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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.

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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.

I learned this perspective on emergent gauge fields from E. Fradkin's book Field Theories in Condensed Matter Physics.

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