# Spinors and the tangent groupoid

## TL;DR

This paper explores Ezra Getzler's approach to the Atiyah-Singer index theorem using Alain Connes' tangent groupoid, constructing a rescaled spinor bundle and linking it to symbol calculus.

## Contribution

It introduces a rescaled spinor bundle on the tangent groupoid and connects Getzler's symbol calculus with Connes' framework, offering a new perspective.

## Key findings

- Construction of a rescaled spinor bundle on the tangent groupoid
- Definition of a convolution operation on sections
- Integration of Getzler's symbol calculus into the algebra

## Abstract

The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a "rescaled" spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how the algebra so-obtained incorporates Getzler's symbol calculus.