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][1] 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!

  [1]: https://arxiv.org/abs/1812.09253