---
tag: X83ECKQ6G8
kind: concept
name: Chern–Simons form
msc: [53C05, 57R20]
sources:
  - url: https://en.wikipedia.org/wiki/Chern%E2%80%93Simons_form
same_as:
  wikidata: Q375489
  wikipedia: Chern–Simons form
---

## Summary

A Chern–Simons form is an odd-degree differential form built from a connection whose exterior derivative is a characteristic form of Chern–Weil theory.
The best-known one is the $3$-form $\mathrm{tr}(A \wedge dA + \frac23 A \wedge A \wedge A)$, whose derivative is $\mathrm{tr}(F \wedge F)$.
Chern–Simons forms give secondary invariants, defined when the primary characteristic forms vanish.
Their integrals over $3$-manifolds are the action of Chern–Simons gauge theory, which is central in low-dimensional topology and physics.

## Explanations

### L1 human

The characteristic forms of [[Chern–Weil theory|82ZPRD39J7]] are closed, and on a trivial bundle they are even exact.
A Chern–Simons form is an explicit form whose derivative is such a characteristic form.
It is like an antiderivative of the curvature expression.
On a closed $3$-manifold the integral of the Chern–Simons $3$-form is a number that changes in a controlled way under gauge transformations.
This number is used to define invariants of knots and of $3$-manifolds.