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In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a quantization of the theory of connections, and the supertrace of the heat kernel of the square of a Dirac operator is the quantization of the Chern character of the corresponding connection", see the introduction of that book. We know that Quillen has three important paper on superconnections and Chern character in 1985, 1986, 1988.

On the other hand, we know that the heat kernel proof of the index theorem had already been achieved by Patodi(1971) and Bismut(1984) etc.

Then can I ask that what is the significance of Quillen's work on superconnections to the heat kernel proof of index theorem?

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    $\begingroup$ Hum... learning something from Quillen does not necessarily imply learning something from reading one of Quillen's papers. It could be that the learning took place in a different setting. Unless the book indicated otherwise, I would not assume that the quote is necessarily referring to any of Quillen's three papers that you indicated. $\endgroup$
    – Willie Wong
    Commented May 7, 2013 at 7:29
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    $\begingroup$ As you said the the heat kernel proof of the index theorem had already been achieved. But not for the familly index theorem... Quillen's formalism gives a strategy for the heat kernel proof of the familly index theorem. This strategy is achieved by Bismut later via Bismut supperconnection. $\endgroup$
    – shu
    Commented May 7, 2013 at 10:47
  • $\begingroup$ @shu Thank you very much for your explain! Now I think I get some of the idea: For a family of manifolds $M\rightarrow B$ and a family Dirac operators we have the horizontal and the vertical direction, therefore Quillen's definition of superconnection is necessary to build the suitable operator on $M$. I think you know a lot about this work and hopefully you can write some survey article on it when you have time. $\endgroup$
    – Zhaoting Wei
    Commented May 7, 2013 at 21:49

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