Solenoidal fields in $D$ spatial dimensions

Question

In the T. Sakurai paper cited below, page 209, the author discusses how the magnetic field can be written in terms to the cross-product $$ \vec B = \vec\nabla \chi \times \vec\nabla \eta,$$ for two scalar potentials $\chi$ and $\eta$.

My question: is this generalizable to higher dimensions and higher forms? In particular, in $D$ dimensions, can I always represent a closed $p$-form $d\omega$ by $$ d\omega = d\chi_1 \wedge\cdots\wedge d\chi_p ,$$ or is there somehow something special about the magnetic field case?

Further if this is not generically true, what are necessary/ sufficient conditions for it to hold?

Edit: I originally came across this claim about magnetic fields here: https://arxiv.org/abs/1005.3977. In this paper, the author cites the work: Sakurai T., "A New Approach to the Force-Free Field and Its Application to the Magnetic Field of Solar Active Regions," Pub. Ast. Soc. Japan, Vol. 31, 209, 1979.

Answer 1

Here is a partial answer. Let $B$ be a closed $p$-form magnetic field strength in $D$ spatial dimensions with $p\geq 1$. OP is then essentially asking if $B$ is can be written locally$^1$ as $$ B~=~\mathrm{d}\chi_1\wedge\ldots \wedge\mathrm{d}\chi_p ~?\tag{1} $$

From Poincare Lemma we know that there exists a locally defined $(p\!-\!1)$-form magnetic gauge potential $A$, so that $B=\mathrm{d}A$.

Eq. (1) is clearly possible with $\chi_1=A$ if $p=1$, so let's assume $p\geq 2$ from now on. Then $A\to A +\mathrm{d}\lambda$ has a $(p\!-\!2)$-form gauge symmetry $\lambda$.

Let us try to count DOF. The $p$-form $B$ has at least $$\begin{pmatrix} D \cr p\!-\!1 \end{pmatrix}-\begin{pmatrix} D \cr p\!-\!2 \end{pmatrix}~=~\begin{pmatrix} D\!-\!1 \cr p\!-\!1 \end{pmatrix}\tag{2}$$ DOF, which should not be bigger than the $p$ DOF on the rhs. of eq. (1). This is generically only possible if

1. $B$ is dual to a vector field, i.e. $p=D-1.$

   • The case $p=2$ follows from Darboux' theorem/the Clebsch representation, cf. an above comment by Cosmas Zachos.

   • In the case $p\geq 3$, the gaugesymmetry $\lambda\to \lambda +\mathrm{d}\xi$ has a $(p\!-\!3)$-form gauge-for-gauge symmetry $\xi$, so a more precise counting shows that $B$ actually has more than $p$ DOF, i.e. eq. (1) is generically not possible.

2. $B$ is a top-form with 1 DOF, i.e. $p=D$.

   • The case $p=2$ follows from Darboux' theorem.

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$^1$ The corresponding global problem can carry additional topological obstructions.