I am interested in learning the Atiyah–Singer theorem, and its version for families of operators. For this purpose, I have tried to read the recent book by D.Bleecker et.al.. However I have difficulty understanding some proofs due to my weak analysis background.
I have learnt the related differential geometric notions (principal bundles, connections, Chern–Weil theory) and the algebraic topological notions (K-theory, characteristic classes) and feel comfortable with Clifford Algebras.
However, my analysis background is very weak. In real analysis, my knowledge is limited to what is covered in baby Rudin. I have not studied any Fourier analysis or measure theory. As regards functional analysis, I know basic notions about Hilbert spaces, but no spectral theory. As a result when concepts such as
- distributions and test functions
- convergence issues with Fourier transforms
- L^p spaces etc.
pop-up or the author proceeds to discuss Sobolev spaces and pseudo-differential operators, I find myself to be very confused and puzzled.
My question is :
- Given my background described above, which analysis books should I study in order to properly understand the analysis background required for the Atiyah–Singer index theorem?
For the time being, I wish to learn only the minimum analysis required for this purpose. Also I prefer to first read up the required pre-requisites rather than learning-on-the-go.
Also :
- Given that I am not strong at analysis, should I consider studying a different book (e.g. Spin Geometry by Lawson–Michelsohn) rather than the book by D.Bleecker et. al. ?
I am aware of a related question on MO which is partially helpful, but most answers there suggest books for learning Index theory rather than its pre-requisites. Thanks so much !
Best Answer
I shall only talk about the heat equation approach in this answer. Also I shall outline the way I read a proof of it which may or may not appeal to you.
In its full generality, the Index Theorem says that if $P: \Gamma(E) \rightarrow \Gamma(F)$ is an elliptic Pseudo-Differential Operator between two complex vector bundles on a compact manifold $M$ then its principal symbol $\sigma_L(P)$ gives an element of $K_{\text{cpct}}(T^*M)$ and then one has $$\operatorname{ind}P=(-1)^{\dim M}\int_{T^*M} \operatorname{ch}(\sigma_L(P))\pi^*\operatorname{Td}(T_{\mathbb C}M)$$
The RHS is called the topological index of the operator and can also be defined axiomatically ( Indeed it is one of the proofs of the Index Theorem )
Clearly, there are some hurdles before you can under-stand the statement. First of all you have to understand what is a Pseudo-Differential Operator. For this, I'd suggest read the first chapter of "Invariance Theory, Heat Equation and the Atiyah-Singer Index Theorem" by Peter Gilkey. This is an excellent source if you have some knowledge of basic Fourier Analysis or are willing to assume certain results from Fourier Analysis. If you really want to do do everything from scratch here's a nice set of notes. This even has a proof of the Geometric Index Theorem
If you feel Fourier Analysis is a bit too much to work with, you can restrict yourself to Differential Operators between vector bundles.
The next step is to understand when is a pseudo-differential operator elliptic (This is easy if you have already studied Pseudo-Differential Operators) and understanding the principal symbol.
The next part is tricky and I don't know of any source which does it in complete details (even the above notes does it only for differential operators). This part is globalizing everything you have done in the local setting to compact manifolds. In my master's thesis, I did however attempt it so feel free to ask me for the notes in case you want to take a look (there may be mistakes and gaps).
After that again it's plain sailing. Show all the elliptic pseudo-differential operators are Fredholm between the appropriate Sobolev spaces (don't worry, Gilkey does whatever is needed) and define its analytic index. You also get the Hodge Decomposition Theorem along the way.
So now you know what is the analytic index. If you do a bit of Chern-Weil theory and $K$-theory with compact supports then the right hand side will make sense to you (after of course you understand the leading symbol) On a side note, I still don't understand the leading symbol very well. In fact see this question.
By now, you should be able to at least understand what the Atiyah-Singer Index Theorem says.
The pure heat equation approach of the theorem only deals with Geometric Dirac Operators. So given a positive self-adjoint elliptic operator $P: E\rightarrow E$, you will have to construct its corresponding heat operator $e^{-tP}:\Gamma(E)\rightarrow\Gamma(E), \ t>0$ and the corresponding heat kernel $k_t(x,y)\in \Gamma(E^*\boxtimes E)$. This should not be difficult if you are already comfortable with $\Psi DO$'s i.e. pseudo-differential operators.
Mc-Kean Singer formula gives the link between heat equation and Index Theory. Once you have analyzed the distribution of eigen-values of $PP^*$ and $P^*P$, you get the formula $$\text{ind }P=\text{tr }e^{-tP^*P}-\text{tr }e^{-tPP^*}= \int_M \left [\text{tr}_xK^{P^*P}_t(x,x)-\text{tr}_xK^{PP^*}_t(x,x) \right ]|\text{dvol}|(x)$$
The heat equation approach then studies the RHS as $t\rightarrow 0$.
For Geometric Dirac Operators, the RHS is especially well-behaved near $0$.
For that again you have to look at the asymptotic expansion of the heat kernel. The standard reference for this is "N. Berline E. Getzler M. Vergne - Heat Kernels and Dirac Operators (Grundlehren Der Mathematischen Wissenschaften) (2003)". However the way I did it was using Dulhamel's Principle as outlined in "(Research Notes in Mathematics Series) John Roe - Elliptic operators, topology and asymptotic methods-Chapman and Hall CRC (1999)"
So you have now reduced the Atiyah-Singer Index Theorem to the evaluation of a limit. At this stage, I'd ask you to see section 4.3 of these notes due to Nicolaescue to complete the proof for Geometric Dirac Operators
What I have described should help you cover all the analysis required since the heat equation approach is after all the most analysis heavy.
What you can do however, is to read up Fourier Analysis from the first link and then start Gilkey's very nice book to get a grasp of $\Psi DO$'s and then decide which approach you want to take (there are broadly three: heat equation, $K$-Theory, co-bordism theory). You might also like this book, given your taste "Translations of Mathematical Monographs 235) Mikio Furuta - Index Theorem. 1-American Mathematical Society (2007)" . I haven't read it fully myself but as far as I can figure out, the author takes a co-bordism approach. If you want to do both the analysis and the topological aspects then you can have a look at "(Texts and Readings in Mathematics) Amiya Mukherjee (auth.) - Atiyah-Singer Index Theorem _ An Introduction-Hindustan Book Agency (2013)"
Hope this helps.