Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.