# Gauge transformation and large gauge transformation

Recently, Strominger posted his lecture notes on the infrared structure of gravity and gauge theory 1703.05448. In section 2.5, the equation (2.5.16) takes the following form

$$e^2\partial_zN=A_z^{(0)}|_{\mathscr I^+_+}-A_z^{(0)}|_{\mathscr I^+_-}$$

Here, $e$ is the electric charge (squared because he used some convention), $z$ is a complex angular coordinates on a 2 sphere at the future null infinity $\mathscr I^+$, $A_z^{(0)}$ is the $z$ component of the vector potential and the superscript means the $l=0$ component of its multiple expansion. $\mathscr I^+_+$ and $\mathscr I^+_-$ are the future and past boundaries of $\mathscr I^+$, respectively.

Strominger interpreted this expression as a certain gauge transformation, which I find difficult to accept. The usual gauge transformation takes the following form,

$$A'_\mu(t,\vec x)-A_\mu(t,\vec x)=\partial_\mu\alpha(t,\vec x).$$

Note that the arguments are all the same. But in Strominger's expression, the two $A$'s on the right hand side are evaluated at different places!

I am not sure how to understand this expression. Is there any other similar examples in physics? Is this because he wants to study the large gauge transformation, which leads him to interpret his expression that way?

• perhaps: when regarded as a function of $z,\bar z$, both $A_z^{(0)}|_{\mathscr I^+_+}$ and $A_z^{(0)}|_{\mathscr I^+_-}$ are evaluated at the same point, $(z,\bar z)$. Does this make sense? – AccidentalFourierTransform Mar 19 '17 at 12:23
• That doesn't look like a gauge transformation, it looks as if it is a transition function between the two local descriptions in $I^+_+$ and $I^+_-$. For another case in which these two are confused in the physics literature, see this recent question and its linked questions. – ACuriousMind Mar 19 '17 at 13:00

You should not interpret that as a gauge transformation. Rather, the claim is that is that as $A_z(u,z,{\bar z})$ evolves along the null generator, the final field $A_z \big|_{ {\mathscr I}^+_+}$ is related to the initial one $A_z \big|_{ {\mathscr I}^+_-}$ via $$A_z \big|_{ {\mathscr I}^+_+} - A_z \big|_{ {\mathscr I}^+_+} \equiv N_z = e^2 \partial_z N ~.$$ You could ask where this came from and why is this true at all? In fact, it is not always true. For instance, if there are magnetically charged particles which pass through ${\mathscr I}^+$, this is no longer true as is evidenced from the structure of the magnetic soft theorem. However, if there are no magnetically charged particles, then it is easy to see that the this must be true from the soft theorem which takes the form $$\langle N_z X \rangle \sim \sum_k \frac{1}{z-z_k} \langle X \rangle = \partial_z \sum_k \log|z-z_k|^2 \langle X \rangle.$$