# Can there be an emergent non-compact gauge field?

Emergent compact gauge fields are ubiquitous in condensed matter theory (like $U(1)$). Are there any examples of an emergent non-compact gauge field, in which case there won't be any quantization conditions and there would be conserved currents and charges which might or might not be physical.

Emergent gauge fields come about in systems with local constraints $j(\vec x) = 0$, since
$$\int DA \exp i \int j \wedge A = \delta (j).$$
Then in a classic physics move we integrate out the matter instead of the Lagrange multiplier and we get an effective gauge theory for $A$.
If $j$ is quantized, an integer for example, then $A$ must be taken to be $U(1)$ valued. In general, $j$ should behave like a current. If $j$ is $\mathbb{R}$ valued, then $A$ will be as well. You could consider a constant density fluid, for instance, and then $A$ would be something like a "dilaton" gauge field.