Temporal vs spatial conservation and memory effect in electrodynamics

We consider the standard Maxwell's theory in 1+3 dimensions in the presence of a timelike boundary. In this context, we show that (generalized) Ampere-Maxwell's charge appears as a Noether charge associated with the Maxwell U(1) gauge symmetry which satisfies a spatial conservation equation. Furthermore, we also introduce the notion of spatial memory field and its corresponding memory effect. Finally, similar to the temporal case through the lens of Strominger's triangle proposal, we show how spatial memory and conservation are related.