## 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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### References:

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