Conformal Edge Currents in Chern-Simons Theories

a r X i v :h e p -t h /9110072v 1 30 O c t 1991SU-4228-487

INFN-NA-IV-91/12

UAHEP 917

October 1991

CONFORMAL EDGE CURRENTS IN CHERN-SIMONS THEORIES A.P.Balachandran,1G.Bimonte 1,2K.S.Gupta,1A.Stern 2,31)Department of Physics,Syracuse University,Syracuse,NY 13244-1130,USA .2)Dipartimento di Scienze Fisiche dell’Universit`a di Napoli,Mostra d’Oltremare pad.19,80125Napoli,Italy .3)Department of Physics,University of Alabama,Tuscaloosa,Al 35487,USA.ABSTRACT

We develop elementary canonical methods for the quantization of abelian and non-abelian Chern-Simons actions using well known ideas in gauge theories and quantum gravity.Our approach does not involve choice of gauge or clever manipulations of func-tional integrals.When the spatial slice is a disc,it yields Witten’s edge states carrying a representation of the Kac-Moody algebra.The canonical expression for the generators of di?eomorphisms on the boundary of the disc are also found,and it is established that they are the Chern-Simons version of the Sugawara construction.This paper is a prelude to our future publications on edge states,sources,vertex operators,and their spin and statistics in 3d and 4d topological ?eld theories.

1.INTRODUCTION

The Chern-Simons or CS action describes a three-dimensional?eld theory of a connec-tion Aμ.In the absence of sources,the?eld equations require Aμto be a zero curvature ?eld and hence to be a pure gauge in simply connected spacetimes.As the dynamics is gauge invariant as well,it would appear that the CS action is an action for triviality in these spacetimes.

Such a conclusion however is not always warranted.Thus,for instance,it is of frequent interest to consider the CS action on a disc D×R1(R1accounting for time)and in this case,as?rst emphasized by Witten[1],it is possible to contemplate a quantization which eliminates degrees of freedom only in the interior of D.In such a scheme,then,gauge transformations relate equivalent?elds only in the interior of D whereas on the boundary ?D,they play a role more akin to global symmetry transformations.The residual states localized on the circular boundary?D are the CS edge states.As they are associated with gauge transformations on?D=the circle S1,it is natural to expect that the loop or the Kac-Moody group[2]of the gauge group will play a role in their description,the latter being a central extension of the former.Witten[1]in fact outlined an argument to show that the edge states form a conformal family carrying a representation of the Kac-Moody group.

Subsequent developments in the quantum theory of CS action have addressed both its formal[3,4,5,6,7]and its physical[8,9,10]aspects.As regards the former,methods have been invented and re?ned for its?xed time quantization[3,4,5]and for the treatment of its functional integral[5,6,7].They yield Witten’s results and extend them as well. An important achievement of all this research beginning in fact with Witten’s work is the reproduction of a large class of two-dimensional(2d)conformal?eld theories(CFT’s) from3d CS theories.

There have been equally interesting developments which establish the signi?cance of the CS interaction for2d condensed matter systems which go beyond phase transition phenomena described by CFT’s[2].It is now well appreciated for instance that the edge states of the Fractional Quantum Hall E?ect(FQHE)are well described by the CS theory and its variants[8,9]and that it is of fundamental importance in the theory of fractional statistics[10].Elsewhere,we will also describe its basic role in the theory of London equations of2d superconductors.

In this paper,we develop a canonical quantization of the CS action assuming for simplicity that spacetime is a solid cylinder D×R1.A notable merit of our approach is that it avoids making a gauge choice or delicate manipulations of functional integrals.It is furthermore based on ideas which are standard in?eld theories with constraints such as QCD or quantum gravity[11]and admits easy generalizations,for example,to4d gauge theories.In subsequent papers,we will extend this approach to certain gauge?eld theories (including the CS theory)with sources.We will establish that an anyon or a Laughlin quasiparticle is not just a single particle,but is in reality a conformal family(a result due to Witten[1])and derive similar results in four dimensions.Simple considerations concerning spin and statistics of these sources will also be presented using basic ideas of European schools[12]on“?elds localized in space–like cones”and generalizing them somewhat.A brief account of our work has already appeared elsewhere[13].

In Section2,we outline a canonical formalism for the U(1)CS action on D×R1and its relation to certain old ideas in gauge theories or gravity.The observables are shown to obey an algebra isomorphic to the U(1)Kac–Moody algebra on a circle[2].The classical canonical expression for the di?eomorphism(di?eo)generators on?D are also found.

In Section3,the observables are Fourier analyzed on?D.It is then discovered that the CS di?eo generators are weakly the same as those obtai ned by the Sugawara construction [2].Quantization is then carried out in a conventional way to?nd that the edge states

and their observables describe a central charge1conformal family[2].We next brie?y illustrate our techniques by quantizing a generalized version of the CS action which has proved important in the theory of FQHE[9].

The paper concludes with Section4which outlines the nonabelian version of the foregoing considerations.

2.THE CANONICAL FORMALISM

The U(1)CS action on the solid cylinder D×R1is

k

S=

δ2(x?y)for i,j=1,2,?12=??21=1(2.2)

k

(using the convention?012=1for the Levi-Civita symbol)and the constraint

?i A j(x)??j A i(x)≡F ij(x)≈0(2.3) where≈denotes weak equality in the sense of Dirac[11].All?elds are evaluated at the same time x0in these equations,and this will continue to be the case when dealing with the canonical formalism or quantum operators in the remainder of the paper.The connection A0does not occur as a coordinate of this phase space.This is because,just

as in electrodynamics,its conjugate momentum is weakly zero and ?rst class and hence eliminates A 0as an observable.

The constraint (2.3)is somewhat loosely stated.It is important to formulate it more accurately by ?rst smearing it with a suitable class of “test”functions Λ(0).Thus we write,instead of (2.3),

g (Λ(0)):=k

2π ?D Λ(0)δA ? D d Λ(0)δA .(2.5)

By de?nition,g (Λ(0))is di?erentiable in A only if the boundary term –the ?rst term –in (2.5)is zero.We do not wish to constrain the phase space by legislating δA itself to be zero on ?D to achieve this goal.This is because we have a vital interest in regarding ?uctuations of A on ?D as dynamical and hence allowing canonical transformations which change boundary values of A .We are thus led to the following condition on functions Λ(0)in T (0):

Λ(0)|?D =0.(2.6)

It is useful to illustrate the sort of troubles we will encounter if (2.6)is dropped.Consider

q (Λ)=

k

of A on?D by canonical transformations.It is a function we wish to admit in our canonical approach.Now consider its PB with g(Λ(0)):

{g(Λ0),q(Λ)}=k

?x i

δ2(x?y) (2.8)

where?ij=?ij.This expression is quite ill de?ned if

Λ(0)|?D=0.

Thus integration on y?rst gives zero for(2.8).But if we integrate on x?rst,treating derivatives of distributions by usual rules,one?nds instead,

? D dΛ0dΛ=? ?DΛ0dΛ.(2.9) Thus consistency requires the condition(2.6).

We recall that a similar situation occurs in QED.There,if E j is the electric?eld, which is the momentum conjugate to the potential A j,and j0is the charge density,the Gauss law can be written as

ˉg(ˉΛ(0))= d3xˉΛ(0)(x)[?i E i(x)?j0(x)]≈0.(2.10) Since

δˉg(ˉΛ(0))= r=∞r2d?ˉΛ(0)(x)?x iδE i? d3x?iˉΛ(0)(x)δE i(x),r=| x|,?x= x

This is di?erentiable in E i even ifˉΛ|r=∞=0and generates the gauge transformation for the gauge group element e iˉΛ.It need not to vanish on quantum states ifˉΛ|r=∞=0,unlike ˉg(ˉΛ(0))which is associated with the Gauss lawˉg(ˉΛ(0))≈0.But ifˉΛ|r=∞=0,it becomes the Gauss law on partial integration and annihilates all physical states.It follows that if (ˉΛ1?ˉΛ2)|r=∞=0,thenˉq(ˉΛ1)=ˉq(ˉΛ2)on physical states which are thus sensitive only to the boundary values of test functions.The nature of their response determines their charge.The conventional electric charge of QED isˉq(ˉ1)whereˉ1is the constant function with value1.

The constraints g(Λ(0))are?rst class since

g(Λ(0)1),g(Λ(0)2) =k

2π ?DΛ(0)1dΛ(0)2

=0forΛ(0)1,Λ(0)2∈T(0).(2.14) g(Λ(0))generates the gauge transformation A→A+dΛ(0)of A.

Next consider q(Λ)whereΛ|?D is not necessarily zero.Since

q(Λ),g(Λ(0)) =?k

2π ?DΛ(0)dΛ=0forΛ(0)∈T(0),(2.15) they are?rst class or the observables of the theory.More precisely observables are obtained after identifying q(Λ1)with q(Λ2)if(Λ1?Λ2)∈T(0).For then,

q(Λ1)?q(Λ2)=?g(Λ1?Λ2)≈0.

The functions q(Λ)generate gauge transformations A→A+dΛwhich do not necessarily vanish on?D.

It may be remarked that the expression for q(Λ)is obtained from g(Λ(0))after a partial integration and a subsequent substitution ofΛforΛ(0).It too generates gauge

transformations like g(Λ(0)),but the test function space for the two are di?erent.The pair q(Λ),g(Λ(0))thus resemble the pairˉq(ˉΛ),ˉg(ˉΛ(0))in QED.The resemblance suggests that we think of q(Λ)as akin to the generator of a global symmetry transformation.It is natural to do so for another reason as well:the Hamiltonian is a constraint for a?rst order Lagrangian such as the one we have here,and for this Hamiltonian,q(Λ)is a constant of motion.

In quantum gravity,for asymptotically?at spatial slices,it is often the practice to include a surface term in the Hamiltonian which would otherwise have been a constraint and led to trivial evolution[14].However,we know of no natural choice of such a surface term,except zero,for the CS theory.

The PB’s of q(Λ)’s are easy to compute:

{q(Λ1),q(Λ2)}=k

2π ?DΛ1dΛ2.(2.16)

Remembering that the observables are characterized by boundary values of test functions, (2.16)shows that the observables generate a U(1)Kac-Moody algebra[2]localized on?D. It is a Kac-Moody algebra for“zero momentum”or“charge”.For ifΛ|?D is a constant, it can be extended as a constant function to all of D and then q(Λ)=0.The central charges and hence the representation of(2.16)are di?erent for k>0and k<0,a fact which re?ects parity violation by the action S.

Letθ(mod2π)be the coordinate on?D andφa free massless scalar?eld moving with speed v on?D and obeying the equal time PB’s

{φ(θ),˙φ(θ′)}=δ(θ?θ′).(2.17)

Ifμi are test functions on?D and?±=?x0±v?θ,then

1v μ2(θ)?±φ(θ) =±2 μ1(θ)dμ2(θ),(2.18) the remaining PB’s being zero.Also???±φ=0.Thus the algebra of observables is

isomorphic to that generated by the left moving?+φor the right moving??φ.

The CS interaction is invariant under di?eos of D.An in?nitesimal generator of a di?eo with vector?eld V(0)is[15]

k

δ(V(0))=?

4π D A L V(0)A≈0(2.21) where L V(0)A denotes the Lie derivative of the one form A with respect to the vector?eld V(0)and is given by

(L V(0)A)i=?j A i V(0)j+A j?i V(0)j.

Next,suppose that V is a vector?eld on D which on?D is tangent to?D,

V i|?D(θ)=?(θ) ?x i

2π 1

4π D A L V A.(2.23) Simple calculations show that l(V)is di?erentiable in A even if?(θ)=0and generates the in?nitesimal di?eo of the vector?eld V.We show in the next Section that it is,in fact,related to q(Λ)’s by the Sugawara construction.

The expression(2.23)for the di?eo generators of observables seems to be new.

As?nal points of this Section,note that

{l(V),g(Λ(0))}=g(V i?iΛ(0))=g(L VΛ(0))≈0,(2.24)

{l(V),q(Λ)}=q(V i?iΛ)=q(L VΛ),(2.25)

{l(V),l(W)}=l(L V W)(2.26) where L V W denotes the Lie derivative of the vector?eld W with respect to the vector ?eld V and is given by

(L V W)i=V j?j W i?W j?j V i.

l(V)are?rst class in view of(2.24).Further,after the imposition of constraints,they are entirely characterized by?(θ),the equivalence class of l(V)with the same?(θ)de?ning an observable.

3.QUANTIZATION

Our strategy for quantization relies on the observation that if

Λ|?D(θ)=e iNθ,

then the PB’s(2.16)become those of creation and annihilation operators.These latter can be identi?ed with the similar operators of the chiral?elds?±φ.

Thus letΛN be any function on D with boundary value e iNθ:

ΛN|?D(θ)=e iNθ,N∈Z.(3.1)

TheseΛN’s exist.All q(ΛN)with the sameΛN|?D are weakly equal and de?ne the same observable.Let q(ΛN) be this equivalence class and q N any member thereof.[q N can also be regarded as the equivalence class itself.]Their PB’s follow from(2.16):

{q N,q M}=?iNkδN+M,0.(3.2)

The q N’s are the CS constructions of the Fourier modes of a massless chiral scalar?eld on S1.

The CS construction of the di?eo generators l N on?D(the classical analogues of the Virasoro generators)are similar.Thus let

be the equivalence class of l(V N)de?ned by the constraint

V i N|?D=e iNθ ?x i

2k M q M q N?M(3.6) which is the classical version of the Sugawara construction[2].

For convenience,let us introduce polar coordinates r,θon D(with r=R on?D) and write the?elds and test functions as functions of polar coordinates.It is then clear that

l N≡l(V N)=k

2π D V l N(r,θ)A l(r,θ)dA(r,θ)(3.7)

where A=A r dr+Aθdθ.

Let us next make the choice

e iMθλ(r),λ(0)=0,λ(R)=1(3.8)

forΛM.Then

q M=q(e iMθλ(r)).(3.9)

Integrating(3.9)by parts,we get

q M=

k

2k M q M q N?M=+k

2π D drdθe iNθλ(r)Aθ(R,θ)F rθ(r,θ)

+

k

2k M q M q N?M≈k2π D drdθe iNθλ(r)Aθ(R,θ)F rθ(r,θ).(3.12)

Now in view of(3.3)and(3.8),it is clear that

V l N(r,θ)A l(r,θ)?e iNθλ(r)Aθ(R,θ)=0on?D.(3.13)

Therefore

l N≈

1

which proves(3.6).

We can now proceed to quantum?eld theory.Let G(Λ(0)),Q(ΛN),Q N and L N denote the quantum operators for g(Λ(0)),q(ΛN),q N and l N.We then impose the constraint

G(Λ(0))|· =0(3.14)

on all quantum states.It is an expression of their gauge invariance.Because of this equation,Q(ΛN)and Q(Λ′N)have the same action on the states ifΛN andΛ′N have the same boundary values.We can hence write

Q N|· =Q(ΛN)|· .(3.15)

Here,in view of(3.2),the commutator brackets of Q N are

[Q N,Q M]=NkδN+M,0.(3.16)

Thus if k>0(k<0),Q N for N>0(N<0)are annihilation operators(upto a normalization)and Q?N creation operators.The“vacuum”|0>can therefore be de?ned by

Q N|0>=0if Nk>0.(3.17) The excitations are got by applying Q?N to the vacuum.

The quantum Virasoro generators are the normal ordered forms of their classical ex-pression[2]:

1

L N=

(N3?N)δN+M,0,c=1.(3.19)

12

When the spatial slice is a disc,the observables are all given by Q N and our quantiza-tion is complete.When it is not simply connected,however,there are further observables

associated with the holonomies of the connection A and they a?ect quantization.We will not examine quantization for nonsimply connected spatial slices here.

The CS interaction does not?x the speed v of the scalar?eld in(2.18)and so its Hamiltonian,a point previously emphasized by Frohlich and Kerler[8]and Frohlich and Zee[9].This is but reasonable.For if we could?x v,the Hamiltonian H forφcould naturally be taken to be the one for a free massless chiral scalar?eld moving with speed v.It could then be used to evolve the CS observables using the correspondence of this ?eld and the former.But we have seen that no natural nonzero Hamiltonian exists for the CS system.It is thus satisfying that we can not?x v and hence a nonzero H.

In the context of Fractional Quantum Hall E?ect,the following generalization of the CS action has become of interest[9]:

k

S′=

K?1IJδ2(x?y),x0=y0(3.21)

k

and the?rst class constraints

k

g(I)(Λ(0))=

2π D dΛA(I)(3.23)

after identifying q(I)(Λ)with q(I)(Λ′)if(Λ?Λ′)|?D=0.The PB’s of q(I)’s are

q(I)(Λ(I)1),q(J)(Λ(J)2) =k

.Letγbe a faithful represen-tation of G

T r TαTβ=δαβ.Let Aμde?ne an antihermitean connection for G with values inγ.We de?ne the real?eld Aαμby Aμ=iAαμTα.With these conventions,the Chern-Simons action for Aμon D×R1is

S=?k

3

A3 ,A=Aμdxμ(4.1)

where the constant k can assume only quantized values for well known reasons.If G= SU(N)andγthe Lie algebra of its de?ning representation,then k∈Z.

Much as for the Abelian problem,the phase space for(4.1)is described by the PB’s Aαi(x),Aβj(y) =δαβ?ij2π

2π D T r Λ(0)(dA+A2) =?k

2π ?D T rΛ01dΛ2(0)=g([Λ1(0),Λ2(0)])(4.4) so that they are?rst class constraints.

Next de?ne

q(Λ)=

k

The PB’s of q(Λ)’s are

{q(Λ1),q(Λ2)}=?q([Λ1,Λ2])?

k

2π D V(0)i T rA i F,V(0)i|?D=0,(4.8) while those generators which also perform di?eos of?D are

l(V)=k

2 D d(V i T rA i A)

=

k

?θ |?D.The PB’s involving l(V)are patterned after(2.24–2.26):

l(V),g(Λ(0)) =g(V i?iΛ(0))=g(L VΛ(0))≈0,(4.10)

{l(V),q(Λ)}=q(V i?iΛ)=q(L VΛ),(4.11)

{l(V),l(W)}=l(L V W).(4.12) We can now conclude that l(V)are?rst class and de?ne observables,all V with the same ?(θ)leading to the same observable.

LetΛαN be any test function with the featureΛαN|?D=e iNθTαand let V i N be de?ned following Section3.As in that Section,let us call the set of?rst class variables weakly equal to q(iΛαN Tα)and l(V N)by q(iΛαN Tα) and l(V N) .[Here there is no sum overαin iΛαN Tα].Let qαN and l N be any member each from these sets.Their PB’s are

qαN,qβM ≈fαβγqγN+M?iNkδN+M,0δαβ,(4.13)

{l N,qαM}≈iMqαN+M,(4.14)

{l N ,l M }≈?i (N ?M )l N +M ,

(4.15)

f αβγbein

g de?ned by [T α,T β]=if αβγT γ.Furthermore,as in Section 3,

l N ≈12k +c V M,α:Q αM Q αN ?M :,(4.18)

(c V being the quadratic Casimir operator in the adjoint representation).The central charge c now is not of course 1,but rather,

c =2k dim G 2k +c V ,dim G .(4.19)

These results about the Kac-Moody and Virasoro algebras are explained in ref.2.Acknowledgement

We have been supported during the course of this work as follows:1)A.P.B.,G.B.,K.S.G.by the Department of Energy,USA,under contract number DE-FG-02-85ER -40231,and A.S.by the Department of Energy,USA under contract number DE-FG05-84ER40141;2)A.P.B.and A.S.by INFN,Italy [at Dipartimento di Scienze

Fisiche,Universit`a Di Napoli];3)G.B.by the Dipartimento di Scienze Fisiche,Univer-sit`a di Napoli.The authors wish to thank the group in Naples and Giuseppe Marmo,in particular,for their hospitality while this work was in progress.They also wish to thank Paulo Teotonio for very helpful comments and especially for showing us how to write equations(2.21),(2.23)and(4.9)in their?nal nice forms involving Lie derivatives.

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