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Hopf Solitons

Hopf solitions (after Heinz Hopf) are topological solitons with a non-trivial hopf invariant (charge). We are interested in their occurence in field theories. Suppose we are concerned with a field theory, taken as a continuous map, $\phi: \mathbb{R}^3 \to S^2$. As in, for example, the O(3) sigma model. We are interested in the effects of imposing the (natural) boundary condition:  $\lim_{\|x \| \to \infty} \phi = \phi_0 .

Now since this boundary condition gives us a homogeneous field at infinity we can perform a single point compactification of three dimensional space to arrive at a map

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explicitly this can be done by e.g. inverse stereographic projection. We can now investigate the topological properties of this map. Specifically our map will represent an element of the group

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What this means is that for any field configuration, we can associate an integer (the Hopf charge). For configurations with this non-zero we say that we have a Hopf soliton. Since these are topological invariants, this charge is a constant of the dynamics provided no singularities are introduced into the field.