---
tag: 31Q0RDPZ00
kind: concept
name: Clifford module
msc: [15A66, 53C27]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_module
same_as:
  wikidata: Q7879936
  wikipedia: Clifford module
---

## Summary

A Clifford module is a module over a [[Clifford algebra|A38B1QFK20]].
It is a vector space on which the vectors of a quadratic space act as linear maps.
The square of each such map is a scalar fixed by the quadratic form: minus its value, or its value, depending on the sign convention.
Spinor spaces are the irreducible Clifford modules.
Bundles of Clifford modules over Riemannian manifolds carry Dirac operators, which makes Clifford modules the algebraic input of index theory.
Their classification shows a periodicity of period $8$ over the reals and $2$ over the complex numbers, closely related to Bott periodicity.

## Explanations

### L1 human

The complex numbers are the real plane with a rule saying that $i \cdot i = -1$.
A Clifford module generalises this: several directions $e_1, \ldots, e_n$ act on a space, each one squaring to $-1$, and different ones anticommuting.
For one direction, the plane $\C$ with $e_1$ acting as multiplication by $i$ is a Clifford module.
For two directions, the quaternions with $e_1$ and $e_2$ acting as left multiplication by $i$ and $j$ form one.
These modules are the spaces on which spinors live.