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Topological defects are of common interest to condensed matter physics, atomic physics, astrophysics and cosmology, as well as algebraic topology. When the symmetry group $G$ spontaneously breaks down to its subgroup $H$, there are continuously connected ground states parametrized by the coset space $G/H$. The homotopy groups of the coset space then tell us what kinds of topological effects are possible. In most cases, non-trivial $\pi_d(G/H)$ implies the existence of $(2-d)$-dimensional topological defect. If the coset space has disconnected pieces ($\pi_0 (G/H) \neq 0$), we expect domain walls. For multiply-connected space ($\pi_1 (G/H)\neq 0$), there are strings/ vortices. If the boundary of space can map non-trivially to the coset space ($\pi_2(G/H)\neq 0$), we expect point-like defects such as monopoles. An exception to the rule is when the whole space is mapped non-trivially to the coset space ($\pi_3(G/H) \neq 0$), where skyrmions are stabilized by non-renormalizable terms in the low-energy effective theory. In this case, it is not the boundary condition that is topologically non-trivial, but the configuration in the bulk.
Topological Dark Matter by Hitoshi Murayama, Jing Shu
Important Topological Defects:
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Topological defects are most easily understood by considering the scalar potential. A non-trivial vev breaks a given group $G$ to some subgroup $H$. Depending on the homotopy class of the vacuum manifold $G/H$ (speak G mod H), we get different topological defects.
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Consider the possibility that φ = +η at x = +∞ and φ = −η at x = −∞. In this case, the continuous function φ(x) has to go from −η to +η as x is taken from −∞ to +∞ and so must necessarily pass through φ = 0. But then there is energy in this field configuration since the potential is non-zero when φ = 0. Also, this configuration cannot relax to either of the two vacuum configurations, say φ(x) = +η, since that involves changing the field over an infinite volume from −η to +η, which would cost an infinite amount of energy
The problem of finding the types of topological defects present in a given model reduces to finding the homotopy groups for a certain symmetry breaking G → H. That is, we need to find πn(G/H) (n = 0, 1, 2, 3) given the groups G and H. In general, this can be quite complicated but there is an immensely useful theorem which is often applicable and simplifies matters.
Suppose we consider a point defect and enclose it by an imaginary sphere S2. If we avoid other defects, all spheres about the point will be equivalent for our purposes, because they can be continuously deformed into one another inside V. Restricted to the sphere, cp defines a map S2+ A. If there is some topologically non-trivial aspect of this map, such as a winding number, the point defect will be topologically stable. It will not be able to just disappear leaving the medium in a uniform state with cp constant because, once it had gone, we could continuously deform the sphere to a point. The winding number, or whatever, would have had to have abruptly vanished, contradicting the assumed continuity. Similarly for a line defect, we take a circle S' about the line and obtain a map S' + A whose topological characteristics could ensure the stability of the defect.
Topological structures in field theories by Goddard and Mansfield
The field theories discussed above fall into two classes from the point of view of their topology. Suppose we are working in d space dimensions. In the first place we have theories, like the Abelian Higgs model of §2.1, where we have a potential function $V(\varphi)$ and $\varphi$ must tend to a zero (i.e. minimum) of V as we approach spatial infinity. In this case, at any given time, $\varphi$ defines a map $$ \varphi_\infty (\hat n) = \lim_{r\to\infty} \varphi(r \hat n)$$ which takes its values in the set of values which minimises V, $$ M=\{ \varphi: V(\varphi)=0 $$ The directions $\hat n$ in which one can approach infinity constitute a (d - 1)-dimensional sphere, the unit sphere in $R^d$. Thus $\varphi_\infty$ defines a map $S^{d-1} \to M$.
The second class of possibilities is not the result of non-trivial boundary conditions. Here we have a field which is always constrained to take its values in some manifold $M$, which is not simply a linear space. This time the boundary conditions are actually supposed to be trivial in the sense that $\varphi$ tends to a limit $\varphi_\infty \in M$ of course, independently of the direction in which we approach spatial infinity. In this case we can compactify space, $R^d$, by adding a point at infinity, to obtain what is topologically a sphere $S^d$ (cf stereographic projection), with $\varphi$ being assigned the value $\varphi_N$ at the point of $S^d$ corresponding to infinity. In this way we obtain a map $\varphi$ : $S^d \to M$.
Topological structures in field theories by Goddard and Mansfield
No topological defects of any type have yet been observed by astronomers, however, and certain types are not compatible with current observations; in particular, if domain walls and monopoles were present in the observable universe, they would result in significant deviations from what astronomers can see. Theories that predict the formation of these structures within the observable universe can therefore be largely ruled out.