---
tag: QMRAVX14BJ
kind: concept
name: Dirac operator on a spin manifold
msc: [53C27, 58J60, 15A66]
specializes: [3DXGJTEQZH]
sources:
  - url: https://en.wikipedia.org/wiki/Spinor_bundle
---

## Summary

The Dirac operator of a Riemannian spin manifold, also called the Atiyah–Singer operator, acts on sections of the [[spinor bundle|PYYQQKAZ0C]].
It combines the spin connection, induced by the [[Levi-Civita connection|CAZYGE1T4E]], with Clifford multiplication by tangent vectors.
It is a first-order elliptic and formally self-adjoint operator whose square is the rough Laplacian plus a quarter of the scalar curvature.
On a closed spin manifold of dimension divisible by $4$, the index of its restriction to positive spinors is the $\hat{A}$-genus.
This ties the existence of metrics of positive scalar curvature to topology.
It is the main example of a [[Dirac operator|3DXGJTEQZH]].

## Explanations

### L1 human

On flat $\R^n$, Dirac found a first-order operator whose square is minus the Laplacian, using matrices that anticommute.
To do the same on a curved manifold one needs a bundle on which these matrices can act consistently from point to point.
A spin structure is exactly the extra data that makes this possible, and the spinor bundle is the resulting bundle.
The Dirac operator differentiates a spinor field along each direction of an orthonormal frame and multiplies by the matching Clifford matrix, then adds the results.