't Hooft–Polyakov monopole

In

${\displaystyle G}$, coupled to a it down to a smaller group ${\displaystyle H}$ via the

Unlike the Dirac monopole, the 't Hooft–Polyakov monopole is a smooth solution with a finite total energy. 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 ${\displaystyle G}$

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 full Yang–Mills–Higgs equations of motion.

Mathematical details

Suppose the vacuum is the

${\displaystyle \Sigma }$. 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 ${\displaystyle \Sigma }$. Otherwise, we would not have a finite energy. In topologically trivial 3 + 1 dimensions, this means spatial infinity is homotopically equivalent to the ${\displaystyle S^{2}}$. So, the are classified by the second homotopy group of ${\displaystyle \Sigma }$, ${\displaystyle \pi _{2}(\Sigma )}$.

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 ${\displaystyle \pi _{d-1}(\Sigma )}$.

Monopole problem

The "monopole problem" refers to the cosmological implications of

remedies the situation by diluting any primordial abundance of magnetic monopoles.

See also

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

References