"Domain wall" from continuous symmetry For a discrete $\mathbb{Z}_2$ symmetry for instance, in some patches of the Universe, it can spontaneously breaks $\mathbb{Z}_2$ and goes in the "right" minimum, while in some others patches it can go in the "left" minimum. This leads to domains walls. Do we have the equivalent of domain walls for continuous symmetries, let's say $U(1)$, because it can go to the minimum $v$ as well as any of the other minima $v\times \exp(i\theta)$?
 A: Yes, when $U(1)$ is spontaneously broken we have cosmic strings (aka vortices), which can be global or local depending on whether the $U(1)$ is global or local symmetry. Then there are magnetic monopoles which arise when a non-abelian symmetry $G$ is spontaneously broken to a smaller symmetry group $H$, such that the quotient space $G/H$ (which is the vacuum manifold of the theory) has non-trivial second homotopy group $\pi_2(G/H)$. For example, when $G=SU(2)$ and $H=U(1)$ the quotient $G/H$ is isomorphic to two-sphere ($S^2$), and
$$\pi_2(SU(2)/U(1))=Z~~~({\rm integers})$$
If $\pi_2(G/H)$ is trivial -- no monopoles.
Similarly, the existence of cosmic strings can be inferred from non-trivial first homotopy group $\pi_1$ of $U(1)$ which is also given by integers. This classification of topological defects can be generalized as
$$
\pi_n(G/H)
$$
which basically represents how many times (if at all) $S^n$ wraps around the vacuum manifold $G/H$. $\pi_1$ tells us about cosmic strings, $\pi_2$ about point defects, or monopoles, $\pi_3$ classifies so-called cosmic textures. There are also higher-dimensional generalizations of these defects.
In addition, another way to think about topological defects, is by looking for non-contractible $S^n$ spheres. In the presence of a cosmic string you cannot continuously contract a circle to a point (if the circle surrounds the string), in the presence of a monopole you cannot contract a two-sphere around the monopole, and so on.
