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) ⊗ U(1). The algebra of generators of smooth gauge transformations in this unbroken theory, however, will not in general contain any subalgebra isomorphic to su(3). This means that global color rotations are not always defined. The resulting inability to classify semiclassical dyon states by their color climinates an apparent paradox related to the vacuum angle θ

Manohar, Aneesh

Nelson, Philip C

English

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsDepartment of Physics Papers Department of Physics3-1983Global Color is not Always DefinedPhilip C. NelsonUniversity of Pennsylvania, nelson@physics.upenn.eduAneesh ManoharFollow this and additional works at: http://repository.upenn.edu/physics_papersPart of the Physics CommonsAt the time of publication, author Philip C. Nelson was affiliated with Harvard University. Currently, he is a faculty member in the Physics &Astronomy Department at the University of Pennsylvania.This paper is posted at ScholarlyCommons. http://repository.upenn.edu/physics_papers/538For more information, please contact repository@pobox.upenn.edu.Recommended CitationNelson, P. C., & Manohar, A. (1983). Global Color is not Always Deﬁned. Physical Review Letters, 50 (1), 943-945. http://dx.doi.org/10.1103/PhysRevLett.50.943Global Color is not Always DefinedAbstractFar away from a fundamental magnetic monopole of the SU(5) grand unified theory, the full theory reducesto one whose gauge group is unbroken SU(3) ⊗ U(1). The algebra of generators of smooth gaugetransformations in this unbroken theory, however, will not in general contain any subalgebra isomorphic tosu(3). This means that global color rotations are not always defined. The resulting inability to classifysemiclassical dyon states by their color climinates an apparent paradox related to the vacuum angle θ.DisciplinesPhysical Sciences and Mathematics | PhysicsCommentsAt the time of publication, author Philip C. Nelson was affiliated with Harvard University. Currently, he is afaculty member in the Physics & Astronomy Department at the University of Pennsylvania.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/538HUTP-82/A061GLOBAL COLOR IS NOT ALWAYS DEFINED*Philip Nelson and Aneesh ManoharLyman Laboratory of PhysicsHarvard UniversityCambridge, Massachusetts 02138ABSTRACTFar away from a fundamental magnetic monopole of the SU(5) grandunified theory, the full theory reduces to one whose gauge group isunbroken SU(3) x U(1). The algebra of generators of smooth gaugetransformations in this unbroken theory, however, will not in generalcontain any subalgebra isomorphic to su(3). This means that globalcolor rotations are not always defined. The resulting inability toclassify semiclassical dyon states by their color eliminates anapparent paradox related to the vacuum angle 0 .This work is supported in part by the National Science Foundation under Grant No. PHY77-22864.tWork supported in part by a NSF graduate fellowship.Recently one of us ' has performed an analysis similiar to Witten's 2of the low-lying states in the fundamental monopole sector of the SU(5)grand unified theory. The result is that in the presence of a non-zero vacuum angle e, both the electric charge and the color hyper-charge (to which it is proportional) take on an arbitrary value. Whilethis result was surprising for the electric charge in Witten's originalanalysis, it at first appears to be a disaster for the hypercharge,which is presumably embedded in the compact color SU(3) symmetry group.The only way out of this dilemma seemed to be the possibilitythat in fact no group of gauge transformations isomorphic to SU(3)exists. While this situation may seem to be no less paradoxical thanthe original paradox, we show in this letter that it indeed obtains.This proof holds for the full quantum theory and gives necessary andsufficient conditions for the unbroken local symmetries of a grandunified theory to be integrable into well-defined global symmetries.We can see the essential details of the problem in an SU(3)theory spontaneously broken to 50(2) x 0(1) by an adjoint Higgs field,whose expectation value in the nonsingular "radial" gauge of Ref. 3can be1taken as <0(0> =1 3 T are the three Pauli matrices.- 2- + 7 T•rClearly the generator of unbroken Abelian tram formations T E.M.1[ 1 3 and one of the nonabelian ones 03 = + 1)are everywhere well-defined; they can be obtained by transforming the1 1constant matrices 1 I -1 J respectively from the sin--2J I 0gular "unitary" gauge of Ref. 3. When we attempt the same thing withthe remaining two independent matrices commuting with 0(z), however,we find that they become singular at the south pole when wetransform to radial gauge:0.1=a2=0 cose/2 sine/2 e-4 —cose/2 0 0sine/2 eio 0 0_0icos0/2isine/2-icos0/2 -isine/2 e-iS0 00 0It seems possible, therefore, that some global gauge trans-formations may not exist. Is there any precedent for such a situa-tion? Yes there is. Consider the problem of defining a Euclideanalgebra of vector fields on the two-dimensional manifold S2. Nearevery point a full algebra of small motions is defined, but it iswell known that not even a single smooth nonvanishing tangent vectorfield exists globally. This is so in spite of the fact that S2 andits tangent bundle can be imbedded in IR3 • which has a perfectly goodalgebra of global translations and rotations. Similarly in gaugetheories it is possible for a twisted unbroken theory to be the re-duction of a trivial unified theory.To proceed beyond this heuristic level it is convenient to4describe the unbroken theory in Wu and Yang's two-patch formalism.The theory is defined on the space 1R x OR3 - B }, where B is a balloccupied by the monopole core. We restrict this space further to asingle sphere S2 at some given time, and divide it into two hemi-spheres Ua a = 1,2. Writing points of S as r, we have zEUI,01 U 02 52, U1 U2 = E, the equator. All local quantities arerepresented as "twisted functions", pairs fa of functions defined onUa respectively and related at the equator by a gauge transformation :f2(6 = = f1(0 = 0 )g(45) where g:E .G and G is SU(2) xrealized as unimodular 3x3 matrices commuting with 0 41 1 .-2Then a local gauge transformation on S2 is represented by ha:Ua G,whose matching condition we obtain as follows:hiOn-1 -1(f2)g (f29 )h1f2 f2 2 whereh2(0= ,0) = g(0)the equator,= ( f )h19 • Hence-1hl (6) = •g (El') •The generators of infinitesimal gauge transformations are likewisetwisted functions 0a:Ua•  g • the Lie algebra of G,condition (1).Suppose we have a(1)with the sameset of such generators 0! i = 1, 4.These act on the various physical quantities, and we can commute themto get new generators. Clearly any smooth basis of "generators ofglobal unbrokenothers, the upper+1k1gauge transformations" must satisfy compatibility, eq. 1, e 2 e.22t.-plus [0i (1), (;) ] = fijk 0 (1), where fijk are the structurea aconstants of g. We can also demand that the 11(1) generator be proper-ly normalized: Tr(04)2 = const. Both these conditions on ip are con-sistent with eq. 1 at the equator. They amount to the requirementthat 02 (;) = Alai(;) 4 (2)morphisms of g.Theorem: All automorphisms of su(n) can be writt2n as i 90g-1or g(01*)g-1, g E U(N). Call the former Aigi.Corollary: The connected component of Aut su(n) containing 1 iswhere A :U Autg, the group of auto-a atransition functions 9(0) = exp(2Q0) where Q =equator Ali (0 = ,0) = Aik2 2 0(0)isomorphic to SU(N)/Z/ N acting by conjugation.To apply all this, we note that the fundamental monopole has0 12) T3 . Thus on thelqj(0 = ,0) Since each Ua is adisk, each A on the equator defines a loop in Aut g which is con-tractible: Aa(0) - 0, where - means "is homotopic to". We have onthe equator 0 - A2 = AgAl Ag. Conversely, if Ag - Owe can definean acceptable set of 0(1 by taking AI = 1 on U1 and A2 = the supposedhomotopy between Ag and the null path. Thus global color can be well-defined if and only if Ag O. The derivation of this conditionevidently did not depend upon the specific gauge group.Proposition: In the presence of the fundamental monopole, nosmooth basis of generators of global unbroken gauge transformationsexists.Proof: g(0) acts trivially on the Abelian generator. On thecorner of g(0) = [1 e-4]EU(2). This projects toESU(2)/E 2, a closed loop. When lifted to the coveringgroup SU(2), however, it is clearly not closed. Hence g(0) is nothomotopically trivial.The above result makes minimal use of the specific monopole vec-tor potential. In fact, closer inspection shows that, for general Qsatisfying the Dirac quantization condition, the issue of color defi-nition is settled completely by the one topological invariant k of9the monopole. k is the number of times g(0) winds around the U(1);color can be globally defined iff k is even. Since the full quantumtheory does not mix different topological sectors, our result isvalid beyond the semiclassical approximation. In particular, itis not spoiled by confinement effects. Of course it is stilllimited to energies low enough for the unbroken theory to be agood approximation to the full one, and to the region far from themonopole core. Similarly the addition of matter fields changesnothing; these live on bundles associated to the principal bundleand derive their definition of gauge transformations from it.The above results can be trivially extended to the generalcase. In the SU(5) SU(3) x U(1) theory, global color is definediff k E 0 mod 3. Hence color is undefined in the presence of thefundamental monopole, resolving the original paradox. On the otherhand, the purely electromagnetic monopole has k = 3 E 0 mod 3and so presents no problem, as expected.We would like to thank Sidney Coleman and David Groisser for 7.helpful discussions. 8.9.References:P. Nelson, "Excitations of SU(5) Monopoles", Harvard Universitypreprint HUTP-82/A062.2. E. Witten, Phys. Lett. 868, 283 (1979).3. C. Dokos and T. Tomaras, Phys. Rev. 021, 2940 (1980).4. T.T. Wu and C.N. Yang, Nucl. Phys. 6107, 365 (1976).5. In a more elevated treatment g is the transition function ofa nontrivial principal fiber bundle over 52, and we are seek-ing an su(2) in the infinite-dimensional algebra of sectionsr(g) where g is the associated bundle e = P gAd '6. This is the only place where the nature of the underlyingunified theory is used.N. Jacobson, Lie Algebras (New York, Wiley, 1962), chap. 9.S. Coleman, "The Magnetic Monopole 50 Years Later", HarvardUniversity preprint HUTP-82/A032 (Erice lectures 1981).This is geometrically obvious: the existence of sections ofthe associated bundle depends on the structure of the prin-cipal bundle; its connection is irrelevant.

Global Color is not Always Deﬁned

Kosmopolis

University of Pennsylvania
ScholarlyCommons
Department of Physics Papers Department of Physics
3-1983
Global Color is not Always Defined
Philip C. Nelson
University of Pennsylvania, nelson@physics.upenn.edu
Aneesh Manohar
Follow this and additional works at: http://repository.upenn.edu/physics_papers
Part of the Physics Commons
At the time of publication, author Philip C. Nelson was affiliated with Harvard University. Currently, he is a faculty member in the Physics &
Astronomy Department at the University of Pennsylvania.
This paper is posted at ScholarlyCommons. http://repository.upenn.edu/physics_papers/538
For more information, please contact repository@pobox.upenn.edu.
Recommended Citation
Nelson, P. C., & Manohar, A. (1983). Global Color is not Always Defined. Physical Review Letters, 50 (1), 943-945. http://dx.doi.org/
10.1103/PhysRevLett.50.943
Global Color is not Always Defined
Abstract
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) ⊗ U(1). The algebra of generators of smooth gauge
transformations in this unbroken theory, however, will not in general contain any subalgebra isomorphic to
su(3). This means that global color rotations are not always defined. The resulting inability to classify
semiclassical dyon states by their color climinates an apparent paradox related to the vacuum angle θ.
Disciplines
Physical Sciences and Mathematics | Physics
Comments
At the time of publication, author Philip C. Nelson was affiliated with Harvard University. Currently, he is a
faculty member in the Physics & Astronomy Department at the University of Pennsylvania.
This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/538
HUTP-82/A061
GLOBAL COLOR IS NOT ALWAYS DEFINED*
Philip Nelson and Aneesh Manohar
Lyman Laboratory of Physics
Harvard University
Cambridge, Massachusetts 02138
ABSTRACT
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) x U(1). The algebra of generators of smooth gauge
transformations in this unbroken theory, however, will not in general
contain any subalgebra isomorphic to su(3). This means that global
color rotations are not always defined. The resulting inability to
classify semiclassical dyon states by their color eliminates an
apparent paradox related to the vacuum angle 0 .
This work is supported in part by the National Science Foundation under Grant No. PHY77-22864.
tWork supported in part by a NSF graduate fellowship.
Recently one of us ' has performed an analysis similiar to Witten's 2
of the low-lying states in the fundamental monopole sector of the SU(5)
grand unified theory. The result is that in the presence of a non-
zero vacuum angle e, both the electric charge and the color hyper-
charge (to which it is proportional) take on an arbitrary value. While
this result was surprising for the electric charge in Witten's original
analysis, it at first appears to be a disaster for the hypercharge,
which is presumably embedded in the compact color SU(3) symmetry group.
The only way out of this dilemma seemed to be the possibility
that in fact no group of gauge transformations isomorphic to SU(3)
exists. While this situation may seem to be no less paradoxical than
the original paradox, we show in this letter that it indeed obtains.
This proof holds for the full quantum theory and gives necessary and
sufficient conditions for the unbroken local symmetries of a grand
unified theory to be integrable into well-defined global symmetries.
We can see the essential details of the problem in an SU(3)
theory spontaneously broken to 50(2) x 0(1) by an adjoint Higgs field,
whose expectation value in the nonsingular "radial" gauge of Ref. 3can be
1
taken as <0(0> =1 3 T are the three Pauli matrices.- 2- + 7 T•r
Clearly the generator of unbroken Abelian tram formations T E.M.1
[ 1 3 and one of the nonabelian ones 03 = + 1)
are everywhere well-defined; they can be obtained by transforming the
1 1
constant matrices 1 I -1 J respectively from the sin-
-2J I 0
gular "unitary" gauge of Ref. 3. When we attempt the same thing with
the remaining two independent matrices commuting with 0(z), however,
we find that they become singular at the south pole when we
transform to radial gauge:
0.1=
a2=
0 cose/2 sine/2 e-4 —
cose/2 0 0
sine/2 eio 0 0_
0
icos0/2
isine/2
-icos0/2 -isine/2 e-iS
0 0
0 0
It seems possible, therefore, that some global gauge trans-
formations may not exist. Is there any precedent for such a situa-
tion? Yes there is. Consider the problem of defining a Euclidean
algebra of vector fields on the two-dimensional manifold S2. Near
every point a full algebra of small motions is defined, but it is
well known that not even a single smooth nonvanishing tangent vector
field exists globally. This is so in spite of the fact that S2 and
its tangent bundle can be imbedded in IR3 • which has a perfectly good
algebra of global translations and rotations. Similarly in gauge
theories it is possible for a twisted unbroken theory to be the re-
duction of a trivial unified theory.
To proceed beyond this heuristic level it is convenient to
4
describe the unbroken theory in Wu and Yang's two-patch formalism.
The theory is defined on the space 1R x OR3 - B }, where B is a ball
occupied by the monopole core. We restrict this space further to a
single sphere S2 at some given time, and divide it into two hemi-
spheres Ua a = 1,2. Writing points of S as r, we have zEUI,
01 U 02 5
2
, U1 U2 = E, the equator. All local quantities are
represented as "twisted functions", pairs fa of functions defined on
Ua respectively and related at the equator by a gauge transformation :
f2(6 = = f1(0 = 0 )g(45) where g:E .G and G is SU(2) x
realized as unimodular 3x3 matrices commuting with 0 41 1 .
-2
Then a local gauge transformation on S2 is represented by ha:Ua G,
whose matching condition we obtain as follows:
hi
On
-1 -1
(f2)g (f29 )h1
f2 f2 2 where
h2(0= ,0) = g(0)
the equator,
= 
( f )h19 
• Hence
-1
hl (6) = •g (El') •
The generators of infinitesimal gauge transformations are likewise
twisted functions 0
a
:U
a
• 
 
g • the Lie algebra of G,
condition (1).
Suppose we have a
(1)
with the same
set of such generators 0! i = 1, 4.
These act on the various physical quantities, and we can commute them
to get new generators. Clearly any smooth basis of "generators of
global unbroken
others, the upper
+1k1
gauge transformations" must satisfy compatibility, eq. 1, e 2 
e.22t.
-
plus [0i (1), (;) ] = fijk 0 (1), where fijk are the structurea a
constants of g. We can also demand that the 11(1) generator be proper-
ly normalized: Tr(04)2 = const. Both these conditions on ip are con-
sistent with eq. 1 at the equator. They amount to the requirement
that 02 (;) = Alai(;) 4 (2)
morphisms of g.
Theorem: All automorphisms of su(n) can be writt2n as i 90g-1
or g(0
1*
)g-1, g E U(N). Call the former Aigi.
Corollary: The connected component of Aut su(n) containing 1 is
where A :U Autg, the group of auto-a a
transition functions 9(0) = exp(2Q0) where Q =
equator Ali (0 = ,0) = Aik2 2 0(0)
isomorphic to SU(N)/Z/ N acting by conjugation.
To apply all this, we note that the fundamental monopole has
0 12) T3 . Thus on the
lqj(0 = ,0) Since each Ua is a
disk, each A on the equator defines a loop in Aut g which is con-
tractible: Aa(0) - 0, where - means "is homotopic to". We have on
the equator 0 - A2 = AgAl Ag. Conversely, if Ag - Owe can define
an acceptable set of 0(1 by taking AI = 1 on U1 and A2 = the supposed
homotopy between Ag and the null path. Thus global color can be well-
defined if and only if Ag O. The derivation of this condition
evidently did not depend upon the specific gauge group.
Proposition: In the presence of the fundamental monopole, no
smooth basis of generators of global unbroken gauge transformations
exists.
Proof: g(0) acts trivially on the Abelian generator. On the
corner of g(0) = [1 e-4]EU(2). This projects to
ESU(2)/E 2, a closed loop. When lifted to the covering
group SU(2), however, it is clearly not closed. Hence g(0) is not
homotopically trivial.
The above result makes minimal use of the specific monopole vec-
tor potential. In fact, closer inspection shows that, for general Q
satisfying the Dirac quantization condition, the issue of color defi-
nition is settled completely by the one topological invariant k of
9
the monopole. k is the number of times g(0) winds around the U(1);
color can be globally defined iff k is even. Since the full quantum
theory does not mix different topological sectors, our result is
valid beyond the semiclassical approximation. In particular, it
is not spoiled by confinement effects. Of course it is still
limited to energies low enough for the unbroken theory to be a
good approximation to the full one, and to the region far from the
monopole core. Similarly the addition of matter fields changes
nothing; these live on bundles associated to the principal bundle
and derive their definition of gauge transformations from it.
The above results can be trivially extended to the general
case. In the SU(5) SU(3) x U(1) theory, global color is defined
iff k E 0 mod 3. Hence color is undefined in the presence of the
fundamental monopole, resolving the original paradox. On the other
hand, the purely electromagnetic monopole has k = 3 E 0 mod 3
and so presents no problem, as expected.
We would like to thank Sidney Coleman and David Groisser for 7.
helpful discussions. 8.
9.
References:
P. Nelson, "Excitations of SU(5) Monopoles", Harvard University
preprint HUTP-82/A062.
2. E. Witten, Phys. Lett. 868, 283 (1979).
3. C. Dokos and T. Tomaras, Phys. Rev. 021, 2940 (1980).
4. T.T. Wu and C.N. Yang, Nucl. Phys. 6107, 365 (1976).
5. In a more elevated treatment g is the transition function of
a nontrivial principal fiber bundle over 52, and we are seek-
ing an su(2) in the infinite-dimensional algebra of sections
r(g) where g is the associated bundle e = P gAd '
6. This is the only place where the nature of the underlying
unified theory is used.
N. Jacobson, Lie Algebras (New York, Wiley, 1962), chap. 9.
S. Coleman, "The Magnetic Monopole 50 Years Later", Harvard
University preprint HUTP-82/A032 (Erice lectures 1981).
This is geometrically obvious: the existence of sections of
the associated bundle depends on the structure of the prin-
cipal bundle; its connection is irrelevant.


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