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The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs. (English) Zbl 0592.58047
In this paper we find two proofs of the Atiyah-Singer index theorem for families of Dirac operators \(\{D_ y\}_{y\in B}\) over closed manifold B where \(D_ y=\left( \begin{matrix} 0,D_-\\ D_+,0\end{matrix} \right)\) is the Dirac operator on closed manifold G with coefficients in an auxiliary vector bundle \(\xi\). There are two reasons that such a direct proof using the heat equation method is possible now. First, the Quillen formalism of superconnections extended by the author to the infinite-dimensional case gives the possibility of finding the most convenient form, which represents in the suitable sense \(Tr_ s \exp (-tD_ y)\) and makes it easy to prove that it also represents \(ch(\ker (D_{+,y})-\ker (D_{- ,y})).\) Secondly the stochastic differential calculus provides us with direct calculations that this form is equal to \(\int_{G}A(R^ G/2)\wedge ch(\xi).\)
Reviewer: K.Wojciechowski

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
58J65 Diffusion processes and stochastic analysis on manifolds
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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