12 INDEX THEOREMS

In physics, we often consider a differential operator defined on a manifold $M$ . Typical examples will be the Laplacian, the d’Alembertian and the Dirac operator. From the mathematical point of view, these operators are regarded as maps of sections

$$D:\Gamma (M,E)\to \Gamma (M,F)$$

where $E$ and $F$ are vector bundles over $M$ . For example, the Dirac operator is a map $F(M,E)\to F(M,E)$ , $E$ being a spin bundle over $M$ . If inner products are defined on $E$ and $F$ , it is possible to define the adjoint of $D$ ,

$$D^{†}:\Gamma (M,F)\to \Gamma (M,E).$$

Since it is a differential operator, $D$ carries analytic information on the spectrum and its degeneracy. In what follows, we are interested in the zero eigenvectors of $D$ and $D^{†}$ ,

$$\ker D\equiv \{s\in \Gamma (M,E)|Ds=0\}$$

$$\ker D^{†}\equiv \{s\in \Gamma (M,F)|D^{†}s=0\}.$$

The analytical index is defined by

$$\mathrm{ind}D=\dim \ker D-\dim \ker D^{†}.$$

Surprisingly, this analytic quantity is a topological invariant expressed in terms of an integral of an appropriate characteristic class over $M$ , which provides purely topological information on $M$ . This interplay between analysis and topology is the main ingredient of the index theorem.

Our exposition follows Eguchi et al (1980), Gilkey (1984), Shanahan (1978), Kulkarni (1975) and Booss and Bleecker (1985). The reader should consult these references for details. Alvarez (1985) contains a brief summary of this subject along with applications to anomalies and strings.

Problems

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正文：英文 · OCR 机器稿 · 待校对

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