---
tag: JJWETW5AT8
kind: concept
name: Yang–Mills equations
msc: [70S15, 81T13, 53C07]
sources:
  - url: https://en.wikipedia.org/wiki/Yang–Mills_equations
same_as:
  wikidata: Q85816613
  wikipedia: Yang–Mills equations
---

## Summary

The Yang–Mills equations are the field equations of a gauge theory: they are nonlinear partial differential equations for a [[connection|NJ21PN5ZGJ]] on a [[principal bundle|8E37E74XVP]].
They are the Euler–Lagrange equations of the Yang–Mills functional, the integral of the squared norm of the [[curvature|ZAJFW1S078]].
For the circle group they reduce to Maxwell's equations in vacuum, and for non-abelian groups they describe the strong and weak forces.
In dimension four their self-dual solutions, the instantons, led to deep results on the topology of four-manifolds.

## Explanations

### L1 human

In electromagnetism the field is described by a potential, and the field strength measures how the potential twists.
Gauge theories replace numbers in this picture by elements of a matrix group, so twists in different directions no longer commute.
The Yang–Mills equations say that the field strength is as balanced as possible, in the sense that it is a critical point of its total energy.
The simplest solutions are flat connections, whose field strength is zero everywhere.