Atiyah Singer

Let's start from the beginning.

Take an order N operator which in local coordinates is given as follows.



M is a multi-index which indicates which coordinates to differentiate again. For example if M=(1,2,0,0,1) which is of order N would correspond to the operator



Now we might as well take the term with the most derivatives. After all that is the most relevant operator by dimensional analysis. Now Fourier transform to get the symbol.

For the Laplacian it is



This is only 0 at the origin. This is the definition of elliptic. We have the operator, we can also take the adjoint because the fields live in spaces with inner products. For example, fermions live in the spinor bundle and there is an inner product on spinors. We can ask that the kernel of the operator and its adjoint are finite dimensional. This operator is Fredholm and it is made of win.

Now what is the difference between the kernel's size and cokernel's size. This measures the mismatch between zero modes. This is where we see the notion of the concept of invariance of measure.



Here a and b's are the coefficients of the eigenfunction expansion. The indices that correspond to nonzero eigenvalues match up perfectly.



Of course we see a pairing between n and n' if and only if the eigenvalue is nonzero. Otherwise we don't know how much they mismatch. We also notice that here n and n' correspond to different chiralities because there are an odd number of gamma matrices in the operation from one to another.

We want the difference between left and right chiralities. How do we calculate this? Next time on If you haven't found something strange.