What is the conserved flux corresponding to center symmetry?

Question

I understand the one-form symmetry of $U(1)$ gauge theories as related to the conservation of magnetic and electric flux, given by integration of two-form electromagnetic field tensor and its dual, respectively. You don't need to consider quantum field theory to see this conservation law. The magnetic flux conservation (in the absence of monopoles) is even seen without considering the dynamics.

In non-Abelian gauge theories you can also define something like a gauge invariant flux using a closed Wilson line or 't Hooft line. This can take any value in the non-Abelian group. But it is said that the one-form symmetry is only associated to the discrete Abelian center of the group.

How do I see that you can only define a conserved flux for the center of the group (preferably classically, or in terms of lattice gauge configurations)?

Answer 1

Meaningful conservation laws must be associated with symmetries between physical states, on the state space after we have quotiented out the gauge redundancy. Regardless of how exactly you treat your gauge theory, the space of distinct classical physical states is such that $A$ and $gAg^{-1} + \mathrm{d}g$ for some gauge transformation $g$ are the same point, i.e. the gauge transformations are do-nothing transformations on the physical state space.

But when $g$ is a constant function valued in the center of the group, then $A = gAg^{-1} + \mathrm{d}g$ already before quotienting out, so central transformations really do act on the physical space of states - they are not gauge. In the Abelian case, the whole group is the center and so you get a global $\mathrm{U}(1)$ symmetry, in the non-Abelian case you get the symmetry of whatever the center of the gauge group is.