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Let some field theory be described by the Lagrangian density ${\cal L}$ on spacetime. Noether's first theorem asserts that given a quasisymmetry $\hat{\delta}\phi$ there is a class of currents $j^\mu$ such that $$\partial_\mu j^\mu =E\hat{\delta}\phi\tag{1}$$ where $E$ are the equations of motion.

Two currents in the same class differ by a trivial current which can be either (1) a current that identically vanishes on-shell, (2) a current which is conserved even off-shell and (3) any combination of these.

Noether's second theorem states that when the quasisymmetry is local, i.e., parameterized by a function $f$, one such current associated to it, verifying (1), is some $S^\mu$ which vanishes on-shell $S^\mu\approx 0$. Therefore any other current in the class $j^\mu$ verifies

$$\partial_\mu(j^\mu - S^\mu)=0\Longrightarrow j^\mu=S^\mu+\partial_\nu k^{[\mu\nu]}\tag{2}.$$

In this paper by G. Barnich & F. Brandt the authors say that this gives rise to a "Noether charge puzzle":

Note that the superpotential is completely arbitrary because it drops out of (1.1) [Eq. (1) of this post] owing to $\partial_\mu\partial_\nu k^{[\nu\mu]}=0$. This implies that the Noether charge corresponding $\delta_f$ is undefined because it is given by the surface integral of an arbitrary $(n-2)$ form.

  1. How the same problem does not happen for a global symmetry for which Noether's second theorem does not apply? I mean, the current class of such symmetry is not the trivial one anymore. Still, if $j^\mu$ is a current in the class we can always add some $\partial_\nu k^{[\mu\nu]}$. How is this any different than the local case?

  2. More importantly if we define the Noether charge by integrating $j^\mu$ over a Cauchy surface $\Sigma$ is the charge, in the global case, well-defined? Because I see the same issue taking place in the global case. Let $j^\mu$ be a current in the class. We get another one by adding $\partial_\nu k^{[\mu\nu]}$, then the charge changes by a boundary term at $\partial \Sigma$.

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The same happens for global symmetries, too. We usually simply define "charge" to be defined as the integration of the current over a surface without boundary. The only difference for the gauge symmetries is that their charge on ordinary closed surfaces in codimension 1 is necessarily zero due to the existence of the current that vanishes on-shell.

Why Barnich and Brandt consider this a "problem" in the gauge case but not in the global case is impossible to tell without reading the paper, but they are certainly aware of this, as e.g. their eq. (2.17) takes care to define the charges of global symmetries over surfaces without boundary. A cursory reading suggests that they do not mean this to be a "problem" in the sense that it is somehow inconsistent, but simply that they are interested in whether or not the "next best thing" for a "gauge charge", namely the integration of the superpotential over a surface of codimension 2, can be made more meaningful than simply being an arbitary choice of some weird superpotential.

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