---
tag: YWBTGC7MFA
kind: concept
name: Instanton
msc: [53C07, 81T13, "58E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Instanton
---

## Summary

In differential geometry, an instanton is a connection on a principal bundle over an oriented Riemannian 4-manifold whose curvature is anti-self-dual for the [[Hodge star|1Y4WX4XQWJ]].
Over a compact 4-manifold, instantons are absolute minima of the Yang–Mills functional among connections on a given bundle, so they solve the Yang–Mills equations.
In physics they are finite-action solutions of the Euclidean field equations that describe tunnelling between vacua.
Their moduli spaces were used by Donaldson to discover new invariants of smooth 4-manifolds.

## Explanations

### L1 human

A connection on a bundle tells how to move internal states, such as the phase of a particle, from point to point, and its curvature measures the twist.
In four dimensions the Hodge star sends 2-forms to 2-forms, so a curvature can equal minus its own star.
Instantons are the connections with this property.
They are the most economical way to twist a bundle with a given topology.
The basic example is the BPST instanton, a connection for the group $SU(2)$ on $\R^4$ whose curvature is concentrated near one point.