# Is there a critical order of the Abelian gauge theory in (2+1)D

In (2+1)D spacetime, it is known that the $U(1)$ gauge theory is always confined (according to Polyakov), while the $\mathbb{Z}_2$ gauge theory can support a deconfined phase. Now consider a generic Abelian gauge theory, i.e. the $\mathbb{Z}_n$ gauge theory (with $n$ being the order of the gauge group), the question is that does the $\mathbb{Z}_n$ theory in (2+1)D spacetime always support a deconfined phase, or there exist a critical $n_c$ such that for $n>n_c$ the $\mathbb{Z}_n$ theory is always confined?

The motivation of this question is that I believe in the $n\to\infty$ limit, the $\mathbb{Z}_n$ theory should behave the same as a $U(1)$ theory (let me know if it is not the case), while when $n=2$, it must reduce to the $\mathbb{Z}_2$ theory. But the $\mathbb{Z}_2$ theory is deconfined and the $U(1)$ theory is confined, so I am wondering if there exist a critical order $n_c$ such that beyond $n_c$ the Abelian gauge theory is always confined.

Also the gauge theory has a string correspondence (in terms of the dynamics of flux tubes, or in terms of the dynamics of membranes covering the Wilson loops). I would like to know if the above question could also be explained in the string theoretical language (in terms of the string tension, string renormalization ect.).

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Can you give some references which you are probably using for these? – user6818 Nov 7 '13 at 0:08
Just some random comments. I guess that all $\mathbb Z_n$ gauge theories have both a confined and deconfined phases, one can for example write down toric code like models for all $\mathbb Z_n$ and consider perturbations (like string tension/magnetic field) which will at some point induce a phase transition. So instead of the existence of a critical $n_c$, isn't it more natural to suspect that the size of the deconfined $\mathbb Z_n$ phase (as a function of perturbations), shrinks as a function of $n$? Such that in the limit $n\rightarrow\infty$, the deconfined phase vanishes gradually? – Heidar Dec 28 '13 at 15:44
This line of thinking seem to imply that for larger $n$, the $\mathbb Z_n$ topological state is less stable against perturbations that induce transition to the trivial (confined) phase. – Heidar Dec 28 '13 at 15:46
@Everett You Dear Everett, I have a naive comment here. I don't know whether the group $Z_n$ has a well-defined limit $\lim_{n\rightarrow \infty}Z_n$ at the mathematical level. For example, if $Z_n\equiv \left \{exp(il\frac{2\pi}{n});l=1,2,...,n\right \}$, then it seems that $\lim_{n\rightarrow \infty}Z_n=U(1)$; while, if $Z_n\equiv \left \{ (-\frac{n}{2})^*,(-\frac{n}{2}+1)^*,...,0^*,1^*,2^*,...,(\frac{n}{2}-1)^*\right \}$($n$ is even for example), where $m^*\equiv \left \{ m,m\pm n,m\pm 2n,...\right \}$, then it seems that $\lim_{n\rightarrow \infty}Z_n=Z$($Z$ represents all the integers). – Kai Li Sep 19 '14 at 9:05
@K-boy I am assuming your first limit. – Everett You Sep 21 '14 at 17:01