Are Dirac Operators Good for Nothing?

and this index is constant on continuous families of such L. Suppose that D is an elliptic operator of order m on sections of a vector bundle E± over a smooth, compact manifold M . Let H (Γ (M,E±)) denote the Sobolev s-norm completion of the space of sections Γ (M,E), with respect to a chosen metric. Then D can be extended to be a bounded linear operator Ds : H s (Γ (M,E)) → Hs−m (Γ (M,E−)) that is Fredholm, and ind (D) := ind ( Ds ) is well-defined and independent of s. In the 1960s, the researchers M. F. Atiyah and I. Singer proved that the index of an elliptic operator on sections of a vector bundle over a smooth manifold satisfies the following formula ([1],[2]):