12.2 The Atiyah-Singer index theorem

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

$$\mathrm{ind}(E,D)=(-1{)}^{m(m+1)/2}{\int }_M\mathrm{ch}\left(\underset{r}{⨁}(-1{)}^r{E}_r\right)\frac{\mathrm{Td}\left(T{M}^{\mathbb{C}}\right)}{e(TM)}{|}_{\text{v o l}}.\text{\hspace{1em}(12.33)}$$

In the integrand of the RHS, only $m$ -forms are picked up, so that the integration makes sense. [Remarks: The division by $e(TM)$ can really be carried out at the formal level. If $m$ is an odd integer, the index vanishes identically, see below. Original references are Atiyah and Singer (1968a, b), Atiyah and Segal (1968).]

The proof of theorem 12.2 is found in Shanahan (1978), Palais (1965) and Gilkey (1984). The proof found there is based on either $K$ -theory or the heat

kernel formalism. In section 13.2, we give a proof of the simplest version of the Atiyah-Singer (AS) index theorem for a spin complex. Recently physicists have found another proof of the theorem making use of supersymmetry. This proof is outlined in sections 12.9 and 12.10. Interested readers should consult Alvarez-Gaume (1983) and Friedman and Windey (1984, 1985) for further details.

The following corollary is a direct consequence of theorem 12.2.

Corollary 12.1

Let $\Gamma (M,E)\stackrel{D}{\to }\Gamma (M,F)$ be a two-term elliptic complex. The index of $D$ is given by

$$\begin{array}{rl}\mathrm{ind}D& =\dim \ker D-\dim \ker D^{†}\\ & =(-1{)}^{m(m+1)/2}{\int }_M(\mathrm{ch}E-\mathrm{ch}F)\frac{\mathrm{Td}\left(T{M}^{\mathbb{C}}\right)}{e(TM)}{|}_{\text{v o l}}.\end{array}\text{\hspace{1em}(12.34)}$$

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