A purely classical analysis of Gauss’s law constraints suffices for our purposes. The quantum treatment, in the presence of topological terms in the action, requires care [77].

Indeed, recall that: (i) the operator $U(\theta ;M^3)$ is constructed from Gauss’s law constraints;

(ii) the latter implement bulk gauge transformations;

(iii) the global $SO(6)$ action originates as a special instance of a gauge transformation, whose parameter is nontrivial at the boundary.

As far as group representations are concerned, dominant integral elements are in 1-to-1 correspondence with isomorphism classes of unitary finite-dimensional irreps of the simply connected $SU(4)$. For $SO(6)$, an analogous statement holds, but the notion of integral element has to be replaced by that of analytically integral element, see e.g. [101]. While all analytically integral elements are integral, the converse is not true. Integral elements that are not analytically integral correspond to irreps of $SU(4)$ which are not irreps of $SO(6)$.

This can be rephrased as follows. The 2-form $N\frac{d\mathcal{A}}{2\pi }$ is cohomologically nontrivial on the base ${\mathbb{C}\mathbb{P}}^2$. If we pull it back to the total space of the Hopf fibration, however, it becomes cohomologically trivial, since we can write $N\frac{d\mathcal{A}}{2\pi }=\frac{N}{2\pi }d{k}_{\psi }$, and $k_{\psi }$ is a globally defined 1-form in the total space.