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A. P. Balachandran,1 G. Bimonte 1,2

arXiv:hep-th/9110072v1 30 Oct 1991SU-4228-487INFN-NA-IV-91/12UAHEP 917October 1991CONFORMAL EDGE CURRENTSINCHERN-SIMONS THEORIESA. P. Balachandran,1 G. Bimonte 1,2K.

S. Gupta,1 A. 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, 80125 Napoli, Italy.3)Department of Physics, University of Alabama,Tuscaloosa, Al 35487, USA.ABSTRACTWe develop elementary canonical methods for the quantization of abelian and non-abelian Chern-Simons actions using well known ideas in gauge theories and quantumgravity.

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 carryinga representation of the Kac-Moody algebra.

The canonical expression for the generatorsof diffeomorphisms on the boundary of the disc are also found, and it is established thatthey are the Chern-Simons version of the Sugawara construction. This paper is a preludeto our future publications on edge states, sources, vertex operators, and their spin andstatistics in 3d and 4d topological field theories.

1. INTRODUCTIONThe Chern-Simons or CS action describes a three-dimensional field theory of a connec-tion Aµ.

In the absence of sources, the field equations require Aµ to be a zero curvaturefield and hence to be a pure gauge in simply connected spacetimes. As the dynamics isgauge invariant as well, it would appear that the CS action is an action for triviality inthese spacetimes.Such a conclusion however is not always warranted.

Thus, for instance, it is of frequentinterest to consider the CS action on a disc D ×R1 (R1 accounting for time) and in thiscase, as first emphasized by Witten [1], it is possible to contemplate a quantization whicheliminates degrees of freedom only in the interior of D. In such a scheme, then, gaugetransformations relate equivalent fields only in the interior of D whereas on the boundary∂D, they play a role more akin to global symmetry transformations. The residual stateslocalized on the circular boundary ∂D are the CS edge states.

As they are associated withgauge transformations on ∂D = the circle S1, it is natural to expect that the loop or theKac-Moody group [2] of the gauge group will play a role in their description, the latterbeing a central extension of the former. Witten [1] in fact outlined an argument to showthat the edge states form a conformal family carrying a representation of the Kac-Moodygroup.Subsequent developments in the quantum theory of CS action have addressed both itsformal [3, 4, 5, 6, 7] and its physical [8, 9, 10] aspects.

As regards the former, methods havebeen invented and refined for its fixed time quantization [3, 4, 5] and for the treatmentof 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 isthe reproduction of a large class of two-dimensional (2d) conformal field theories (CFT’s)from 3d CS theories.1

There have been equally interesting developments which establish the significance ofthe CS interaction for 2d condensed matter systems which go beyond phase transitionphenomena described by CFT’s [2]. It is now well appreciated for instance that the edgestates of the Fractional Quantum Hall Effect (FQHE) are well described by the CS theoryand its variants [8, 9] and that it is of fundamental importance in the theory of fractionalstatistics [10].

Elsewhere, we will also describe its basic role in the theory of Londonequations of 2d superconductors.In this paper, we develop a canonical quantization of the CS action assuming forsimplicity that spacetime is a solid cylinder D × R1. A notable merit of our approach isthat it avoids making a gauge choice or delicate manipulations of functional integrals.

Itis furthermore based on ideas which are standard in field theories with constraints such asQCD or quantum gravity [11] and admits easy generalizations, for example, to 4d gaugetheories. In subsequent papers, we will extend this approach to certain gauge field theories(including the CS theory) with sources.

We will establish that an anyon or a Laughlinquasiparticle is not just a single particle, but is in reality a conformal family (a resultdue to Witten [1]) and derive similar results in four dimensions. Simple considerationsconcerning spin and statistics of these sources will also be presented using basic ideasof European schools [12] on “fields localized in space–like cones” and generalizing themsomewhat.

A brief account of our work has already appeared elsewhere [13].In Section 2, we outline a canonical formalism for the U(1) CS action on D × R1 andits relation to certain old ideas in gauge theories or gravity. The observables are shown toobey an algebra isomorphic to the U(1) Kac–Moody algebra on a circle [2].

The classicalcanonical expression for the diffeomorphism (diffeo) generators on ∂D are also found.In Section 3, the observables are Fourier analyzed on ∂D. It is then discovered that theCS diffeo generators are weakly the same as those obtai ned by the Sugawara construction[2].

Quantization is then carried out in a conventional way to find that the edge states2

and their observables describe a central charge 1 conformal family [2]. We next brieflyillustrate our techniques by quantizing a generalized version of the CS action which hasproved important in the theory of FQHE [9].The paper concludes with Section 4 which outlines the nonabelian version of theforegoing considerations.2.

THE CANONICAL FORMALISMThe U(1) CS action on the solid cylinder D × R1 isS = k4πZD×R1 AdA,A = Aµdxµ,AdA ≡A ∧dA(2.1)where Aµ is a real field.The action S is invariant under diffeos of the solid cylinder and does not permit anatural choice of a time function. As time is all the same indispensable in the canonicalapproach, we arbitrarily choose a time function denoted henceforth by x0.

Any constantx0 slice of the solid cylinder is then the disc D with coordinates x1, x2.It is well known that the phase space of the action S is described by the equal timePoisson brackets (PB’s){Ai(x), Aj(y)} = ǫij2πk δ2(x −y) for i, j = 1, 2,ǫ12 = −ǫ21 = 1(2.2)(using the convention ǫ012 = 1 for the Levi-Civita symbol) and the constraint∂iAj(x) −∂jAi(x) ≡Fij(x) ≈0(2.3)where ≈denotes weak equality in the sense of Dirac [11]. All fields are evaluated atthe same time x0 in these equations, and this will continue to be the case when dealingwith the canonical formalism or quantum operators in the remainder of the paper.

Theconnection A0 does not occur as a coordinate of this phase space. This is because, just3

as in electrodynamics, its conjugate momentum is weakly zero and first class and henceeliminates A0 as an observable.The constraint (2.3) is somewhat loosely stated. It is important to formulate it moreaccurately by first smearing it with a suitable class of “test” functions Λ(0).

Thus wewrite, instead of (2.3),g(Λ(0)) : = k2πZD Λ(0)(x)dA(x) ≈0 . (2.4)It remains to state the space T (0) of test functions Λ(0).For this purpose, we recallthat a functional on phase space can be re lied on to generate well defined canonicaltransformations only if it is differentiable.

The meaning and implications of this remarkcan be illustrated here by varying g(Λ(0)) with respect to Aµ:δg(Λ(0)) = k2πZ∂D Λ(0)δA −ZD dΛ(0)δA. (2.5)By definition, g(Λ(0)) is differentiable in A only if the boundary term – the first term –in (2.5) is zero.

We do not wish to constrain the phase space by legislating δA itself tobe zero on ∂D to achieve this goal. This is because we have a vital interest in regardingfluctuations of A on ∂D as dynamical and hence allowing canonical transformations whichchange boundary values of A.

We are thus led to the following condition on functionsΛ(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.Considerq(Λ) = k2πZD dΛA(2.7)It is perfectly differentiable in A even if the function Λ is nonzero on ∂D.

It createsfluctuationsδA |∂D= dΛ |∂D4

of A on ∂D by canonical transformations.It is a function we wish to admit in ourcanonical approach. Now consider its PB with g(Λ(0)):{g(Λ0), q(Λ)} = k2πZd2xd2yΛ(0)(x)ǫij [∂jΛ(y)]" ∂∂xi δ2(x −y)#(2.8)where ǫij = ǫij.

This expression is quite ill defined ifΛ(0) |∂D̸= 0.Thus integration on y first gives zero for (2.8). But if we integrate on x first, treatingderivatives of distributions by usual rules, one finds instead,−ZD dΛ0dΛ = −Z∂D Λ0dΛ .

(2.9)Thus consistency requires the condition (2.6).We recall that a similar situation occurs in QED. There, if Ej is the electric field,which is the momentum conjugate to the potential Aj, and j0 is the charge density, theGauss law can be written as¯g(¯Λ(0)) =Zd3x¯Λ(0)(x) [∂iEi(x) −j0(x)] ≈0 .

(2.10)Sinceδ¯g(¯Λ(0)) =Zr=∞r2dΩ¯Λ(0)(x)ˆxiδEi −Zd3x∂i ¯Λ(0)(x)δEi(x), r =| ⃗x |, ˆx = ⃗xr(2.11)for the variation δEi of Ei, differentiability requires¯Λ(0)(x) |r=∞= 0. (2.12)[dΩin (2.11) is the usual volume form of the two sphere ].

The charge, or equivalentlythe generator of the global U(1) transformations, incidentally is the analogue of q(Λ). Itis got by partial integration on the first term.

Thus let¯q(¯Λ) = −Zd3x∂i ¯Λ(x)Ei(x) −Zd3x¯Λ(x)j0(x) . (2.13)5

This is differentiable in Ei even if ¯Λ |r=∞̸= 0 and generates the gauge transformation forthe gauge group element ei¯Λ. It need not to vanish on quantum states if ¯Λ |r=∞̸= 0, unlike¯g(¯Λ(0)) which is associated with the Gauss law ¯g(¯Λ(0)) ≈0.

But if ¯Λ |r=∞= 0, it becomesthe Gauss law on partial integration and annihilates all physical states. It follows that if(¯Λ1 −¯Λ2) |r=∞= 0, then ¯q(¯Λ1) = ¯q(¯Λ2) on physical states which are thus sensitive onlyto the boundary values of test functions.

The nature of their response determines theircharge. The conventional electric charge of QED is ¯q(¯1) where ¯1 is the constant functionwith value 1.The constraints g(Λ(0)) are first class sinceng(Λ(0)1 ), g(Λ(0)2 )o=k2πZD dΛ(0)1 dΛ(0)2=k2πZ∂D Λ(0)1 dΛ(0)2=0for Λ(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) where Λ |∂D is not necessarily zero. Sincenq(Λ), g(Λ(0))o=−k2πZD dΛdΛ(0)=k2πZ∂D Λ(0)dΛ = 0 for Λ(0) ∈T (0),(2.15)they are first class or the observables of the theory.

More precisely observables are obtainedafter 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+ dΛ which do not necessarilyvanish on ∂D.It may be remarked that the expression for q(Λ) is obtained from g(Λ(0)) after apartial integration and a subsequent substitution of Λ for Λ(0).

It too generates gauge6

transformations like g(Λ(0)), but the test function space for the two are different. Thepair q(Λ), g(Λ(0)) thus resemble the pair ¯q(¯Λ), ¯g(¯Λ(0)) in QED.

The resemblance suggeststhat we think of q(Λ) as akin to the generator of a global symmetry transformation. It isnatural to do so for another reason as well: the Hamiltonian is a constraint for a first orderLagrangian such as the one we have here, and for this Hamiltonian, q(Λ) is a constant ofmotion.In quantum gravity, for asymptotically flat spatial slices, it is often the practice toinclude a surface term in the Hamiltonian which would otherwise have been a constraintand led to trivial evolution [14].

However, we know of no natural choice of such a surfaceterm, except zero, for the CS theory.The PB’s of q(Λ)’s are easy to compute:{q(Λ1), q(Λ2)} = k2πZD dΛ1dΛ2 = k2πZ∂D Λ1dΛ2 . (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 if Λ |∂D is a constant,it can be extended as a constant function to all of D and then q(Λ) = 0. The centralcharges and hence the representation of (2.16) are different for k > 0 and k < 0, a factwhich reflects parity violation by the action S.Let θ (mod 2π) be the coordinate on ∂D and φ a free massless scalar field movingwith speed v on ∂D and obeying the equal time PB’s{φ(θ), ˙φ(θ′)} = δ(θ −θ′) .

(2.17)If µi are test functions on ∂D and ∂± = ∂x0 ± v∂θ, then1vZµ1(θ)∂±φ(θ), 1vZµ2(θ)∂±φ(θ)= ±2Zµ1(θ)dµ2(θ),(2.18)the remaining PB’s being zero. Also ∂∓∂±φ = 0.

Thus the algebra of observables isisomorphic to that generated by the left moving ∂+φ or the right moving ∂−φ.7

The CS interaction is invariant under diffeos of D. An infinitesimal generator of adiffeo with vector field V (0) is [15]δ(V (0)) = −k2πZD V (0)iAidA. (2.19)The differentiability of δ(V (0)) imposes the constraintV (0) |∂D= 0 .

(2.20)Hence, in view of (2.4) as well, we have the resultδ(V (0)) = −k4πZD ALV (0)A ≈0(2.21)where LV (0)A denotes the Lie derivative of the one form A with respect to the vector fieldV (0) and is given by(LV (0)A)i = ∂jAiV (0)j + Aj∂iV (0)j.Next, suppose that V is a vector field on D which on ∂D is tangent to ∂D,V i |∂D (θ) = ǫ(θ) ∂xi∂θ!|∂D,(2.22)ǫ being any function on ∂D and xi |∂D the restriction of xi to ∂D. V thus generates adiffeo mapping ∂D to ∂D.

Consider nextl(V )=k2π 12ZD d(V iAiA) −ZD V iAidA=−k4πZD ALV A. (2.23)Simple calculations show that l(V ) is differentiable in A even if ǫ(θ) ̸= 0 and generatesthe infinitesimal diffeo of the vector field V .

We show in the next Section that it is, infact, related to q(Λ)’s by the Sugawara construction.The expression (2.23) for the diffeo generators of observables seems to be new.As final points of this Section, note that{l(V ), g(Λ(0))} = g(V i∂iΛ(0)) = g(LV Λ(0)) ≈0 ,(2.24)8

{l(V ), q(Λ)} = q(V i∂iΛ) = q(LV Λ),(2.25){l(V ), l(W)} = l(LV W)(2.26)where LV W denotes the Lie derivative of the vector field W with respect to the vectorfield V and is given by(LV W)i = V j∂jW i −W j∂jV i.l(V ) are first class in view of (2.24). Further, after the imposition of constraints, they areentirely characterized by ǫ(θ), the equivalence class of l(V ) with the same ǫ(θ) definingan observable.3.

QUANTIZATIONOur strategy for quantization relies on the observation that ifΛ |∂D (θ) = eiNθ ,then the PB’s (2.16) become those of creation and annihilation operators. These lattercan be identified with the similar operators of the chiral fields ∂±φ.Thus let ΛN be any function on D with boundary value eiNθ:ΛN |∂D (θ) = eiNθ, N ∈Z .

(3.1)These ΛN’s exist. All q(ΛN) with the same ΛN |∂D are weakly equal and define the sameobservable.

Let ⟨q(ΛN)⟩be this equivalence class and qN any member thereof. [qN canalso be regarded as the equivalence class itself.] Their PB’s follow from (2.16):{qN, qM} = −iNkδN+M,0 .

(3.2)The qN’s are the CS constructions of the Fourier modes of a massless chiral scalar fieldon S1.9

The CS construction of the diffeo generators lN on ∂D (the classical analogues of theVirasoro generators) are similar. Thus let< l(VN) >be the equivalence class of l(VN) defined by the constraintV iN |∂D= eiNθ ∂xi∂θ!|∂D,N ∈Z,(3.3)(x1, x2) |∂D (θ) being chosen to be R(cos θ, sin θ) where R is the radius of D. Let lN beany member of< l(VN) > .It can be verified that{lN, qM} = iMqN+M ,(3.4){lN, lM} = −i(N −M) lN+M .

(3.5)These PB’s are independent of the choice of the representatives from their respectiveequivalence classes. Equations (3.2), (3.4) and (3.5) define the semidirect product of theKac-Moody algebra and the Witt algebra (Virasoro algebra without the central term) inits classical version.We next show thatlN ≈12kXMqM qN−M(3.6)which is the classical version of the Sugawara construction [2].For convenience, let us introduce polar coordinates r, θ on D ( with r = R on ∂D )and write the fields and test functions as functions of polar coordinates.

It is then clearthatlN ≡l(VN) = k4πZ∂D dθeiNθA2θ(R, θ) −k2πZD V lN(r, θ)Al(r, θ)dA(r, θ)(3.7)where A = Ardr + Aθdθ.10

Let us next make the choiceeiMθλ(r), λ(0) = 0 ,λ(R) = 1(3.8)for ΛM. ThenqM = q(eiMθλ(r)).

(3.9)Integrating (3.9) by parts, we getqM = k2πZ∂D dθeiMθAθ(R, θ) −ZD drdθλ(r)eiMθFrθ(r, θ)(3.10)where Frθ is defined by dA = Frθdr ∧dθ. Therefore12kXMqMqN−M =+k4πZ∂D dθeiNθA2θ(R, θ)−k2πZD drdθeiNθλ(r)Aθ(R, θ)Frθ(r, θ)+k4πZD drdθdr′λ(r)λ(r′)eiNθFrθ(r, θ)Frθ(r′, θ)(3.11)where the completeness relationXNeiN(θ−θ′) = 2πδ(θ −θ′)has been used.The test functions for the Gauss law in the last term in (3.11) involves Frθ itself.

Wetherefore interpret it to be zero and get12kXMqMqN−M ≈k4πZ∂D eiNθA2θ(R, θ)dθ −k2πZD drdθeiNθλ(r)Aθ(R, θ)Frθ(r, θ). (3.12)Now in view of (3.3) and (3.8), it is clear thatV lN(r, θ)Al(r, θ) −eiNθλ(r)Aθ(R, θ) = 0on ∂D.

(3.13)ThereforelN ≈12kXMqMqN−M11

which proves (3.6).We can now proceed to quantum field theory. Let G(Λ(0)), Q(ΛN), QN and LN denotethe quantum operators for g(Λ(0)), q(ΛN), qN and lN.

We then impose the constraintG(Λ(0))|·⟩= 0(3.14)on all quantum states. It is an expression of their gauge invariance.Because of thisequation, Q(ΛN) and Q(Λ′N) have the same action on the states if ΛN and Λ′N have thesame boundary values.

We can hence writeQN|·⟩= Q(ΛN)|·⟩. (3.15)Here, in view of (3.2), the commutator brackets of QN are[QN, QM] = NkδN+M,0 .

(3.16)Thus if k > 0 (k < 0), QN for N > 0 (N < 0) are annihilation operators ( upto anormalization ) and Q−N creation operators. The “vacuum” |0 > can therefore be definedbyQN | 0 >= 0 if 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] :LN = 12k :XMQMQN−M :(3.18)They generate the Virasoro algebra for central charge c = 1 :[LN, LM] = (N −M)LN+M + c12(N3 −N)δN+M,0 , c = 1 . (3.19)When the spatial slice is a disc, the observables are all given by QN and our quantiza-tion is complete.

When it is not simply connected, however, there are further observables12

associated with the holonomies of the connection A and they affect quantization. We willnot examine quantization for nonsimply connected spatial slices here.The CS interaction does not fix the speed v of the scalar field in (2.18) and so itsHamiltonian, a point previously emphasized by Frohlich and Kerler [8] and Frohlich andZee [9].

This is but reasonable. For if we could fix v, the Hamiltonian H for φ couldnaturally be taken to be the one for a free massless chiral scalar field moving with speedv.

It could then be used to evolve the CS observables using the correspondence of thisfield and the former. But we have seen that no natural nonzero Hamiltonian exists forthe CS system.

It is thus satisfying that we can not fix v and hence a nonzero H.In the context of Fractional Quantum Hall Effect, the following generalization of theCS action has become of interest [9]:S′ = k4πKIJZD×R1 A(I)dA(J). (3.20)Here the sum on I, J is from 1 to F, A(I) is associated with the current j(I) in the IthLandau level and K is a certain invertible symmetric real F × F matrix .

By way offurther illustration of our approach to quantization, we now outline the quantization of(3.20) on D × R1.The phase space of (3.20) is described by the PB’snA(I)i (x), A(J)j (y)o= ǫij2πk K−1IJ δ2(x −y),x0 = y0(3.21)and the first class constraintsg(I)(Λ(0)) = k2πZD Λ(0)dA(I) ≈0 ,Λ(0) ∈T (0) . (3.22)with zero PB’s.The observables are obtained from the first class variablesq(I)(Λ) = k2πZD dΛA(I)(3.23)13

after identifying q(I)(Λ) with q(I)(Λ′) if (Λ −Λ′) |∂D= 0. The PB’s of q(I)’s arenq(I)(Λ(I)1 ), q(J)(Λ(J)2 )o= k2πK−1IJZ∂D Λ(I)1 dΛ(J)2 .

(3.24)Choose a Λ(I)N by the requirement Λ(I)N |∂D (θ) = eiNθ and let q(I)N be any member ofthe equivalence class < q(I)(Λ(I)N ) > characterized by such Λ(I)N . Thennq(I)N , q(J)Mo= −iK−1IJ NkδN+M,0 .

(3.25)As K−1IJ is real symmetric, it can be diagonalized by a real orthogonal transformationM and has real eigenvalues λρ (ρ = 1, 2, ..., F). As K−1IJ is invertible, λρ ̸= 0.

SettingqN(ρ) = MρIq(I)N(3.26)we have{qN(ρ), qM(σ)} = −iλρNkδρσδN+M,0 . (3.27)(3.27) is readily quantized.

If QN(ρ) is the quantum operator for qN(ρ),[QN(ρ), QM(σ)] = λρNkδρσδN+M,0 . (3.28)(3.28) describes F harmonic oscillators or edge currents.

Their chirality, or the chiralityof the corresponding massless scalar fields, is governed by the sign of λρ.The classical diffeo generators for the independent oscillators qN(ρ) and their quantumversions can be written down using the foregoing discussion. The latter form F commutingVirasoro algebras, all for central charge 1.4.

THE NONABELIAN CHERN-SIMONS ACTIONLet G be a compact simple group with Lie algebra G. Let γ be a faithful represen-tation of G. Choose a hermitian basis {Tα} for γ (more precisely iγ) with normalization14

Tr TαTβ = δαβ. Let Aµ define an antihermitean connection for G with values in γ. Wedefine the real field Aαµ by Aµ = iAαµTα.

With these conventions, the Chern-Simons actionfor Aµ on D × R1 isS = −k4πZD×R1 TrAdA + 23A3,A = Aµdxµ(4.1)where the constant k can assume only quantized values for well known reasons. If G =SU(N) and γ the Lie algebra of its defining representation, then k ∈Z.Much as for the Abelian problem, the phase space for (4.1) is described by the PB’snAαi (x), Aβj (y)o= δαβ ǫij2πk δ2(x −y),x0 = y0(4.2)and the Gauss lawg(Λ(0)) = −k2πZD TrnΛ(0)(dA + A2)o= −k2πZD Tr (Λ(0)F) ≈0(4.3)where F = Fijdxidxj is the curvature of A, Λ(0) = iΛ(0)αTα and Λ(0)α ∈T (0).

This testfunction space for Λ(0) ensures that g(Λ(0)) is differentiable in Aαi . The PB’s between g’sareng(Λ1(0)), g(Λ2(0))o= g([Λ1(0), Λ2(0)]) −k2πZ∂D Tr Λ01dΛ2(0) = g([Λ1(0), Λ2(0)])(4.4)so that they are first class constraints.Next defineq(Λ) = k2πZD Tr (−dΛA + ΛA2),Λ = iΛαTα .

(4.5)It is differentiable in Aαi even if Λ|∂D ̸= 0. But if Λ|∂D is zero, it is equal to the Gauss lawg(Λ).

Further, q(Λ) is first class for any choice of Λ sincenq(Λ), g(Λ(0))o= −g([Λ, Λ(0)]) ≈0 . (4.6)Thus (with Λ|∂D free), q(Λ)’s define observables, the latter being the same if their testfunctions are equal on ∂D.15

The PB’s of q(Λ)’s are{q(Λ1), q(Λ2)} = −q([Λ1, Λ2]) −k2πZ∂D Tr (Λ1dΛ2)(4.7)which can be recognized as a Kac-Moody algebra for observables.The diffeo generators can also be constructed following Section 3. The generators ofdiffeos which keep ∂D fixed and vanish weakly areδ(V (0)) = k2πZD V (0)i TrAiF,V (0)i|∂D = 0 ,(4.8)while those generators which also perform diffeos of ∂D arel(V )=k2π ZD V i TrAiF −12ZD d(V i TrAiA)=k4πZD TrALV A(4.9)where V i|∂D(θ) = ǫ(θ)∂xi∂θ|∂D.

The PB’s involving l(V ) are patterned after (2.24–2.26):nl(V ), g(Λ(0))o= g(V i∂iΛ(0)) = g(LV Λ(0)) ≈0 ,(4.10){l(V ), q(Λ)} = q(V i∂iΛ) = q(LV Λ) ,(4.11){l(V ), l(W)} = l(LV W) . (4.12)We can now conclude that l(V ) are first class and define observables, all V with the sameǫ(θ) leading to the same observable.Let ΛαN be any test function with the feature ΛαN|∂D = eiNθTα and let V iN be definedfollowing Section 3.

As in that Section, let us call the set of first class variables weaklyequal to q(iΛαNTα) and l(VN) by ⟨q(iΛαNTα)⟩and ⟨l(VN)⟩. [ Here there is no sum over αin iΛαNTα].

Let qαN and lN be any member each from these sets. Their PB’s arenqαN, qβMo≈fαβγqγN+M −iNkδN+M,0 δαβ ,(4.13){lN, qαM} ≈iMqαN+M ,(4.14)16

{lN, lM} ≈−i(N −M)lN+M,(4.15)fαβγ being defined by [Tα, Tβ] = ifαβγTγ. Furthermore, as in Section 3,lN ≈12kXM,αqαMqαN−M .

(4.16)We next go to quantum field theory. In quantum theory, the operators for g(Λ(0)), qαNand lN are denoted by G(Λ(0)), QαN, LN and all states are subjected to the Gauss lawG(Λ(0))|· >= 0 .

(4.17)As a consequence, all the weak equalitites can be regarded as strong for the quantumoperators. We are thus dealing with a Kac-Moody algebra for a certain level [2].

A suitablehighest weight representation for it can be constructed in the usual way [2], therebydefining the quantum theory. The expression for the Virasoro generators normalized tofulfill the commutation relations (3.16) is not the normal ordered version of (4.16), butas is well known, it isLN =12k + cVXM,α: QαMQαN−M : ,(4.18)(cV being the quadratic Casimir operator in the adjoint representation).The centralcharge c now is not of course 1, but rather,c = 2k dim G2k + cV, dimG ≡dimension of G.(4.19)These results about the Kac-Moody and Virasoro algebras are explained in ref.

2.AcknowledgementWe 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-0 2-85ER -40231, and A. S. by the Department of Energy, USA under contract numberDE-FG05-84ER40141; 2) A. P. B. and A. S. by INFN, Italy [at Dipartimento di Scienze17

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, inparticular, for their hospitality while this work was in progress.

They also wish to thankPaulo Teotonio for very helpful comments and especially for showing us how to writeequations (2.21), (2.23) and (4.9) in their final nice forms involving Lie derivatives.18

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