Editing 't Hooft–Polyakov monopole (section)

==Mathematical details==
Suppose the vacuum is the [[vacuum manifold]] <math>\Sigma</math>. Then, for finite energies, as we move along each direction towards spatial infinity, the state along the path approaches a point on the vacuum manifold <math>\Sigma</math>. Otherwise, we would not have a finite energy. In topologically trivial 3&nbsp;+&nbsp;1 dimensions, this means spatial infinity is homotopically equivalent to the [[topological sphere]] <math>S^2</math>. So, the [[superselection sector]]s are classified by the second [[homotopy group]] of <math>\Sigma</math>, <math>\pi_2(\Sigma)</math>.

In the special case of a Yang–Mills–Higgs theory, the vacuum manifold is isomorphic to the quotient space <math>G/H</math> and the relevant homotopy group is <math>\pi_2(G/H)</math>. This does not actually require the existence of a scalar Higgs field. Most symmetry breaking mechanisms (e.g. technicolor) would also give rise to a 't&nbsp;Hooft–Polyakov monopole.

It is easy to generalize to the case of <math>d+1</math> dimensions. We have <math>\pi_{d-1}(\Sigma)</math>.