research level - Boundary currents for Asymptotic Symmetry Group (ASG)

In the context of asymptotic symmetry groups, what is a boundary current? Why is it called a "current"?

Context: I'm reading Strominger's recent paper on Asymptotic symmetry group of Yang-Mills (link here) and he has a section on the boundary current (section 2.3). I can follow the math completely fine, but some of the words are confusing to me.

Answer

In this context, a "current" is an object obeying an affine Lie algebra, also called current algebra and a special case of a Kac-Moody algebra. It is an algebra formed by unit weight operators: take for example a current $J^a(z)$, where $a$ is a label and $z$ is a complex coordinate. The algebra is given by

$$[J^a_n,J^b_m]=i{f^{ab}}_cJ^c_{n+m}+mkd^{ab}\delta_{n+m},$$

where

$$J^a_n=\frac{1}{2\pi i}\oint dz \, z^{-(n+1)}J^a(z).$$

The integer $n$ denotes the mode number, the integer $k$ is the level and $d^{ab}=(t^a,t^b)$ defines the inner product between generators.

The word "boundary" refers to the fact that the symmetry group underlying the algebra preserves a certain structure at the boundary of the geometry at infinity. In the case of the paper you are reading, the symmetry group is $U(1)$ and the boundary is given by $\mathcal{I}^+$.

Additional information:

Affine Lie algebras play a role in string theory/conformal field theory, where they can be used to generate states in certain representations of a group. For example, the state

$$J^a_{-1}\tilde{\alpha}^{\mu}_{-1}|0\rangle$$

corresponds to a massless vector $A^{\mu a}$ in the adjoint representation of the underlying group ($\tilde{\alpha}^{\mu}_{-1}$ is a creation operator).