The purpose of this talk is to explain a connection between the tangent groupoid, used by Alain Connes to conceptualize the K-theory proof of the Atiyah-Singer index theorem, and Ezra Getzler’s local proof of the index theorem for the Dirac operator. I shall first give an introduction to the tangent groupoid, and then I shall construct a “rescaled” spinor bundle on the tangent groupoid. The existence and equivariance of this bundle encode the major features of Getzler’s proof of the index theorem. This is joint work with Zelin Yi.

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