Is there any (locally) conserved charges associated to gauge symmetries? I'm currently in my second year of master.
From what I understand, in QFT, Noether's first theorem implies that for any continuous symmetry (i.e. associated to a $n$-dimensional Lie group $G$, $n\geq 1$), there are $n$ corresponding conserved currents and thus $n$ conserved charges.
From this question, I understand that, for gauge symmetries, the symmetry group associated to $G$ is an infinite-dimmensional Lie group. So, can we apply Noether's theorem this symmetry group? Does it give an infinite number of conserved charges?
 A: Since most of the answers in the post mentioned by Qmechanic are either too technical or come to the wrong conclusion, I will try to give a simple yet precise answer. I will consider Maxwell theory as an example.
Let's try to apply the Noether's theorem to gauge symmetries of the Maxwell theory, i.e. $A_\mu\to A_\mu +\partial_\mu\lambda$ for arbitrary function $\lambda(x)$. This symmetry is also called a local symmetry, as you can take $\lambda$ to be non-vanishing inside any arbitrary region and zero outside.
Now you may start with the Lagrangian ${\cal L}=-\frac14 F_{\mu\nu}F^{\mu\nu}$ and compute the standard Noether current for the gauge transformation parametrized by the function $\lambda$. You find
\begin{align}
J^\mu_\lambda=F^{\mu\nu}\partial_\nu\lambda
\end{align}
You can check that the current is conserved $\partial_\mu J_\lambda^\mu=0$ only after using the equations of motion $\partial_\mu F^{\mu\nu}=0$. Now the charge Noether charge over a hypersurface $\Sigma$ (which can be a constant time surface) is given as usual by
\begin{align}
Q_\lambda=\int_\Sigma t_\mu J^\mu_\lambda=\int_\Sigma t_\mu F^{\mu\nu}\partial_\nu\lambda
\end{align}
where $t_\mu$ is the unit normal to $\Sigma$. Now the crucial step is that upon integration by parts and using the field equations we can write the charge as a boundary integral
\begin{align}
Q_\lambda=\oint_{\partial\Sigma} t_\mu n_\nu \big(F^{\mu\nu}\lambda\big)
\end{align}
where $\partial\Sigma$ is the boundary of $\Sigma$ and $n_\mu$ is the unit normal tangent to $\Sigma$ and normal to its boundary $\partial \Sigma$. Note that if $\Sigma$ is the constant time hypersurface, then we have $t_\mu n_\nu F^{\mu\nu}=n\cdot E$, the normal electric field to the boundary.
Now if $\Sigma$, has no boundary, e.g. if it is a 3 sphere $S^3$, then the charge is identically zero. This is what usually people refer to when talking about the charge of gauge symmetries. However, if the space has a boundary, then the charge is non-zero. This can naturally happen if your problem is defined in a finite region of space (e.g. in Casimir effect), or if you consider flat spacetime with natural boundary conditions $A_\mu\sim 1/r$, then you have nontrivial charges if $\lambda\big\vert_{\partial \Sigma}\neq 0$.
The conclusions above are true for any field theory with local symmetries (including gauge theory and gravity). The charges reduce to boundary integrals. This is proven in Barnich, Brandt 2001 (here)
To read more about the latter case, you may see this reference as an starting point and continue with more advanced topics in the references therein.
