quantum field theory - Compact QED and Non-compact QED - Polyakov textbook

This question is related with Polyakov, "Gauge Fields and Strings" section 4.3

Firstly, Polyakov define a QED on a lattice

1. 
   
   

   Compact QED \begin{align} S = \frac{1}{2} \sum_{x, \alpha, \beta} (1-\cos(F_{x,\alpha\beta}) ) \end{align} where $F_{x,\alpha\beta} = A_{x,\alpha} + A_{x+\alpha,\beta} - A_{x+\beta, \alpha} - A_{x,\beta}$ with $- \pi \leq A_{x,\alpha} \leq \pi$

   
   


2. 
   

   Non compact QED \begin{align} S = \frac{1}{4 e_0^2} \sum_{x,\alpha\beta} F_{x,\alpha\beta}^2 \end{align} here $- \infty \leq A_{x,\alpha} \leq \infty$

   
   

In the textbook, Compact version (periodic) is Natural version of QED which is related with charge quantization.

Here i have a few basic question.

1 : Why periodicity gives compactness?

2 : What is the physical difference or usefulness of compact or Non-compact QED? (Is this compact concept is only related with the lattice theory?)

3 : Is periodicity of $A_{x,\alpha}$ really related with charge conservation?

My guess from equation 4.32 in textbook, which is \begin{align} q_0 = \frac{1}{2\pi} \oint_{L} A_{x, \delta} \end{align} The periodicity of $A_{x,\alpha}$ gives some number after loop integral. $i.e$, (ration for certain loop)

Answer

1 : When $A$ is restricted to $[-\pi,\pi)$, the group of gauge transformations is the compact $\mathrm{U}(1)$ group. (Roughly speaking, to be compact, a group needs to be bounded.) In the nonperiodic theory, the gauge group is the group of real numbers under addition, which is noncompact.

2 : The most important distinction between the compact and noncompact theories is that the compact one allows ("Polyakov") monopoles, which are topological configurations of $A$. (For the same reason, the XY model has topological defects, called vortices, while a real scalar field does not.) Polyakov monopoles are very similar to Dirac monopoles, but Dirac's argument was that the Dirac string is unobservable because charges are quantized, while Polyakov's is that the string has no energy cost, because $F_{\alpha\beta} = 2\pi$ along its length.

You could define a compact $\mathrm{U}(1)$ gauge theory in the continuum, but monopoles are no longer possible. (Again, the same is true of vortices in the XY model; they have a core at which $\theta$ is undefined.) There could still be a distinction for the model on a space with nontrivial topology.

3 : How is compactness related to discreteness of charge?

Suppose we have a matter field $\psi$ with charge $q$ defined on the sites of the lattice. If you apply a gauge transformation $\psi \rightarrow \psi \mathrm{e}^{\mathrm{i} q \theta}$ on one site, then $A$ changes by $\pm\theta$ on all links connected to this site. If we choose $\theta = 2\pi$, this transformation of $A$ is redundant in the compact theory. Because $A$ is effectively unchanged, in order for the gauge transformation to be a symmetry, $\psi$ must also be unchanged, so we require $q$ to be an integer.

Another way to see this is to use the Dirac argument that the existence of magnetic monopoles requires discreteness of electric charges.

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If you have access to it, you might want to look at J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51, 659 (1979) for a detailed review of these ideas.