What precisely and mathematically does it mean to have $W$ bosons carry electric charges?

We know from Wikipedia that experiments say that $W$ bosons carry electric charges:
$W^\pm$ carry $+$ and $-$ of electric charges.

However, the gauge fields do not sit at any representation of the gauge group, but the gauge fields transform under the adjoint action of the gauge group. See eg https://physics.stackexchange.com/a/191982/42982.

So how can $W^\pm$ carry $+$ and $-$ of electric charges, which are charge 1 and -1 representation of $U(1)$ electromagnetic gauge group? But gauge fields do not sit at any representation of the gauge group?

Some may say, "well it is easy, we have lagrangian terms (which need to be neutral) like
$$
\bar{\nu}_e W^{+ \mu} \partial_{\mu} e^-.
$$
$$
\bar{\nu} W^{- \mu} \partial_{\mu} e^+.
$$
So these explain why $W^\pm$ carry $+$ and $-$ of electric charges."

But again gauge fields do not sit at any representation of the gauge group (say $U(1)$?), precisely and mathematically.

This post imported from StackExchange Physics at 2020-12-13 12:44 (UTC), posted by SE-user annie marie heart