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?

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. – Willie Wong May 7 '13 at 7:29