dorsal/arxiv
View SchemaTowards a third-order topological invariant for magnetic fields
| Authors | Gunnar Hornig, Christoph Mayer |
|---|---|
| Categories | |
| ArXiv ID | physics/0203048 |
| URL | https://arxiv.org/abs/physics/0203048 |
| DOI | 10.1088/0305-4470/35/17/309 |
| Journal | J. Phys. A: Math. Gen., Vol. 35, No. 17 (2002) 3945-3959 |
Abstract
An expression for a third-order link integral of three magnetic fields is presented. It is a topological invariant and therefore an invariant of ideal magnetohydrodynamics. The integral generalizes existing expressions for third-order invariants which are obtained from the Massey triple product, where the three fields are restricted to isolated flux tubes. The derivation and interpretation of the invariant shows a close relationship with the well-known magnetic helicity, which is a second-order topological invariant. Using gauge fields with an SU(2) symmetry, helicity and the new third-order invariant originate from the same identity, an identity which relates the second Chern class and the Chern-Simons three-form. We present an explicit example of three magnetic fields with non-disjunct support. These fields, derived from a vacuum Yang-Mills field with a non-vanishing winding number, possess a third-order linkage detected by our invariant.
{
"annotation_id": "4259e642-045a-49c1-9373-c3dc1a1b84d1",
"date_created": "2026-03-02T18:00:39.420000Z",
"date_modified": "2026-03-02T18:00:39.420000Z",
"file_hash": "8c192518a930abad79e5e1ab55a3820db05da6ff581f30e39754170a4bd496f4",
"private": false,
"record": {
"abstract": "An expression for a third-order link integral of three magnetic fields is\npresented. It is a topological invariant and therefore an invariant of ideal\nmagnetohydrodynamics. The integral generalizes existing expressions for\nthird-order invariants which are obtained from the Massey triple product, where\nthe three fields are restricted to isolated flux tubes. The derivation and\ninterpretation of the invariant shows a close relationship with the well-known\nmagnetic helicity, which is a second-order topological invariant. Using gauge\nfields with an SU(2) symmetry, helicity and the new third-order invariant\noriginate from the same identity, an identity which relates the second Chern\nclass and the Chern-Simons three-form. We present an explicit example of three\nmagnetic fields with non-disjunct support. These fields, derived from a vacuum\nYang-Mills field with a non-vanishing winding number, possess a third-order\nlinkage detected by our invariant.",
"arxiv_id": "physics/0203048",
"authors": [
"Gunnar Hornig",
"Christoph Mayer"
],
"categories": [
"physics.plasm-ph"
],
"doi": "10.1088/0305-4470/35/17/309",
"journal_ref": "J. Phys. A: Math. Gen., Vol. 35, No. 17 (2002) 3945-3959",
"title": "Towards a third-order topological invariant for magnetic fields",
"url": "https://arxiv.org/abs/physics/0203048"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "04d862ed-99c3-48de-a8e3-d56c639788b0",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}