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**A short proof of the local Atiyah-Singer index theorem.**
*(English)*
Zbl 0607.58040

This paper presents a simplification of the author’s proof of the local form of the Atiyah-Singer index theorem [Commun. Math. Phys. 92, 163-178 (1983; Zbl 0543.58026)]. It was based on physical ideas of Witten involving super-symmetric quantum field theory. The method involved obtaining an approximate heat kernel, \(K_ t(x,y)\), for the twisted Dirac operator on \({\mathbb{R}}^ n\) which has the property that \(\lim_{t\to 0}\) \(Tr_ S(K_ t(x,x))\) was easily seen to be exactly the combination of characteristic classes needed for the index formula. The author then shows that the approximation is good enough to yield the index theorem. The present paper obtains the approximation to the heat kernel by more elementary methods which avoid much of the formalism used in the first paper. This paper is valuable in its own right, as well as being useful for understanding the first version.

Reviewer: J.Kaminkar

### MSC:

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

58J20 | Index theory and related fixed-point theorems on manifolds |