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Complex Monopoles in the Georgi-Glashow-Chern-Simons Model

a r X i v :h e p -t h /9808045v 2 23 S e p 1998September 23,1998(corrected.)

UMN-TH-1626/98,TPI-MINN-98/11-T

hep-th/9808045

Complex Monopoles in the Georgi-Glashow-Chern-Simons Model Bayram Tekin,Kamran Saririan and Yutaka Hosotani School of Physics and Astronomy,University of Minnesota Minneapolis,MN 55455,U.S.A.

Abstract

We investigate the three dimensional Georgi-Glashow model with a Chern-Simons

term.We ?nd that there exist complex monopole solutions of ?nite action.They dominate the path integral and disorder the Higgs vacuum,but electric charges are not con?ned.Subtleties in the gauge ?xing procedure in the path integral and issues related to Gribov copies are noted.

PACS:11.10.Kk;11.15.Kc;https://www.wendangku.net/doc/fd16685906.html,

Keywords:Monopoles,Chern-Simons theory,Con?nement

1.Introduction

Years ago Polyakov showed that in the three-dimensional Georgi-Glashow model,or more generally in the compact QED3,monopole con?gurations in the Euclidean space make dominant contributions in the functional integral for the con?nement of electric charges [1].The logarithmic potential between two electric probe charges is converted to a linear potential in the background of a monopole gas,leading to the linear con?nement.The Higgs vacuum is disordered.The expectation value of the triplet Higgs?eld vanishes ( φ =0),whereas all components of the SU(2)gauge?elds acquire masses.The long range order in the Higgs vacuum is destroyed by monopole con?gurations.

Gauge theory in three dimensions can accommodate a purely topological term,the Chern-Simons term,which a?ects the equations of motion.It gives a topological mass to gauge?elds[2].In the Georgi-Glashow model,even in perturbation theory,the unbro-ken U(1)gauge?eld also becomes massive,and thus the issue of the linear con?nement disappears in the presence of the Chern-Simons term;there is no long-range force in the Georgi-Glashow-Chern-Simons(GGCS)model to start with.The electric?ux is not con-served.It does not matter for the issue of the con?nement whether or not monopole con?gurations dominate in the functional integral.

Once the Chern-Simons term is added the action becomes complex in the Euclidean space.Many authors have shown that there is no real monopole-type?eld con?guration of a?nite action which solves the Euler equations[3]-[7].This fact has been interpreted as indicating the irrelevance of(real)monopole con?gurations in the model.

There remain a few puzzles.Although there is no con?nement in the presence of the Chern-Simons term,there remains the issue of the long-range order in the Higgs vacuum. How can the Higgs vacuum be disordered if monopole con?gurations are totally irrelevant? Does the expectation value φ become nonvanishing and the long-range order is restored once the Chern-Simons term is added?Also,if the theory allows complex monopole con-?gurations,does their contribution to the partition function vanish as in the real case? There are also subtle questions related to the gauge invariance as well as Gribov copies in various gauges in the GGCS model which,to our knowledge,have not yet been answered.

We shall re-examine these considerations in the context of complex monopole solutions to show that the Higgs vacuum remains disordered.Although there are no real monopole con?gurations which solve the Euler equations,there exist complex monopole con?gura-tions which extremize the Euclidean action in a?xed gauge.They dominate the functional integral in quantum theory and destroy the long-range order in the Higgs vacuum.The e?ect of Gribov copies is also re-examined.What we mean by complex monopoles will be clear in the text but for now we should state that these are complex-valued solutions to the equations of motion.The non-abelian?eld strengths are complex but the U(1)’t Hooft ?eld strengths are real and exactly those of a magnetic monopole.With the U(1)projec-tion our complex monopoles can be interpreted as the topological excitations characterized by the the groupΠ2(SO(3)/U(1))=Z.

It is worthwhile to recall the correspondence between compact QED3and the Joseph-son junction system in the superconductivity[8].The normal barrier region sandwiched by two bulk superconductors becomes superconducting due to supercurrents?owing through the barrier.The three-dimensional compact QED is related to the Josephson junction sys-tem by the electro-magnetic dual transformation.The U(1)?eld strengths(E1,E2,B)in the Georgi-Glashow model correspond to(B1,B2,E3)in the Josephson junction.Electric charges in the Georgi-Glashow model are magnetic charges inserted in the barrier in the Josephson junction.If there were no supercurrents,the magnetic?ux between a pair of magnetic monopole and anti-monopole inserted in the barrier forms dipole?elds,giving a logarithmic potential between the pair.However,due to supercurrents the magnetic?ux is squeezed to form a Nielsen-Olesen vortex giving rise to a linear potential.

Monopoles(instantons)in compact QED3are supercurrents in the Josephson junction. Polyakov introduced a collective?eldχwhich mediates interactions among monopoles.The ?eldχcorresponds to the di?erence between the phases of the Ginzburg-Landau order parametersΨGL in the bulk superconductors on both sides of the barrier in the Josephson junction.Bothχandδ(argΨGL)satisfy the same sine-Gordon type equation.

Now we add the Chern-Simons term in the Georgi-Glashow model.At the moment we haven’t understood what kind of an additional interaction in the Josephson junction

system corresponds to the Chern-Simons term in the Georgi-Glashow model.It could be a

θF μν?F

μνterm in the superconductors on both sides.Normally a θF μν?F μνterm is irrelevant in QED.However,if the values of θon the left and right sides are di?erent,this term may result in a physical consequence,which may mimic the e?ect of the Chern-Simons term in the Georgi-Glashow model.

If monopoles are irrelevant in the presence of the Chern-Simons term,it would im-ply that suppercurrents cease to ?ow across the barrier in the corresponding Josephson junction.Although we have not found the precise analogue in the Josephson system yet,and therefore we cannot say anything de?nite by analogy,we feel that it is very puzzling if suppercurrents suddenly stop to ?ow.Monopoles should remain important even in the presence of the Chern-Simons term.The correspondence between the Chern-Simons theory and the Josephson junction arrays has been discussed in ref.[9].

As we shall discuss below,there is a subtle issue in quantizing a Chern-Simons theory.The arguments below are based on the Chern-Simons theory in a ?xed gauge.In the path integral a gauge condition restricts functional space to be integrated.Within this subspace complex monopole solutions are found.This is a delicate issue as the Chern-Simons term is not gauge invariant.

2.The model

The action for the scalar ?elds interacting with the gauge ?elds in the three-dimensional Euclidean space is given by S =S 0+S cs +S h where

S 0=?1

g 2

d 3x?μνλtr A μ?νA λ+2g 2

d 3x 14(h a h a ?v 2)2 (2.1)W

e adopt the notation A μ=i 2h a τa ,and D μh =?μh +[A μ,h ].The classical equations o

f motion are

D μF μν+i

DμDμh=?λ(tr h2+m2)h.(2.2)

Notice that S cs is pure imaginary in the Euclidean space for real gauge?eld con?gu-rations.Equations of motion become complex.To de?ne a quantum theory,one needs to ?x a gauge.We would like to?nd dominant?eld con?gurations.Polyakov,in the theory whereκ=0,showed that monopole con?gurations are essential in removing the degen-eracy of the vacuum and lead to the linear con?nement of electric charges.Forκ=0a real monopole con?guration is shown to have an in?nite action which makes its contribu-tion vanish in the vacuum.Pisarski[3]interpreted this as the con?nement of monopole. Fradkin and Schaposnik[6]consider a theory where Chern-Simons term is induced as a one loop e?ect after integrating fermions.In this case the t’Hooft-Polyakov real monopole [1,10]minimizes the classical equations of motion but not at the one loop level.Fradkin and Schaposnik deform the real t’Hooft-Polyakov monopole in the complex con?guration space.They have concluded that a monopole-antimonopole pair is bound together by a linearly growing potential.

It is clear that more careful analysis is necessary to?nd the absence or existence of complex monopole con?gurations and their implications.In the context of path integrals, stationary points of the exponent in the integrand may be located generically at complex points,though the original integration over?eld con?gurations is de?ned along the real axis. In the saddle-point method for integration such saddle-points give dominant contributions in the https://www.wendangku.net/doc/fd16685906.html,plex monopoles can be vital in disordering the vacuum.

When v=0in(2.1),the perturbative vacuum manifold is SO(3)/U(1).Following t’Hooft and Polyakov,we make the spherically symmetric monopole ansatz which breaks SO(3)gauge×SO(3)rotation to SO(3).

h a( x)=?x a h(r)

A aμ( x)=

r2?μνb?x a?x b(φ21+φ22?1)+

+(δaν?xμ?δaμ?xν)

r2d

(φ21+φ22)h=0(2.5)

for the Higgs?eld and

φ1φ′2?φ2φ′1?A(φ21+φ22)+

(1?φ21?φ22)+iκ(φ′2?Aφ1)?h2φ1=0(2.7) (φ′2?Aφ1)′?A(φ′1+Aφ2)+

φ2

κ(1?φ21?φ22)=const(2.9) Eq.(2.6)gives an additional information that the constant in Eq.(2.9)is0.

However,due caution is necessary in writing down equations of motion in quantum theory.The Chern-Simons term is not gauge invariant.Even if the action is varied in a pure gauge direction,it may change.In other words the action need not be stationary under such variations.In the monopole ansatz one combination of Eqs.(2.6)-(2.8)corresponds to such variations.If one?xes the gauge?rst,this particular equation does not follow from the gauge-?xed action.

This becomes clearer when equations are derived from the action written in terms of the monopole ansatz.The action is then written as follows(using eq.(2.3)):

S=4π

2r2

(φ21+φ22?1)2

+iκ φ′1φ2?φ′2(φ1?1)+A(φ21+φ22?1)

(h2?v2)2 (2.10)

The regularity at the origin and the?niteness of the action place boundary conditions at r=0:h=0,φ1=1,φ2=0

at r =∞:h =v ,φ1=0,φ2=0,A =0.(2.11)

In the original ’t Hooft-Polyakov [1,

10]

monopole

solution,

φ2(r )=A (r )=0.

For v =0the unbroken U (1)is parametrized by residual gauge transformations gen-erated by ?=exp i

g 2 tr d (A ∧d ??1?)+iκ

g 2f (∞).(2.14)

If the action is viewed as a functional of all h ,A ,φ1,and φ2and varied with respect to those ?elds,then one would obtain Eqs.(2.5)-(2.8).It is obvious,however,that Eq.(2.6)and the boundary condition (2.11)are incompatible at r =∞.There would be no solution of a ?nite action.Instead,one might ?x a gauge ?rst and vary the action within the gauge chosen.Indeed,this is what is done in quantum theory either in the canonical formalism or in the path integral formalism.In the path integral we start with

Z = D A D h ?F (A )δ[F (A )]e ?S .

(2.15)

What we are looking for is a?eld con?guration which extremizes S within the gauge chosen;

δS|

F(A)=0

.If the action is manifestly gauge invariant,the order of two operations,?xing a gauge and extremizing I,does not matter.However,in the presence of the Chern-Simons term,the action is not manifestly gauge invariant.As we shall see explicitly in the following

section,two con?gurations A(j)(j=1,2)determined byδS|

F j(A)=0

are physically di?erent in general,i.e.A(1)is not on a gauge orbit of A(2).

Before closing this section,we would like to remark that there are three approaches in de?ning the path integral.The issue is how to de?ne the con?guration space Aμin (2.15).The?rst possibility is to restrict gauge orbits such that transformation function?be S3-compatible,namely?|r=∞=?∞.For(2.12)it implies that f(∞)=2πp where p is an integer.With this e?S becomes gauge invariant.However,this restriction leads to con?ict with gauge?xing.Suppose that a gauge con?guration is given.Now one?xes a gauge.However,a gauge orbit of the given con?guration may not intersect the gauge?xing condition.For instance,if ∞0dr A=2πp,then the con?guration cannot be represented in the radial gauge.Ifφ2/φ1|r=∞=0,then the con?guration cannot be represented in the unitary gauge.If there is no f(r)satisfying

f′′+2

{φ1sin f+φ2(1?cos f)}=A′+

,(2.16)

then the con?guration cannot be represented in the radiation gauge.Put it di?erently,the gauge?xing procedure removes a part of physical gauge con?gurations.This approach is not acceptable.

The second possibility is to impose no restriction on?.In the case e?S is not gauge invariant in general.In this paper we adopt this approach to?nd consequences.

The third possibility is to restrict gauge?eld con?gurations Aμto be S3compatible. This excludes monopole ansatz(2.3)entirely.With this restriction the total monopole charge must vanish.This is certainly a legitimate approach,and there occurs no problem of the gauge invariance of the theory.One has to address a question why the spacetime needs to be compacti?ed from R3to S3in de?ning a theory.We leave this possibility for future consideration.

3.Gauge choices

There are four gauge choices which are typically considered:

(i)Radial gauge (A =?x a ?x μA a μ=0).

As is obvious from (2.12),this gauge choice is always possible.h ,φ1and φ2are independent ?elds.Equations derived by extremizing the action (2.10)are Eq.(2.5)and

φ′′

1+1

r 2(1?φ21?φ22)φ2?iκφ′1?h 2φ2=0(3.2)

Eq.(3.1)and (3.2)are obtained by naively setting A =0in (2.7)and (2.8).Eqs.(3.1)and (3.2)imply (2.9)but not necessarily (2.6).The left hand side of (2.6)does not vanish in general.

(ii)Unitary gauge (φ2=0).

This gauge was employed in ref.[3].Equations which follow are (2.5)and

φ21A ?i

r 2

(1?φ21)φ1?iκAφ1=0(3.4)Eq.(3.3)follows from (2.8)by setting φ2=0.

Eq.(3.3)is incompatible with the boundary condition (2.11)for κ=0.Even for con?gurations which do not satisfy the Euler equations,the unitary gauge may not be possible.Suppose that a con?guration (h,φ1,φ2,A )satis?es the boundary condition (2.11),yielding a ?nite action.To bring it to the unitary gauge one has to choose tan f =(φ2/φ1)in (2.12).The action changes by a ?nite amount

4πκφ1 r =∞r =0.(3.5)

The new A satis?es the boundary condition only if f ′=(φ1φ′

2?φ′

1φ2)/(φ21+φ22)vanishes

at r =∞,which is not generally satis?ed.

(iii)Radiation gauge (?μA a μ=0).

The gauge condition implies that

A′+2

r =0,A(r)=2

(1?φ21?φ22)?iκ(φ′1+Aφ2)?h2φ2

?2 ∞r du12κ(φ21+φ22?1) =0(3.7) Here A is expressed in terms ofφ2by(3.6).

There is residual gauge invariance.The gauge condition is respected by the transfor-mation(2.12),provided f(r)obeys the Gribov equation[11]:

f′′+2

r2 φ1sin f+φ2(1?cos f) =0(3.8)

¨f+˙f?2 φ1sin f+φ2(1?cos f) =0.(3.9) Here a dot represents a derivative with respect to t=ln r.

As was pointed out by Gribov,Eq.(3.9)is an equation for a point particle in a potential V=2{φ1cos f?φ2(f?sin f)}with friction force.Forφ1=1andφ2=0,which corresponds to the trivial vacuum con?guration A aμ=0in(2.3),there appear three types of solutions.With the initial condition f|r=0=0,(i)f(r)=0,(ii)f(∞)=π,or(iii) f(∞)=?π.The last two are the Gribov copies(see Fig.1).

It is interesting to consider the relevance of the’t Hooft-Polyakov monopole con?gu-rationsφ2=0butφ1=0(φ1(0)=1andφ1→0as r→∞).1In terms of the Gribov equation,this case corresponds to a particle moving in a time-dependent potential.As the potential becomes exponentially small for large t,there can be a continuous family of solutions parameterized by the value of f(∞).The asymptotic value f(∞)depends of the

00.5

1.52

2.5

3.5

05101520

f (r )r f ’(0) BPS Vacuum 0.2

Figure 1:Solutions f (r )to the Gribov Eq.(3.8).Solid lines correspond to the BPS monopoles with various values of f ′(0)whereas points correspond to the vacuum.For the vacuum f (∞)=πfor positive f ′(0).

initial slope f ′(0).In the BPS limit it ranges from 0to 3.98.See Fig.2.For |f ′(0)|?1,|f (r )|remains small.For |f (r )|?1,f (r )approaches an asymptotic value before φ1and φ2make sizable changes,i.e.,f (r )behaves as in the vacuum case.The maximum value for |f (∞)|is attained for f ′(0)=±2.62.

These Gribov copies in the radiation gauge lead to an important consequence in the Chern-Simons theory.Under the gauge transformation the action changes as

S →S +i 4πκ

In ref.[5],the parameter f (∞)is interpreted as the collective coordinate of the monopole con?guration,and is therefore integrated over.Here,we alternatively associate f (∞)with the Gribov copies within a given gauge,corresponding to the solutions of eq.(3.8).In this interpretation,one sums over the Gribov copies,i.e.,one integrates over the parameter f (∞).

00.5

1.5

2.5

3.5

0.010.11101001000

f ( )8f’(0)π

Figure 2:f (∞)for BPS monopole vs.initial slope (f ′(0)).

monopole ranges from ?3.98to 3.98,depending on f ′(0).There seems no reason for expecting the cancellation.

(iv)Temporal gauge (A a 3=0)

This gauge destroys spherical symmetry in the Euclidean space,but allows physical interpretation in the Minkowski spacetime.

We are going to show that there are complex monopole solutions in certain gauges.We argue also that in the case of the radiation gauge Gribov copies do not lead to cancellation.When φ2is complex in (3.8),a solution f (r )to Eq.(3.8)necessarily becomes complex.In other words the transformation speci?ed with f (r )is not a gauge transformation.However,one can show that there are solutions f (r )to Eq.(3.8)in which f (0)=0and f (∞)is real.

In terms of a gauge invariant quantity

=(φ1+iφ2)e ?i r A (r )dr η

?=(φ1?iφ2)e +i r A (r )dr (3.11)the action (2.10)simpli?es to

S =4π

2(1?ηη?)2?κ

2(h ′)2+h 2ηη?+λr 2

Note that ∞0dr A(r)implicitly depends onηandη?.

https://www.wendangku.net/doc/fd16685906.html,plex monopole solutions

As discussed in the previous section,the gauge is?xed in quantum theory.The aim of this section is to show the existence of complex monopole solutions in certain gauges. We shall see a subtle relation among di?erent gauges.A solution in one gauge can be transformed into another gauge by a“complex”gauge transformation,but the transformed con?guration may not be a solution in the new gauge.This is a re?ection of the gauge non-invariance of the original theory in the presence of the Chern-Simons term.It implies that relevant?eld con?gurations in the path integral may appear di?erent,depending on the gauge.This raises a question if all gauges are equivalent as they should.A related issue has been analyzed in the temporal gauge in ref.[12].It has been shown that implementation of constraints before quantization does not yield the correct physics in Chern-Simons theory.

We note that the t’Hooft U(1)?eld strength is de?ned by

Fμν=h a

?abc h a(Dμh)b(Dνh)a(4.1)

With our ansatz it becomes

Fμν=?

?μνa?x a

r ,φ1(r)=

In this gauge there is no Gribov copy.Equations to be solved are Eqs.(2.5),(3.1), and(3.2).There is a solution in which h andφ1are real,butφ2is pure imaginary.An appropriate ansatz is

φ1(r)=ζ(r)cosh κr

.(4.4)

The non-Abelian?eld strengths are

F aμν=1

(?aμν??μνb?x a?x b)ζ(r)sinh

κr

(δaν?xμ?δaμ?xν)ζ(r)cosh

κr r2

h ?λ(h2?v2)h?2

(ζ2?1)ζ? h2+κ2

ζ2+

(h′)2+

λr2

00.20.40.6

0.8

0246810

r BPS φ1

BPS h

φ1-i φ2

Figure 3:The solution in the radial gauge for v =1,κ=0.5and λ=0.5.The solutions at arbitrary values of κand λare numerically generated using the exact known BPS solutions for φ1,BPS and h BP S .In this plot we show the solutions for ζ(r )and h (r ),and also φ1(r )and ?iφ2(r ).

(iii)Radiation gauge (?μA a μ=0)

Again we look for a solution in which φ2is pure imaginary:φ2=i ˉφ

2and A =i r 2

dr h ?λ(h 2?v 2)h ?2dr Dφ1D ˉφ2 ? 2 D ˉφ2Dφ1 ?

14 φ1ˉφ2 ? 01

∞r du 22 =0(4.8)

where Dφ1D ˉφ2 = φ′

1ˉφ′2 ?

2 ˉφ2φ1 ,r 2 r 0du ˉφ2(u ).(4.9)

It is not easy to ?nd a simple ansatz for φ1and ˉφ

2which solves the equations.Note that the equations in (4.8)are in a gauge covariant form except for the last term in the second equation.If one makes a complex gauge transformation,promoting f in

(2.12)to a complex f=iˉf,

φ1ˉφ2 → +coshˉf?sinhˉf

?sinhˉf+coshˉf φ1ˉφ2 ,A?ˉf′,(4.10)

then the equations remain the same except that the matrix factor(0,1)in the second equation in(4.8)is replaced by(sinhˉf,coshˉf).Even if one choosesˉf such that the new

r f′R?

r f′I?

In ref.[3],the unitary gaugeφ2=0was adopted.As remarked before,the boundary conditions cannot be satis?ed in this gauge forκ=0.In other words,solutions obtained in this gauge after relaxing the boundary conditions necessarily have an in?nite action for κ=0.This is what Pisarski found.If one considers a monopole-antimonopole pair,the action can be made?nite.The action is proportional to the distance between the pair, which leads to the con?nement picture of monopole-antimonopole pairs.

We also remark again that Pisarski’s solution in the unitary gauge is di?erent from the con?guration obtained from the solution in the radial gauge in Section3-(i)by a gauge transformation.

(ii)A?eck,Harvey,Palla,and Semeno?’s work

It has been recognized in ref.[5]that gauge copies of monopole solutions can lead to cancellation in the path integral.As expressed in(2.14),gauge copies yield an extra factor exp{(4πiκ/g2)f(∞)}=exp{inf(∞)}where the quantization condition forκhas been employed.In ref.[5]the factor f(∞)was promoted to a collective coordinateΛ.It was argued then that the integration over the collective coordinateΛfrom0to2πgives a vanishing contribution when the monopole number is non-vanishing.

However,this argument is incomplete.As explained in Section2-(iii),the possible range for f(∞)in the radiation gauge for the monopole in the BPS limit is[?3.98,3.98]. No cancellation is expected.

(iii)Fradkin and Schaposnik’s work

In ref.[6],the authors start with a gauge invariant theory with massive fermions in the Abelian theory.The integration of fermion degrees induces a Chern-Simons term, which leads to the decon?nement of charges.The authors have argued that there appears a linear con?ning potential between monopoles and antimonopoles so that a monopole gas becomes a dipole gas exhibiting no Debye screening and destroying the con?nement picture.Further,they argue that the lack of the gauge invariance of the Chern-Simons term makes gauge copies of monopole con?gurations leads to the cancellation in the path integral.

Their argument is unsatisfactory in the context of our work in two respects.First they consider monopole con?gurations obtained in the absence of the Chern-Simons term and insert them into the Chern-Simons term to?nd implications.The fact that the con-?gurations do not solve the equations in the presence of the Chern-Simons term leads to a linear potential among monopoles and antimonopoles.This argument is not consistent; one should examine solutions in the full theory including the Chern-Simons term.

Secondly their argument is given for the Abelian theory with monopole con?gurations carrying singularity at their cores.Although the authors claim that their argument goes through for the non-Abelian case as well,there are important di?erences between the two cases.In the Abelian theory with monopole background?elds,the Chern-Simons term, after a gauge transformationω( x),yields an extra factor exp{i a n aω( x a)/2}where n a and x a are magnetic charge and position of the a-th monopole.Hence Fradkin and Schaposnik argue that the integration ofω( x)eliminates contributions of monopoles completely.The factorω( x a)results from the singularity of the Abelian monopole con?guration at x= x a. In the non-Abelian theory,however,monopole con?gurations are regular everywhere and the Chern-Simons term produces a factor exp{inf(∞)}(see(2.14)).Only the value of the gauge potential f(r)at r=∞is important.We have observed that f(∞)takes values in a limited range and no cancellation is expected.

(iv)Diamantini,Sodano and Trugenberger’s work

Diamantini et al.have formulated the compact Maxwell-Chern-Simons theory in the dual theory on the lattice,in which the dual?eld variable fμ=?μνρFνρ/2becomes fundamental.[7]In this compact U(1)theory monopoles naturally arise on the lattice.Dia-matini et al.have found a complex solution for a monopole-antimonopole pair.The dual ?eld fμhas a string singularity between the two poles.In the presence of the Chern-Simons term the string,carrying a magnetic?ux,becomes electrically charged.As Henneaux and Teitelboim pointed out,[13]such a string becomes observable and has a?nite energy density so that the monopole-antimonopole pair is linearly con?ned.

Although the con?nement picture of monopoles in ref.[7]is consistent with refs. [3,5,6],due caution is necessary in extending the picture to non-Abelian theory.The dual

theory in[7]is entirely Abelian,consisting of only gauge?elds.In the Georgi-Glashow-Chern-Simons model,the solution is regular everywhere.There is no place where an observable string singularity enters.It is not clear if the role of the Higgs?eld in the continuum non-Abelian theory can be completely mimiced by the lattice structure in the dual theory.

(v)Jackiw and Pi’s work

Jackiw and Pi[14]have argued that the addition of the Chern-Simons interaction destroys the topological excitations such as monopoles.They parameterizeφ1=ρcosθandφ2=ρsinθ.The boundary conditions(2.11)areρ(0)=1,θ(0)=1andρ(∞)=0. Under a gauge transformation(2.12),θ→θ?f and A→A?f′.θ′?A is gauge invariant.

The authors employ one of the classical equations of motion,Eq.(2.6),to reduce the action,which is subsequently minimized.However,as remarked near the end of Section2, Eq.(2.6)is incompatible with the boundary condition ensuring the?niteness of the action. Consequently all monopole con?gurations have an in?nite action in their formalism.

Indeed,Eq.(2.6)readsρ2(θ′?A?1

iκ.Upon utilizing this equation,the e?ective action(2.10)is reduced to

S JP=4π

4 ρ2+1

2r2

(ρ2?1)2+

(h2?v2)2 (5.1)

The presence of the1/ρ2,h2ρ2,and r2(h2?v2)2terms makes it impossible to have a con?guration of a?nite action.

In quantum theory a gauge is?xed.In the radial gauge A=0,for instance,one of the classical equation,Eq.(2.6),which is derived by varying A,does not follow.Hence the relationρ2(θ′?A?1

iκshould not be used to simplify the action.

(vi)More subtleties

There remains subtle delicacy in de?ning the quantum theory of the Chern-Simons theory.We have started with the Faddeev-Popov formula(2.15).If one picks a radial gauge F(A)=xμA aμ=0,then there is a complex monopole solution which extremizes

the restricted action,namely the action in the given gauge slice.What happens,say,in the temporal gauge?The answer is not clear.As explained in Section4,solutions look di?erent,depending on the gauge chosen.Two operations in the path integral,?xing a gauge and?nding con?gurations which extremize the action,do not commute with each other in the Chern-Simons theory.In the above papers by Pisarski and by Jackiw and Pi the action is extremized with respect to arbitrary variations of gauge?elds,and then a gauge is picked.This procedure yields one more equation to be solved,and in general this equation turns out incompatible with the?niteness of the action for monopoles.

In QED or QCD the order of the two operations does not matter.For instance, instanton solutions in QCD can be found in any gauge.The apparent noncommutability of the two operations in the Chern-Simons theory is traced back to the gauge non-invariance of the Chern-Simons term particularly in the monopole background.In the original derivation of the Faddeev-Popov formula(2.15)the gauge invariance of the action was assumed.The formula(2.15)needs to be scrutinized in the Chern-Simons theory.There is also ambiguity in the de?nition of the Chern-Simons term as pointed out by Deser et al.[18]Further investigation is necessary.

6.Chern-Simons-Higgs theory

In the absence of Yang-Mills term one obtains the Chern-Simons-Higgs theory.Al-though the pure Chern-Simons theory de?nes a topological?eld theory,one obtains a dynamical theory after matter couplings are introduced.The equations of motion for the gauge?elds are?rst order in derivatives which makes the theory easier to handle at least at the classical level.The issue whether this theory makes sense or not at the quantum level was addressed by Tan et.al.in ref.[15].Using the two loop e?ective potential in the dimensional regularization scheme,it has been shown that one has to start with the Yang-Mills term,but the limit of vanishing Yang-Mills term exists after renormalization. In this section we are mainly interested in the Chern-Simons-Higgs theory at the tree level.

Monopoles in CS-Higgs theory was discussed in references[16]and[17].Lee showed that instantons of an in?nite action induce an e?ective vertex which break the global