Embedded Monopoles

Using the embedded defect method, we classify the possible embeddings of a 't Hooft-Polyakov monopole in a general gauge theory. We then discuss some similarities with embedded vortices and relate our results to fundamental monopoles.


Introduction
When describing classical solutions to spontaneously broken Yang-Mills theories, there are several available methods. If the solutions are topologically stable, there is a topological classification [1]; while underneath this structure, there are finer classifications relating to the degeneracy of classical solutions. Such degeneracy is important for properties other than stability -for instance, Goddard, Nuyts and Olive use such a classification of non-Abelian monopoles to conjecture a dual gauge group [2].
For non-Abelian monopoles, a finer than topological classification can be achieved by using fundamental monopoles [3]. Any monopole can be described as a composite of several fundamental monopoles, each of which is associated with a simple root of the gauge group. Furthermore, in the BPS limit each fundamental monopole is known to take an Prasad-Sommerfeld form on the su(2) algebra defined by its root [3].
For defects in general (BPS and non-BPS), a good formalism to describe a finer than topological classification is the embedded defect method of Barriola,Vachaspati,and Bucher [4]. This method describes how a solution in one gauge theory is embedded in a larger theory and gives some constraints for when this embedding is allowed. Of particular interest are embedded vortices and embedded monopoles.
For embedded vortices, the embedding conditions of ref. [4] can be expressed in a very simple way that directly relates to the geometry of the vacuum manifold [5]. It can then be shown that embedded vortices separate into classes under the action of the residual gauge symmetry and from their embeddings easily specify their gauge degeneracy.
Such a picture has not been applied to embedded monopoles. For that reason, this letter aims to: (a) Apply embedded defect methods to monopoles. (b) Find similarities between embedded monopoles and embedded vortices. (c) Describe how these results on embedded monopoles relate to fundamental monopoles.
In addition, this gives some new results and a new picture of non-Abelian monopoles, both of which may be useful when examining their properties.

Formalism
Consider a spontaneously broken Yang-Mills theory with a compact semi-simple gauge group G and a scalar field Φ ∈ V in the D representation of G Here the covariant derivative is , where X is in the Lie algebra G In this letter, we use a coordinate independent notation, which we think better reflects the geometry behind most of our results [5,6]. Then the gauge kinetic term in (1) is defined by an inner product on G with f a a coupling constant for each simple subgroup G a ⊆ G. Likewise, the scalar kinetic term is defined by a Euclidean inner product Φ 1 , Φ 2 on V [7]. In this notation, the field equations are If the potential V [Φ] is minimized at some value Φ 0 , the residual gauge symmetry is Here Ad refers to the adjoint representation of G on G, where Ad(g)X = gXg −1 and the derived representation is ad(X)Y = [X, Y ] for X, Y ∈ G.
A central feature of this and some related papers [5,6] is the reduction of M into irreducible subspaces under the adjoint action of H. These subspaces correspond to irreducible representations of H on M: Physically, each M a defines a gauge multiplet of massive gauge bosons (for instance, the W and Z-bosons in electroweak theory).
Embedded defects [4] are (2 − k)-dimensional topological defects that remain a solution when they are embedded into a larger gauge theory (with k = 0, 1, 2 for domain walls, vortices and monopoles). They are defined by an inclusion of their gauge symmetry breaking in that of the full theory In the original reference [4], these embedded defects have fields fully embedded over the spatial domain Here we find the constraint (9) too restrictive (see the discussion around (35) below). Therefore we also consider a more general set of asymptotically embedded defects that as These asymptotically coincide with an embedded defect, but may differ from that form (8) elsewhere. An embedded defect is a solution of the full theory if the field equations reduce to consistent field equations on its embedded theory [4]. This requirement gives four constraints from the two field equations in (3): (a) The current, evaluated from Φ emb and A µ emb , satisfies While this is the case when (9) is satisfied, it is otherwise very restrictive and generally only holds asymptotically for certain parameter values. (c) The scalar potential, evaluated for Φ emb , satisfies This condition constrains the potential -for instance it holds in the BPS limit.
(d) The gauge kinetic terms satisfies Equation (14) always holds by algebraic closure of G emb , as observed in [4].

Embedded monopoles
By (7) an embedded monopole is defined by embedding an su(2) → u(1) symmetry breaking in that of the full theory Then the embedded monopole has a 't Hooft-Polyakov form [8] on the su(2) subtheory (16) with su(2) basis [t a , t b ] = ǫ abc t c and g(r) = e ϕt3 e ϑt2 e −ϕt3 in spherical polars. Likewise, an asymptotically embedded monopole only has its asymptotic fields similar to (16), where [9] H . (17) Here m S and m V are the scalar and gauge boson masses in the embedded su (2) subtheory. Elsewhere the fields of an asymptotically embedded monopole may differ from the embedded monopole.
Both an embedded and asymptotically embedded monopole have a long range magnetic field [10] The scalar field asymptotically tends to the vacuum with Φ ∼ Φ 0 in thex 3 -direction. Then the magnetic generator M = M (ẑ) is an element of H and satisfies the topological quantization exp(4πM ) = 1 [11].
To simplify the following calculations, we express (16) in a unitary gauge (so V emb = R Φ 0 ). This expression is achieved with a gauge transformation which takes the embedded monopole (16) to [12,13] Here A D =φ(1 − cos ϑ)/r sin ϑ is the Dirac gauge potential (∇ · A D = 0, ∇ ∧ A D =r/r 2 ) and η 1 = sin ϕθ + cos ϕφ,η 2 = − cos ϕθ + sin ϕφ (21) are two orthonormal unit vectors orthogonal tor. To prove (20), we write the gauge field as then use the following identities Ad(g −1 )θ · t = cos ϕ t 1 + sin ϕ t 2 , and evaluate Now, our task is to find when such a monopole solution is fully or asymptotically embedded. This is determined by the constraints (11) and (12).
To examine the first constraint J ν , G ⊥ emb = 0, we start by rewriting (15) as Substituting J ν below (3) into (11), we find Now D 0 Φ = 0, while spatially in a unitary gauge Using this and noting d(G)Φ 0 , Φ 0 = 0, we see that (11) is an algebraic constraint upon the embedding (15) To this we apply a result proved in [5]: Therefore if λ a = λ b the monopole embedding in (15) is given by with M a a gauge family in (6). If λ a = λ b the embedding can also be between gauge families (see ref. [5]). For the second constraint D µ D µ Φ emb , V ⊥ emb = 0, we evaluate the scalar kinetic term in a unitary gauge. As observed above D 0 Φ emb = D 0 D 0 Φ emb = 0 -then we only need to consider the spatial components In evaluating this, we used the identity Therefore by (32), condition (12) is satisfied when Note (35) is a very constraining assumption. Therefore if both conditions (31) and (35) hold, there are embedded monopole solutions like (16). Towards the core their scalar fields vanish and these monopoles are similar to a 't Hooft-Polyakov monopole.
A more usual situation is when (35) does not hold. Before we discussed asymptotically embedded monopoles, which coincide with an embedded monopole only at infinity. Consequently, the question arises whether condition (12) could be satisfied only in the asymptotic region.
To answer this question, we consider the asymptotic form of (32). Substituting (17) into (32), we find Then condition (12) is satisfied if the second term in (32) is negligible compared to the first. Exponentials beat powers, so this occurs when the scalar mass m S is less than twice the gauge mass m V . When so, the asymptotically embedded monopoles are classified only by (31). From now on we examine the first constraint (31).

Solution sets of gauge equivalent monopoles
Because each monopole is defined from an su(2) embedding and these embeddings generally form degenerate sets, we expect the monopoles to have some degeneracy. Our tactic for determining this degeneracy is to consider the action of the residual gauge symmetry H upon the su(2) embedding.
To start, consider a rigid H transformation (no spacetime dependence) of a monopole's asymptotics By (16) this simply takes t → Ad(h)t, from which the su(2) embedding moves to Therefore each monopole is an element in a set of gaugeequivalent monopoles. This set is represented by the Hequivalent su(2) embeddings where C H (su (2)) ⊂ H is the centralizer of su (2) in H (which acts trivially on every element in su (2)). This set can be expressed more transparently by noting that the denominator in (40) satisfies Next, we note that This expression follows from again using C H (N ) ⊆ C H (u(1)), which implies [u(1), C H (N )] = 0. Then, because any X ′ ∈ N is proportional to Ad(h)X for some h ∈ U(1), we infer C H (X ′ ) = C H (X) and obtain (42). Therefore by (41,42) each fully or asymptotically embedded monopole is an element in a set For embedded monopoles, (43) is exact and quantifies their H-degeneracy. For asymptotically embedded monopoles, (43) refers only to their degeneracy at infinity and further degeneracy may occur in the core. Finally, we note there is part of the orbit (43) that corresponds to spatial rotations. This part is found by considering the action of h(χ) = exp(t 3 χ) on an su(2) embedding By (16) this action is entirely equivalent to a spatial rotation. In passing, we note that this structure relates to the angular momentum of the monopole [12,14].

Similarities with embedded vortices
Similar arguments have also been applied to embedded vortices [5]. Such vortices are U(1) → 1 Nielsen-Olesen vortices [15] embedded according to (7) with U(1) = exp(Xϑ) and X ∈ M. In [5], it was found that each vortex has its embedding constrained by with M a one of the irreducible subspaces in equation (6).
By similar arguments to those in sec. 4 these have degenerate solution sets For further discussion, we refer to reference [5]. An interesting point is that the solution sets of embedded vortices and monopoles are very similar: with obvious parallels between equations (31) and (46) and the orbits (43) and (47). Such parallels arise from the following sequence of embeddings Also note that similar considerations apply to whether vortices are fully embedded or asymptotically embedded -this will be discussed elsewhere [16].

Relation to fundamental monopoles
In the following discussion, we restrict the scalar field to an adjoint representation of G. Then rank(G) = rank(H) = r and one may choose a maximal Abelian subgroup T ⊂ H with orthonormal generators {T 1 , · · · , T r }.
Recall that the magnetic generator M of a non-Abelian monopole satisfies a topological constraint exp(4πM ) = 1. It can then be shown that these generators have a general form [2,11] where each n a is an integer and {β (1) , · · · , β (r) } are simple roots. These simple roots span the set of all roots α ∈ Φ(G) i ad(T )E α = αE α (50) with each {E α } a different root space. A fundamental monopole is a monopole that has its magnetic generator M associated with a simple root, so that M = β * (a) · T in (49). They are called fundamental because any non-Abelian monopole can be decomposed into several such monopoles [3] -this is supported both by (49) and index theory methods in the BPS limit.
In reference [3], fundamental monopoles were seen to take an asymptotic Prasad-Sommerfeld form on the su(2) α subtheory that has generators (Here each E ±α pair is normalized to [E α , E −α ] = i α · T so that [t a α , t b α ] = ǫ abc t c α .) Clearly, this structure is related to the su(2) embeddings in (15), but how does the classification of these fundamental monopoles relate to the embedding condition N ⊆ M a in (31)?
To answer this question, we define