6 April 2000

Ž .Physics Letters B 478 2000 373–378

Dual gluons and monopolesin 2q1 dimensional Yang–Mills theory

Ramesh Anishetty 1, Pushan Majumdar 2, H.S. Sharatchandra 3

Institute of Mathematical Sciences, C.I.T campus Taramani. Madras 600-113, India

Received 11 January 2000; accepted 14 February 2000Editor: M. Cvetic

Abstract

2q1-dimensional Yang–Mills theory is reinterpreted in terms of metrics on 3-manifolds. The dual gluons are related todiffeomorphisms of the 3-manifolds. Monopoles are identified with points where the Ricci tensor has triply degenerateeigenvalues. The dual gluons have the desired interaction with these monopoles. This would give a mass for the dual gluonsresulting in confinement. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 11.15.-q; 11.15.Tk

1. Introduction

Quark confinement is well understood in 2q1Ž .dimensional compact U 1 gauge theory. It is a

consequence of the existence of a monopole plasmaw x w x1,2 . Duality transformation 3 turned out to be veryuseful in this context. It is of interest to know howfar these ideas can be extended to non-abelian gaugetheories. For this reason, duality transformation for2q1-dimensional Yang–Mills theory was obtained

w xin lattice gauge theory in both hamiltonian 4 andw xpartition function 5 formulations. The dual theory

exhibits close relationship to 2 q 1-dimensional

1 E-mail: ramesha@imsc.ernet.in2 E-mail: pushan@imsc.ernet.in3 E-mail: sharat@imsc.ernet.in

gravity, but without diffeomorphism invariance. Thisalso indicates a way of describing the dynamicsusing local gauge invariant variables.

In this paper, we consider duality transformationŽ .for 2q1-dimensional continuum Yang–Mills the-

Ž .ory in close analogy to the case of compact U 1w xlattice gauge theory 3 . We reinterpret the Yang–

Mills theory as a theory of 3-manifolds, as in grav-ity, but without diffeomorphism invariance. We usethis relation for identifying the dual gluons and theirinteractions. The dual gluons are related to diffeo-morphisms of the 3-manifold. We also identify the

w xmonopoles in the dual theory. ’t Hooft 6 has advo-cated the use of a composite Higgs to locate themonopoles. Here we propose to use the orthogonal

Žset of eigenfunctions of a gauge invariant, symmet-.ric local, matrix-valued field for this purpose. Iso-

lated points where the eigenvalues are triply degen-

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00247-1

( )R. Anishetty et al.rPhysics Letters B 478 2000 373–378374

erate have topological significance and they locatethe monopoles. We use the Ricci tensor to constructa new coordinate system for the 3-manifold. Themonopoles are located at the singular points of thiscoordinate system and they have the expected inter-actions with the dual gluons. We expect that theseinteractions lead to a mass for the dual gluons and

Ž .result in confinement as in the U 1 case.w xLunev 7 has pointed out the relationship of

2q1-dimensional Yang–Mills theory with gravity.He uses a gauge invariant composite B aB a as ai j

metric, and rewrites the classical Yang–Mills dy-namics for it. The corresponding formulation of thequantum theory is somewhat involved. Our metric isin a sense dual of Lunev’s choice. As we makeformal transformations in the functional integral, thequantum theory is simpler and has a nicer interpreta-tion.

There are also approaches to relate 3 q 1-dimensional Yang–Mills theory to a theory of a

w xmetric 8 . On the other hand, the dual theory inŽ .3q1-dimensions can also be related to a new SO 3

w xgauge theory 9 .In Section 2 we briefly review duality transforma-

tion and confinement in 2q1-dimensional compactŽ .U 1 lattice gauge theory. In Section 3 we obtain the

dual description of 2q1-dimensional Yang–Millstheory in close analogy to Section 2. We point outthe close relationship to gravity and identify the dualgluons and their interactions. In Section 4 we pro-vide a new characterization of monopoles usingeigenfunctions of the symmetric matrix B aB b. Ini i

Section 5 we use the Ricci tensor to construct apreferred coordinate system for 3-manifolds. We re-late the monopoles to singularities of this coordinatesystem. We also identify their interactions with thedual gluons. Section 6 contains our conclusions.

2. Review of confinement in 2H1-dimensional( )compact U 1 lattice gauge theory

In this section we briefly review duality transfor-w x w xmation 3 and confinement 1 in 2q1 dimensional

Ž .compact U 1 lattice gauge theory. This provides aparadigm for our analysis of 2q1 dimensionalYang–Mills case.

The Euclidean partition function in the Villainformulation is given by

`

Zs dA nŽ .Ý ŁH iy`nihi j

=1

exp y n A nŽ .Ý i j2ž 4k nij

2yn A n qh n . 1Ž . Ž . Ž .j i i j /

Ž . Ž .Here A n g y`,` are non-compact link vari-iˆ Ž .ables on links joining the sites n and nq i. h n si j

0,"1,"2 . . . are integer variables corresponding tothe monopole degrees of freedom and are associated

ˆŽ .with the plaquette nij . n is the difference opera-iˆŽ . Ž . Ž .tor, n f n sf nq i yf n . We may introduceiŽ .an auxiliary variable e n to rewrite Z asi

` `

Zs dA n de nŽ . Ž .Ý ŁH Hi iy` y`nihi j

=2

exp y e nŽ .Ý ižni

2 i1q e e n n A n q h n .Ž . Ž . Ž .Ý i jk k i j i j2 /k nij

2Ž .

Ž .Integration over A n gives the d function con-j

straint

e n e n s0 3Ž . Ž .i jk j k

ˆ Ž . Ž .for each n and i. The solution is e n sn f n .i i

Thus we get the dual form of the partition function

`

Zs df nŽ .Ý ŁHy`nihi j

=2

exp y n f nŽ .Ý ižn

iq f n r n , 4Ž . Ž . Ž ./2k

1Ž . Ž .where r n s e n h n . This has the followingi jk i jk2

interpretation. The field f describes the dual photon.ŽIn 2q1 dimensions, the photon has only one trans-

( )R. Anishetty et al.rPhysics Letters B 478 2000 373–378 375

verse degree of freedom and this is captured by theŽ ..scalar field f n . The monopole number at site n is

Ž .given by r n . It takes integer values and the dualphoton couples locally to it with strength 1rk .

If we sum over the monopole degrees of freedom,Ž . w xwe get a mass term for f n 1,3 . The reason for

this is that the monopole plasma is screening thelong range interactions between the monopoles. AWilson loop for the electric charges in this systemwould correspond to a dipole sheet in this plasma.This gives an area law and hence a linear confiningpotential between static electric charges.

The advantage of this formal duality transforma-tion is that it gives a precise separation of the ‘spinwave’ and the ‘topological’ degrees of freedom.Therefore it provides a stepping stone for goingbeyond semi-classical approximations.

We use this approach for 2q1 dimensionalYang–Mills theory in the next section.

3. Dual gluons in 2H1-dimensional Yang–Millstheory

In this section we point out the close relationshipbetween Yang–Mills theory and Einstein–Cartan

Ž .formulation of gravity in 2q1 or 3 Euclideandimensional space. We use this analogy extensivelythroughout the paper.

The Euclidean partition function of 2q1 dimen-sional Yang–Mills theory is

1a 3 a aZs DDA x exp y d xB x B xŽ . Ž . Ž .H Hi i i2ž /2k

5Ž .� aŽ . Ž .4where A x , i,as1,2,3 is the Yang–Mills po-i

tential and1a a a abc b cB s e E A yE A qe A A 6Ž .Ž .i i jk j k k j j k2

is the field strength. As in Section 2, we rewrite Z asw x5

Zs DDAa x DDea xŽ . Ž .H i i

=213 aexp d x y e xŽ .H i2½ ž

ia aq e x B x . 7Ž . Ž . Ž .i i 5/k

The second term in the exponent is precisely theŽ .Einstein–Cartan action for gravity in 3 Euclidean

aŽ . ab abc cdimensions. e x is the driebein and v se Ai i i

the connection 1-form.In contrast to Section 2, we do not get a d

function constraint on integrating over Aa in thisi

case. Since A appears at most quadratically in theexponent, the integration over A may be explicitlyperformed. This integration is equivalent to solvingthe classical equations of motion for A as a func-tional of e and replacing A by this solution:

ac b w x ce E d qe A e e x s0. 8Ž . Ž .Ž .i jk j abc j k

Ž .Now 8 is precisely the condition for a driebein e tobe torsion free with respect to the connection 1-formAc.i

If we assume the 3=3 matrix ea to be non-sin-iw xgular, then this solution A e can be explicitly given

w x10 . In this case, no information is lost by multiply-Ž . aing 8 by e and summing over a. We get,la a < <Ž y1.m bw xe e E e q e e e e A e s 0. Definingi jk l j k b k lm i jk j

bŽ y1 .m w x w xA e s A , we get, A e y d A e sj b jm l i l i m mŽ < <. a a1r e e e E e . Taking the trace on both sides,i jk l j k

w x Ž < <. a a bw xA e s y 1r2 e e e E e . Then, A e sm m i jk i j k le

b1i a a a ae e E e y d e e E e .Ž .< < i jk l j k l i m jk m j ke 2

w x XBy a shift of A, AsA e qA , the integrationover A reduces to

i 1X X X

DDA exp A e A sH H i a ia , jb jb 1r2ž /k det eŽ .i a , jb

1s , 9Ž .3r2 adet eŽ .i

where e se e abcec.i a, jb i jk k

B a is related to the Ricci tensor R as follows:i i k

jab a y1R sF e e 10Ž . Ž .bi k i j k

where F ab se e abcBc. Thus an integration over Ai j i jk k

gives

i1 'Zs DDgexp y g q g R 11Ž .H i i2ž /k

( )R. Anishetty et al.rPhysics Letters B 478 2000 373–378376

where the metric g seaea and RsR g k i. Noteji i j i ky3r2Ž a.that DDgsDDedet e , as required. The configu-i

rations where e is singular is naively a set of mea-< <sure zero, so that the assumption e /0 is reason-

able.Ž .Eq. 11 provides a reformulation of 2q1-

Ždimensional Yang–Mills theory classical or quan-.tum in terms of gauge invariant degrees of freedom.

It is now a theory of metrics on 3-manifolds; whichhowever is not diffeomorphism invariant because ofthe term g in the action. As a result, not only thei i

geometry of the 3-manifold, but also the metric gi j

of any coordinate system chosen on the manifold isrelevant.

Ž .For 3 dimensional Euclidean gravity, an integra-Ž .tion over e 7 would give the d-function constraint

w xR s0, resulting in a topological field theory 12 .i j

There are no massless gravitons as a consequence.Now however, the diffeomorphisms provide mass-less degrees of freedom corresponding to gluons.They may be described as follows. The 3 manifoldsare described by the metric g in the coordinatei j

system x. We may choose a new coordinate systemAŽ . Ž .f x As1,2,3 , with a standard form of the met-

w xric G f . We haveA B

Ef A Ef B

w xg x s G f . 12Ž . Ž .i j A Bi jE x E x

This gives the form of the action as

A BEf Ef3 w xSs d x y G fH A Bi jž /E x E x

Ai Efw x w x(q G f R f , 13Ž .i2k E x

A AEf Ef AŽ . Žwhere sdet . We identify f x Asi iŽ .E x E x

.1,2,3 as the dual gluons. A simple way of seeingthis is as follows. Note that the second term comes

'with a factor is y1 , whereas the first term doesnot. In this sense it is analogous to the u-term in

'QCD which continues to have the factor is y1in the Euclidean version.. Consider a random phaseapproximation to Z. The extrema of the phase factorcorrespond to solutions of the the vacuum Einstein

Ž .equations. In this case 3 dimensions , this meansthat the space is flat. Now we may choose thestandard form G sd . f A now represent arbi-A B A B

trary curvilinear coordinates for that manifold. ThenŽ . Ž A.2the first term in 13 is just =f . This describes

three massless scalars. As in Section 2 they representthe one transverse degree of freedom for each color.Thus the gluons are now described in terms of gaugeinvariant, local, scalar degrees of freedom.

In the general case R/0, consider normal coor-AŽ .dinates f x at a given point. The metric has the

standard form,

w x w x C DG f sd qR f f f q . . . . 14Ž .A B A B A BC D

f A represents the dual gluons and R the geometricaspects of the manifold. Both are degrees of freedomof 2q1 dimensional Yang–Mills theory. f A areinvariant under the Yang–Mills gauge transforma-

Ž .tions. Thus Eq. 13 describes Yang–Mills dynamicsin terms of gauge invariant degrees of freedom.

4. Monopoles

We now identify the monopoles of Yang–Millstheory in terms of the dual variables. Monopoles are

� aŽ .4related to Yang–Mills configurations A x with aiŽ . w xnon-trivial U 1 fibre bundle structure 11 . In such

configurations, the monopoles are characterized byw xpoints with the following property 13 . Consider a

surface enclosing a point and a set of based loopsspanning it. Consider eigenvalues of the correspond-ing Wilson loop operator. As one spans the sphere,the eigenvalue changes continuously from zero to 2p

instead of coming back to zero. Thus such pointshave topological meaning. Moreover a small changein their position can produce a large change in theexpectation value of the Wilson loop. Therefore wemay expect that such points are relevant for confine-ment, even though a semi-classical or dilute gasapproximation may not be available. Therefore it isimportant to provide a characterization of thesemonopoles and their interactions with the dual glu-ons.

In case of ’t Hooft–Polyakov monopole, the loca-tion of the monopoles is given by the zeroes of the

w xHiggs field 14 . In pure gauge theory we do notw xhave such an explicit Higgs field. ’t Hooft 6 has

proposed use of a composite Higgs for this case.

( )R. Anishetty et al.rPhysics Letters B 478 2000 373–378 377

We follow a different procedure here. Considerthe eigenvalue equation of the positive symmetric

aŽ . bŽ . abŽ .matrix B x B x s I x for each x:i i

I ab x x A x sl A x x A x . 15Ž . Ž . Ž . Ž . Ž .a b

AŽ . Ž .The eigenvalues l x , As1,2,3 are real and theAŽ . Žcorresponding three eigenfunctions x x , Asa

.1,2,3 form an orthonormal set. The monopoles inaŽ .any Yang–Mills configuration A x can be locatedi

AŽ .in terms of x x . We will illustrate this explicitlyaw xin case of the Prasad–Sommerfield solution 15 . For

this I ab has the tensorial form

I ab x sP r d ab qQ r x a x b 16Ž . Ž . Ž . Ž .Ž .with P 0 /0 and finite. At rs0, the eigenvalues

are triply degenerate. Away from rs0, two eigen-values are still degenerate, but the third one is dis-tinct from them. The corresponding eigenfunctionŽ . 1Ž . alabelled As1, say is x x sx . This preciselyˆa

has the required behavior for the composite Higgs atw xthe center of the monopole 6 .

AŽ .We may regard x x as providing three inde-aŽ .pendent triplets of normalized Higgs fields. Using

them, we may construct three abelian gauge fields,

1A A a abc A A Ab x sx x B x y e e x D x D xŽ . Ž . Ž .i a i i jk a j b k c3

17Ž .

We have

1A A abc A A Ab x se E a y e e x E x E x 18Ž . Ž .i i jk j k i jk a j b k c3

where the three abelian gauge potentials are given byAŽ . AŽ . aŽ .a x sx x A x . For each As1,2,3, the sec-i a i

ond part of the right hand side is the topologicalw xcurrent for the Poincare–Hopf index 14 . It is the

contribution of the magnetic fields due to themonopoles. These monopoles are located at pointswhere this index is non-zero.

Since, x a se e abcx Bx C, we may rewrite ourA A BC b c

abelian fields as

b A x se E a A x qe A BCc BcC 19Ž . Ž . Ž .Ž .i i jk j k j k

where c A se A BCx BE x C has the form of a ‘purei a i a

gauge’ potential, but is not, because of the singular-Ž A.ity in x .a

aŽ .Thus for any configuration A x of the Yang–i

Mills potential, monopoles may be characterized asthe points where the eigenvalues of the symmetric

aŽ . bŽ .matrix B x B x become triply degenerate. Wei i

may use the corresponding eigenfunctions to con-struct three abelian gauge fields with respectivemonopole sources. Instead of I ab, we may also usethe gauge invariant symmetric tensor field

aŽ . aŽ . AŽ .B x B x and it’s eigenfunctions x x . Thisi j i

provides a gauge invariant description of themonopoles.

We may also use the Ricci tensor R j siŽ . k jŽ .R x g x for this purpose. The three eigenfunc-i k

AŽ . Ž . Žtions x x , As1,2,3 Ricci principal directionsiw x.16 provide three orthogonal vector fields for the3-manifold. In regions where eigenvalues of R j arei

degenerate, the choice of the vector fields is notunique. One can make any choice requiring continu-ity. However isolated points where R j is triplyi

degenerate are special, and have topological signifi-cance. At such points the vector fields are singular.Thus the monopoles correspond to the singular pointsof these vector fields. The index of the singular pointis the monopole number.

We emphasize that the centers have a topologicalinterpretation which is independent of the way weconstruct them.

5. Interaction of dual gluons with monopoles

Dual gluons are identified with a coordinate sys-AŽ . Ž .tem f x As1,2,3 on the 3-manifold Eqs. 13 ,

Ž .14 . We now consider special coordinate systemswhich are singular at the location of the monopole.In case of the Prasad–Sommerfield monopole, the

Ž .correspond to the spherical coordinates r,u ,f withthe monopole at the origin. In the general case, wemay construct the coordinate system as follows. Atthe site of the monopole, one of the eigenfunctions

1Ž .x x say, has the radial behavior. Then we mayi

construct the integral curves of this vector field bysolving the equations

dx1 dx 2 dx 3

s s . 20Ž .1 1 1x x x x x xŽ . Ž . Ž .1 2 3

We may choose these integral curves to be theequivalent of the r-coordinate, i.e. we identify thesecurves with usconstant, fsconstant curves of the

( )R. Anishetty et al.rPhysics Letters B 478 2000 373–378378

new coordinate system. Consider closed surfacessurrounding the monopole which are nowhere tan-gential to these integral curves. A simple choice isjust the spherical surfaces. We may identify them

Žwith the surfaces rsconstant. We have not speci-fied the u ,f coordinates completely, but this is not

.required for our purpose. We thus have a coordinateAŽ .system x x whose coordinate singularities corre-

spond to the monopoles. In this coordinate system,3 3 A BC A B C 'Ž . Ž .'Hd x g RsHd x e e E x E x E x GG x Ri jk i j k

Ž .x where GG is the metric in this coordinate system.i jŽ 2 3.Now E e E x E x is non-zero at xsx andi i jk j k 0

is related to the monopole charge at x as follows.0A Ž . A Ž . Ž . A Ž .Let x x y x x s r x x x whereˆ0

AŽ . AŽ .x x x x s1. We see that there is a coupling ofˆ ˆ3Ž . Ž .(the field combination G x R x r x to theŽ .

Ž . 3Ž .monopole charge density E k x sm d x , wherei i i 0Ž . A BC A B Ck x se e x E x E x . Thus a certain combi-ˆ ˆ ˆi i jk j k

AŽ .nation of the dual gluon f x and the geometricŽ .degree of freedom R x couples to the monopoles.

Ž .In analogy to the compact U 1 lattice gauge theoryŽ .Section 2 , this may be expected to give a mass forthe dual gluon and hence confinement. There areother interactions which are not of topological originand these are to be interpreted as self interactions.

6. Conclusion

We have argued that the duality transformationfor 2q1 dimensional Yang–Mills theory can becarried out in close analogy to the abelian case. Thedual theory has geometric interpretation in terms of3-manifolds. We identified the dual gluons with thecoordinates of the 3-manifolds and monopoles with

the coordinate singularities. We expect that this willprovide a new approach for understanding quarkconfinement.

Acknowledgements

Ž .One of us P.M. wishes to thank Dr. ElizabethGasparim and Dr. Mahan Mitra for explanation ofseveral mathematical concepts.

References

w x Ž .1 A.M. Polyakov, Phys. Lett. B 59 1975 82.w x Ž .2 A.M. Polyakov, Nucl. Phys. B 120 1977 429.w x Ž .3 T. Banks, R. Myerson, J. Kogut, Nucl. Phys. B 129 1977

493.w x Ž .4 R. Anishetty, H.S. Sharatchandra, Phys. Rev. Lett. 65 1990

813.w x5 R. Anishetty, S. Cheluvraja, H.S. Sharatchandra, M. Mathur,

Ž .Phys. Lett. B 314 1993 387.w x Ž .6 G. ’t Hooft, Nucl. Phys. B 190 1981 455.w x Ž .7 F.A. Lunev, Phys. Lett. B 295 1992 99.w x8 M. Bauer, D.Z. Freedman, P.E. Haagensen, Nucl. Phys. B

Ž .428 1994 147; P.E. Haagensen, K. Johnson, Nucl. Phys. BŽ .439 1995 597.

w x9 Pushan Majumdar, H.S. Sharatchandra, imscr98r05r22,hep-thr9805102.

w x10 Pushan Majumdar, H.S. Sharatchandra, imscr98r04r14,hep-thr9804128.

w x Ž .11 T.T. Wu, C.N. Yang, Phys. Rev. D 12 1975 3845.w x Ž .12 E. Witten, Nucl. Phys. B 311 1988 46.w x Ž .13 P. Goddard, D.I. Olive, Rep. Prog. Phys. 41 1978 1357.w x14 J. Arafune, P.G.O. Freund, C.J. Goebel, J. Math. Phys. 16

Ž .1975 433.w x Ž .15 M.K. Prasad, C.M. Sommerfield, Phys. Rev. Lett. 35 1975

760.w x16 L.P. Eisenhart, Riemannian Geometry, Princeton University

Press, 1949.