---
tag: CKGDR9P9PA
kind: concept
name: Yang–Mills functional
msc: ["58E15", 53C07, 81T13]
sources:
  - url: https://en.wikipedia.org/wiki/Yang–Mills_equations
---

## Summary

The Yang–Mills functional assigns to a [[connection|NJ21PN5ZGJ]] on a [[principal bundle|8E37E74XVP]] over a compact Riemannian manifold the integral of the squared norm of its [[curvature|ZAJFW1S078]].
It does not change under gauge transformations.
Its critical points are the solutions of the [[Yang–Mills equations|JJWETW5AT8]], and its absolute minima include the flat connections.
In dimension four its minima in a given bundle are often the instantons, which led to deep results on four-manifolds.

## Explanations

### L1 human

A connection tells how to compare the fibres of a bundle at nearby points, and its curvature measures how much this comparison twists around small loops.
The Yang–Mills functional adds up the square of this twisting over the whole manifold, like the total energy of a field.
For the circle group the functional is the energy of an electromagnetic field, half the integral of the squared field strength.
Connections that make it as small as possible, or that balance it to first order, are the natural fields of gauge theory.