Roadmap to study Atiyah–Singer index theorem

Question

I am a physics undergrad and want to pursue a PhD in maths (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non departmental courses, thought I will be able to take them is a couple of semesters. Anyway, So I was thinking about taking a mini-project of sorts which I can self-study or maybe ask a prof. I had heard about this theorem while studying topological solitons in physics.

What are the prerequisites for studying the Atiyah–Singer theorem. I had a look online, but I couldn't figure out exactly, in what field this is, or what are the prerequisites. Wikipedia says it is a theorem in differential geometry, but obviously what differential geometry I know is insufficient. I know about manifolds, differential forms, lie derivatives, lie groups, and some killing vectors stuff. A look at the material online also talks about some operators in DG which I have never come across. Are there an algebraic topology prerequisites?

Please could you recommend me some references which can take me there?

Thanks in advance!

Answer 1

The Atiyah-Singer index theorem involves a mixture of algebra, geometry/topology, and analysis. Here are the main things you'll want to understand to be able to know what the index theorem is really even saying.

1. Algebra: The most important concept here is Clifford algebras. For example, Dirac operators arise from combining covariant differentiation and Clifford multiplication. You'll also want to learn about the associated spin groups.

2. Geometry/Topology: The most fundamental idea here is understanding vector bundles. The Atiyah-Singer index theorem is about elliptic differential operators between sections of vector bundles, so you won't get anywhere without a firm understanding of bundles. Next, you'll want to understand the basics of spin geometry and Dirac operators, especially if your interests are more physics-based. One nice form of the index theorem is the cohomological formula for the index in terms of characteristic classes. I would advise you to get familiar with cohomology and obtain a basic knowledge of characteristic classes (Chern-Weil theory is nice if you already have a geometry background). If you want to understand the original index theorem or how the cohomological formula for the index is derived from it, you'll need to learn some topological K-theory.

3. Analysis: Here the big topic is differential operators in the context of manifolds. Hence you'll want to know what a differential operator between vector bundles is. You'll need to know what the symbol of such a differential operator is, and also what it means for such a differential operator to be elliptic.

As for a reference, the classic text is Spin Geometry by Lawson and Michelsohn. An easier but less detailed introduction can be found in the relevant sections of Geometry, Topology, and Physics by Nakahara. There's a fair amount of other books but these are the two I know best. There are some good notes on spin geometry here. These notes by Nicolaescu may also be of interest to you.

Answer 2

There are two main approaches to the index theorem: $K$-theory and heat kernels. The latter approach treats the index of Dirac operators, which is actually more general than it sounds since for common situations all operators (in terms of index theory) are twisted Dirac operators.

Based on your background I would recommend the heat kernel approach since it is more geometric, physicsy, and you'd have to learn a decent amount of algebraic topology for the $K$-theory approach.

There is some algebraic topology that enters into both approaches and this is the theory of characteristic classes. In the heat kernel approach, one takes the Chern-Weil viewpoint, which is pretty easy to pick up as long as you are familiar with de Rham cohomology.

I found the most useful books to be Berline, Getzler, and Vergne's Heat Kernels and Dirac Operators and John Roe's Elliptic Operators, Topology, and Asymptotic Methods.

For the $K$-theory approach (which I think is useful to at least get a bit of an understanding of) I like the original papers as well as expository works of Nigel Higson e.g. Lectures on Operator K-Theory and the Atiyah-Singer Index.

Answer 3

There is a new wonderful, big and deep book dealing precisely with everything you need (assuming knowledge of "basic" differential geometry):

• Bleecker; Booß-Bavnek - Index Theory with Applications to Mathematics and Physics, International Press 2013, (792 pages!).

These are the same authors of the following old classic, but the new book is NOT a new edition but a completely NEW title (also with modern TeX typeset, figures and additional topics and details):

• Bleecker; Booß-Bavnek - Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics, Springer Verlag 1985.

You may also find in this other answer a very brief overview of the proof of Atiyah-Singer and why pseudo-differential operators are needed. There you will find links to lectures and other titles regarding the theorem.