In theoretical physics, the 't Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without the Dirac string.
It arises in the case of a Yang–Mills theory with a gauge group
, coupled to a Higgs field which spontaneously breaks it down to a smaller group
It was first found independently by Gerard 't Hooft and Alexander Polyakov.
gauge symmetry is unbroken and the solution is non-singular also near the origin.
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.
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
So, the superselection sectors are classified by the second homotopy group of
In the special case of a Yang–Mills–Higgs theory, the vacuum manifold is isomorphic to the quotient space
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 Hooft–Polyakov monopole.
The "monopole problem" refers to the cosmological implications of grand unification theories (GUT).
Since monopoles are generically produced in GUT during the cooling of the universe, and since they are expected to be quite massive, their existence threatens to overclose it[clarification needed].
This is considered a "problem" within the standard Big Bang theory.
Cosmic inflation remedies the situation by diluting any primordial abundance of magnetic monopoles.