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Magnetic monopoles are solitons, i.e. field configurations on space (which is 3 dimensional). In pure $SU(N)$ gauge theory, magnetic monopoles can be constructed via 't Hooft's abelian projection (https://arxiv.org/abs/hep-th/0010225).

However, there is also a result that states that principle $SU(N)$ bundles over 3-manifolds are trivial. Doesn't this contradict the existence of monopoles?

Edit: Maybe it has to do with the fact that the gauge group should be $SU(N)/\mathbb Z_N$ in which case there are non-trivial bundles?

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    $\begingroup$ What does the triviality of SU(N) bundles have to do with magnetic monopoles? The magnetic monopole is a non-trivial U(1)-bundle (since the U(1) is the electromagnetic gauge group). If you think your reference about 't Hooft monopoles states anywhere that the SU(N) configuration before SSB has to be a non-trivial SU(N) bundle, please be more explicit where you think it does so. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 10, 2023 at 15:30
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    $\begingroup$ @ACuriousMind I don't think the abelian projection has much to do with SSB. $\endgroup$
    – dennis
    Commented Jan 10, 2023 at 17:17
  • $\begingroup$ Why do you say so? Chapter 5.2 of your reference is explicitly about the different phases of such a projected theory under a Higgs-like mechanism (and "Higgs mechanism" is essentially just another word for SSB). In any case, whether this is essential to the notion of the Abelian projection is beside the point, I really just want to know where the statement about non-trivial SU(N) bundles having to be involved comes from. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 10, 2023 at 17:33
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    $\begingroup$ @ACuriousMind. Well it comes from this: Monopoles are thought to have some topological structure. Therefore it is natural to think that this refers to the topology of the underlying bundle. However, if all bundles are trivial, I don't see how a monopole can have a topology. $\endgroup$
    – dennis
    Commented Jan 10, 2023 at 17:45
  • $\begingroup$ But as I said in my first comment, the "monopole" refers to something charged under a U(1) (or more generally some $\mathrm{U}(1)^k$ or whatever), i.e. an Abelian group. What bearing does the (non-)existence of non-trivial bundles for the non-Abelian SU(N) have on this? The monopole can be topologically non-trivial as a U(1) configuration even if the SU(N) configuration it comes from is trivial, can it not? $\endgroup$
    – ACuriousMind ♦
    Commented Jan 10, 2023 at 17:51

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