This is a basic question I haven't see answered anywhere and I can't seem to figure out.

The usual statement of the 1+1D chiral anomaly Ward identity is that the divergence of the chiral current is the background field strength:

$\partial_\mu \langle j^\mu\rangle = \epsilon^{\mu \nu} F_{\mu \nu}/2\pi.$

I want to rewrite this in terms of the covariant chiral current $J^\mu = \epsilon^{\mu \nu}\langle j_\nu\rangle$. I believe it says $dJ = F/2\pi$. I am worried about this expression on a compact spacetime, however, since $F/2\pi$ may have a nonzero surface integral, while the integral of a divergence over a closed surface is zero. Must it be that somehow the covariant chiral current $J$ is not gauge invariant? I don't see a mechanism for this to happen though.

Thanks!