which might intrigue you. From a modern point of view, this is often expressed by encapsulating Bott periodicity in the construction of K-homology (dual to K-theory). An elliptic operator gives rise to a K-homology class, and the index arises as a natural pairing between K-theory and K-homology. The proof of the index theorem comes down to using the product structure in K-homology to "compute" the index. One advantage of this approach (of many) is that $ind_a$ and $ind_t$ are constructed in very similar ways, and one is more likely to expect that they are the same. | |
| May 5, 2010 at 12:20 | comment | added | Paul Siegel | As for why $ind_t = ind_a$, the key idea is expressed by a compatibility between $ind_a$ and the Thom map. Specifically if $i: M \to N$ is an inclusion then the map $i_!: K(T^*M) \to K(T^*N)$ described above in the case $N = \mathbb{R}^n$ commutes with $ind_a$. This basically expresses the close relationship between Bott periodicity and the Thom map in K-theory. In the IEO papers, A-S characterized $ind_t$ according to two axioms: $ind_t$ is the identity if $M$ is a point, and $ind_t$ commutes with $i_!$. They then proved the index theorem by showing that $ind_a$ satisfies the axioms. | |
| May 5, 2010 at 11:47 | comment | added | Paul Siegel | As I mentioned, I can't give great intuition for why the Todd class specifically rears its head (it isn't hard to prove that the Todd class does the job, but the geometric meaning of the proof eludes me), but I hope my argument at least explains why the chern character of the symbol is relevant. It also more or less explains where the integral comes from: the map $H(U) \to H(T^* \mathbb{R}^n) \to \mathbb{R}$ which parallels the corresponding map in K-theory is just given by integration. I didn't do a great job of explaining the map $K(U) \to K(T^* \mathbb{R}^n)$, but it all works out. | |
| May 5, 2010 at 0:50 | comment | added | Daniel Moskovich | This is a really helpful answer! One thing I still don't understand with this approach (besides the passage from K-theory to homology and (in particular) how the topological index becomes the integral of a certain local quantity) is why one might expect the analytic index and the topological index to coincide. Why should these two integers be related? | |
| May 4, 2010 at 19:26 | history | answered | Paul Siegel | CC BY-SA 2.5 |