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If magnetic monopoles exist, they explain why electric charge is quantized. Moreover, magnetic monopoles appear automatically in extensions of the standard model, such as grand unified theories.
I am quite certain that magnetic monopoles really exist. How, when and if they will be found is another matter. Polyakov
There are two types of monopoles. One type is called Dirac monopoles and should be considered as a toy model, because there are many problems with such objects. Nevertheless, Dirac monopoles are great to get familiar with many important notions that are needed to describe more complicated topological objects. Dirac monopoles can be added to a pure $U(1)$ Yang-Mills theory, i.e. a theory with only one gauge boson present. Dirac monopoles are not topological solitons, because the electromagnetic field is singular at the location of the monopole and therefore the field energy is infinite.
The other type is called 't Hooft-Polyakov monopole. These are topological solitons and correspond to a field configuration with finite energy. They exist in gauge theories with, for example $SU(2)$ gauge symmetry, where the scalar field breaks the symmetry.
Further Details:
A Dirac monopole does not care about time and thus we are only interested in its description in space $\mathbb{R}^3$. The magnetic point charge introduces a singularity, which means that the electromagnetic field becomes infinite at the location of the monopole. Thus, we must restrict our description to $\mathbb{R}^3 - \{0 \} \simeq S^2$. The effect of the monopole is that we can't define our vector potential consistently everywhere on $S^2$. Instead, we must use (at least) two local patches, for example, two hemispheres. In the region where the patches overlap, we must have a function, called a transitions function, that transforms the description of the vector potential in one patch into the description in the other patch. In the most minimalistic case, the overlap is topologically simply a circle on the sphere $S^2$, i.e. $S^1$. Our transition function is therefore a map $g(x)$ that yields for each point of the overlap region, i.e. on the spatial circle $S^1$, the gauge transformation that correctly connects our two patches. Thus, the transition function is a map from $S^1$ to $U(1)$.
Each Lie group is also a manifold, and $U(1)$ is simply the set of unit complex numbers and therefore simply the unit circle $S^1$. Therefore, we can say that the transition function is a map from the spatial overlap region $S^1$ to the gauge group $S^1$. Now depending on how many monopoles are present in our system, the transition function changes. This means directly that all information about the monopole or the monopoles is encoded in the transition function, which is a map $S^1 \to S^1$.
When no monopole is present in the system, our transition function is simply the trivial map, that maps each point of the spatial circle onto the same point on $S^1$. If there is exactly one monopole present, we find instead that our transition function is a one-to-one map, i.e. each point of the spatial circle is mapped onto exactly one point of the gauge circle $S^1$. When two monopoles are present in our system, our transition function is a two-to-one map. This means that when we move once around the spatial circle $S^1$, we meet every element of the gauge circle $S^1$ twice. In this sense, the number of monopoles is characterized by an integer number, called the winding number.
In topology terms, we say that a Dirac monopole is characterized by the first homotopy $\pi_1$ group of $U(1) \simeq S^1$. This homotopy group is $\pi_1(S^1)=\mathbb{Z}$, the integers, which is exactly what we use to label monopoles.
There are many good reason to not introduce magnetic monopoles in the standard $U(1)$ gauge theory of electromagnetism. However, Dirac did it anyway and realized that we don't need a globally well defined vector potential. Instead it is sufficient if $\vec A$ is a connection.
For an explicit calculation of the potential on both hemispheres, see page 243 in Topological Solitons by Manton, Sutcliffe
Because of the singularity at the location of the monopole it is not a topological soliton. Classically it has "infinite mass".
So far, no QFT of Dirac monopoles exists yet. However for monopoles that are topological solitons, like the 't Hooft-Polyakov monopole there is no such problem.
The occurrence of Gribov horizons points to a more general problem in the gauge fixing procedure. Unlike in electrodynamics, global gauge conditions may not exist in non-abelian gauge theories [52]. In other words, it may not be possible to formulate a condition which in the whole space of gauge fields selects exactly one representative. This difficulty of imposing a global gauge condition is similar to the problem of a global coordinate choice on e.g. $S^2$. In this case, one either has to resort to some patching procedure and use more than one set of coordinates (like for the Wu–Yang treatment of the Dirac Monopole [53]) or deal with singular fields arising from these gauge ambiguities (Dirac Monopole). Gauge singularities are analogous to the coordinate singularities on non-trivial manifolds (azimuthal angle on north pole).
Topological Concepts in Gauge Theories by F. Lenz
The electromagnetic potential $a_\mu$ in the presence of a monopole must be singular. Indeed, it must be singular at at least one point on every closed surface surrounding a monopole. That this is the case can be seen as follows. Let us calculate the total magnetic flux leaving - say, for simplicity - a sphere. Suppose the potential $a_\mu$ is nonsingular everywhere on the sphere. Then, using Stokes' theorem, one can write the flux from the northern hemisphere as […] Similarly, the flux from the southern hemisphere is: […] Clearly, on adding the two contributions together, one gets zero for the total magnetic flux, in contradiction with the assertion that there is a magnetic monopole inside, which should give a total flux of $e \pi \tilde e$ for a monopole of charge $\tilde e$. One concludes therefore that the supposition above was wrong, or in other words that $a_\mu$ must have a singularity somewhere on the sphere.
page 28 in Some Elementary Gauge Theory Concepts by Hong-Mo Chan, Sheung Tsun Tsou
The 't Hooft-Polyakov monopole also does not care about time and thus we again investigate its behaviour in spatial $\mathbb{R}^3$ only.
We assume that spontaneous symmetry breaking happens, which means some scalar field $\phi$ gets a non-zero vacuum expectation value $\langle \phi \rangle \neq 0$. Now, the important thing is that there is, in general, not a unique vacuum expectation value that minimizes the scalar potential. Instead, there is an infinite number of possible vev orientations that also minimize the potential, because we can act on the scalar field with gauge transformations. The Lagrangian is invariant under gauge transformations and thus when some scalar field configuration $\langle \phi \rangle$ minimizes the scalar potential, the gauge transformed field configuration $g\langle \phi \rangle$ also minimizes the potential.
Finite energy means for the scalar field only that at spatial infinity, we must have
$$ |\langle \phi \rangle |= v^2 . $$
This condition is fulfilled with $\langle \phi \rangle = (0,0,v)$, but also, for example, with $\langle \phi \rangle = (v,0,0)$. These two scalar field configurations are connected by a gauge transformation.
The set of all possible vevs is called the vacuum manifold:
$$ M \equiv \{ \phi : V(\phi)=0 \}. $$
The structure of the vacuum manifold and thus of all possible vevs can be understood as follows.
As mentioned above, we can in general transform a given scalar field configuration $\phi_0$ that minimizes the scalar potential with a gauge transformation $g\phi \equiv \phi^g_0$ and get again a scalar field configuration that minimizes the scalar potential. However, not all $\phi^g_0$ are necessarily distinct. In general, there is some subgroup $H$ that does not change $\phi^g_0$ at all. (This subgroup $H$ characterizes the unbroken subgroup.) Explicitly: For some $h \in H$, we have
$$ \phi^h_0 =\phi_0.$$
An explicit example would be when our scalar field configuration is $(0,0,v)$, all gauge transformation that only affect the first two components, have no effect.
To understand the structure of the vacuum manifold, we want to get rid of these redundant field configurations. Mathematically, this is done by "modding out" the gauge transformations that do not change anything at all. This idea comes about as follows:
Given an element $g$, we know that it has exactly the same effect as the combination of gauge transformations $g h$, for all $h \in H$, because, well, by definition $H$ is the subgroup of transformations that have no effect on $\phi_0$:
$$ g h \phi_0 = g \phi_0 = \phi_0^g \quad \forall h \in H. $$
Thus, we get those gauge transformations that have a truly distinct effect by grouping together each $g$ with those gauge transformations that we can get from $g$ by acting on it from the right with some element of $H$. As explained above, we do this because the effect of some $g$ and each other gauge transformation that we can get from $g$ by acting on it with some element of $H$ has exactly the same effect on $\phi_0$ and thus does not yield anything new. We will then treat each of these sets of gauge transformations as a single element of the vacuum manifold.
Mathematically, this is called the coset space $G/H$:
$$ \{gH : g \in G \} \equiv G/H. $$
Each element of this coset space is called a coset and is a set of gauge transformations that has exactly the same effect on $\phi_0$. Each coset is a point on our vacuum manifold. Therefore, when a group $G$ breaks to a subgroup $H$ the vacuum manifold $M$ is given by
$$ M = G/H $$
Explicitly, for the breaking of $SU(2)$ to $U(1)$, we have $SU(2)/U(1)=S_{vac}^2$, i.e. our vacuum manifold is two-sphere.
To get a field configuration with finite energy, the only thing that we require is that the scalar field is in a configuration at spatial infinity that minimizes the scalar potential. Thus a possible field configuration with finite energy is the trivial configuration
$$ \phi^a = (0,0,v). $$
everywhere at spatial infinity (i.e. on the sphere $S^2$ with radius $r = \infty$). However, equally possible are more complicated and interesting configurations, for example:
$$ \phi^a \to^{r\to \infty} \frac{v r^a}{|r|}. $$
This scalar field configuration looks as follows
and is thus called "the hedgehog".
Again, the scalar field at spatial infinity takes only values in the vacuum manifold (i.e. field configuration that minimize the scalar potential): $\phi^a \phi^a = v^2$. When we now talk a walk on the sphere $S^2$ at spatial infinity such that we touch each point on this sphere exactly ones, we notice that on this path the scalar field configurations also takes each value on the possible vacuum manifold $S_{vac}^2$ exactly once, too. In this sense, this field configuration is defined by a the winding number $1$. This field configuration defines a one-to-one map from the spatial sphere at infinity $S^2$ to the vacuum manifold $S_{vac}^2$. In contrast, for the "trivial" field configuration, where $ \phi^a = (0,0,v) $ everywhere on $S^2$, all elements of the spatial sphere at infinity get mapped to the same element on $S_{vac}^2$ and thus this map has winding number $0$.
The important topological idea is now that there is no global gauge transformation that transforms the scalar field configuration with winding number $1$ into the trivial one with winding number $0$. This is known as the statement that "you can't comb the hedgehog". This means that we can't find a gauge transformation that is defined everywhere on $S^2$ and transforms the hedgehog field configuration into the unitary gauge, where the scalar field points in the same direction everywhere on $S^2$. There is a gauge transformation that almost achieves this, however it is singular at the southpole and is therefore not globally defined. In this sense the field configurations $ \phi^a \to^{r\to \infty} \frac{v r^a}{|r|}$ and $ \phi^a = (0,0,v)$ are truly distinct and describe different physical phenomena.
By inspecting the Hedgehog field configuration further, one can see that it is the source of a Coulomb like potential that looks exactly like the potential of a Dirac monopole.
The similarities between this topological non-trivial scalar field configuration and the Dirac monopole go even further.
1.) Recall that we investigate a model where an $SU(2)$ gauge symmetry breaks to the abelian subgroup $U(1)$. At spatial infinity, the scalar field takes only values that minimize the scalar potential and thus there the gauge symmetry is only the unbroken $U(1)$ symmetry. The $SU(2)$ symmetry gets broken by the non-zero vev. Now one similarity is that the Dirac monopole appeared in a $U(1)$ pure gauge theory. One says that outside the "monopole core" the Dirac and the 't Hooft-Polyakov monopole are indistinguishable.
2.) We are able to bring the Hedgehog field configuration into the unitary gauge, where the scalar field points in the same direction everywhere, except for the monopole. There, the gauge transformation that achieves this "combing" of the Hedgehog is singular. This singularity is exactly the same singularity that we call the Dirac string for the Dirac monopole. One can show (see Shnir page 154-155) that in the unitary gauge, the monopole potential is exactly the Dirac potential embedded in an $SU(2)$ group.
Now this can be really confusing because the Dirac monopole appears in a pure $U(1)$ gauge theory, whereas for the 't Hooft-Polyakov monopole, so far, we were mainly concerned with the configuration of a scalar field.
The thing is that for the t Hooft-Polyakov monopole, we actually have the freedom to shift our perspective. By performing the singular gauge transformation that combs the Hedgehog scalar field configuration (except at the south pole), the scalar field configuration at infinity becomes trivial. However, this gauge transformation has an effect on the other parts of the theory, too!
This can be best understood by defining a gauge-invariant electromagnetic field strength tensor (see: Topology of Higgs fields by J. Arafune, P. G. O. Freund, and C. J. Goebel)
$$ F_{\mu\nu} = \partial_\mu A_\nu -\partial_nu A_\mu + \frac{1}{v^3e} \epsilon_{abc} \phi^a \partial_mu \phi^b \partial_\nu \phi^c .$$
In the unitary gauge, where we have "combed" the Hedgehog, the scalar field configuration at spatial infinity is trivial and thus carries winding number $0$. However, now the gauge field configuration becomes non-trivial and suddenly carries winding number $1$. The singular gauge transformation (eq. 5.3 in Shnir) that brings the scalar field into the unitary gauge "transfers" the magnetic charge from the scalar field to the gauge field!
Now, we want to understand this situation using again notions from topology. The scalar field at spatial infinity defines a map $S^2 \to S^2_{vac}$. Such maps are defined by the second homotopy class $\pi_2(G/H)$, which in our case is $\pi_2(SU(2)/U(1))= \pi_1(S^2_{vac})= \mathbb{Z}$.
In contrast, a Dirac monopole was characterized by the first homotopy class of the gauge group $\pi_1(U(1))$.
The clou is now that $\pi_2(G/H)$ and $\pi_1(H)$ are isomorphic!!!
We can't rotate the Hedgehog field configuration smoothly to the trivial configuration $\phi_0$ globally on the sphere at infinity $S^2$. However, we can transform it into the $\phi$ on the two hemispheres separately. To get a consistent description this way, we need to define a transition function in the area where the two descriptions overlap, i.e. , for example, at the equator. (It doesn't matter where we choose that our descriptions (our two patches) overlap).
By definition, the scalar field is everywhere on $S^2$ in a configuration that minimizes the scalar potential and thus the gauge symmetry is broken here from $SU(2)$ to $U(1)$. Therefore, the gauge transformation that transforms the description on one hemisphere into the description on the other hemisphere, must be an element of the unbroken subgroup $U(1)$.
The scalar field yields a field value for each point on the sphere at spatial infinity, i.e. is a map from $S^2$ to the vacuum manifold. By combining this fact, with the patched description and the transition function idea from above, we get a correspondence between the map from $S^2$ to $S^2_{vac}= G/H$ and the map from $S^1$, (the equator) to $U(1)$. It can be shown that this map is an isomorphism (See Shnir page 167)!
The isomorphism $\pi_2(G/H) = \pi_1(H)$ means that the topological charge that we used to characterize the Hedgehog scalar field configuration is equivalent to the topological charge that we use the characterize a Dirac monopole!
Magnetic monopoles are topologically stable solutions of a non-abelian gauge theory with spontaneous symmetry breaking. […] The solution with this topological property in (3+1)-dimensional spacetime is the monopole, which has the form as |r| → ∞ that $$ \vec Φ(r) = A \frac{\vec r}{| \vec r |} \quad (| \vec r| → ∞), $$ with A a constant. The thing to notice here (as with the vortex) is the distinction between internal and real space components. The vector Φ lives in internal isospace but the radial vector r lives in real space. However, eqn 49.12 links these two together. In components we have, for example $φ^1(| \vec r| → ∞) = A \frac{x}{|\vec r|}$, that is, along the spatial 1-direction (also known as the x-direction) the field points along the internal 1- direction. Similarly along the y-direction the field points along the internal 2-direction and so on. In this sense, the field points radially outwards at infinity, which is why Polyakov called this the hedgehog solution.
page 453 in Quantum Field Theory for the gifted Amateur by T. Lancaster
The "core" structure of a ’t Hooft–Polyakov monopole is complicated, but the long range electromagnetic fields are the same as for a Dirac monopole. In this sense,we can say that ’t Hooft–Polyakov monopole are Dirac monopoles with the singularity smoothed out. They are stable, because the magnetic charge can not change through smooth deformations of the field. (Source: page 249 in Topological Solitons by Manton, Sutcliffe)
Through numerical analysis it was shown that any small deformation of the monopole solution increases the Yang-Mills energy. Therefore they correspond to local minima. However, there is no proof that they are global minima! (Source: page 259 in Topological Solitons by Manton, Sutcliffe)
A ’t Hooft–Polyakov monopole has spin $0$ because it is spherically symmetric.
The energy density of a BPS monopole depends only on the Higgs field. (Source: page 10 in http://www.itp.uni-hannover.de/~lechtenf/Events/Lectures/manton.pdf )
From the viewpoint of fibre bundle theory one faces the following problem. Since the fields in the 't Hooft-Polyakov Monopole are regular everywhere, they come from a trivial bundles, whereas the potential of the Dirac monopole (with $m\neq 0$) possessing a string singularity determines a nontrivial principal bundle. How is it then possible to recover the Dirac monopole from the 't Hooft-Polyakov monopole? With the help of the above gauge transformation we got the string singularity by "brute force" admitting a singularity $\gamma$. The answer within the framework of fibre bundles will be given in section 10.6.
Differential Geometry, Gauge Theories, and Gravity by M. Göckeler, T. Schücker
Recommended Resources
Ever since Dirac has proposed the Dirac monopole generalizing the Maxwell’s theory, the monopole has become an obsession theoretically and experimentally [1]. After the Dirac monopole we have had the Wu-Yang monopole [2], the ’t Hooft-Polyakov monopole [3], the grand unification (Dokos-Tomaras) monopole [4], and the electroweak (Cho-Maison) monopole [5–9]. But none of them except for the electroweak monopole might become realistic enough to be discovered.
Around a magnetic monopole the total space is not simple $S^2 \times U(1)$ (the trivial bundle), where $S^2$ is a spatial sphere around the monopole and $U(1)$ is our gauge group. Instead the monopole alters the global structure such that the total space is instead $S^3$ (the principal bundle). This means, through the presence of the magnetic monopole the fibers get glued differently together. This is reflected through a different connection on the fibre bundle, which is what we call in the physics a different gauge potential.
$S^3$ and $S^2 \times U(1)$ are locally the same but are globally different.
The best explanation for these things can be found in chapter 0 of the book Topology, Geometry and Gauge fields by G. Naber.
See also Hopf Bundle.
It can be proved [2.25] that 't Hooft-Polyakov magnetic monopole solutions occur for any gauge theory in which a simple (or semi-simple) gauge group is broken down spontaneously to a factor group $H=H' \times U(1)$, where $H'$ is a simple or semi-simple gauge group and $U(1)$ is identified with the electromagnetic gauge group.
page 590 in Conceptual Foundations of Modern Particle Physics by Robert E Marshak
When the gauge group is compact, which means that the charge is quantized, monopoles exist. See, the great discussion chapter 2 of Some Elementary Gauge Theory Concepts by Hong-Mo Chan,Sheung Tsun Tsou
Take in particular theories with gauge algebra $\mathfrak{su}(n)$. It was noted in Section 1.4 that if we deal with pure Yang-Mills theory containing only gauge bosons, the gauge group is $SU(N)/Z_N$, but if the theory contains other fields belonging to the fundamental representation, such as quarks in QCD, then the gauge group is $SU(N)$. We shall now show that whereas monopole charges are admitted by the former theory, no monopole charge exists in the latter theory. It is sufficient to illustrate with the simplest example, namely the theory with the gauge algebra $\mathfrak{su}(2)$. […] [A] theory with gauge group $SU(2)$ cannot admit monopoles in the sense defined above since any closed curve on $\Gamma_0$ on $S^3$ can be continuously deformed to a point, as illustrated in Figure 2.5. On the other hand, if we are dealing with the pure Yang-Mills theory, the gauge group is no longer $SU(2)$ but $SU(2)/Z_2 \simeq SO(3)$, which is obtained from $SU(2)$ by identifying pairs of elements with opposite signs. […] Theories with gauge group $SU(N)$ have no monopoles, but the pure gauge theories with gauge group $SU(N)/Z_N$ will have monopoles whose charges take values in $Z_N$. Thus, in particular, full QCD containing both gluons and color triplet quarks has no monopoles while pure QCD with only gluons admit monopoles with charges labelled by a"triality", i.e. integers modulo 3, or equivalently the cube roots of unity: $\zeta_r= exp(i2\pi r/3)$, $r=0,1,2$.
page 24 in Some Elementary Gauge Theory Concepts by Hong-Mo Chan, Sheung Tsun Tsou
The condition for a nonsingular monopole is that the topological charge is in the kernel of the mapping $\pi_1(H) \to \pi_1(G)$. [..] The fact that the condition is both necessary and sufficient makes it the jewel of monopole theory, well worthy of being honored with a box. It enables us to tell instantly, without solving a single differential equation, whether a given theory admits non- singular monopoles, and what kinds it admits.
section 4.2 "Making Monopoles" in The Magnetic Monopole Fifty Years Later by Sidney R. Coleman
"Gauge theories in which a simple group G is spontaneously broken at a mass scale $M_X$ to a subgroup $G_1 \times U(1)$ containing an explicit $U(1)$ factor necessarily possess t'Hooft-Polyakov magnetic monopoles of mass $m \sim M_X /g^2$ . (These are topologically stable classical configurations of the gauge and Higgs fields of the theories). All grand unified theories possess such monopole solutions since $U(1)_{EM}$ is unbroken. In particular, $M_X \sim 10^{14}$ Gev in 3-2-1 desert theories, so that $m \sim 10^{16}$ GeV. These monopoles may have been produced prolifically in the early universe, especially at the time of the phase transition below which the grand unified symmetry was broken. " Langacker - Grand Unified Theories
"But upon grand unification into SU(5) [or more generally any group without U(1) factors] electric charge is quantized. The result here is deeply connected to Dirac’s remark (chapter IV.4) that electric charge is quantized if the magnetic monopole exists. We know from chapter V.7 that spontaneously broken nonabelian gauge theories such as the SU (5) theory contain the monopole." from Quantum Field Theory in a Nutshell by A. Zee
No! This was found by Wu and Yang through their description of the magnetic monopole, which makes it possible to avoid a string of singularities.
The above arguments hold for any sphere surrounding the monopole. Hence, by increasing continuously the radius of the sphere, the singularity on it will trace out a continuous curve. In other words, we have deduced that a monopole must be attached to a whole line of singularities stretching all the way to infinity. Such a line of singularities, first noted by Dirac, is called a Dirac string. […] From the above example, then, one sees that the Dirac string is actually not a physical singularity at all, but merely a singularity in our representation of the potential in a particular gauge choice. It is similar in nature to the coordinate singularity in the cartographer's zenithal projection of the globe. There, if the north pole is chose as the zenith, the the coordinates of the north pole is chose as the zenith, then the coordinates of the north pole ar singular, and if the south pole is chosen as the zenith, then the coordinates of the south pole are singular, although there is no actual singularity on the globe itself. To give a full representation of the globe, the cartographer usually uses two zenithal projections, say one from the north pole and one from the south pole; then specifies which point in one projection corresponds to which point on the other in the region where the two projections overlap, e.g. by giving the longitudes of points in the equatorial region on both charts. In a very similar fashion, one can give a fully nonsingular representation of the gauge potential $a_\mu$ over the whole surface surrounding a monopole provided that one pays the price of using more than one coordinate patch and specifying in the overlap regions of these patches the relation between the potentials defined in the different patches. […] In the presence of a monopole the electromagnetic potential $a_\mu(x)$ has to be patched.
page 29 in Some Elementary Gauge Theory Concepts by Hong-Mo Chan, Sheung Tsun Tsou
In the sphere example, if we use such a patched description, the gauge field is nonsingular in both patches. However, now the two expression for the magnetic flux through the two hemispheres do no longer cancel. By using Stokes' theorem, we only integrate over the boundary of both patches, which is just the equator
\begin{align} \int_{S^2} \text{ Flux} &= \int_{\text{ Northern Hemisphere}} da^{(1)} + \int_{\text{ Southern Hemisphere}} da^{(1)} \notag \\ &= \int_{\text{ Equator: } 0 \to 2 \pi} a^{(1)} + \int_{\text{ Equator: } 2 \pi \to 0} a^{(2)} \\ &= \int_{\text{ Equator: } 0 \to 2 \pi} a^{(1)} - \int_{\text{ Equator: } 0 \to 2 \pi} a^{(2)} \\ \tag{1} \end{align}
The description on the two hemispheres are related by a gauge transformation. This gauge transformation is also called the transition function:
\begin{align} a_\mu^{(1)} &= a_\mu^{(2)} - d\alpha^{(21)} \notag \\ \phi^{(2)} &= e^{-i \alpha^{(21)} } \phi^{(1)} . \tag{2} \end{align}
The transition function must be well defined in its region and therefore be single valued. This means especially that
$$ e^{-i \alpha^{(21)}(\theta, 2\pi) } = e^{-i \alpha^{(21)}(\theta, 0) } \tag{3}$$ where the interval $\{0,2\pi\}$ is meant around the equator, holds.
Now, from Eq. 2 it follows that the integral in Eq. 1 is simply
\begin{align} \int_{S^2} \text{ Flux} &= \int_{\text{ Equator}} a^{(1)} - \int_{\text{ Equator}} a^{(2)} \notag \\ &= \int_{\text{ Equator}} d\alpha^{(21)} = \alpha^{(21)} \left( \frac{\pi}{2}, 2\pi \right) - \alpha^{(21)} \left( \frac{\pi}{2}, 0 \right) \, . \end{align}
The magnetic flux is therefore the increase in $\alpha^{(21)} $ as we move around the equator. Because of Eq. 3 this must be an integer multiple of $2 \pi$.
This is discussed nicely in section 3.4 at page 65 in Topological Solitons by Manton and Sutcliffe.
The Wu-Yang approach not only eliminates the “Dirac string” but, in the process, does three things: (1) it provides a derivation of Dirac's quantization condition in a geometric context; (2) it exhibits the close relationship between a "global" (large) gauge transformation - in contrast to a "local" gauge transformation - to topological winding numbers (see §10.4a); and finally (3) it puts in proper perspective the relation between the Dirac magnetic monopole and the 't Hooft-Polyakov magnetic monopole (see §10.2c).
page 260 in Conceptual Foundations of Modern Particle Physics by Robert Eugene Marshak
The crucial difference with the original Dirac proposal was that, because these monopoles appear as regular, soliton-like solutions to the classical field equations, they are unavoidable and cannot be left out.
The Dirac monopole can be added by hand to a pure $U(1)$ gauge theory. In contrast, the t' Hooft-Polyakov monopole appears in a $SU(2)$ or $SO(3)$ Yang-Mills theory, where additionally scalars are present.
Dirac monopoles are classified by $\pi_1(S^1)$ (source) and ’t Hooft–Polyakov monopoles by $\pi_2(G/H)=\pi_2(S^2)$, where $G$ is the gauge group before and $H$ after symmetry breaking.
Comparing the 't Hooft-Polyakov monopole with the Dirac monopole it will seem as if the have almost nothing in common; to be more precise, nothing at all except that they both possess magnetic charge. This is not quite true […] From Equation (10.59) we may say that the gauge transformation transfers responsibility for the monopole from the first (Dirac) term, to the second (topological, Higgs) one. […] It is shown how a gauge transformation relates the Dirac and 't Hooft-Polyakov monopoles.
page 412 and page 414 and page 424 in Quantum Field Theory by L. Ryder
The Georgi-Glashow model is capable of predicting a magnetic monopole of the non-singular "'tHooft-Polyakov" type which acquires the property of a Dirac monopole as $r \to \infty$. […] There are other important differences between the "'t Hooft-Polyakov" and Dirac monopoles: in particular, the finite core spatial structure of the "'t Hooft-Polyakov" monopole is quite different from the "string" singularities associated with the Dirac monopole.
page 587 in Conceptual Foundations of Modern Particle Physics by Robert E Marshak
Notice how the ’t Hooft–Polyakov monopole and Dirac monopole are very different beasts. Dirac monopoles are singular mathematical solutions of electrodynamics which necessitate the introduction of point particles as the sources of the magnetic flux. These particles have arbitrary spin and mass. In contrast ’t Hooft–Polyakov monopoles are nonsingular solutions arising from the interaction of a non-abelian gauge theory and a scalar field. All of their properties, such as their mass, are determined by the original theory.
page 455 in Quantum Field Theory for the gifted Amateur by T. Lancaster
Each of the apparently independent lines of thought followed in the two preceding sections has led us to consider a gauge theory with the symmetry group, G, spontaneously broken by the vacuum to a subgroup H. From §2 we expect to find electric charge quantisation (at least if the electric charge, Q, generates H= U(1)) and magnetic monopoles, whilst according to §3 we expect to find extended 'soliton' solutions to such theories with long-range magnetic fields. In this way we are led to anticipate a non-singular extended solution which at large distances looks like a Dirac monopole, and further that its magnetic charge will be related to the topological quantum number specified by the boundary conditions on the scalar fields. This section will show how these expectations are indeed realised.
Magnetic monopoles in gauge field theories by P GODDARD and DI OLIVE
In general, the mass of of the resulting monopole $M_{mp}$ is fixed - in analogy to the Georgi-Glashow model - by the symmetry breaking scale $M_X$ for $G \to H' \times U(1)$ and is of order $M_{mp} \sim M_X/g^2$ [see Eq. (10.58)].
page 590 in Conceptual Foundations of Modern Particle Physics by Robert E Marshak
’t Hooft estimated the mass of these beasts as MW/α ≈ 137MW ≈ 11 TeV, making them very heavy indeed and outside the range of our experiments. Their existence therefore remains an open question.
page 455 in Quantum Field Theory for the gifted Amateur by T. Lancaster
See also section 4.4 "Why Monopoles are Heavy" in The Magnetic Monopole Fifty Years Later by Sidney R. Coleman
There are three known solutions to the GUT monopole over-abundance problem. The first is that the GUT phase transition that produces magnetic monopoles is followed by a period of inflation that dilutes the monopole density to acceptable levels [Guth, 1981]. The second is that the GUT model includes a period during which electromagnetism is broken. During this period, the magnetic monopoles will get connected by strings, leading to rapid annihilation and dilution [Langacker and Pi, 1980]. The third possibility is that the GUT phase transition never occurred and the universe was always in a state of broken GUT symmetry [Dvali, Melfo and Senjanovic, 1995]. All three solutions of the monopole over-abundance problem require fine tuning of parameters and/or model building solely for the purpose of eliminating magnetic monopoles.
See also The search for magnetic monopoles by Arttu Rajantie
Unfortunately - in terms of physical interest - the SSB of the standard electroweak group $G=SU(2)_L \times U(1)_Y$ to $H=U(1)_{EM}$ does not yield a 't Hooft-Polyakov-type monopole. With the knowledge that the electric charge Q is the sum of the $I_{3L}$ generator in $SU(2)_L$ and the $U(1)_Y$ generator, this can be seen as follows: the $U(1)_{EM}$ group does not lie entirely in the $U(1)_Y$ group and hence any non-contractible closed loop in $H=U(1)_{EM}$ [remember that $G/H=[SU(2)_L \times U(1)_Y]/U(1)_{EM}$ is the coset space] may be deformed in $G$ to stay non-contractible in $U(1)_Y$. Thus, unless the deformation of the closed loop in $U(1)_Y$ is trivial, it cannot be deformed to a point in $G$ and, consequently, the condition for a 't Hooft-Polyakov monopole [i.e. a contractible loop in $G$ and non-contractible in $H$] is not satisfied. The "exact sequence" statement of this negative result is: $$ \Pi_2[SU(2)_L \times U(1)_Y/U(1)_{EM}]\simeq \Pi_2[U(1)]_{SU(2)_L \times U(1)_Y} =0 \, ,$$ which tells us that no "'t Hooft-Polyakov" monopole can be generated from the SSB of the standard electroweak group.
page 590 in Conceptual Foundations of Modern Particle Physics by Marshak
Take note that some authors argue there could be topological objects in the standard model: https://ncatlab.org/nlab/show/monopole#ElectroweakMonopoles