Local Index Theory and the Tangent Groupoid

Here Index(D) is the Fredholm index of the Dirac operator acting on smooth sections of the spinor bundle S over an even dimensional compact Riemannian spin manifold M , e−tD 2 is the solution operator for the heat equation ∂s ∂t+D s = 0 for D, and Trs is the supertrace. The goal of this essay is to explicitly compute the righthand side in terms of local information on M , thereby proving the Atiyah-Singer index theorem. As we shall see e−tD 2 is given by a smoothing kernel kt, called the heat kernel of D, which admits an asymptotic expansion as t→ 0. Indeed, we can express Trs(e −tD2) asymptotically as: