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May 7, 2013 at 21:49 comment added Zhaoting Wei @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.
May 7, 2013 at 10:47 comment added shu 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.
May 7, 2013 at 7:29 comment added Willie Wong 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.
May 7, 2013 at 5:52 history edited Zhaoting Wei CC BY-SA 3.0
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May 7, 2013 at 3:51 history asked Zhaoting Wei CC BY-SA 3.0