A short proof of the local Atiyah-Singer index theorem

Abstract

(Received 24 April 1985) IN THIS paper, we will give a simple proof of the local Atiyah-Singer index theorem first proved by Patodi [9]; in fact, his earlier proof of the Gauss-Bonnet-Chern theorem (Patodi [8]) is quite close to ours. (Perhaps he did not find the proof for Dirac operators given in this paper because he was unaware of the symbol calculus for Clifford algebras.) A paper of Kotake [S] contains a proof of the Riemann-Roth theorem for Riemann surfaces along similar lines, and a recent paper of Bismut [3] is also very closely related. It might be helpful to give a short history of this theorem. As explained in Atiyah et al. [l], all of the common geometric complexes, namely, the twisted Dirac operators, &operators, signature operators and the De Rham complex, are, locally, Dirac operators. We shall refer to all of these operators as Dirac operators, although this is not globally correct on non-spin manifolds. The index theorem for Dirac operators was first proven, at least for Kahler manifolds, by Hirzebruch using cobordism theory. A few years later, McKean and Singer gave their famous formula for the index of the Dirac operator: Index (D) = Str efo2( = Tr erp-e’ - Tr erpcm-)