[Nakahara GTP] 12.2 The Atiyah–Singer index theorem

    This article is one of Manifold, Differential Geometry, Fibre Bundle.

12.2.1 Statement of the theorem

Theorem 12.2 (Atiyah-Singer index theorem). Let $(E,D)$ be an elliptic complex over an $m$-dimensional compact manifold $M$ without a boundary. The index of this complex is given by $$\begin{align}\mbox{ind}(E,D)=(-1)^{m(m+1)/2}\int_M\mbox{ch}\left(\oplus_r(-1)^rE_r\right)\left. \frac{\mbox{Td}(TM^\mathbb{C})}{e(TM)}\right|_{vol}.\end{align}$$ In the integrand of the RHS, only $m$-forms are picked up, so that the integration makes sence.

Corollary 12.1. Let $\Gamma(M,E)\stackrel{D}{\rightarrow}\Gamma(M,F)$ be a two-term elliptic complex. The index of $D$ is given by $$\begin{align}\mbox{ind}D=\dim\ker D-\dim\ker D^\dagger\nonumber\\ (-1)^{m(m+1)/2}\int_M (\mbox{ch}E-\mbox{ch}F)\left. \frac{\mbox{Td}(TM^\mathbb{C})}{e(TM)}\right|_{vol}.\end{align}$$