In one pdf on solitons, I am finding the following written
For the sine-Gordon theory, it is much better to think of $\phi$ as a field modulo $2\pi$, i.e. as a function $\phi: R \rightarrow S_{1}$. In this language, the boundary condition is that $\phi(x) = 0 \mod 2\pi$ as $x \rightarrow\pm\infty$. Thus, instead of thinking of the kink as joining two vacua, we can think of it as “winding around the circle” instead. "
I am unable to understand what does the boundary condition being $\phi(x) = 0 \operatorname{mod} 2π \,\text{as}\, x \rightarrow\pm\infty$ and “winding around the circle” means. Can anyone help?
