{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "AV0Q1VSJ88"
    },
    {
      "field": "uses",
      "target": "AV0Q1VSJ88"
    },
    {
      "field": "defines",
      "target": "B6K478R9XE"
    }
  ],
  "body": "\n## Summary\n\nThe 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]].\nIt does not change under gauge transformations.\nIts critical points are the solutions of the [[Yang–Mills equations|JJWETW5AT8]], and its absolute minima include the flat connections.\nIn dimension four its minima in a given bundle are often the instantons, which led to deep results on four-manifolds.\n\n## Explanations\n\n### L1 human\n\nA 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.\nThe Yang–Mills functional adds up the square of this twisting over the whole manifold, like the total energy of a field.\nFor the circle group the functional is the energy of an electromagnetic field, half the integral of the squared field strength.\nConnections that make it as small as possible, or that balance it to first order, are the natural fields of gauge theory.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58E15",
      "53C07",
      "81T13"
    ],
    "name": "Yang–Mills functional",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Yang–Mills_equations"
      }
    ],
    "tag": "CKGDR9P9PA"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/CKGDR9P9PA/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/CKGDR9P9PA.md",
  "msc": [
    "58E15",
    "53C07",
    "81T13"
  ],
  "name": "Yang–Mills functional",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CKG/CKGDR9P9PA.md",
  "status": null,
  "tag": "CKGDR9P9PA",
  "uri": "m2://CKGDR9P9PA",
  "url": "https://tamnd.github.io/math2/wiki/yang-mills-functional/",
  "writing": {
    "tag": "CKGDR9P9PA",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}