Generalized helicity and Beltrami fields

TL;DR

This paper introduces covariant non-abelian generalizations of magnetic helicity and Beltrami equations, exploring their gauge invariance, variational principles, and conserved quantities within Yang-Mills theory.

Contribution

It develops a covariant, non-abelian framework for helicity and Beltrami fields, linking extremals of the Yang-Mills action to solutions of the Beltrami equation.

Findings

Any extremal of the Yang-Mills action with a specific boundary constraint satisfies the covariant non-abelian Beltrami equation.

The paper establishes gauge invariance and conserved currents for the generalized helicity.

It provides a variational principle underpinning the non-abelian Beltrami fields.

Abstract

We propose covariant and non-abelian generalizations of the magnetic helicity and Beltrami equation. The gauge invariance, variational principle, conserved current, energy-momentum tensor and choice of boundary conditions elucidate the subject. In particular, we prove that any extremal of the Yang-Mills action functional $\tfrac{1}{4}\int_\Omega\tr{F_{\mu\nu}F^{\mu\nu}}\,d^4x$ subject to the local constraint $\epsilon^{\mu\nu\alpha\beta}\tr{F_{\mu\nu}F_{\alpha\beta}}=0$ satisfies the covariant non-abelian Beltrami equation.