Gauge Parameterization of the $n$-Field

We propose a gauge parameterization of the three-dimensional $n$-field using an orthogonal $\mathbb {SO}(3)$-matrix, which, in turn, is defined by a field taking values in the Lie algebra $\mathfrak {so}(3)$ (rotation-angle field). The rotation-angle field has an additional degree of freedom, which corresponds to the gauge degree of freedom of rotations around the $n$-field. As a result, we obtain a gauge model with local $\mathbb {SO}(2)\simeq \mathbb U(1)$ symmetry that does not contain a $\mathbb U(1)$ gauge field.