VOLUME 50, +UMBER 13 PHYSICAL REVIEW LETTERS 28 MARcH 1983

Global Color Is Not Always Defined

Philip Nelson. and Aneesh ManoharIyman Laboratory of Ikysics, IIarvard University, Cambridge, Massachusetts 02l38

(Received 27 December 1982)

Far away from a fundamental magnetic monopole of the SU(5) grand unified theory,the full theory reduces to one whose gauge group is unbroken SU(3) SU(1). The algebraof generators of smooth gauge transformations in this unbroken theory, however, willnot in general contain any subalgebra isomorphic to su(3). This means that global colorrotations are not always defined. The resulting inability to classify semiclassical dyonstates by their color eliminates an apparent paradox related to the vacuum angle 0.

PACS numbers: 12.10.En, 11.15.Ex, 14.80.Hv

Recently one of us' has performed an analysissimilar to Witten's' of the low-lying states in thefundamental monopole sector of the SU(5) grandunified theory. ' The resul. t is that in the presenceof a nonzero vacuum angl. e 8, both the electriccharge and the color hypercharge (to which it isproportional. ) take on an arbitrary value. Whilethis result was surprising for the electric chargein Witten's original. anal. ysis, it at first appearsto be a disaster for the hypercharge, which ispresumably embedded in the compact color SU(3)symmetry group.

The only way out of this dilemma seemed to bethe possibility that in fact no group of gauge trans-formations isomorphic to SU(3) exists. Whilethis situation may seem to be no less paradoxicalthan the original. paradox, we show in this Letterthat it indeed obtains. This proof holds for theful. l quantum theory and gives necessary and suf-ficient conditions for the unbroken local symme-tries of a grand unified theory to be integrabl. einto wel. l-def ined global symmetries.

We can see the essential details of the probl. emin an SU(3) theory spontaneously broken to SU(2)(3 U(1) by an adjoint Higgs field, whose expecta-tion value in the nonsingular "radial. " gauge ofRef. 3 can be taken as

'T are the three Paul. i matrices. Cl.early the gen-erator of unbroken Abelian transformations

(1 0TE M

(0 2~ r -2)and one of the non-Abel. ian ones

(0 —2(v ~ x+ 1

are everywhere well defined; they can be ob-tained by transforming the constant matrices

1 0 0 1 0 0

0 1 0

(o 2 o

0 —1 0

0 0 —2

respectively, from the singular "unitary" gaugeof Ref. 3. When we attempt the same thing withthe remaining two independent matrices commut-ing with y(z), however, we find that they becomesingular at the south pole when we transform toradial gauge:

cos(S/2) sin(S/2)e 'e)cr, = cos(8/2)

sin(&/2) e'~

0', = i cos(9/2)

i sin(6/2) e~

—i cos(tt/2) —e sin(S/2) e 'e)0 0

It seems possible, therefore, that some globalgauge transformations may not exist. Is thereany precedent for such a situation? Yes there is.Consider the problem of defining a Euclideanalgebra of vector fields on the two-dimensional.manifold S'. Near every point a ful. l. algebra ofsmall. motions is defined, but it is well knownthat not even a single smooth nonvanishing tan-gent vector field exists globall. y. This is so inspite of the fact that S' and its tangent bundle canbe imbedded in R', which has a perfectly goodalgebra of global translations and rotations. Sim-il.arl. y in gauge theories it is possible for a twist-ed unbroken theory to be the reduction of a trivialunified theory.

To proceed beyond this heuristic level it is con-venient to describe the unbroken theory in Wu

1983 The American Physical Society 943

VOLUME 50, NUMBER 13 PHYSICAL REVIEW LETTERS 28 MARcH 1983

and Yang's two-patch formalism. The theoryis defined on the space R x(R' 8-), where 8 isa ball occupied by the monopole core. We re-strict this space further to a single sphere S'at some given time, and divide it into two hemi-spheres U, o. = 1,2. Writing points of S' as x,we have z H U„U, U U, = S', U, A U, = F., the equa-tor. All local. quantities are represented as"twisted functions, " pairs f of functions definedon U„, respectively, and related at the equatorby a gauge transformation'. f,(9=,~, ~p) =f, (e= —,'m, q)~'+ where g: E -G and G is SU(2) S U(1),realized as unimodular 3x 3 matrices' commut-ing with

1 0 0

0 1 0

0 0 —2

Then a local gauge transformation on S' is rep-resented by k: U —6, whose matching conditionwe obtain as follows:

f, -(f,)".On the equator,

g g hy hy g

Aut su(n) containing 1 is isomorphic to SU(N)/Z„acting by conj ugation.

To apply al. l. this, we note that the fundamentalmonopole has transition function' g(~p) = exp(2Q~p),where

0 0

0 ——'i 7

Thus on the equator A, "(&=2~, ip) = A«~&'" A, "'(~=-',~,p). Since each U is a. disk, eachA on theequator defines a loop in Aut9 which is contrac-tibl. e: A (~p)-0, where"-" means "is homotopicto." We have on the equator O-A, =At Ay A, .Conversely, if A. ,—0 we can define an acceptableset of g

' by taking A, = 1 on U, and A, equal tothe supposed homotopy between A, and the nullpath. Thus global color can be well defined ifand only if A, - 0. The derivation of this condi-tion evidently did not depend upon the specificgauge group.

ProPosition. —In the presence of the fundamen-tal monopole, no smooth basis of generators ofglobal unbroken gauge transformations exists.

Proof. g(cp) acts—trivially on the Abelian gen-erator. On the others, the upper corner of

Hence

where

f h2

g(q ) =~ ~

& U(2).( I 0

e '~/

This projects to

h. (~=-', V) =g4)h, (~=-', V')g 4)The generators of infinitesimal gauge transforma, -tions are likewise twisted functions it„:U„- 9,the I ie algebra of G, with the same condition (1).

Suppose we have a set of such generators g ',i =1,. . ., 4. These act on the various physicalquantities, and we can commute them to get newgenerators. Clearly any smooth basis of "gen-erators of global unbroken gauge transforma-tions" must satisfy compatibil. ity, Eq. (1), plus[g '(r), („'(r)]=f, ,„( "(r), where f;» are thestructure constants of A. We can also demandthat the U(1) generator be properly normalized:.Tr(P ')'= const. Both these conditions on g areconsistent with Eq. (1) at the equator. Theyamount to the requirement that ~t '(r) = A„"(r)xg, '(z), where A„:U„-Autg, the group of auto-morphisms of 9.

Theorem. All automorphis—ms of su(n) can bewritten as ('-gg'g ' org(('*)g ', gaU(&).Call the former A,".

Coro/lazy. —The connected component of

(+i ql2

~C SU(2)/Z„

0 e iP/2J

a closed loop. When lifted to the covering groupSU(2), however, it is clearly not closed. Henceg(cp) is not homotopically trivial.

The above result makes minimal use of the spe-cific monopole vector potential. In fact, closerinspection shows that, for general Q satisfyingthe Dirac quantization condition, the issue of col-or definition is settled completely by the one top-ological invariant k of the monopole. ' k is thenumber of times g(p) winds around the U(1); col-or can be globally defined iff 4 is even. Since thefull quantum theory does not mix different topo-logical sectors, our result is valid beyond thesemiclassical approximation. In particular, itis not spoiled by confinement effects. Of courseit is still limited to energies low enough for theunbroken theory to be a good approximation to thefull one, and to the region far from the monopolecore. Similarly the a.ddition of matter fieldschanges nothing; these live on bundles associated

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VOLUME 50, +UMBER 13 PHYSICAL REVIEW LETTERS 28 MARCH 1983

to the principal bundle ana cterive their definitionof gauge transformations from it.

The above results can be trivially extended tothe general case. In the SU(5) - SU(3) @U(l) theo-ry, global color is defined iff k=—Omod3. Hencecolor is undefined in the presence of the funda-mental monopole, resolving the original paradox.On the other hand, the purely electromagneticmonopole' has 4 = 3= 0 mod3 and so presents no

problem, as expected.As mentioned in Ref. 1, this work was partial-

ly motivated by Abouelsaood's discussion of chro-modyons. " The present Letter provides a topo-logical basis for understanding his result thatcertain excitations do not exist.

We would like to thank Sidney Coleman andDavid Groisser for helpful discussions. Thiswork was supported in part by the National Sci-ence Foundation under Grant No. PHY77-22864.One of us (P.N. ) was the recipient of a NationalScience Foundation graduate fellowship.

'P. Nelson, preceding Letter [Phys. Rev. Lett. 50,939 (1983)j .

E. Witten, Phys. Lett. 86B, 283 (1979).3C. Dokos and T. Tomaras, Phys. Rev. D 21, 2940

(198O).4T. T. Wu and C. N. Yang, Nucl. Phys. 8107, 365

{1976).In a more elevated treatment g is the transition func-

tion of a nontrivial principal fiber bundle over S, andwe are seeking an su(2) in the infinite-dimensional al-gebra of sections E(S), where b is the associated bundleh=~x g

Ad

This is the only place where the nature of the under-lying unified theory is used.

N. Jacobson, Lie A/gebras (Wiley, New York, 1962),Chap. 9.

S. Coleman, Harvard University Report No. HUTP-82/A082 (to be published).

This is geometrically obvious: The existence of sec-tions of the associated bundle depends on the structureof the principal bundle; its connection is irrelevant.

A. Abouelsaood, Harvard University Report No.HUTP-82/A088 (to be published).

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