3d QED in IR can be described in terms of dual scalar field $\varphi$ have trivially conserved current with two indices, associated with U(1) one-form symmetry:

$$ J_{\mu\nu} = \epsilon_{\mu\nu\rho}\partial^\rho \varphi $$

In Baryons as Quantum Hall Droplets there are two statements about this current, which are unclear to me (before (2.4)):

It is conserved simply because the space of $\varphi$ configurations is a circle and $\pi_1(S^1) = \mathbb{Z}$.

I dumbfounded by this statement, in my opinion conservation current doesn't related to any topology..

The charged objects are strings.

I have one unsatisfactory argument for this: because string sweep 2d surface, we can simply integrate this current over 2d surface, and in such way define coupling of current to string.

I will be very appreciate for any answers!