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Towards a third-order topological invariant for magnetic fields

Gunnar Hornig·Christoph Mayer
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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.

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Gunnar Hornig, Christoph Mayer. 2002-03-15. Towards a third-order topological invariant for magnetic fields. https://doi.org/10.1088/0305-4470%2F35%2F17%2F309

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