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!
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.
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.
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.
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.