In which representation are monopoles of grand unifying theories classified? In the context of grand unification theories A.Zee's book states that $SU(5)$ (or $SO(10)$ if $SU(5)$ is considered as outdated as GUT candidate) as GUT and as spontaneously broken non-abelian gauge theory contains the monopole.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? 
 A: 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.


*

*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.

*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.

*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.


*

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

*Geometric definition of gauge invariance, Elihu Lubkin.

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

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