Dark Monopoles in Grand Unified Theories

We consider a Yang-Mills-Higgs theory with gauge group $G=SU(n)$ broken to $G_{v} = [SU(p)\times SU(n-p)\times U(1)]/Z$ by a Higgs field in the adjoint representation. We obtain monopole solutions whose magnetic field is not in the Cartan Subalgebra. Since their magnetic field vanishes in the direction of the generator of the electromagnetic group $U(1)_{em}$, we call them Dark Monopoles. These Dark Monopoles must exist in some Grand Unified Theories (GUTs) without the need to introduce a dark sector. We analyze the particular case of $SU(5)$ GUT, where we obtain that their mass is $M = 4\pi v \widetilde{E}(\lambda/e^{2})/e$, where $\widetilde{E}(\lambda/e^{2})$ is a monotonically increasing function of $\lambda/e^{2}$ with $\widetilde{E}(0)=1.294$ and $\widetilde{E}(\infty)=3.262.$ We also give a geometrical interpretation to their non-abelian magnetic charge.

field, we need not to introduce a dark sector. This is an interesting feature, since we can have these monopoles as dark matter candidates in the standard Grand Unified Theories.
Monopoles with a magnetic flux in a non-abelian direction have been constructed for a Yang-Mills-Higgs theory with G = SU (3) broken to "SU (2)×U (1)" [16] (see also [4,17,18]). They were associated to the su(2) subalgebra generated by the Gell-Mann matrices λ 2 , λ 5 and λ 7 and an ansatz was constructed using some general arguments of symmetry. On the other hand, in the present work we consider a Yang-Mills-Higgs theory with an arbitrary gauge group SU (n) broken to by a scalar field in the adjoint representation and we use a general procedure [19,20] to construct the monopole asymptotic configuration, associated to some su(2) subalgebras. We consider su(2) subalgebras with generators M a , which are linear combinations of some step operators. Then, the asymptotic form of the gauge and magnetic fields are linear combinations of the generators M a , while the asymptotic form of the scalar field is a linear combination of generators S and Q a , a = 0, ±1, ±2 , which form, respectively, a singlet and a quintuplet under the su(2) subalgebra.
From these asymptotic configurations, we construct an ansatz for the whole space and calculate the Hamiltonian. Then, we obtain the second order differential equations for the profile functions. We also find the numerical solution for these equations in the case G = SU (5), for some particular coupling constant values. Moreover, we show that the mass of a Dark Monopole is a monotonically increasing function of λ/e 2 , and for G = SU (5), the mass range at the classical level is where E(0) = 1.294 and E(∞) = 3.262. It is interesting to note that due to the fact that for the Dark Monopoles B i and D i φ are linear combinations of different generators, the Bogomolny equation B i = D i φ does not have a non-trivial solution.
We also construct a Killing vector ζ associated to an asymptotic symmetry of the Dark Monopole and show that these monopoles have a conserved current in a non-abelian direction. The associated magnetic charge Q M is quantized in multiples of 8π/e and we give a geometrical interpretation to this charge. Although Dark Monopoles are associated to the trivial sector of Π 1 (G v ), the conservation of Q M could prevent them to decay. Our construction is quite general and, in principle, it could be generalized to other gauge groups.
This paper is organized as follows: in section 2 we review a general procedure to construct the asymptotic configuration for the fields of a monopole. Then, in section 3 we show the specific construction of the asymptotic configuration of a Dark Monopole for the gauge group SU (n) and we propose the ansatz. We also show that our solution is not equivalent to any other solution whose magnetic field lies in the Cartan subalgebra. In section 4 we get the Hamiltonian for our Dark Monopoles and the radial equations for the profile functions. We also obtain the numerical solution for these equations, for some particular coupling constant values, and the mass range for the SU (5) Dark Monopole.
Finally, in section 5 we construct a Killing vector associated to an asymptotic symmetry of the Dark Monopole and the corresponding current and conserved charge. We conclude with a summary of the results.

Magnetic monopoles in non-Abelian theories
In this section we will fix some conventions and review a general construction of the asymptotic form of monopole solutions. We will consider a Yang-Mills-Higgs theory in 3 + 1 dimensions with gauge group G of rank r, which is simple and simply connected, and with a real scalar field φ = φ a T a in the adjoint representation. The generators T a form an orthogonal basis for the Lie algebra g of G which satisfy T r (T a T b ) = y δ ab , where ψ 2 y is the Dynkin index of the representation and ψ is the highest root of g. We will also use the Cartan-Weyl basis with Cartan elements H i , which form a basis for the Cartan subalgebra H, and step operators E α , satisfying the commutation relations Moreover, For an arbitrary root α we define the generators which form an su(2) subalgebra. We will denote by α i , i = 1, 2, . . . , r, the simple roots, and by λ i , i = 1, 2, . . . , r the fundamental weights of g, which satisfy the relation Let φ 0 be the vacuum configuration of the theory which spontaneously breaks the gauge group G to G v . By a gauge transformation, the vacuum configuration φ 0 , can be made to lie in the Cartan subalgebra H, that is φ 0 = u · H, where u is a vector. For a vacuum in the adjoint representation, all the generators of G v must commute with φ 0 and form a Lie algebra which we will call g v . Since φ 0 commutes with itself and all other generators of G v , it will generate an invariant subgroup U (1) of G v . In order for this U (1) to be compact, the vector u must be proportional to a fundamental weight of G [5]. Then, in this case, the symmetry breaking by φ 0 in the adjoint representation, G v will have the general form [3,5] where K is semisimple group, Z is a discrete subgroup of the center of K, Z(K), which belongs to U (1) and K, i.e., Z = U (1)∩K. We shall call this a minimal symmetry breaking. Then, from the condition that the vacuum φ 0 fulfill, we obtain that the vacuum manifold G/G v satisfies [4], For a static configuration with W 0 = 0, D 0 φ = 0 and G i0 = 0, the energy is where we define In order for the monopole solution to have finite energy, at r → ∞, The first condition implies that asymptotically, φ must lay in the vacuum manifold. We can then consider that, asymptotically, φ is a gauge transformation of φ 0 , that is, By similar arguments the gauge field has the form [19,20] where and M 3 is a generator of g v , in order for D i φ = 0. Let us consider that there exist two other generators, M 1 and M 2 of g, which do not belong to g v , and which together with M 3 form a su(2) algebra We will call M i by the monopole generators. Then, in order to remove the Dirac string singularity from W (0) µ in the string-gauge and for the configuration to be spherically symmetric, we will consider that (2.11) The asymptotic gauge field (2.8) can be written in Cartesian coordinates as with n j = x j /r. The gauge configuration gives rise to the asymptotic magnetic monopole field (2.13) The group element (2.11) is single-valued, except at θ = π, where [20] g(π, ϕ)g(π, 0) −1 = exp(−2iϕM 3 ) = h(ϕ) . (2.14) Since M 3 is a su(2) generator, it has integer or half-integer eigenvalues, and therefore h(ϕ), 0 ≤ ϕ ≤ 2π, provides a closed loop in G v which is associated to sectors of Π 1 (G v ) and the monopole solutions are associated to these topological sectors.
For the symmetry breaking G → G v , with G v given by (2.4), we can recover the asymptotic form of the 't Hooft-Polyakov monopole [1,2] and generalizations to larger gauge groups [21,22], considering the su(2) subalgebras formed by the generators M i = T α i , for roots α such that α · u = 0, for φ 0 = u · H. Since T α 3 ∈ H, the magnetic field (2.13) for these monopoles is in the Cartan subalgebra H, up to conjugation by g(θ, ϕ).

The Dark Monopoles
Now we want to construct monopoles with asymptotic magnetic field which is not in the Cartan subalgebra H, that is M 3 / ∈ H, in theories with φ in the adjoint representation, which are relevant to some GUTs. We will call them Dark Monopoles, since their magnetic field vanishes in the direction of the generator of the electromagnetic group U (1) em , which we consider to be in H. Since [M 3 , φ 0 ] = 0 and M 3 is hermitian, it is usually considered that M 3 belongs to the same Cartan subalgebra as φ 0 . However, this is not necessary when G v is a non-abelian gauge group. In fact, more monopole solutions can be obtained if we do not impose this condition. A nice analysis of this problem in the SU (3) → U (2) case can be found in [23,24]. In the case of Z 2 monopoles, for theories where φ is not in the adjoint representation, one can have solutions with M 3 in the direction of some step operators [20,25,26]. Also note that string-vortex solutions with magnetic fields as combinations of step operators have been constructed for Yang-Mills-Higgs theories for various gauge groups [27][28][29][30][31][32][33]. For simplicity, we will consider that the gauge group is G = SU (n) and that where λ p is an arbitrary fundamental weight of su(n). This vacuum, spontaneously breaks SU (n) to [5] G It is useful to recall that the roots of the algebra su(n) in the basis of the simple roots have the form where e i are orthonormal vectors in a n-dimensional vector space, and therefore the roots of su(n) have the same length square, which is equal A simple way to obtain the commutators between the step operators E α in an arbitrary representation of su(n) is to use the fact that in the n-dimensional representation of su(n), the step operator E α associated to the root α = (e i − e j ), is represented by the n × n matrix (E ij ) kl = δ ik δ jl and that the commutator of two generators is the same in any representation. Let us also recall that for the Cartan involution of an arbitrary semisimple Lie algebra g [34], and g can be decomposed as g = g (0) ⊕ g (1) where Then g (0) forms a subalgebra of g and the generators of g (1) form a representation of g (0) . For example, for g = su (3), there are three generators in g (0) which form a su(2) subalgebra and there are five generators in g (1) which form a quintuplet of this su(2) subalgebra. In order to construct Dark Monopole solutions, we shall consider that the monopole generators M i , which form a su(2) subalgebra, belong to g (0) . Then, φ 0 , which is in g (1) , will be in a representation of this su(2), as we will see later on. Using the definition of eq.(2.3), we will consider that M 3 = 2T α 2 , M 1 = 2T β 2 and M 2 = 2T γ 2 where α, β, γ are roots of su(n). Then, the condition that they form a su(2) algebra implies that α + β + γ = 0. Now, since M 3 ∈ g v , then [M 3 , φ 0 ] = 0, which implies that α · λ p = 0, and therefore α does not have the simple root α p in its expansion in the simple root basis. Thus, for α = e i − e j , if i < j, either i > p or j ≤ p, and if i > j, either i ≤ p or j > p. On the other hand, since M 1 and M 2 do not belong to g v , then β · λ p = 0 and γ · λ p = 0, which implies that β and γ have the simple root α p in their expansion in the simple root basis. Then, denoting by T ij a , a = 1, 2, 3, the generators defined in Eq.(2.3) for α = e i − e j , we can conclude that the possible monopole generators, for α positive are where there are two possibilities: a) 1 ≤ i < j ≤ p and j < k, with p < k ≤ n; b) p < i < j ≤ n and k < j with 1 < k ≤ p. Each of these su(2) subalgebras can be labeled by these three numbers i, j, k. On the other hand, when α is a negative root, i > j, which can be seen as an exchange between i ↔ j in the cases above. We should also remark that there may be other su(2) subalgebras, with M 3 being a combination of step operators, from which we could construct other Dark Monopole solutions. However, for simplicity, in this work we will only consider the su(2) subalgebras related to positive roots, given by eq.(3.4).
For each su(2) subalgebra, we can construct a monopole solution. And in order to obtain the asymptotic configuration of the scalar field (2.7) for each of them, it is convenient to decompose φ 0 as Therefore, S is a singlet. On the other hand, one can check that Q 0 belongs to a quintuplet together with the generators (3), in the three dimensional representation, these generators correspond to the Gell-Mann matrices λ7, −λ5, λ2.
satisfying the commutation relations where c ± l,m = l(l + 1) − m(m ± 1) with l = 2. Although for any su(2) subalgebra M i , the generators Q m always form a quintuplet and therefore l = 2, we will continue to write l to keep track of this constant. It can also be useful for possible generalizations of Dark Monopole construction with different l for other gauge groups.
Since M i ∈ g (0) and Q m ∈ g (1) , then, Therefore, we can conclude that Now, since Q m ∈ g (1) , then [Q m , Q p ] ∈ g (0) . Thus, where A mp , B ± mp , D δ mp are constants and T δ 2 are other possible generators of g (0) . Then, taking the trace of this commutator with M 3 , M ± and T −δ 2 , and using the previous results, we can conclude that This set of generators M i , Q m form an su(3) subalgebra of su(n), since they are linear combinations of the generators T ij a , T ik a , T jk a , a = 1, 2, 3. In order to construct the asymptotic form for the scalar field, let us recall that in a (2j + 1) irreducible representation of a su(2) algebra with generators J i , i = 1, 2, 3, and with eigenstates |j, m , the spherical harmonics can be written as [35], From Eqs. (2.7), (2.11) and (3.5), the asymptotic form for the scalar field can be written as From the commutation relations (3.6), eq.(3.7) can be written as Hence, Therefore, the asymptotic configuration for the scalar field is with α = 2v √ 6 |λ p | 4π 2l + 1 (3.11) and l = 2. From this asymptotic configuration, we can propose an ansatz for the whole space as φ(r, θ, ϕ) = φ s + φ q (r, θ, ϕ) where f (r) is a radial function such that f (r = 0) = 0 and f (r → ∞) = 1.
From the asymptotic gauge field configuration (2.12), one can propose the ansatz with the radial function u(r) satisfying the conditions, u(r = 0) = 1 and u(r → ∞) = 0.
From this gauge field we obtain the magnetic field where P ik T = δ ik − n i n k , P ik L = n i n k and u (r) stands for du/dr. Using the fact that it is direct to verify that our solution is spherically symmetric with respect to which means that (3.12) and (3.15) satisfies (3.20)

On the equivalence between solutions
After constructing the ansatz for our Dark Monopoles, we must discuss the reason why our solution is not equivalent to any other solution whose magnetic field lies in the Cartan subalgebra H. We recall that the arguments we present here are valid for monopoles in the case of minimal symmetry breaking. First, let us denote by (φ M , W M i ) a field configuration at infinity in the positive z−direction and, therefore, φ M = φ 0 . At this point, the gauge field takes values in the su(2) subalgebra of the generators M i , i = 1, 2, 3. However, note that this configuration is not unique, since we can obtain an equally valid solution by means of a global gauge transformation P . Under this transformation, we have that [23,24] φ P = P φ M P −1 , while M P i = P M i P −1 , are also generators of an su(2) subalgebra of G. Now, since we want to preserve the symmetry breaking, i.e., φ P = φ M = φ 0 , we see that P must belong to G v . However, note that since P is position independent, for other directions than the positive z−direction this global G v action does not leave the Higgs field at infinity invariant. This follows from the fact that for a general directionr this P is not an element of the unbroken group G v (r), which is position dependent. So, even if two monopole solutions are related by the conjugation of an element P ∈ G v in the north pole, this global gauge transformation cannot be implemented to the whole asymptotic configuration because P will not belong to the local unbroken gauge group G v (r).
In fact, we can not even define a gauge transformation P (θ, ϕ) = g(θ, ϕ) P g −1 (θ, ϕ), with g(θ, ϕ) given by eq.(2.11), that takes values in G v (r) for every directionr at infinity. This happens because the generators of G v which do not commute with the magnetic field, which is proportional to M 3 in the north pole, cannot be globally well-defined. This situation is the well-known problem of "Global Color" [36][37][38][39][40][41][42][43][44] and happens to some monopole solutions for theories with a non-abelian unbroken symmetry (NUS).
Then, in our specific case, there are indeed global gauge transformations that take our magnetic field in the north pole to an usual one lying in H. The simplest of such transformations is of the form P = exp −iπT ij 1 /2 . But from the considerations above we see that such a transformation cannot be globally implemented, which implies that our monopole solution is distinct from those with a magnetic flux in the Cartan subalgebra. We also add that there is an example [45] of a similar situation in the SU (3) → U (2) symmetry breaking, where two distinct monopole solutions can be related in the north pole by the global action of the SU (2) subgroup of U (2), while we cannot move between these solutions dynamically, implying the solutions are physically distinct.

Hamiltonian and equations of motion
In this section we shall obtain the Hamiltonian for our Dark Monopole, as well as the equations of motion (EoMs) for the profile functions. It is important to note that the "traditional" BPS bound for this monopole is zero, since Tr(B i φ) = 0 and therefore the magnetic charge associated to the U (1) group vanishes. However, since B i is a linear combination of M a and D i φ is a linear combination of Q m , then the Bogomolny equation [46] B i = D i φ does not have a non-trivial solution. Hence, there is no solution associated to this vanishing bound.
Let us start with the kinetic term of the scalar field. Since the component φ s is such that ∂ i (φ s ) = 0 and [φ s , M i ] = 0, it implies that D i φ s = 0. Then, from eq.(3.12) one can obtain that Making use of eq.(3.17), eq.(4.1) can be written as From eq.(3.8) and the fact that Y lm = (−1) m Y * lm , one can obtain that Moreover, using the properties of Vector Spherical Harmonics (VSH) [47] we obtain that From the magnetic field it follows that (4.5) Finally, we use eqs. (3.12), the fact that and Tr(SQ 0 ) = 0 to obtain that Joining all the contributions and making the change of variables ξ = evr the Hamiltonian (2.5) for the Dark Monopole will be where u (ξ), f (ξ) denote derivatives with respect to ξ. The conditions for E to be stationary with respect to f (ξ) and u(ξ) provide the equations of motion for the ansatz of the Dark Monopole: The appropriate boundary conditions for a non-singular finite-energy solution are Before looking for numerical solutions to eqs. (4.8a) and (4.8b), we shall analyze the behavior of the profile functions when ξ ≈ 0 and also when ξ → ∞.

Approximate Solutions
When ξ 1, eq.(4.8a) remains non-linear, since the dominant contribution is of the form u = u(u 2 − 1)/ξ 2 . However, since we are looking for approximate solutions, it is reasonable to series expand (4.8a) about ξ = 0 to order ξ 2 . Then, it is a trivial task to see that with c 1 ∈ R, gives the behavior of u(ξ), subject to the boundary conditions (4.9), near the origin. We do not bother to fix the constant c 1 , since we are only interested in the behavior of the solution.
With regard to eq.(4.8b) one can see that the dominant contribution is of the form where we used the approximation u 2 (ξ → 0) ≈ 1. This equation is in the form of the Euler-Cauchy equation. Then, the solution which satisfies (4.9) is where c 2 ∈ R is also an arbitrary constant. It is important to stress that solutions (4.11) and (4.12) agree with the fact that we are looking for non-singular monopole solutions. One can explicitly check that the expression of φ, W i and B i are regular at the origin. At this point, we can make an important comparison between the 't Hooft-Polyakov monopole and our Dark Monopoles. While the behavior of the profile function in the gauge field ansatz (u(ξ)) is the same for both, in the case of the Higgs field (f (ξ)) we see a distinct behavior. In the 't Hooft-Polyakov case, f (ξ) ∼ ξ, although in our construction f (ξ) ∼ ξ 2 .
Finally we analyze how the asymptotic values (4.10) are approached. In order to do so, it is convenient to substitute f = (h/ξ) + 1 in the eqs.(4.8a) and (4.8b) and take ξ → ∞, which results in Thus, the solutions behave as Therefore, for distances larger than the monopole core R core = 1 ev 6 |λ p | 2 l(l + 1) , the gauge field configuration (3.15) reduces to the asymptotic form (2.12) and the magnetic field (3.16) takes the form of a hedgehog as in eq.(2.13).

Numerical Solution
From the fact that we cannot find an analytical solution to the set of equations (4.8a) and (4.8b), it is reasonable to look for numerical solutions. We numerically solved the problem making use of the MATLAB R program bvp4c, which implements the solution of boundary value problems (BVPs). In order to do so, the system of equations (4.8a) and (4.8b) were recast as a system of first order equations of the form where u, v, f and w are considered to be independent. Once more, we stress that in the case of our Dark Monopoles l = 2 and one can obtain several distinct solutions by choosing different SSB patterns through the choice of λ p in the Lie algebra of G. These solutions must satisfy the constraints in the behavior imposed by the approximate solutions (4.11) and (4.12). Figure 1 shows the solution for the case of the SU(5) Dark Monopole, where the symmetry breaking is of the form SU (5) → "SU (3) × SU (2) × U (1)", where the quotation marks refer to the local structure of the unbroken gauge group, only. In the SU(5) case we can take the fundamental weight λ p to be λ 2 or λ 3 , since both of them generate the desired SSB. Then, |λ p | 2 = 6/5. One can see that this solution agrees with the expected behavior, since u − 1 ∼ −ξ 2 and f ∼ ξ 2 near zero, while they both reach the asymptotic values rather fast. The total energy of the solution, which is interpreted as the classical mass, is given by eq.(4.7) and to simplify the analysis we use the rescaled mass,Ẽ, Performing an analysis similar to [48], we can obtain the mass range for the Dark Monopoles. Note first thatẼ is a monotonically increasing function of λ/e 2 , since The lower bound for the mass happens when λ = 0, and numerical integration shows that for the SU (5) monopoleẼ(0) = 1.294.
Similar to the case of the 't Hooft-Polyakov monopole [49], in the limit λ → ∞ the mass of the monopole stays finite and it is given by since f (ξ) ≡ 1 ∀ ξ > 0 but f (0) = 0. Then, the only radial equation of motion is Solving eq.(4.18) and performing the integration in (4.17) gives us the upper bound for the monopole mass. In the SU (5) case, the upper bound isẼ(λ → ∞) = 3.262. For comparison, for the 't Hooft-Polyakov monopole in the SU (2) case,Ẽ(λ = 0) = 1 [50] and E(λ → ∞) = 1.787 [49]. Note that for a given SSB, where λ p is fixed, the value of the mass is the same for all the Dark Monopole solutions associated to the the su(2) subalgebras (3.4). This follows directly from the fact that the hamiltonian is independent of the indices i, j, k that label those su (2) subalgebras. Moreover, these are classical results. To determine the properties of the Dark Monopoles at the quantum level, one could use for example semi-classical quantization.

Non-abelian magnetic charge
One of the main properties of the Dark Monopole solution is that its magnetic field is in a direction outside the Cartan subalgebra H. Thus, as we mentioned before, this monopole has vanishing abelian magnetic charge, since Tr(B i φ) = 0. However, from eq.(2.13) we see that far from the monopole core it has a non-abelian magnetic flux in the direction g(θ, ϕ) M 3 g −1 (θ, ϕ), with M 3 given by eq.(3.4). We shall define ζ( r) = a(r) g(θ, ϕ) M 3 g −1 (θ, ϕ) = a(r) n a M a , (5.1) which is in the direction of the monopole non-abelian magnetic flux, where a(r) ∈ R is a radial function such that ζ is regular everywhere. This implies that when r → 0, a(r) ∼ r.
On the other hand, when r R core , we consider that a(r) = 1. Then, using the fact that in this asymptotic region the gauge and the scalar fields assume the form (2.12) and (2.7), respectively, it is easy to verify that asymptotically ζ satisfies the conditions Recalling the infinitesimal form of a gauge transformation for W µ and φ, we can conclude that the asymptotic configuration of the monopole is invariant under a gauge transformation of the form exp(iζ). Therefore, ζ is a Killing vector which is associated to a symmetry of the asymptotic fields of the monopole. According to [51] and [52], from the existence of a Killing vector ζ for an asymptotic symmetry one can associate a conserved charge. It is interesting to note that ζ satisfies the same equations as the scalar field φ for the 't Hooft-Polyakov monopole, outside the monopole core. Therefore, in this special case φ can be identified with the Killing vector ζ. Note that if we perform an arbitrary gauge transformation U on the monopole's fields then, from eqs. (5.2) and (5.3), we obtain that ζ must transform as in order to be a Killing vector of the transformed fields. Moreover, since ζ and W i take values in the su(2) subalgebra formed by M a , we can expand them as where (D i ζ) a = ∂ i ζ a −e abc W ib ζ c . We shall also introduce the notation ζ for the asymptotic configuration of ζ. Then, it follows from eq.(5.1) that ζ a = n a is a unitary vector. Note that ζ 2 a = 1 defines a 2-sphere, which we will denote by Σ. Now, let us define a gauge-invariant magnetic current by taking a projection of * G µν in the direction of the Killing vector ζ as where |ζ| ≡ ζ a ζ a = 1. Besides that, * G 0i = B i and * G ij = − ijk E k . The conservation of the current J µ M follows from its definition as a divergence of an antisymmetric tensor and from the fact that Tr B i ζ is twice differentiable.
Thus, the conserved non-abelian magnetic charge is Note that eq.(5.6) is just a measure of the non-abelian flux in the normalized ζ(θ, ϕ) direction. Furthermore, we must emphasize that the introduction of the radial function a(r) has no contribution to the magnetic charge. This artifact was introduced so that we could define a regular magnetic current for the Dark Monopole. Besides that, as pointed out by [53] there is no unambiguous way to measure the charge density of a monopole. Only the total charge makes sense. Let us now analyze the geometric meaning of the magnetic charge (5.6). From the asymptotic condition (5.2) it follows that Then, from eq.(5.4) and using vector notation, as well as the fact that |ζ| = 1 when r → ∞, eq.(5.7) can be written as Now, using eq.(3.9a) the expression of the non-abelian magnetic charge (5.6) can be written as and from eq.(5.8) we conclude that As it is well-known, this integral is a topological quantity which is an integer and has the geometrical interpretation [54] which is to measure the number of timesζ covers Σ asr covers S 2 ∞ once. For our particular Dark Monopole construction, whereζ a = n a , N ζ = 1. However, in principle, one could obtain higher magnetic charges, generalizing our construction, considering for example a gauge transformation g(θ, ϕ) = exp (−iϕkM 3 ) exp (−iθM 2 ) exp (iϕkM 3 ) , k ∈ Z , which would be associated toζ covering Σ k times asr covers S 2 ∞ once. It is important to remark that for the Dark Monopole, the magnetic charge is not the usual one (in the abelian direction), associated to the homotopy classes of the scalar field, like in the 't Hooft-Polyakov case. Therefore, from the results above we can conclude that the non-abelian magnetic charge of the Dark Monopole is conserved and quantized in multiples of 8π/e. And even though they are associated to the trivial sector of Π 1 (G v ), the conservation of Q M could prevent them to decay, at least classically. However, it is necessary to analyze in more detail the stability of the Dark Monopole.

Conclusion
In this work we have obtained a general procedure to construct magnetic monopole solutions, which we call Dark Monopoles, since their magnetic field vanishes in the direction of the generator of the U (1) em electromagnetic field. In order to do that, we considered theories with gauge group SU (n) and a scalar in the adjoint representation. These Dark Monopoles must exist in some Grand Unified Theories and we analyzed some of their properties for the SU (5) case. In particular, we obtained their mass range.
We also have shown that our monopole solution has a conserved magnetic current J µ M in the direction of the Killing vector ζ. The associated charge is quantized and it measures the number of timesζ covers Σ asr covers S 2 ∞ once. In principle, the conservation of this non-abelian magnetic charge could prevent the Dark Monopoles to decay. However, the stability should be analyzed in more detail in the future.
Finally, we expect that this construction can be generalized to other gauge groups and it could be interesting to analyze some of the phenomenological implications of these Dark Monopoles.