Conserved charge at null infinity associated with Large gauge transformation I am reading Strominger's lecture notes "Lectures on the infrared structure of gravity and gauge theory" (https://arxiv.org/abs/1703.05448). At some point, following (I guess) the authors of the paper "New symmetries in massless QED" (https://arxiv.org/abs/1407.3789), the author of the lecture notes defines future and past conserved charges at near spatial infinity. The one for the future is
$$Q_{\varepsilon}=\frac{1}{e^2} \int_{\mathcal{I}^+_-} d^2z\gamma_{z\bar{z}}\varepsilon(z,\bar{z})F_{ru}^{(2)}$$
(the same definition was made in the paper the author follows, but with a different notation). I am trying to understand how or why this is a sensibly defined charge, in the sense of it being the zeroth (i.e. time)-component of a conserved current associated with the large gauge transformations: $A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\varepsilon(x)$. So, my guess (and please correct me if I am wrong) is that I have to start with this current, which at Minkowski space is $J^{\mu}=\partial_{\nu}[\varepsilon(x)F^{\nu\mu}]$ and transform to retarded coordinates with the next step being to integrate over space (that is over the coordinates $\{r,z,\bar{z}\}$). The result I get is
$$Q_{\varepsilon}=\frac{1}{e^2}\int d^2zdr\ r^2\gamma_{z\bar{z}}\nabla^i[\varepsilon(x)F_{i0}]$$
How do I proceed from here, such that I obtain an expression identical to the one above?? What considerations/assumptions must I make?
P.S.: The space of integration in my attempt of deriving the expression for the charge is still all the space. How do I go from there to being the past of future lightlike infinity ($\mathcal{I}^+_-$)?
 A: The current under consideration is
$$
J \sim \star d ( \varepsilon \star F ) 
$$
It is clear that this current is conserved $d \star J = 0$. The Noether charge is then given by
$$
Q_\Sigma = \int_\Sigma \star J \sim \int_\Sigma d ( \varepsilon \star F )  \sim \oint_{\partial\Sigma} \varepsilon \star F .
$$
where $\Sigma$ is any Cauchy slice of the theory. We see that the Noether charge corresponding to this current is a boundary term evaluated on the boundary of the Cauchy slice.
In the paper, the authors take $\Sigma = {\cal I}^+ \cup i^+$. The boundary of this Cauchy slice is $\partial \Sigma = {\cal I}^+_-$. It follows that
$$
Q_\varepsilon^+ =  \oint_{{\cal I}^+_-} \varepsilon \star F 
\sim \int_{\mathcal{I}^+_-} d^2z\gamma_{z\bar{z}}\varepsilon(z,\bar{z})F_{ru}^{(2)}(-\infty,z,{\bar z}).
$$
In your question, you are constructing the charge on a constant $t$ slice instead. The integrand $\nabla^i[\varepsilon(x)F_{i0}]$ is a total derivative so it reduces to a simple boundary integral. The only boundary in your case is the one located at $r \to +\infty$ (namely, spatial infinity) so the charge in your question reduces to
$$
Q_{\varepsilon}(t) \sim \frac{1}{e^2}\int d^2z \gamma_{z\bar{z}} \varepsilon(t,z,{\bar z})F^{(2)}_{tr}(t,z,{\bar z})
$$
This is the charge at a constant $t$ slice. The charges constructed in the paper are then obtained by taking $t \to \pm\infty$,
$$
Q_\varepsilon^\pm = \lim_{t \to \pm\infty} Q_{\varepsilon}(t) .
$$
PS - I have not kept track of any overall factors in this answer. That's for you to work out!
