---
tag: PYYQQKAZ0C
kind: concept
name: Spinor bundle
msc: [53C27, 15A66]
sources:
  - url: https://en.wikipedia.org/wiki/Spinor_bundle
same_as:
  wikidata: Q1049915
  wikipedia: Spinor bundle
---

## Summary

The spinor bundle of a spin manifold is the complex vector bundle associated with its [[spin structure|RSPZ9TCFQ8]] and the spin representation of the spin group.
Its sections are the spinor fields.
It carries a Hermitian metric, a connection induced by the Levi-Civita connection and a Clifford multiplication by tangent vectors, and these define the Dirac operator.

## Explanations

### L1 human

On a spin manifold, each point has a small complex vector space of spinors, and these spaces fit together into a bundle.
A tangent vector acts on spinors, and acting twice by a unit vector gives minus the identity.
On flat space $\R^4$ the spinor bundle is the trivial bundle with fibre $\C^4$, matching the four complex components of a spinor in Dirac's equation for the electron.