I understand (i think) that for a magnetic monopole to exist as the result of a gauge group $G$ being spontaneously broken to a subgroup $H$ by the Higgs mechanism, that certain criteria must be fulfilled. One of these is that there must be a non-trivial second homotopy. Which i believe means that the resultant vacuum manifold must be non-trivial.

So for example if the vacuum manifold is a 2-sphere, the second homotopy classifies the ways you can map a 2-sphere onto this manifold. A 2-sphere cannot be deformed to a point, and thus we introduce winding numbers which can be associated with topological charge = magnetic monopole. [I may be wrong here]

So, in many papers discussing the electroweak monopole, the following statements appear in all of them:

'it was thought that the Weinberg-Salam model possesses no non-trivial second homotopy' (i.e. no monopoles exist)

followed by

'However, the Weinberg-Salam model with the hypercharge $U(1)$, could be viewed as a gauged $CP^1$ model in which the (normalized) Higgs doublet plays the role of the $CP^1$ field'

I confess that I am completely lost by this last statement. If anybody could shed any light as to what a gauged $CP^1$ field/$CP^1$ model is (or a good book that explains it) it would be great,


1 Answer 1


Your question is answered in the paper Monopoles in Weinberg-Salam Model by Cho and Maison, from which I think the quotes are taken. What the gauged $CP^1$ model is exactly is not really relevant to the answer, which is purely mathematical (it is a type of model to which the authors reduce the bosonic sector of the standard Weinberg-Salam model with extra hypercharge added).

The authors derive an ansatz for the general solutions to the model, a.k.a. "dyon" or "Higgs doublet", and it turns out that with the extra $U(1)$ in the picture this ansatz can be spherically symmetric, which was thought impossible for topological reasons. $CP^1$ is a fancy notation for the 2D sphere, and 2D spheres have non-trivial second homotopy group, so the topological obstruction is removed. Here is the explanation in context:

"The basis for this “non-existence theorem” is, of course, that with the spontaneous symmetry breaking the quotient space $SU(2)\times U(1)/U(1)$ allows no non-trivial second homotopy. This has led many people to conclude that there is no topological structure in the Weinberg-Salam model which can accommodate a magnetic monopole... In the following we establish the existence of a new type of monopole and dyon solutions in the standard Weinberg-Salam model, and clarify the topological origin of the magnetic charge.

[...] So the above ansatz describes a most general spherically symmetric ansatz of a $SU(2)\times U(1)$ dyon. Here we emphasize the importance of the non-trivial $U(1)$ degrees of freedom to make the ansatz spherically symmetric. Without the extra $U(1)$ the Higgs doublet does not allow a spherically symmetric ansatz. This is because the spherical symmetry for the gauge field involves the embedding of the radial isotropy group $SO(2)$ into the gauge group that requires the Higgs field to be invariant under the $U(1)$ subgroup of $SU(2)$. This is possible with a Higgs triplet, but not with a Higgs doublet. In fact, in the absence of the hypercharge $U(1)$ degrees of freedom, the above ansatz describes the $SU(2)$ sphaleron which is not spherically symmetric. The situation changes with the inclusion of the extra hypercharge $U(1)$ in the standard model, which can compensate the action of the $U(1)$ subgroup of $SU(2)$ on the Higgs field."


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