In which representation are monopoles of grand unifying theories classified?

Question

In the context of grand unification theories A.Zee's book states that $SU(5)$ (or $SO(10)$ if $SU(5)$ is considered as outdated as GUT candidate) as GUT and as spontaneously broken non-abelian gauge theory contains the monopole. I am wondering in which representation of $SU(5)$ it should be classified, may be as a singlet? Or is this question not meaningful since the monopole is perhaps a soliton? So if it were a soliton, how would it be classified?

Answer 1

Monopoles are somewhat subtle, and there are different layers towards their classification, each being more correct than the previous one.

^{Quick remark: what is usually called the $\text{SO}(10)$ model should more properly be called the $\text{Spin}(10)$ model, because it contains spinors. One has $\pi_1\text{Spin}(n)=0$ and $\text{Spin}(2n)^\vee=\text{PSO}(2n)$ and $\text{Spin}(2n+1)^\vee=\text{PSp}(2n)$. Also, one should point out that the correct GUT embedding reads $\text{SU}(3)\times \text{SU}(2)\times \text U(1)/\mathbb Z_6\hookrightarrow \text{SU}(5)$ and so, strictly speaking, the formulas below only make sense for $n=6$.}

The ABC of magnetic monopoles.

1. Goddard-Nuyts-Olive: the monopoles of a gauge theory with gauge group $G$ are classified by the representations of the GNO/Langlands dual group $G^\vee$ (cf. Ref. 1). For example, the dual of $\text{SU}(N)$ is $\text{PSU}(N)$, whose representations are basically the adjoint and its tensor powers. Similarly, the dual of $\text{SO}(2n)$ is $\text{SO}(2n)$ itself, and that of $\text{SO}(2n+1)$ is $\text{Sp}(2n)$. The representations of these are all well-known.

2. Lubkin: the GNO monopoles are typically unstable unless there is some topological charge that protects them. This topological charge takes values in $\pi_1G$ (cf. Ref. 2), and so one may have non-trivial monopoles if and only if $G$ is not simply-connected. For $\text{SU}(N)$ one has $\pi_1=0$ and so there are no monopoles. On the other hand, $\pi_1 \text{SO}(n)=\mathbb Z_2$, so here one does expect a (unique) non-trivial monopole.

3. The SSB monopole: if the theory undergoes a Higgs mechanism $G\to H$ one has slightly more freedom in making stable monopoles (cf. Refs. 3,4). In particular, these are classified not by $\pi_1G$ or $\pi_1H$ but by $\pi_2(G/H)=\operatorname{ker}(\pi_1 H\to\pi_1 G)$. For example, if $G$ is simply-connected one has $\operatorname{ker}(\pi_1 H\to\pi_1 G)=\pi_1H$ and so the classification reduces to that of Lubkin. Indeed, if $$ \text{SU}(5)\to \text{SU}(3)\times \text{SU}(2)\times \text U(1)/\mathbb Z_n,\qquad n\in\{1,2,3,6\} $$ one has a $$ \operatorname{ker}(\mathbb Z\times\mathbb Z_n\to 0)=\mathbb Z\times\mathbb Z_n $$ classification. Similarly, for an $\text{SO}(10)$ model one has a $$ \operatorname{ker}(\mathbb Z\times\mathbb Z_n\to \mathbb Z_2)=\mathbb Z\times\mathbb Z_{\lceil n/2\rceil} $$ classification (this last equality is an educated guess; ask your friend the mathematician to be sure).

References.

1. Gauge theories and magnetic charge, P.Goddard, J.Nuyts, D.Olive.

2. Geometric definition of gauge invariance, Elihu Lubkin.

3. Magnetic monopoles in unified gauge theories, G.'t Hooft.

4. Particle Spectrum in the Quantum Field Theory, Alexander M. Polyakov.