This post is a tentative work-in-progress reply to Aires Ferreira's question:
"One can also write generalised conservation laws that involve the internal degrees of freedom (DOF) of a quantum theory (e.g., a conservation law relating spin density and spin current density). Such "covariant conservation laws" look formally the same but the differential operators then acquire knowledge of the underlying non-Abelian vector potentials acting on the internal DOFs. Such laws can be easily derived by writing down the action and minimising it w.r.t. infinitesimal gauge transformations (SU(2) for the example above of the spin DOF). I wonder how a formal reasoning in terms of topology as the one in your post would proceed in this more abstract (but physically very relevant) case."
Let us recall first that the action functional for a non-Abelian $\mathrm{U}(n)$-gauge field $A$ is the Yang-Mills action functional
$$S_{YM}=(1/2)\int_{M}|F_A|^2\sqrt{g}d^{n+1}x=-(1/2)\int_{M} \mathrm{tr}\left( F_A\wedge * F_A\right),$$
where $M$ is an $n+1$ dimensional oriented smooth manifold, $A$ is the local representative of a connection $\omega$ on a smooth principal $\mathrm{U}(n)$-bundle $P\to M$ (obtained by pulling back by a local section), $F_A=dA+A\wedge A$ is the local representative of the curvature $\Omega=d\omega+\omega\wedge \omega$ and $*$ is the Hodge dual. The fields take values in the Lie algebra $\mathfrak{u}(n)$ which is interpreted as the real vector space of skew-Hermitian matrices. The integrand is gauge invariant, meaning it patches to a well-defined $n+1$ form on $M$. The space of connections on a principal bundle is an affine space and we can consider variations of the conection on $P$ locally determined by $A_t= A+t\delta$, $t\in [0,1]$, where $\delta$ is a Lie algebra valued one-form and determines a section of the associated bundle $P\times_{\mathrm{Ad}} \mathfrak{u}(n)$ (where $\mathrm{Ad}$ denotes the adjoint representation)---i.e., under the gauge transformation $g(x)$ we have $A\to g^{-1}Ag +g^{-1}dg$ and $\delta\to g^{-1}\delta g$ so that $A_t\to g^{-1}A_tg +g^{-1}dg$.
Now the variation of $S_{YM}$ can be understood from
$$F_{A+t\delta}=F_A + td_A\delta +t^2\delta\wedge \delta,$$
where $d_A\delta=d\delta +[A,\delta]=d\delta + A\wedge \delta +\delta\wedge A$ is the covariant (exterior) derivative of $\delta$ (observe that the commutator is the *graded* commutator which takes into account the degree of the form). To compute the variation of $S_{YM}$ we only need to consider $|F_{A+t\delta}|^2$ to first order in $t$, and that is given by $2\langle d_A\delta,F\rangle$. In other words
$$\frac{d}{dt}\left(S_{YM} (A_t)\right)\big|_{t=0}=\int_{M} \langle d_A\delta , F_A\rangle \sqrt{g}d^{n+1}x=-\int_{M}\mathrm{tr} \left( d_A \delta \wedge *F_A\right).$$
Assuming $\delta$ vanishes at infinity (compact support variations), we can bring $d_A$ to act on $*F_A$ to obtain
$$\frac{d}{dt}\left(S_{YM} (A_t)\right)\big|_{t=0}=-\int_{M}\mathrm{tr} \left( \delta \wedge d_A(*F_A)\right).$$
Observe that the sign doesn't change in integration by parts because $\delta$ is one-form valued. If there were no matter fields one could write the Yang-Mills equation by imposing that the action is stationary for all variations $\delta$, so that
$$d_A(*F_A)=0.$$
Now if there are matter fields, we need to supplement the action $S_{YM}$ with $S_{\text{matter}}$. The matter fields are sections of an associated bundle $P\times_{\rho} V$ where $\rho: \mathrm{U}(n)\to \mathrm{U}(V)$ is some unitary representation of the gauge group. The matter fields couple to $A$ through the covariant derivative $d_A=d+A$ (where $A$ acts in the appropriate induced $\mathfrak{u}(V)$ representation). Now the point is that we can write, after integration by parts,
$$\frac{d}{dt}\left(S_{\text{matter}} (A_t)\right)\big|_{t=0}=-\int_{M}\mathrm{tr} \left( \delta \wedge j_A\right),$$
where $j_A$ is an $n$-form depending on $A$. Gauge-invariance implies that $j_{A}$ transforms under gauge transformations as $j_{A}\to j_{A'}=g^{-1}j_A g$, with $A'=g^{-1}(d+A)g$. Putting everything together yields the Yang-Mills equation with a source term
$$ d_A(*F_A) =-j_A.$$
Both objects in the left and right hand side determine sections on $\left(P\times_{\mathrm{Ad}}\mathfrak{u}(n)\right)\otimes \Lambda^{n} T^*M$. To obtain ordinary differential $n$-forms, i.e., sections of $\Lambda^{n} T^*M$, we can take a trace with an arbitrary section $\delta$ of $P\times_{\mathrm{Ad}}\mathfrak{u}(n)$:
$$\mathrm{tr}\left( \delta d_A(*F_A)\right) =-\mathrm{tr}\left(\delta J_A\right).$$
We can then observe that
$$-\mathrm{tr}\left(d_A(\delta)\wedge *F_A\right)+d\left[\mathrm{tr}\left( \delta *F_A\right)\right] =-\mathrm{tr}\left(\delta J_A\right).$$
We can take the quantity on the right-hand side to be our candidate of $-J$ in the previous post. If we can guarantee that $d_A(\delta)=0$, i.e., $\delta$ determines a covariantly constant section of $P\times_{\mathrm{Ad}}\mathfrak{u}(n)$. If this happens $J=\mathrm{tr}\left(\delta J_A\right)=d\left[\mathrm{tr}\left(\delta *F_A\right)\right]$ and hence $dJ=0$. In that case, for decomposable manifolds $M=\mathbb{R}\times N$, we have that for $N_t=\{t\}\times N$
$$ Q(N_t)=\int_{N_t} J$$
is independent of the $t$, by the same argument of Stokes' theorem. In Abelian gauge theories things are much simpler because $d_A(*F_A)=d(*F_A)$ (because the commutator part is trivial). In that case we just need a section of $P\times_{\mathrm{Ad}}\mathfrak{u}(1)$ which is naturally a trivial bundle (it can be seen as the homomorphism bundle of the electromagnetic line bundle over $M$ which is trivializable by the identity homomorphism). Hence we can take $\delta=1$, which gives
$$J=\mathrm{tr}\left(\delta J_A\right)=J_A.$$
So in the Abelian case the topological argument for the conserved current is naturally available since $J$ is closed (in fact, exact). In the non-Abelian case, if there is a covariantly constant section of $P\times_{\mathrm{Ad}}\mathfrak{u}(n)$ then we can get one. Supposing such object exists, the curvature $F_A$ in the adjoint representation is an obstruction to its local existence [the curvature of a connection on a bundle is an obstruction to the local existence of parallel sections]. Observe that the adjoint representation of $\mathrm{U}(1)$ is trivial so this is not incompatible with the above argument. In short, in the non-Abelian case, the reasoning of the previous post breaks down because of the nontriviality of $P\times_{\mathrm{Ad}}\mathfrak{u}(n)$ and the associated connection $d_A$ which does not allow us to see $J$ as a closed $n$-form.
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