Intheoretical physics, the't Hooft–Polyakov monopole is atopological soliton similar to theDirac monopole but without theDirac string. It arises in the case of aYang–Mills theory with agauge group${\displaystyle G}$, coupled to aHiggs field whichspontaneously breaks it down to a smaller group${\displaystyle H}$ via theHiggs mechanism. It was first found independently byGerard 't Hooft andAlexander Polyakov.^[1][2]

Unlike the Dirac monopole, the 't Hooft–Polyakov monopole is a smooth solution with a finite totalenergy. The solution is localized around${\displaystyle r=0}$. Very far from the origin, the gauge group${\displaystyle G}$ is broken to${\displaystyle H}$, and the 't Hooft–Polyakov monopole reduces to the Dirac monopole.

However, at the origin itself, the${\displaystyle G}$gauge symmetry is unbroken and the solution is non-singular also near the origin. The Higgs field${\displaystyle H_i(i=1,2,3)}$,is proportional to${\displaystyle x_{i}f(|x|)}$,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 fullYang–Mills–Higgs equations of motion.

Suppose the vacuum is thevacuum manifold${\displaystyle \mathrm{\Sigma }}$. 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${\displaystyle \mathrm{\Sigma }}$. Otherwise, we would not have a finite energy. In topologically trivial 3 + 1 dimensions, this means spatial infinity is homotopically equivalent to thetopological sphere${\displaystyle S^2}$. So, thesuperselection sectors are classified by the secondhomotopy group of${\displaystyle \mathrm{\Sigma }}$,${\displaystyle {\pi }_2(\mathrm{\Sigma })}$.

In the special case of a Yang–Mills–Higgs theory, the vacuum manifold is isomorphic to the quotient space${\displaystyle G/H}$ and the relevant homotopy group is${\displaystyle {\pi }_2(G/H)}$. 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.

It is easy to generalize to the case of${\displaystyle d+1}$ dimensions. We have${\displaystyle {\pi }_{d-1}(\mathrm{\Sigma })}$.

The "monopole problem" refers to the cosmological implications ofgrand 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 standardBig Bang theory.Cosmic inflation remedies the situation by diluting any primordial abundance of magnetic monopoles.

• 't Hooft anomaly
• 't Hooft loop
• 't Hooft symbol

• Magnetic monopoles
• Gauge theories

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• Wikipedia articles needing clarification from May 2019