This question is related with Polyakov, "Gauge Fields and Strings" section 4.3
Firstly, Polyakov define a QED on a lattice
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 $- \pi \leq A_{x,\alpha} \leq \pi$
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)