't Hooft-Polyakov monopole

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In theoretical physics, the 't Hooft-Polyakov monopole (Hedgehog) is a topological soliton similar to the Dirac monopole but without any singularities. It arises in the case of a Yang-Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks it down to a smaller group H via the Higgs mechanism.

Unlike the Dirac monopole, the Gerard 't Hooft-Polyakov monopole is a smooth solution with a finite total energy. The solution is localized around $r = 0$ . Very far from the origin, the gauge group G is broken to H, and the 't Hooft-Polyakov monopole reduces to the Dirac monopole.

However, at the origin itself, the G gauge symmetry is unbroken and the solution is non-singular also near the origin. The Higgs field

$H_i \qquad (i=1,2,3) \,$

is proportional to

$x_i f(|x|) \,$

where the adjoint indices are identified with the three-dimensional spatial indices. The gauge field at infinity is such that the Higgs field's dependence on the angular directions is pure gauge. The precise configuration for the Higgs field and the gauge field near the origin is such that it satisfies the full Yang-Mills-Higgs equations of motion.

[edit] Mathematical details

Suppose the vacuum is the vacuum manifold Σ. 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 Σ. Otherwise, we would not have a finite energy. In topologically trivial 3 + 1 dimensions, this means spatial infinity is homotopically equivalent to the topological sphere S². So, the superselection sectors are classified by the second homotopy group of Σ, π₂(Σ).

In the special case of a Yang-Mills-Higgs theory, the vacuum manifold is isomorphic to the quotient space G/H and the relevant homotopy group is π₂(G/H). Note that this doesn't actually require the existence of a scalar Higgs field. Most symmetry breaking mechanisms (e.g. technicolor) would also give rise to a 't Hooft-Polyakov monopole.

It's easy to generalize to the case of d + 1 dimensions. We have π_d−1(Σ).