No, I’m not crazy. I just want to know more about the Atiyah-Singer Index Theorem.
I’m reading through S.K. Donaldson’s Floer Homology Groups in Yang-Mills Theory, in which he invokes the theorem with almost no explanation. I’ll use a mix of TeX and the board-supplied super/subscripts.
On an oriented Riemannian 4-manifold X and a principal fiber bundle P over it with group G, one wants to find “instantons”, i.e.: connections whose curvature is anti-self-dual. Near one such solution, the nearby connections A+a are given by the equation d[sup]+[/sup][sub]A[/sub]a = 0, where d[sup]+[/sup][sub]A[/sub] is the standard map from 1-forms to 2-forms induced by A composed with the projection onto the self-dual 2-forms. Since the instanton condition is gauge invariant, we can also require d[sup]*[/sup][sub]A[/sub] to vanish (mapping 1-forms to 0-forms as in deRham theory). Now he defines
D[sub]A[/sub] = -d[sup]*[/sup][sub]A[/sub]a \oplus d[sup]+[/sup][sub]A[/sub]a
Mapping from \Omega[sup]1/sup to \Omega[sup]0/sup \oplus \Omega[sup]+/sup: pairs of a 0-form and a self-dual 2-form. He states that adding the d[sup]*[/sup][sub]A[/sub] term makes this operator elliptic, thus giving it a Fredholm index. He then states that the ASIT lets us say
ind D[sub]A[/sub] = c(G)\kappa§ - (dim G)(1 - b[sub]1[/sub] + b[sup]+[/sup])
where c(G) is “a normalising constant”, \kappa§ is “a characteristic class number obtained by evaluating a characteristic class on the fundamental cycle ”, b[sub]1[/sub] is the normal first Betti number of X, and b[sup]+[/sup] is the rank of a maximal positive subspace for the intersection form on H[sub]2/sub.
Now, the questions.
a) My standard example of an elliptic operator is the Laplacian on a Riemannian manifold, which maps a space to itself. In what sense is D[sub]A[/sub] elliptic, if its domain and range are different spaces? Also, why does adding the d[sup]*[/sup][sub]A[/sub] term make the operator elliptic?
b) How does the ASIT determine c(G)?
c) What the hell is \kappa§? I know some basic Chern class stuff, but the language used here (“a characteristic class”) is very confusing.
d) How is the intersection form defined on homology?
e) What does the ASIT really say? Is it basically just this statement, so understanding b), c), and d) will explain it all, or is this more of a special case?