Monopoles, vortices and the geometry of the Yang–Mills bundles

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  • Monopoles, vortices and the geometry of the YangMills bundlesZ. F. Ezawa and H. C. Tze

    Citation: J. Math. Phys. 17, 2228 (1976); doi: 10.1063/1.522869 View online: View Table of Contents: Published by the AIP Publishing LLC.

    Additional information on J. Math. Phys.Journal Homepage: Journal Information: Top downloads: Information for Authors:

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  • Monopoles, vortices and the geometry of the Yang-Mills bundles*

    Z. F. Ezawa Laboratoire de Physique Theorique et Particules Elemcntaires,f Unh'crsite de Paris X I. 91405 Orsay. France

    H. C. Tze Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305 (Received 16 March 1976)

    A topological classification of monopoles and vortices is formulated in terms of fibre bundles. The distinction between Dirac and 't Hooft monopoles is made in the light of the energy finiteness problem. Finite-length vortices with Dirac monopoles at the end points are also discussed.

    I. INTRODUCTION AND CONCLUSION Originally introduced to formulate and solve global

    topological problems, fibre bundlesl have provided us with an ideal language for discussing relativity, in-variance, and gauge transformations. 2 They may well become the standard mathematical baggage for particle physicists. In this paper, we apply the topology of fibre bundles to classify monopoles and vortices in non-Abelian gauge theories. The bundle formulation allows a compact, unified treatment of the 't Hooft-Polyakov monopoles, 3,4 the Dirac monopoles, 5,6 and the Nielsen-Olesen vortices. 7,8

    Here we summarize the results of our analysis. The 't Hooft and Dirac monopoles are classified by "1 (li) and "1 (G), respectively, where G is the global symmetry group of the physical system and li is an isotropy sub-group of G. In spontaneously broken gauge the theories, 't Hooft monopoles have finite ~nergy while Dirac mono-poles in general have infinite energy, possibly corre-sponding to unobservable quarks. Dirac monopoles are accompanied by Nielsen-Olesen vortices, the latter being also characterized by 711 (G). A system of n Dirac monopoles of the same type has finite energy if n is the dimension of "l(G). This implies that in theories with G = SU(3)/ 2'3 a three-monopole system is physically realizable as well as monopole-antimonopole system. They would be field-theoretical candidates for extended baryons and mesons. 9,6

    II. THE YANG-MILLS BUNDLES AND MONOPOLES In a geometrical approach to local gauge invariance,

    a fibre bundle can be uniquely associated with a physical system of fields, as we now explain. The base space x is the (3 + I)-dimensional space-time manifold cover-ed by a set of coordinate patches {Vi}' With each point x c:: Vi' we associate a set of fields { 11 2" .. }, which we will make cross-sections of the fibre bundle. All the realizable sets in the system make up a field manifold Y. The field is defined with respect to a certain holo-morphic representation T[Gj of a Lie group G, the sym-metry group of the system. The Yang-Mills potential A" is the connection which assumes value in the Lie algebra of G. Let G be n-dimensional. A set of n con-tinuous mappings {Xk } from Vi is chosen to form a base in the Lie algebra of G. G acts on it through the adjoint representation Ad[Gj. Under the base transformation

    2228 Journal of Mathematical Physics, Vol. 17, No. 12, December 1976

    the fields and the connection transform as

    S: (x) - '(x) = T[S(x)j (x),

    A" (x) - A: (x) = Ad[S(x)jA" (x) + (i/ e)a "S(X) S-I(X),

    respectively. We also refer to (1) as a gauge transformation.




    Consider two overlapping coordinate patches Vi' VI n V2 ;, 0. The fields are defined in each patch as

  • Y the fibre and G the structure group. The field is nothing but a cross section of the fibre bundle.

    For simplicity we assume that G is connected. Then, there exist the universal covering group G* and a sub-group C of the center Z* of G* so that the isomorphism

    G*/C=G. CcZ* (3)

    fOllows .10 By definition G* is simply connected. Next, we define the isotropy group H (u) at a point u E: Y by H(u)={hE G; T(h)u=u}. By the equivalence relation that u-v if and only if G/H(u)=G/H(v), all the elements of Yare grouped into the equivalence classes {YJ Then, by fixing u E Yj arbitrarily, the homeomorphism

    Yj=G/H(u), uEYp (4)

    is proved. 10 G acts on Yj transitively.

    We explain this. Since G is the symmetry group, if u(x) is a solution then T[S(x)]u(x) is a solution. But it is not true that any solution v(x) is obtainable from u(x) by a gauge transformation. We define an homogeneous space Y(u) by Y(u) = {v E: Y; l' = T(S)u, 'fI S E: G}. the fibre Y is the union of all such nonoverlappmg homogeneous spaces Y .. Y is the set of all possible solutions to the system, ~ith the boundary condition picking out one of the Y;,s. By taking an element u from Yj , the topological structure of Y. is determined by (4). In the spontaneous-ly symmetry b'roken theory, H(u) is the unbroken sym-metry at the point u. Especially, if the symmetry is completely broken, H(u) ={e} for all uE Y, and the fibre bundle is the principal bundle. 1

    In field theory, we consider an obj ect occupying a finite domain in X. Either it arises from the dynamics or must be introduced explicitly by hand. The extended , t Hooft monopole3 is an example of the former and the point-like Dirac monopole5 is one of the latter. The object interacts with the fields and we Seek to categorize it through this interaction. 11

    We take an element u(x,) E Y. We perform the equiva-lent transport of u(xo) along a loop l. At a point XE: l, it gives



    P being the ordering operator along the loop l. In (5b) we have assumed that the loop l exists in a single co-ordinate patch, but the modification in the general case is straightforward with the USe of the gauge transformation. 12

    The operator g/ (x, x o) draws a curve l* in G as x moves from Xo along the loop 1 in X. It is to be noticed that the curve l* is not necessarily a loop in G. Now we sweep the loop lover a 2-sphere S at a fixed time in X. The corresponding curve l* traces out a surface g(S) in G. Its boundary og(S) is a loop in G and traced by g/(xo, xo). We denote by R the subset of G which is swept by all possible boundaries og(S). Later we shall show how to determine R. In this way, with any sphere S in X, we can associate a member of the second relative homotopy group 1f2 (G, R, e). Accordingly, if two spheres

    2229 J. Math. Phys., Vol. 17, No. 12, December 1976

    Sl and Sz are mapped into the same member, we can find a continuous gauge transformation so that the con-nections are the same at the corresponding points on the spheres Sl and S2' During this transformation, the energy of the system changes only continuously. We are thus led to the follOwing definition.

    Definition: Two spheres Sl and Sz are said to enclose the same type of monopoles if they are mapped into the same homotopy class of 1fz(G,R,e).

    From this definition we get the ensuing theorem.

    Theorem: Monopoles are classified by the fundament-al homotopy group 1f1(R).

    Proof: There is an exact homotopy sequence1 : 1fz(G)-1f2(G,R)~ 1f1 (R). By the construction of R, the boundary operator 0 is a surjection. Since G is a compact Lie group, 1fz(G) =0. Hence, 1f2 (G,R,e)=1f 1(R) by the homeomorphism o.

    Thus far we have yet made no analysiS of the bound-ary conditions on the fields. Of utmost importance to our work is the energy finiteness condition. From the Lagrangian (2) it is seen that the kinetic energy con-tribution of the field if>k at infinity is negligible if and only if the condition

    \7"if>k=O (6) is obeyed asymptotically. 9 In general, the energy finite-ness condition for a field is that the connection becomes flat with respect to the field at infinity.

    If the connection is flat, then we obtain k (xo) = T[g/(xO,XO)]if>k(XO) with (5b). This expression is in fact the integrated form of (6). Since og(S) is the trace of g/ (xo, xo), we have

    (7) turns out to determine R, the aggregate of all boundaries ilg(S). The types of monopoles are in fact determined by this energy finiteness condition (6).

    The condition (6) is purely a physical requirement.


    In classical electrodynamics we treat a charged test particle moving in the electromagnetic field. If the field represents such a test particle, there is no reason to put (6) for the field. Yet, the energy of the system is finite since the field represents a single particle. On the contrary, if we treat a spontaneously broken gauge theory, the vacuum is a medium. for instance, the condensed phase of the Higgs fields. The system has finite energy if and only if (6) holds asymptotically for all fieldS that participate in making the vacuum. De-pending on the physical requirements, we have to im-pos e the appropriate energy finiteness condition.

    At present, we are most interested in finding extended obj ects in spontaneously broken gauge theories. Thus, we require (6) for all the fields eventually. However, there are two different categories. (A) It is satisfied for a single monopole, or (B) it is not satisfied by a single monopole. In the case (A), any monopole system has finite energy and is realizable. In the case (B), only certain sets of monopoles are realizable. To the categories (A) and (B) correspond the' t Hooft mono-ple and the Dirac monopole, respectively.

    Z.F. Ezawa and H.C. Tze 2229

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  • A. The 't Hooft monopoles

    We require all fields k to satisfy the condition (6) asymptotically. Because the asymptotic behaviors of {k} determine the solution uniquely, the fibre over x at infinity is enough to characterize the object. In the Lagrangian theory, the set {k} which gives the absolute minimum of the potential constitutes the classical vac-uum. The set of all classical vacua Y is the fibre. Y is decomposed into the union of the homogeneous spaces {YJ.

    Since (7) holds all the fields we deduce ag(S)E li(u) , H(u) being the isotropy group at UE Vi' This is neces-sary and sufficient for finite energy of the system. Hence, we have

    Corollary A: 't Hooft monopoles are classified by 1f1 (li(u for each homogeneous space Y i where u E Yi

    The isotropy group lieu) is the unbroken symmetry at a point u. 1f1 (li(u does not depend on u but only on the homogeneous space Y i that contains u. If the system is subj ect to no additional boundary conditions, it allows all types of monopoles given by 1T1 (H) for all homo-geneous spaces Y i of Y.

    Here we recall the classification of 't Hooft mono-pole proposed in Ref. 4. The equivalent transport maps a sphere S into a sphere g(S)u(xo) in Y(u) because of (7). Therefore, the solutions are also classified by 1T2 (Y(U). Due to (3) and (4) we get 1T2 (Y) = 1T1 (H i ) , and the two classifications are equivalent. However, this result is only applicable to 't Hooft monopoles. A few illustrations are in order.

    A trivial example is the case where the symmetry is completely broken. The bundle is the principal bundle. There are no 't Hooft monopoles. Physically, this is also clear since there is no long range gauge field remaining.

    The next simplest example is the case where the sym-metry is completely broken except for a U(l) subgroup Hem which we identify with "electromagnetism."3 The monopoles are classified by 1T 1 (Hem)=Zoo, the additive group of integers. The topological spectrum is the same as the well-known Abelian U(l) monopoles. In gauge theories with G = SU(N)/ Z N we introduce a suf-ficient number of fields in the adjoint representation until the only unbroken symmetry is given by the generator

    Aem = (1/ N)diag(l, 1, ... ,1 - N) (8) regarded as an element of the universal covering group G* = SU(N). As the loop 1 = as covers the sphere S, g/

  • hand, Dirac monopoles are introduced by hand: The masses, spins and other internal degrees of freedom are free parameters we can assign arbitrarily.


    The system of a single Dirac monopole has infinite energy in a spontaneously broken vacuum since the en-ergy finiteness condition (6) is not satisfied asymptoti-cally. Without violating the analysis of Sec. II B, we can require (6) outside of a tube starting at the mono-pole and ending at infinity. The obj ect that is confined in the tube is a Dirac monopole accompanied by the vortex. It is easy to see that, far enough away from the point monopole, its accompanying vortex is indistin-guishable from the Nielsen-Olesen vortex. 7 This point has been discussed in ample detail in a recent work. 8

    Here we define the Nielsen-Olesen vortices in our terminology. We take the same fibre bundle, the fibre Y being the aggregate of all classical vacua. Any loop I around the tube is mapped into a path gl(xo,xo) in G by the equivalent transport (5). Since the connection is flat outside the tube, we get T[gl(xo,xo)]u=u. The path gl connects the unit element e and an element of the isot-ropy subgroup H(u). Just as in the definition for topo-logically distinct monopoles, we are led to the following classification.

    Definition: Two loops 11 and 12 are said to enclose the same type of vortices if they are mapped into the same homotopy class of lT1 (G, lip e).

    From this we deduce:

    Theorem: Vortices are formed and classified by lT 1 (G) if and only if Hi is embedded in a simply connected domain of G.

    .* Proof: There is an exact sequence1 : lTl(H;l~lT1(G) ~lTl(G,lii)~lTo(Hi)=O. Here lT1(G)=lT1 (G,H i ) if and only if j* sends lT1(H i ) to the neutral element of lT1(G).

    The classification is the same as that of the Dirac monopoles. Nielsen-Olesen vortices are such objects that terminate at Dirac monopoles. 8 The isotropy group H(Il) is the unbroken symmetry at a point U E Y. A suf-ficient condition for the vortex formation is to break the symmetry completely, i.e., H(u)={e}. In the U(l) theory,7 one Higgs field is enough for this purpose. In the SU(2)/22 theory, 7 two Higgs fields are introduced in the regular representation. In the SU(3)/23 theory, we take two Higgs fields u={