# Atiyah-Singer index theorem

Every year or so I make an attempt to "really" learn the Atiyah-Singer index theorem. I always find that I give up because my analysis background is too weak -- most of the sources spend a lot of time discussing the topology and algebra, but very little time on the analysis. Question : is there a "fun" source for reading about the appropriate parts of analysis?

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It doesn't really address the question you ask, but I'm adding a link to a related answer, just in case it's useful: mathoverflow.net/questions/14714/… –  Minhyong Kim Jun 20 '10 at 21:22

I found Booss, Bleecker: "Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics" (review) very beautifull and had read it just for fun. It is a very nice piece of exposition, motivates everything and demands from the reader only very little preknowledge.

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There's an English translation of that book? I saw the book listed in the library catalogue, and went to check it out, only to find that it was in (the original) German. Now I need to try to find the English version. –  Willie Wong Jul 28 '10 at 14:28
There's definitely an English translation! If you look on one of the author's websites (I forget which one), you can also find a chapter or two of an unfinished second edition in English. –  Andy Putman Jul 28 '10 at 16:47

I know it may seem rather "old", but the notes from the IAS "Seminar on the Atiyah-Singer Index Theorem from back in 1965 (published by Princeton Univ. Press) may be just what you are looking for, since it covers all the analytic machinery in great detail. It was written to be easily accessible to a math graduate student who had a basic analysis course.

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That book is just as amazing today as it was, I imagine, when it was written! –  Mariano Suárez-Alvarez Feb 24 '11 at 7:23

You need to understand pseudodifferential operators if you want to understand the original statement of the full Atiyah-Singer index theorem. However, in most applications to differential geometry, only the theorem for twisted Dirac operators is needed. (One of the main results of Atiyah and Singer is that the Bott periodicity theorem - or rather, its generalization to vector bundles, the Thom isomorphism theorem for K-theory - reduces the general case to that of twisted Dirac operators.)

If you want to learn the theory of pseudodifferential operators, I recommend the original papers of Kohn and Nirenberg and Hörmander. This theory is not needed to prove the Atiyah-Singer index theorem: you can get away with the existence of an asymptotic solution of the heat equation. To see this in action, see the paper of McKean and Singer.

One advantage of the heat-kernel approach is that it is well-adapted to study the generalizations of the theory, such as the theory of analytic torsion and the family index theorem.

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Personally, I would have written... "To see this in action, see the book of Berline, Getzler, and Vergne" :) As I recall, however, one needs to do a little bit better than just the existence of an asymptotic expansion for the heat kernel - one also needs the expansion to be compatible with Clifford degree in an appropriate sense. I've always found that aspect of the analysis to be very subtle. –  Paul Siegel Feb 24 '11 at 10:26

I first learnt about the Atiyah-Singer index theorem from Shanahan's Springer notes (638). I liked it because while developing the main theory, it went through the standard examples (Dirac, Dolbeaut, de Rham, signature) in some detail. At the time I was primarily interested in using index theory so wasn't so bothered about the details of the proof, but it does at least sketch the proof (it may do more, I don't remember and don't have a copy on my shelves) but I do remember that the words "pseudo-differential operator" occur which suggests that even if sketchy, the main points are all there. He also covers the equivariant theory.

Another place where it's put in context is Spin Geometry by Lawson and Michelsohn. That's quite nice because the whole theory of Dirac operators and Clifford algebras is developed from scratch so there are lots of "entry points" depending on whether or not you're more of an algebraist or geometer or other.

MR numbers:

• The Atiyah-Singer Index theorem: MR487910
• Spin Geometry: MR1031992
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Chapter 4 of Wells's "Differential Analysis on Complex Manifolds" is titled "Elliptic Operator Theory" and is, I think, close to what you want. It certainly explains why elliptic operators have finite-dimensional kernels and cokernels.

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The exposition in Wells book contains much, much more than the Fredholm property. In fact, if you understand that chapter, you are ready for reading Atiyah and Singer's original paper. –  Johannes Ebert Feb 12 '11 at 14:21

I'm also rather weak in analysis background, but I found the original paper rather readable.

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Andy: Nutshell version. In $R^n$ an order d linear diff op can be described by: Fourier transform, then multiply by symbol, then transform back. The symbol is polynomial of degree d. Composition of operators corresponds, modulo lower order stuff, to multiplication of symbols. Elliptic means that the symbol is invertible modulo lower order. (Sometimes "symbol" means symbol modulo lower order.) Introducing non-polynomial symbols, enlarge the class of operators. These pseudo-diff ops can have negative order. For every elliptic op $P$ of order d there is an approximate inverse $Q$ of order -d. –  Tom Goodwillie Jul 28 '10 at 15:49
Continued: Ops of order <0 make things smoother. The approximate inverse of the elliptic $P$ can be fixed up so that $PQ-1$ and $QP-1$ have order $-\infty$ rather than just $<0$, and so make all distributions $C^\infty$. This implies that everything in kernel of $P$ is smooth and a similar statement for cokernel. All of these phenomena can be translated to a manifold, of course. –  Tom Goodwillie Jul 28 '10 at 15:59
Continued: Technically degrees of smoothness are kept track of using Sobolev norms; roughly a distribution is in the Sobolev space $H_{-s}$ (or is it called $H_s$?) if you can differentiate it $s$ times and still have an $L^2$ function. Order d ops take $H_s$ boundedly to $H_{s+d}$. $PQ-1$ and $QP-1$ thus take $H_s$ boundedly to $H_{s-1}$, which means that $P$ and $Q$ as ops $H_s\to H_{s-d}$ and back are inverses modulo compact operators. That implies finite ker and coker. This passes to the limit as $s\mapsto -\infty$. By now I'm using compactness of the manifold (didn't in last comment). –  Tom Goodwillie Jul 28 '10 at 16:06
In some sense the index ker-coker is a virtual vector space depending continuously on the elliptic op, so it really doesn't care about the whole op (or the whole symbol) -- just the symbol modulo lower order stuff -- the rest is a contractible choice. The index theorem says exactly how it depends on that. I see now that this thread is from last October, so I feel a little silly about jumping in, but I thought it would be fun to write this sketch. –  Tom Goodwillie Jul 28 '10 at 16:09
I still noticed it. Thanks Tom! –  Andy Putman Jul 28 '10 at 16:32

There is also a "physicist's" proof of the index theorem. (No, really, keep reading!) Attached to every supersymmetric quantum field theory (or even quantum mechanics) there is something called the Witten index. There are quantum mechanical systems for which the Witten index coincides with the index of an elliptic operator (made out of the supercharges of the theory). It is not difficult to argue that the Witten index has a homotopy invariance property which allows one to compute it in different "temperature" regimes. At infinite temperature it simply counts the difference between the dimensions of the kernel and cokernel of the elliptic operator, whereas at zero temperature it can be evaluated to give an integral formula for the index in terms of characteristic classes. This can be made rigourous and the details can be found this paper of Getzler's.

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There's a more classical related-to-physics proof, due to Gilkey. "The index theorem and the heat equation," Publish or Perish Press, Boston Mass (1974), 125pp. –  Ryan Budney Dec 3 '09 at 16:41
Indeed! I had forgotten that one. Thanks. (I found the supersymmetric proof more intuitive, although at the end of the day it's very similar and possibly isomorphic to the one based on the heat kernel.) –  José Figueroa-O'Farrill Dec 3 '09 at 17:16
This seems to be rather a physical interpretation of the heat kernel proof... –  Johannes Ebert Oct 17 '10 at 11:18
Well, don't be fooled by words like "temperature",... The proof sounds physical because that is the scientific context in which it first appeared, but it's essentially isomorphic to the heat kernel proof. The word "heat" is not an accident :) –  José Figueroa-O'Farrill Oct 17 '10 at 13:27

I agree with T. Booss and Bleecker is a very good book to learn about the index theorem from. It begins very gently indeed. (I'd have simply voted T's response up, but I don't have the permissions to do that yet.)

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Ålthough written from the K-homology point of view, the book "Analytic K-homology" by Higson and Roe should be quite useful (both for basics about elliptic differential operators and index theory; iirc they sketch the proof of the index theorem for Spin^c manifolds).

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I'm reading the original paper right now and finding it a little terse in some places, but a good guide. "Spin Geometry" by Michelson and Lawson has a chapter that covers in great detail all the analysis you need for index theory as well as complete proofs of the index theorem in all of its forms. It also has some decent chapters on K-theory (What you really need is to understand the Thom Isomorphism in K-theory. Segal's "Equivariant K-Theory" also has a nice description.)

Hope this helps.

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If you have access to Atiyah's collected works at your library, try taking a look at those. There are a few transcribed lectures and short expository papers where he explains the context and motivation of the theorems. (If I remember correctly, they are classified as miscellanea and appear in the first volume, but there might also be some in the second volume. I don't have it at hand to check.)

He writes beautifully, and for myself, I didn't feel like I "got" the index theorem until I read these.

EDIT: Oops, I didn't read your question carefully enough: you are mostly looking for the analysis part. In that case I would just second the recommendation of the relevant chapter in Warner or Wells.

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Antony Wasserman has some course notes including the index theorem itself, at least in special cases, but also with some very readable notes on background aspects of op alg and op theory. IMHO the style is pretty compressed, but the tools used are relatively accessible.

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Perhaps you will also like

"Heat kernels and Dirac operators" by Nicole Berline,Ezra Getzler,Michèle Vergne

Perhaps not the "full" Atiyah-Singer index theorem, but that for Dirac-Operators. So if you are more a differential-geometer than an "analysis-guy", I think this is an appropriate reference.
(EDIT: As José Figuera-O'Farril remarked, AS for Dirac Operators should be enough to get AS for elliptic operators)

In addition another book by Gilkey (compare the post of José Figueroa-O'Farrill and the comment by Ryan Budney) should be very readable:
Invariance theory, the heat equation, and the Atiyah-Singer index theorem by Peter B. Gilkey

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But the general index theorem is the index theorem for some Dirac operator, so can't this be turned into a general proof? –  José Figueroa-O'Farrill Dec 4 '09 at 7:03
Hmm, I think you're right. With "full" I meant AS for elliptic operators (in fact weak elliptic is enough). But it should be enough to proof the index theorem for (generalized) Dirac operators. –  Spinorbundle Dec 4 '09 at 13:27
Isn't the general statement about elliptic pseudodifferential operators? As far as I know Dirac operators are elliptic differential operators, and even of a special kind. So I think I'm missing something very basic here; I always thought that the proof for Dirac operators was rather special. Pardon my ignorance, but I'm actually not of the field. –  Andrea Ferretti Dec 4 '09 at 14:08
As far as I know the AS index theorem is an index theorem for elliptic differential operators, but several proofs use pseudodifferential operators. According to wikipedia: "Most version of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators." So perhaps you're right, but for more differential-geometric guys (like me) the most interesting cases are (generalized) Dirac operators. (My comment above is perhaps incorrect, but I had a kind of approximation in mind; perhaps something can point out wheter this is true or not) –  Spinorbundle Dec 4 '09 at 21:51

Hi,

If you're still interested in improving your background for understanding the ASIT, you can try with these lecture notes, from a course given at the Utrecht University by E.P van den Ban and M. Crainic.

The course spent time in the analytic part of (this) proof, going through pseudo-differential operators and it's symbols, and showing the role of Fourier Analysis in it's construction.

Greetings.

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Liviu Nicolaescu's "Lectures on the Geometry of Manifolds" has a long chapter on differential operators on manifolds including elliptic operators.

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Liviu Nicolaescu has also the following set of lecture notes on the subject: nd.edu/~lnicolae/ind-thm.pdf –  Giuseppe Tortorella Feb 5 '11 at 10:32

I have also good experience with the book of Lawson and Michelsohn. The theory of pseudo-elliptic differential operators is also well explained in the book of Wells, named "Analysis and complex geometry" (or something like that).

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