Towards a third-order topological invariant for magnetic fields

Gunnar Hornig, Christoph Mayer

    Research output: Contribution to journalArticlepeer-review

    18 Citations (Scopus)


    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 show 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 3-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.
    Original languageEnglish
    Pages (from-to)3945-3959
    Number of pages15
    JournalJournal of Physics A: Mathematical and General
    Issue number17
    Publication statusPublished - 2002


    • Magnetohydrodynamics (MHD)
    • Topological invariants
    • MHD


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