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On the classification of quantum W-algebras

arXiv:hep-th/9111062v1 28 Nov 1991EFI 91-63DTP-91-63On the classification of quantum W-algebrasP. BOWCOCK1 2Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, U.S.A.andG.

M. T WATTS3Department of Mathematical Sciences, University of Durham, South Road, Durham, DH1 3LE,U.K.ABSTRACTIn this paper we consider the structure of general quantum W-algebras.Weintroduce the notions of deformability, positive-definiteness, and reductivity of aW-algebra. We show that one can associate a reductive finite Lie algebra to eachreductive W-algebra.

The finite Lie algebra is also endowed with a preferred sl(2)subalgebra, which gives the conformal weights of the W-algebra.We extend thisto cover W-algebras containing both bosonic and fermionic fields, and illustrate ourideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov Hamiltoniansystems. We then discuss the possibilities of classifying deformable W-algebras whichfall outside this class in the context of automorphisms of Lie algebras.

In conclusionwe list the cases in which the W-algebra has no weight one fields, and further, thosein which it has only one weight two field.Nov. 19911Email BOWCOCK@EDU.UCHICAGO.CONTROL2Supported by U.S. DOE grant DEFG02-90-ER-40560 and NSF grant PHY9000363Email G.M.T.WATTS@UK.AC.DURHAM

1IntroductionIn the last few years remarkable progress has been made in the understanding of two-dimensional field theories that are conformally invariant. A key to completing this programis the classification of extended conformal algebras, or W-algebras.

The first examples ofW-algebras were the conformal algebra, various superconformal algebras and Kac-Moodyalgebras [1]. Later it was realised that a wider variety of algebras could be constructed fromGKO coset theories [2–5].

Interest then moved to Toda theories, which provided many ex-amples of W-algebras [6]; later these were re-incorporated into the Drinfeld-Sokolov schemeof Hamiltonian reductions [7–10]. Most recently it has been shown that the generalisedDrinfeld-Sokolov reduction of WZW models can be used to produce a new class of alge-bras.

In this construction one gauges a WZW model associated with a Lie algebra g by thecurrents associated with some nilpotent subalgebra. For bosonic W-algebras with fields ofinteger conformal weight, the nilpotent subalgebra can be labelled by an (integral) su(1, 1)embedding in g [11].

This latter method has the advantage that the properties of the al-gebra obtained are easily related to the finite Lie algebraic ingredients of the construction,while the corresponding relationship in the GKO case is much more mysterious for thepresent.However illuminating the examples cited above may be, we cannot hope to obtain aclassification scheme for W-algebras if we tie ourselves to any one construction. It is thisthat motivates us to adopt a more general standpoint in this paper.

The study of W-algebras is hampered by their infinite-dimensional nature. Worse still, their commutationrelations are generally non-linear in the generating fields.

Some progress has been madeby looking for examples using algebraic computing techniques [12, 13], but the calculationsinvolved are complex, and so the searches are restricted to examples containing two orthree fields of low conformal weight.In this paper we shall restrict our attention to ‘deformable’ algebras. By deformablewe mean that the algebra satisfies the Jacobi identity for a continuous range of values ofthe central charge c. In general the structure constants of the algebra are allowed to befunctions of c. The algebras excluded by this restriction are a very complicated set ofobjects, which in principle include, for example, a large number of W-algebras which canbe constructed from lattices using vertex operators [14, 15].

There is some hope that thesealgebras are extensions of deformable algebras, occurring when a generically non-integerweighted primary field becomes integer weighted for particular values of c.The main result of our paper is contained in section 2, where we demonstrate theexistence of a finite subalgebra associated with each classical W-algebra. Perhaps of moreimportance is that we can extend this result to the quantum case if we demand that thequantum algebra have a ‘good classical limit’.

After discussing what precisely this means,we demonstrate the existence of a similar finite subalgebra in the limit that the centralcharge c →∞. This provides us with an easily computable characteristic for W-algebras.A special role is played by an su(1, 1) subalgebra of this finite algebra, which correspondsto the modes L1, L0, L−1 of the Virasoro algebra which generate M¨obius transformations.The characteristic we have derived is a finite dimensional Lie algebra, and an su(1, 1)embedding.

This is precisely the data used in the generalised Drinfeld-Sokolov reductionmethod. In sections 3 and 4 we clarify this connection.

After a review of the generalised1

Drinfeld-Sokolov construction of W-algebras, we calculate the structure constants of the W-algebra obtained by this method up to linear order in the fields, and use this to demonstratethat in this case the finite subalgebra is simply the Lie algebra g associated with the WZWmodel that we are reducing and that, further, the su(1, 1) embedding is the same as thatused in the construction. This provides us with a proof of the existence of the classicalW-algebras associated with each finite Lie algebra g, and su(1, 1) embedding.Armed with these results we discuss certain features of the W-algebras that are con-structed in this way in section 5.

In particular we give a complete list of such algebraswhich have no Kac-Moody components, and those with only one spin-2 field. We alsocomment on the possible use of automorphisms of the finite Lie algebra to generate homo-morphisms of the W-algebra.

The resulting algebras will be deformable, but will not havea good classical limit.Finally, we conclude with some comments on the relevance of our approach to theclassification of deformable W-algebras.2Finite algebras from W-algebrasIf we want to classify extended conformal symmetries, or W-algebras, we should like toattribute to them some easily computable characteristics which specify the algebra com-pletely.In this section we shall construct a finite Lie algebra associated with classicalW-algebras and their quantum counterparts. Although we do not prove that this specifiesthe W-algebra, it does reveal something of its structure, and may ultimately form partof some classification scheme.

To start with we shall consider the relationship between ageneral quantum W-algebra and its classical counterpart. After discussing the ‘vacuumpreserving algebra’ (vpa) for both types of algebra, we show that in the classical case thiscontains a finite subalgebra if we define a ‘linearised’ Poisson bracket.

We then extend thisresult to the quantum case by showing that the corresponding finite algebra decouples inthe limit that the central charge goes to infinity.Let us begin by discussing the relationship between quantum W-algebras and theirclassical counterparts. A quantum W-algebra comprises a set of modes W am of some simplefields W a(z), a notion of normal ordering, and a Lie Bracket.

W-algebras are usually pre-sented in the form of an operator product expansion, which we may represent schematicallyasW a(z)W b(z′)=gab(z −z′)−∆a−∆b+Xcf ab(1)c(z −z′)∆c−∆b−∆a[W c(z′) + gabc (z −z′)∂W c(z′) + . .

. ](2.1)+Xc,df ab(2)cd(z −z′)∆c+∆d−∆a−∆b[ ◦◦W c(z′)W d(z′) ◦◦+ .

. .] + .

. .

|z| > |z′|Here we have arranged the right hand side according to degree in W. Since we are interestedin conformal field theories, we assume that the algebra contains the Virasoro algebraL(z)L(z′) = c2(z −z′)−4 + 2L(z′)(z −z′)−2 + ∂L(z′)(z −z′)−1 + O(1) ,(2.2)2

as a subalgebra. We also assume that the algebra is generated by a finite number of primaryfields W a(z) which obeyL(z)W a(z′) = ∆aW a(z′)(z −z′)−2 + ∂W a(z′)(z −z′)−1 + O(1) ,(2.3)where ∆a is the weight of W a.

The commutation relations of the modes W am which aregiven by W a(z) =P W amz−∆a−m can be deduced in the standard manner from the operatorproduct expansion by a double contour integral. These take the form[W am, W bn]=gabP(∆a, ∆b, 0, m, n) + f ab(1)cP(∆a, ∆b, ∆c, m, n)W cm+n+f ab(2)cdP(∆a, ∆b, ∆c + ∆d, m, n) ◦◦W cW d ◦◦m+n + .

. .

,(2.4)where P is some known polynomial. In terms of modes (2.3) becomes[Lm, W an] = [(∆a −1)m −n]W am+n .

(2.5)Any field that obeys (2.5) for all m is called primary, and any field that obeys it form = −1, 0, 1 is called quasi-primary. The modes of a quasi-primary field Oim form anindecomposable representation of the su(1, 1) algebra generated by L−1, L0, L1.

One canuse the representation theory of su(1, 1) to show that the quasi-primary fields and theirderivatives span the space of fields in the algebra, and that the polynomials P are relatedto Clebsch-Gordan coefficients.We define the hermitian conjugate by(W am)† = W a−m,(2.6)and this induces a natural inner product on the states of the quantum theory. The re-quirement that this inner product be positive definite and the representation theory of theVirasoro algebra requires that the central charge c > 0 and that the fields all have positivedefinite weight.

This then implies that the metric gab is only non-vanishing of fields ofequal weight, that it is positive definite, and that a basis of fields satisfying (2.6) can bechosen for which the metric is diagonal. We call such a W-algebra positive-definite.

Withthis choice of basis, the algebra (2.4) takes the form[W am, W bn]=(c/(2∆a −1)!∆a)δabm(m2 −1) . .

. (m2 −(∆a −1)2)δm+n,0(2.7)+f ab(1)cP(∆a, ∆b, ∆c, m, n)W cm+n + f ab(2)cdP(∆a, ∆b, ∆c + ∆d, m, n) ◦◦W cW d ◦◦m+n + .

. .where f are constants, and we have used that P(∆a, ∆a, 0, m, n) = m(m2 −1) · · ·(m2 −(∆a −1)2)δm+n,0 .

We do not as yet require that the algebra be defined for more thanone value of c. Examples of algebras which are not positive-definite include Kac-Moodyalgebras based on non-compact groups, and algebras including ‘ghost’ fields with strangestatistics, such as the bosonic N = 2 superalgebra recently considered in [16].We should remark on the definition of normal ordering◦◦◦◦which we use here.Inmeromorphic conformal field theory, we assign a field uniquely to each state byφ(z) ↔φ(0)|0⟩= |φ⟩= φ−∆φ|0⟩. (2.8)3

We can define the normal ordered field××φφ′ ×× by××φφ′ ×× ↔φ−∆φφ′−∆φ′|0⟩. (2.9)However this is not the only possible normal ordering.

Following Nahm, [17, 13], we haveintroduced the normal ordering ◦◦◦◦, by◦◦φφ′ ◦◦= P ××φφ′ ×× ,(2.10)where P is the projector onto su(1, 1) highest weight fields so that the resulting compositefields are quasi-primary. Further ambiguities arise when we try to normal order more thantwo fields, since this product is not associative.

One can, for example, decide to order thefields by conformal weight and index a, and then always nest the normal orderings fromthe left. However, there are many choices of basis.

We shall call a particular choice ofbasis a presentation of the W-algebra as a commutator algebra. The underlying structure,which is that of a meromorphic conformal field theory, is the same for each presentation,but the structure constants will be different.

The point that we should like to stress isthat the classical limit of different presentations are identical. This is a consequence ofthe observation that the difference in two orderings can be written as a commutator, andthus must be an O(ℏ) term which vanishes in the classical limit.

(In fact, the projectionoperator P does not simply amount to a reordering, but the difference between the twonormal orderings can be seen to be the Virasoro descendents of commutators, so that theresult is true in this case too. )Let us now consider classical W-algebras.

This is a Poisson bracket algebra of fieldsW a(x) of one variable which closes on (differential) polynomials and central terms. Wecan represent the Poisson bracket schematically as{W a(x), W b(y)} = gab∂∆a+∆b−1δ(x −y)+Xcf ab(1)c[∂∆a+∆b−∆c−1δ(x −y)W c(y) + gabc ∂∆a+∆b−∆c−2δ(x −y)∂W c(y) + .

. .

]+Xc,df ab(2)cd∂∆a+∆b−∆c−∆d−1δ(x −y)[W c(y)W d(y) + . .

.] + .

. .

,(2.11)where the right hand side has been ordered according to degree in W, and f ab(i)c..z may befunctions of the central charge c. Alternatively, if we take the space on which the fields aredefined to be the unit complex circle, we can expand the fields in modes exactly as in thequantum case. Identical mode algebras are generated if, in the equations (2.1), (2.11) weuse the correspondence(z −z′)−N →(−1)N−12πiδN−1(z −z′)/(N −1)!

,(2.12)although if we use identical structure constants we do not expect that both quantum andclassical commutator algebras satisfy the Jacobi identity if they are non-linear. We shallassume that the classical algebras have a number of properties that they would inheritautomatically as the classical limit of the quantum algebras.They contain a classicalversion of the Virasoro algebra12π{L(x), L(y)} = −c12δ′′′(x −y) −2L(y)δ′(x −y) + ∂L(y)δ(x −y) ,(2.13)4

and the generating fields obey the classical version of (2.5). Further, the only terms inthe Poisson bracket algebra which are independent of W a are taken to be of the formm(m2 −1) · · ·(m2 −(∆a −1)2)δm+n,0δab.

By analogy, we refer to such algebras as positive-definite classical algebras.We now discuss the relationship between a classical W-algebra and its quantised version.We shall see that an extremely important criterion for a quantum algebra to have a classicallimit is that it is well-defined for all values of the central charge c, with the exceptionperhaps of a few isolated values or closed intervals.By this we mean that that thereis an operator product algebra for a set of fields W a(z) of fixed conformal weights ∆a,with structure constants f(i) which are continuous functions of c, which is associative fora continuous range of c values.We call such algebras deformable.We shall see thatdeformability is however not sufficient for a quantum algebra to have a classical limit.As an example let us first consider the classical Virasoro algebra, (2.13). Quantisingthis algebra yields[L′m, L′n] = ℏc′12m(m2 −1)δm+n,0 + (m −n)ℏL′m+n ,(2.14)where the prime ′ indicates that we have the normalisation inherited from the classicalPoisson bracket structure.

To recover the standard normalisation we must substituteL′ = ℏL,c′ = ℏc . (2.15)Similarly, for a general quantum W-algebra, we can re-introduce ℏby the substitutionsL →L′/ℏ, c →c′/ℏ, W a →W a′/ℏαa ,(2.16)where the constants αa are to be determined.

The classical limit is given by the usualcorrespondence{W a, W b} = limℏ→01iℏ[W ′a, W ′b]. (2.17)For this limit to make sense we require that the quantum operator product algebra remainassociative as ℏ→0, or equivalently, as c →∞.

This is why the W-algebra must bedeformable. Substituting (2.16) into (2.7) we obtain schematically[W ′a, W ′b] = c′ℏ2αa−1δ + f(1)W ′cℏαa+αb−αc + f(2)◦◦W ′eW ′f◦◦ℏαa+αb−αf −αe + O(W 3) .

(2.18)For this to be the quantisation of a classical W-algebra, we require that the right-hand sidebe O(ℏ). The ℏdependence comes both explicitly from W →W ′ℏα, but also implicitly,from f(i)(c) →f(i)(c′/ℏ).

If we havef ab(i){ci} = O(ℏγi(a,b,{ci})) as ℏ→0 ,(2.19)and all the fields ◦◦(W ′)p ◦◦are O(1), then we must impose that for each term in the singularpart of the operator product algebra expansion of W a and W bmin(a,b,ci)(γi + αa + αb −iXj=1αci) ≥1 . (2.20)5

If it is not possible to find such constants αa, then we say that the W-algebra has noclassical limit. The restriction (2.20) only restricts the couplings to fields which appearin the commutation relation, or equivalently to fields in the singular part of the operatorproduct of W a and W b.

It is obvious that we have no restriction on the regular terms sinceW a(z)W b(z′) = . .

. + ◦◦W a(z′)W b(z′) ◦◦+ .

. .

,(2.21)where the coupling to ◦◦W aW b ◦◦is O(1).If the classical limit of a W-algebra is positive-definite, we call the quantum algebrareductive. To examine what restrictions this implies, we must consider the central termsin (2.18).

Fixing the behaviour of the central term we requireαa = 1 for all a . (2.22)For this choice of αa the requirement (2.20) becomes, for each term which appears in thesingular part of the operator product expansion,f(i) = O(c1−i) as c →∞.

(2.23)If it is not possible to impose (2.20), then the only possibility of recovering a classicalW-algebra is that the normal ordered products are no longer O(1).Generically for aW-algebra which comes from the quantisation of a classical algebra we have◦◦W a(x)W b(y) ◦◦= W a(x)W b(y) + O(ℏ) . (2.24)It is possible for the first term to vanish if the bosonic fields W a, W b can be written ascomposite fermionic fields,W a = d(x)f(x),W b = d(x)e(x) ,(2.25)where classically d(x)d(x) = 0, and quantum mechanically ◦◦d(x)d(y) ◦◦= O(ℏ).

A simpleexample of this possibility may be seen by considering the first two W(4,6) algebras of ref[12]. The coupling constants of these algebras do not meet the requirements (2.23) or even(2.20).

In fact one of these two algebras can be constructed as the bosonic ‘reduction’ ofa fermionic W-algebra [18], the N = 1 superconformal algebra. This yields a W-algebrawith fields of spins 4 and 6, with zero central charge classically.

We shall discuss suchreductions further in section 5.Let us now turn to the question of constructing the advertised finite Lie algebra fromclassical and quantum W-algebras.Although the full set of modes of a quasi-primaryoperator Oi only form an indecomposable representation of su(1, 1), the subset of modes{Oim : |m| < ∆(Oi)}(2.26)form an irreducible representation of su(1, 1). The set of all such modes for a W-algebraforms a closed subalgebra.We call this the vacuum-preserving algebra (vpa), since inthe quantum case these are precisely the modes which annihilate both the right and leftsu(1, 1) invariant vacua.

Although this algebra involves only a finite number of modes ofeach field, for a non-linear algebra it will only close on the modes associated with an infinite6

number of such quasi-primary fields, so that it is not a finite Lie algebra. For the linearVirasoro algebra the vpa is the set {L1, L0, L−1} which form the algebra su(1, 1) = sl(2, R);for the superconformal algebra, the vpa is the algebra osp(1, 2).

These subalgebras giveuseful information about the structure constants of fields in conformally invariant andsuperconformally invariant theories respectively and we would like to define a similar finiteLie algebra associated to a general W-algebra.Let us consider first a classical W-algebra. The assumption ofpositive-definitenesssays that the algebra takes the form{W am, W bn}=(c/((2∆a −1)!∆a)m(m2 −1) .

. .

(m2 −(∆−1)2)δabδm+n+f ab(1)cP(∆a, ∆b, ∆c, m, n)W cm+n + . .

. (2.27)If we restrict attention to the vpa, we see immediately that the central terms are absent.

Wecan now consider a new bracket {., . }P on the modes in the vpa, which consists simply of thelinear term in (2.27).

We can easily check that the Jacobi identity is satisfied by this newbracket when we restrict to the vpa. This is because for |m| < ∆a P(∆a, ∆b, 0, m, n) = 0and consequently{W am,XpW bn+pW c−p}=Xpf ab(1)dP(∆a, ∆b, ∆d, m, n + p)W dm+n+pW c−p+ f ac(1)dP(∆a, ∆c, ∆d, m, −p)W dm−pW bn+p+ O(W 3) ,(2.28)so that in the classical case the contributions to the Jacobi identity from the quadraticand higher order terms which we have neglected do not contribute, and so the restrictedbracket{W am, W bn}P = f ab(1)cP(∆a, ∆b, ∆c, m, n)W cm+n .

(2.29)is a closed Lie algebra, g. Since we have, by assumption, included the modes L±1, L0 inthe vpa, we see that we automatically have an su(1, 1) embedding su(1, 1) ⊂g given bya classical W-algebra. In the case of the Zamolodchikov algebra WA2, we have the modes{Q±2, Q±1, Q0, L±1, L0} forming the algebra sl(3) with su(1, 1) in the maximal regularembedding.We should like to attempt the extension of this argument to the quantum case.

Inthe classical case, the contribution from composite terms to the linear terms in the doublecommutator vanished for the vpa because the only possible ‘contraction’ from three fieldsto one arose from the central term which decoupled precisely for these modes. However, inthe quantum case there are other contributions to this term which arise from the need tonormal order composite fields.

As an illustrative example we return to the Zamolodchikovalgebra WA2, this time in its quantised version. The quantum commutation relations are[Lm, Qn]=(2m −n)Qm+n(2.30)[Qm, Qn]=c3m(m2 −1)(m2 −4)5!δm+n+(m −n)30(2m2 −mn + 2n2 −8)Lm+n + β(m −n)Λm+n ,(2.31)7

whereβ=1622+5c ,Λ(z)=◦◦T(z)T(z) ◦◦. (2.32)The only non-trivial double commutators are [Lp[Qm, Qn]], [Qp[Qm, Qn]] and the only com-posite field appearing in the intermediate channel is Λ, so we need only consider the linearterms in [Lp, Λm+n], [Qp, Λm+n].

These are[Lp, βΛm+n] = 165p(p2 −1)3!Lm+n+p + · · ·(2.33)[Qp, βΛm+n] = β4(5p3−5p2(m+n)+3p(m+n)2−(m+n)3−17p+9(m+n))35Qp+m+n + · · ·(2.34)The first of these commutators vanishes when we take p = −1, 0, 1. In the second commu-tator there is a contribution from the Λ term in [Q, Q] which does not vanish even whenwe restrict to the vpa, and a more careful consideration of this term shows that it arisesfrom the need to normal order composite fields.

Instead we can ensure that this term doesnot violate the consistency of the ‘linearised’ Jacobi identity by taking the limit c →∞.In this case β →0 and the vpa linearises, again to give sl(3, R). Note that we need tocombine the limit c →∞and the restriction to the vpa to ensure that both commutatorsvanish.We are now in a position to prove this feature, namely that the vpa linearises to givea finite Lie algebra as c →∞, assuming that the W-algebra is reductive.

Let us denotea generic composite field composed of i basic fields as ◦◦(W)i ◦◦. The contribution to theJacobi identity from the coupling through such terms in [W a, [W b, W c]] is[W am, [W bn, W cp]]=Xi[W am, f bc(i)P(∆b, ∆c, ∆(i), n, p) ◦◦(W)i ◦◦n+p] .

(2.35)Let us also write the linear part of the contribution from the commutator of W a with W (i),[W am, ◦◦(W)i ◦◦n] = gae(i)P(∆a, ∆(i), ∆e, m, n)W em+n . (2.36)At this point we must split our argument into two cases, depending on whether ◦◦(W)i ◦◦appears in the singular or regular part of the operator product expansion of W a with W e.If◦◦(W)i ◦◦appears in the singular part of the operator product expansion then we cancalculate the order of g(i) by taking the three point functionCabi = ⟨W aW e ◦◦(W)i ◦◦⟩.

(2.37)This can be written in two ways. We have thatCabi = f ae(i)⟨◦◦(W)i ◦◦◦◦(W)i ◦◦⟩= O(c) ,(2.38)using (2.23) and evaluating the leading contribution in c to ⟨0|(W)i∆(i)(W)i−∆(i)|0⟩.

We alsoobtainCabi = g(i)⟨W aW e⟩∼gae(i) c ,(2.39)using (2.7) and (2.36), and suppressing non-zero constants. Thus we see that gae(i) = O(1)and so the contribution to the Jacobi identity of three basic fields [W a, [W b, W c]] to the field8

W e from the term in [W b, W c] of form ◦◦(W)i ◦◦is O(gae(i)f bc(i)) = O(c1−i), if the field ◦◦(W)i ◦◦appears in the singular part of the operator product expansion of W a with W e. However, ifthe field ◦◦(W)i ◦◦does not, then we cannot apply (2.23) to deduce the order of the couplinggae(i). In this case we have that ◦◦(W)i ◦◦has conformal weight ∆(i) ≥∆a + ∆e.

However, thepolynomials P(∆a, ∆(i), ∆e, m, n) vanish identically if |m| < ∆a, ∆(i) ≥∆a + ∆e (see ref. [19]), and we can use this fact to bypass our ignorance of the coupling f ae(i).This shows that if we consider Jacobi’s identity for the vpa modes of the generatingfields in the limit c →∞, then all contributions to linear terms from composite fields inthe intermediate channel drop out.

Since the commutator algebra is a Lie algebra for all cvalues by the assumption of deformability, the only obstruction to the vpa algebra restrictedto the generating fields satisfying the Jacobi identity was from such contributions. Thus,this algebra in the c →∞limit of a reductive W-algebra is a finite Lie algebra.We have now shown how to recover finite Lie algebras from positive-definite classical W-algebras and reductive quantum W-algebras.

We shall call this the linearised vpa algebra.By the Levi-Malcev theorem, the most general form for a finite Lie algebra would be thesemidirect product of a semi-simple Lie algebra with its radical, which is its maximalsolvable ideal. However, we can use the positive-definitness of the classical algebra to showthat the maximal solvable ideal of the finite Lie algebra we have constructed is in fact itscentre, or, in other words, that the linearised vpa is the direct sum of a semisimple Liealgebra with an abelian Lie algebra (For results on the structure of Lie algebras used here,see e.g.

[20]).To do this, let us consider the maximal solvable ideal a of the linearised vpa of areductive W-algebra. Let us suppose that a particular mode W am ∈a.

The modes L±1, L0are always in the linearised vpa, and we know the commutation relations of W am with Lmto be of the form[Lm, W an] = ((∆a −1)m −n)W am+n . (2.40)Since a is an ideal, eqn.

(2.40) implies that W am ∈a ⇒W an ∈afor all |n| < ∆a. Withthe standard normalisation for a positive-definite quantum W-algebra we haveW a(z)W a(z′) = (c/∆)(z −z′)−2∆a + 2L(z′)(z −z′)−2∆a+2 + .

. .

. (2.41)Using the Virasoro Ward identities, (see e.g.

appendix B of ref. [21]), we can deduce thecoefficients of all the terms ∂iL(z′) in this operator product and we can deduce that[W a∆−1, W a1−∆] = 12(∆−1)/(∆(2∆−1))L0 + .

. .

(2.42)For ∆a > 1 this is non-zero, and so, for ∆a > 1 we have L0 ∈a. If L0 ∈a, then weimmediately we get that L0, L±1 ∈a.

which is a contradiction since a solvable ideal cannotcontain a semi-simple algebra.So, if W am ∈a, where a is a solvable ideal, then ∆a ≤1. If ∆a < 1 then it contributesno modes to the vpa; if ∆a = 1 then m = 0, and we can thus denote the elements of aas Ui0, the zero modes of a set of weight one fields Ui(z).

These zero modes Ui0 form asolvable Lie algebra. However, it has been known for a long time (see e.g.

[22]) that therequirement that the inner product on the primary fields of weight one is positive definiteforces them to have a Kac-Moody algebra based on a compact semi-simple Lie algebra plussome u(1)n current algebra. Thus the zero modes of weight one fields in cft form a finite9

dimensional Lie algebra which is the direct product of a semisimple Lie algebra with anabelian algebra, and we see that the ideal a is abelian.The only possibility left open to us now is that the linearised vpa has the structure of asemi-simple Lie algebra semidirect product with an abelian algebra. Suppose that U0 ∈a.Then [U0, W am] ∈a for all W am in the vpa.

If [U0, W am] = Xm, then the operator productexpansion of U(z) with W a(z) must be of the formU(z)W a(z′) = . .

. + X(z′)(z −z′)−1 + O(1) ,(2.43)since U0 =R dz/(2πi)U(z).However, from (2.43) we see that the field X must haveconformal weight equal to that of W a.

Since Xm ∈a, we see that X must have weightone, and so W a must have weight one. We already know that a commutes with the zeromodes of the spin one fields, so in fact a is the centre of the vpa.

This completes the proofthat the linearised vpa of the classical limit of a reductive W-algebra is the direct sum ofa semisimple Lie algebra with an abelian algebra.The above discussion has been for a purely bosonic W-algebra. If we wish to includefermions then we must also consider Lie superalgebras, since the vpa of fermionic fieldswill contain anti-commutators of the modes of fermionic fields.

We shall use the notationof [23] for Lie superalgebras, with the algebra decompositiong = g¯0 ⊕g¯1(2.44)where the bosonic generators are in g¯0 and the fermionic in g¯1. We define the grade of agenerator X to be g(X) = j if X ∈g¯.

The (super)Lie bracket then takes the form[X, Y ] = XY −(−1)g(X)g(Y )Y X(2.45)The bosonic fields have modes in g¯0 and the fermionic fields have modes in g¯1. It will alsobe the case that the bosonic fields will have integral conformal weight and the fermionicfields half-integral conformal weight, for unitarity.

A fermionic field of weight ∆will havemode decompositionψ(z) =Xn∈Z+1/2ψnz−n−∆(2.46)As for bosonic fields, the vpa contains the modes of the fermionic fieldsψm : |m| < ∆(2.47)We see that for a free fermionic field of conformal weight 1/2 , there are no modes in thevpa. Thus analysis of the vpa will yield no information on the free fermion content ofa theory.

However, this is not obstacle since free fermions have already been shown tofactorise from the Hilbert space by Goddard and Schwimmer [24].Since the fermionic fields will have half integer modes, the su(1, 1) decomposition mustbe compatible in the sense that the decomposition takes the formg¯0 = ⊕j∈ZDj , g¯1 = ⊕j∈Z+1/2Dj(2.48)where Dj is the representation of dimension 2j+1. We can also prove that the superalgebraconsists of the direct sum of simple (super)-algebras and an abelian Lie algebra in ananalogous manner to that above for purely bosonic W-algebra vpa’s.10

This means that the field content of any positive-definite W-algebra which is definedfor all c values must comprise a set of free fermion fields of weight one half (which donot contribute to the vpa; such fields have already been shown to factorise [24]), a set ofbosonic free fields of weight one (u(1)n current algebra) and a set of fields whose weightsare given by an sl(2) embedding in a semisimple Lie superalgebra which is compatible withthe grading of the superalgebra as in (2.48).For a purely bosonic W-algebra, the field content will comprise a set of weight one fieldsand a set of bosonic fields whose conformal spins are given by an integral su(2) embeddingin a semi-simple Lie algebra.We shall now go on to show that this is indeed the case for the generalised DrinfeldSokolov constructions mentioned earlier, and then to consider various cases of particularinterest. The rest of this paper will be concerned only with the case of bosonic algebrasfor simplicity.3Hamiltonian systems and co-adjoint orbitsThe analysis of the previous section showed that to each reductive W-algebra one couldassociate a finite Lie algebra with some su(1, 1) embedding specified.

This is reminiscentof the data that is required for a generalisation of the Drinfeld-Sokolov construction ofHamiltonian structures that have been studied recently [11, 25, 26]. In the next sectionwe shall show that this data is recovered as the finite Lie algebra we constructed.

Thisprovides us with an existence proof for the classical W-algebras associated with each Liealgebra and su(1, 1) embedding. In this section we give a brief review of this construction.The classical Hamiltonian systems of Drinfeld and Sokolov [8] are based on a Poissonbracket structure on g∗, the dual to the Lie algebra g, and the extension of this to ˆg, thecentrally-extended Kac-Moody algebra related to g. An element of ˆg consists of a pair,(j(z), c) ,(3.1)where c is a number and j is a field on S1 valued in g. The coordinate on S1, we denoteby z, with 0 ≤z < 2π.

With this definition, the Lie bracket of two elements of ˆg is givenby[(j1(z), c1), (j2(z), c2)] = ([j1(z), j2(z)], kZTr{∂j1(z)j2(z)} dz) ,(3.2)where the second term corresponds to the cocycle of ˆg. An element of ˆg∗is given by a pair(q, λ), where q is a g-valued field on S1 and λ is a number.

The action of this element on(j, c) in ˆg is given by⟨(q, λ), (j, c)⟩=ZTr{qj} dz + cλ . (3.3)With this, we may identify ˆg and ˆg∗, and we obtain a canonical action of ˆg on ˆg∗, thecoadjoint action ad∗.

If (q, λ) ∈ˆg∗, (ji, ci) ∈ˆg, then we havead∗(j1,c1)·(q, λ)[(j2, c2)]=(q, λ)[ad(j1,c1)·(j2, c2)]=⟨(q, λ), ([j1, j2], kZTr{(j1)′j2})⟩11

=Ztr{[q, j1]j2 + kλ∂j1j2} .Thus we obtainad∗(j1,c1)·(q, λ)=([q + kλ∂, j1], 0) .This is simply an infinitesimal gauge transformation of q. This phase space also has acanonical action of ˆG, the coadjoint action Ad∗, given byAd∗U·q(h)=q(AdU·h) ,(3.4)Ad∗U·(q, λ)=(U−1qU + kλU−1∂U, λ) .

(3.5)There is a canonical Poisson bracket structure on ˆg∗, the Berezin-Kirilov-Kostant-Lie-Poisson bracket. If U, V are two functionals on ˆg∗, then their Poisson bracket is also afunctional on ˆg∗.

When evaluated on q ∈ˆg∗it is explicitly given by{U, V }q=⟨q, [dqU, dqV ]KM⟩,(3.6)where djU, djV are any elements of ˆg, such that for all δj ∈ˆg∗,U(j + δj) = U(j) + ⟨djU, δj⟩+ O(δj2) . (3.7)We shall usually suppress the j suffix if it is clear from context.We may accordingly evaluate the Poisson brackets of the components of ˆg∗.

If {T i}form a basis of the generators of ˆg with Tr{T iT j} = gij, then we can define the functionalsˆT i(x), ˆe on (q, λ) ∈ˆg∗byˆT i(x)[(q, λ)]=Tr{T iq(x)} ,ˆe[(q, λ)]=λ . (3.8)We haved ˆT i(x)=(T iδ(x −y), 0) ,dˆe=(0, 1) .

(3.9)Thus we can evaluate the Poisson brackets of these functionals{ ˆT i(x), ˆT j}(q,λ)=Zdz Tr{q[T iδ(x −z), T jδ(y −z)] + kλ∂z(T iδ(x −z))T jδ(y −z)} .Thus we see that{ ˆT i(x), ˆT j}(q,λ)=f ijk ˆT k(y)δ(x −y) −kˆegijδ′(x −y) . (3.10)This is the Kac-Moody algebra ˆg.In particular we shall often denote the zero gradesubalgebra functional ˆJi by ji and ˆe by 1, with the Poisson brackets{ji(x), jj(y)} = f ijkjk(y)δ(x −y) −k2δijδ′(x −y) .

(3.11)The method of hamiltonian reduction involves constraining currents associated withnilpotent elements of the algebra. In the traditional reduction associated with Toda the-ory or the standard KdV hierarchy one gauged the maximal nilpotent algebra associated12

with, say, all the positive roots of g. It was then realised that one could generalise thisconstruction by gauging some smaller set of currents, and moreover, that this set could besuccinctly labelled by some su(1, 1) embedding. Since we are interested in bosonic positive-definite W-algebras, we may assume that the su(1, 1) embedding is integral.

Non-integralembeddings result in bosonic fields of half-integral weight and the resulting W-algebras arenot positive-definite.Let us consider some modified Cartan-Weyl basis for g,g = g−⊕h ⊕g+ . (3.12)Hereg± = ⊕CE±α , h = ⊕CHi ,(3.13)with the commutation relations[Eα, E−α]= (2/α2)αiHi , [Hi, Eβ]= βiEβ .

(3.14)One can always conjugate any su(1, 1) subalgebra of g so that I+ ∈g+, I0 ∈h, I−∈g−where I+, I−, I0 are the usual raising, lowering and diagonal basis of su(1, 1). We maywrite I0 = ρ∨· H. If we use the standard normalisation for the su(1, 1) algebra,[I0, I±] = ±I± , [I+, I−] =√2I0 ,(3.15)then we may define the characteristic of the su(1, 1) embedding to be (ρ∨· e1, ..., ρ∨· ei),where ej are the simple roots of g. It is a fact that the entries of the characteristic are0, 1/2, 1.

For integral embeddings they must either be 0 or 1. The standard reduction isassociated with the principal embedding whose characteristic contains all ones.We may grade g with respect to the ρ∨·H eigenvalue asg = ⊕mgm .

(3.16)The elements of g which are highest weight states for this su(1, 1) action form a commutingsubalgebra of g. We denote these highest weights by E(ei), and the corresponding lowestweights by E(−ei). The highest weights are annihilated by I+ and the lowest weights by I−.We denote the subalgebra ⊕n≥0gn by p+ and the subalgebra ⊕n>0gn by n+ ⊂g+.

Similarlyfor p−, n−. For the standard reduction associated with the principal reduction, n± = g±.Since ˆG acts on ˆg∗, we may perform a classical Hamiltonian reduction [27] with respectto the subgroup ˆN−, where N−is the subgroup of G which has the nilpotent subalgebran−as its Lie algebra.

In this procedure one chooses an image of the momentum map π andthe phase space consists of equivalence classes under the residual symmetry of the inverseimage of π. Here π is essentially the projection map g 7→n+.

We can choose the image ofπ in such a way that the inverse image consists of elements of ˆg∗of the form(b(z) + I+, 1) ,(3.17)where b(z) ∈p−. We call this space M.The action of ˆN−is now an equivalence relation on M. From the form of M, we maychoose coordinates on this space to be gauge invariant differential polynomials of the entries13

in the matrices b(z). In particular, there is a unique gauge transformation Ad∗N = exp(ad∗n)with n ∈ˆn−which givesN−1(b + I+)N + kN−1∂N = I+ +XnWnE(n) ,(3.18)where E(n) span the kernel of I−.To show this gauge transformation is unique, takecomponents in gm.

As a result, the entries of n are uniquely determined polynomials inthe entries of b(z).The Poisson bracket structure on gauge invariant functionals φ, ψ on classes of q ∈˜Mis given by{φ, ψ}q = ⟨q, [∇qφ, ∇qψ]KM⟩,(3.19)where j = ∇qφ is any element of ˆg such thatφ(q + δq) = φ(q) + ⟨δq, ∇qφ⟩+ O(δq2) ,(3.20)for all q ∈M.Thus ∇qφ is determined up to an element of ˆn−; one such choice is∇qφ = dqφ. It is easy to show that this bracket is well defined and satisfies the Jacobiidentity [8].Further, we can imbed this structure in the Lie-Poisson bracket structure by consider-ing the map µ from ˆG0, corresponding to zero-graded Lie algebra g0, to gauge invariantfunctionals on M given byµ : (g0 + I+, 1) 7→{Wi(g0)} ,(3.21)whereN−1(g0 + I+)N + kN−1∂N = I+ +XnWnE(n) .

(3.22)Since this gauge transformation is unique, the polynomials Wn(g0) are a choice of coor-dinates on the manifold ˜M = M/ ˆN−. It can also be shown [8] that the Poisson bracketstructure on ˆg and ˜M are compatible in the sense that{µ∗φ, µ∗ψ} = µ∗{φ, ψ} ,(3.23)where µ∗φ is a functional on ˆg∗.

µ∗is called the generalised Miura transformation. For thestandard reduction corresponding to the principal embedding, g0 = h, and so the Miuratransformation provides a free field representation of the Poisson bracket algebra (3.19).For more general reductions we obtain a construction in terms of the currents of the zero-graded (non-abelian) algebra.

Since the polynomials Wn(g0) are a choice of coordinates on˜M, this algebra is closed, although not necessarily on linear combinations of the originalcoordinates.4Linearised Poisson brackets for classical W-algebrasWe are now in a position to calculate the Poisson brackets of the W-algebra given by theDrinfeld-Sokolov reduction associated with some Lie algebra g and a particular integralsu(1, 1) embedding. The purpose of this section is to show that the finite subalgebra and14

su(1, 1) embedding of section 2 associated with this W-algebra coincide with those chosento specify the reduction. This will demonstrate the existence of the classical W-algebraassociated to each such pair.

We can check this in this case by using the expressions wededuced in the previous section for the W-algebra Poisson brackets. The finite subalgebrais simply the ‘linearised’ vpa for the generating fields, so we will only need to calculatethe Poisson brackets to linear order in the fields.

This is the feature which makes thecalculation tractable.We need to calculate the Poisson brackets of the functionals H,H =Zdz2πif a(z)W a(z) ,(4.1)where W a is a W-algebra field and now z is a complex coordinate.If we denote theW-algebra fields of weight ∆a = a + 1 by W a, then the modes W am are given by H forf(z) = zm+a. We shall not differentiate between fields of equal conformal weight, to avoidproliferation of indices.

From (3.18) we know that the gauge invariant fields correspondto the lowest weights of the sl(2) embedding. If we wish to identify a particular field of agiven weight, then we shall use the notation W[X], where X is a generator of g which is alowest weight of the sl(2).The results of the last section tell us that the Poisson bracket algebra is given by{W am, W bn}=Zdz2πi Tr(j(z)[dW am, dW bn]) + kZdz2πi Tr(dW am(dW bn)′) .

(4.2)If we are interested in the term in this Poisson bracket which is linear in the fields W, thenwe clearly only need to calculate dW am to linear order in the fields W.Consider the arbitrary element of the space M to be of the formj = b(z) + I+ . (4.3)We may decompose g with respect to the sl(2) subalgebra to find the highest weights ofthis sl(2) which we shall denote by Ea.

Then bases for g, g−and n−are given byg=⊕a ⊕2a+1i=0 CEi,a ,(4.4)g−=⊕a ⊕a+1i=0 CEi,a ,(4.5)n−=⊕a ⊕ai=0 CEi,a . (4.6)whereEi,a = adi(I+) ◦Ea .

(4.7)Then we may writej = I+ + j0 + I+ +XaaXi=0ji,a(z)Ei,a . (4.8)We shall now consider a gauge transformation of the formjl = exp(ad(l)) ◦(j + k∂) ,(4.9)15

where l is some current in n−. If l is the transformation which puts j into the highestweight gauge, then l is defined implicitly as a polynomial function in the components of j0and their derivatives, byjl = jW = I+ +XaW aEa .

(4.10)An important step in the argument is to decompose l into components which are homoge-neous in the components of j0, and further into components of Ei,a:l=Xjl(j)(4.11)=XjXaa−1Xi=0lj|iaEi,a ,(4.12)where lp|rb is homogeneous in j0 of degree p. If we decompose the gauge invariant functionalsW a into homogeneous pieces W p|a of degree p, then upon substituting (4.12) into eqn. (4.10), we obtainW 1|a=j0,a −k(l1|0a)′(4.13)W 2|a=TrE2a,a[l(1), (j0 + 12[l(1), I+])]−k(l2|0a)′ ,(4.14)wherel1|i,a=aXp=i+1(−k∂)p−i−1jp,a(4.15)l2|i−1,a=a−1Xp=i(k∂)p−i(−1)pTrE2a−p,a[l(1), (j0 + 12[l(1), I+])].

(4.16)We have chosen a normalisation for the sl(2) highest weight vectorsTr(E2a,aE0,a) = 1 ,(4.17)For simplicity we have used notation which assumes that there is a unique field of eachweight, but the generalisation is straightforward. The normalisation (4.17) means that thegenerators I±, [I+, I−] obey su(1, 1) commutation relations with nonstandard normalisationfactors.We can now deduce dH for H =R W a(z)f(z) dz2πi.

This will in general be a functionof the entries jia, and since we are interested in the Poisson brackets of gauge-invariantquantities, we may simply substitute j by jW, after we have evaluated dW. This makesthe evaluation of dW very easy since li|ja = 0, jia = δi0W a.

We can thus identify the termswhich will contribute to dH where H =R fW a. Using the notation abcjkl= Tr{Ej,a[Ek,b, El,c]} ,(4.18)we havedH=ZaXp=0(k∂)pfdji,a+Za−1Xp=0Xb,cb−1Xq=0bXr=q+1(k∂)r−q−1 [W c(−k∂)pf]abc2a −pq0djr,b .

(4.19)16

Using Tr(jdji,a) = ji,a, we can easily see that dji,a = (−i)iE2a−i,a. We are really onlyinterested in the case f = za+m where H = W am, and so finally we obtaindW am=aXi=0(a + m)i(−k)izm+a−iE2a−i,a+Xb,ca−1Xi=0c−1Xq=0c−qXj=1(−1)q+j∂j−1(W b∂izm+a) bac02a −imE2c−j−q,cki+j−1+O(W 2) ,(4.20)where we denote(b)(b −1) .

. .

(b −c + 1) by (b)c . (4.21)We are now in a position to evaluate the Poisson brackets of the modes W am to linearorder in the fields.

We shall decompose dW into its homogeneous pieces, asdW am = dW a(0)m+ dW a(1)m+ O(W 2) . (4.22)Using (4.2) we see that the terms which contribute to the linear piece of the Poisson bracketare{W am, W bn}=kZdz2πi Tr(dW a(0)m(dW b(0)n)′) +Zdz2πi Tr(jW(z)[dW a(0)m, dW b(0)n])(4.23)+kZdz2πi Tr(dW a(0)m(dW b(1)n)′) + kZdz2πi Tr(dW a(1)m(dW b(0)n)′) + O(W 2) .O’Raifeartaigh et al.

in [25, 11] have shown that L = αW[I−] + Pa βaW[Ua]2 is a Virasoroalgebra for the system we have described, where W[I−] is the field corresponding to therepresentation of the embedded sl(2) itself, and W[Ua] are fields corresponding to thesinglets in the decomposition of g with respect to the embedded sl(2). The fields W[E0,a]transform as primary fields of weight a + 1 w.r.t.

this Virasoro algebra. The central termin the Virasoro algebra is generated purely by the field W[I−], since we have already shownthat there are no central terms in the Poisson brackets of composite fields.

In the caseW = W[I−], eqn. (4.20) becomes exact,dW[I−]m = zm+1[I+, [I+, I−]] −k(m + 1)zm[I+, I−] .

(4.24)We can now evaluate the Poisson bracket{W[I−]m, W[I−]n} = kλ(m −n)W[I−]m+n + k3m(m2 −1)δm+n ,(4.25)which corresponds to a rescaled Virasoro algebra. In (4.25) λ is defined by[I+, [I+, I−]] = λI+ .

(4.26)Thus, if we re-scale W[I−] to return (4.25) to the standard normalisation, we recoverc = 12k/λ2 = 6k(ρ∨)2 where ρ∨defines the su(1, 1) embedding.17

Thus the Poisson brackets (4.24) represent a W-algebra. To establish that this is apositive-definite W-algebra, note that the first term in (4.24) corresponds to the centralterm and is easy to compute;Zdz2πiaXi=0bXj=0(a + m)i(b + n)(j + 1)(−k)i+j+1zm+n+a+b−i−j−1 Tr(E2a−i,aE2b−j,b) .The trace in this term gives δabδai δbj(−1)a and so we get now=Zdz2πiδab(a + m)a(a + n)a+1zm+n−1k2a+1(−1)a=−k2a+1m(m2 −1) .

. .

(m2 −a2)δabδm+n,0(4.27)Thus the algebra (4.24) is a positive-definite W-algebra since the central term is non-degenerate. Following the theoretical framework we laid out before, we can now restrictour attention to the algebra of the modes of the vpa.

For the vpa the central term vanishesand the second term in (4.24) is now easy to compute.Zdz2πi Tr(jW(z)[dW a(0)m, dW b(0)n])=Zdz2πi Tr( XcW c(z)Ec!×aXi=0(a + m)i(−k)izm+a−iE2a−i,a,bXj=0(b + n)j(−k)jzn+b−jE2b−j,b)ki+j=XcW cm+naXi=0bXj=0(a + m)i(b + n)j(−k)i+j Tr(Ec[E2a−i,a, E2b−j,b])ki+j ,=XcW cm+nmin(a+b,b+c)Xλ=max(b,c)(a + m)(a+b−λ)(b + n)(λ−c) cba0a −b + λ2b + c −λka+b−c ,remembering that the trace gives a delta function δ(c + a + b, 2a + 2b −j −i).The third term is more complicated,T3≡kZdz2πi Tr(dW a(0)m(dW b(1)n)′)=Zdz2πiaXl=0za+m−l(a + m)l(−1)l×Xd,cb−1Xi=0c−1Xp=0c−pXj=1(−1)p+j∂j(W d(z)∂izp+b) dbc02b −ipTr(E2a−l,aE2c−j−p,c)ki+j+lThe last trace gives us δ(l, a), δ(a, c), δ(j, c −p) and so we get nowT3=Zdz2πiXdb−1Xi=0a−1Xp=0(−1)azm∂a−p(W d(z)∂izp+b) dba02b −ipki+2a−p(a + m)a=Zdz2πiXdb−1Xi=0a−1Xp=0(−1)m(∂a−pzm)(∂izp+b)W d(z) dba02b −ipki+2a−p(a + m)a18

=Zdz2πiXdb−1Xi=0a−1Xp=0(−1)mzm+p−a+p+b−i× (a + m)a(m)(a−p)(n + b)(i)W d(z) dba02b −ipki+2a−nThis last term in curly braces represents a trace, which gives us δ(i, b + p −d −c), and sowe get, with λ = 2b + d −p,T3=XdW dm+n2b+dXλ=b+d+1(a + m)(a+b−λ)(b + n)(λ−d) dab0a −b + λ2b + d −λka+b−d=XdW dm+nb+d+nXλ=b+d+1(a + m)(a+b−λ)(b + n)(λ−d) dab0a −b + λ2b + d −λka+b−d , (4.28)where we used the fact that for λ > b + d + n the second of the two Pochhammer symbolsvanishes.By similar reasoning we can evaluate the fourth term of (4.24) and when we put themall together we obtain for the vpa terms {W am, |m| ≤a},{W am, W bn}=XcW cm+nmin(b+c+n,a+b)Xλ=max(c,b−m)(a + m)(a+b−λ)(b + n)(λ−c)× cab0a −b + λ2b + c −λka+b−c+ O(W 2)(4.29)This establishes the linearised vpa commutation relations. We can now compare them withthe commutation relations of g in a particular basis.

We know that W[I−]±1,0 form thesl(2) embedded inside g, and so it is convenient for us to take the modes of the (quasi-)primary fields W am to correspond to the representation of sl(2) with highest weight Ea. Ifwe take the correlation to beW am ∼= Ea−m,af(a, m)(4.30)where f(a, m) are some constants, then the fact that W[I−]±1,0 are the embedded sl(2)tells us that f(a, m) is given byf(a, m) = µ(a)(−kλ)a+m(a + m)!/(2a)!

,(4.31)where λ is defined in equation (4.26). We may now evaluate the commutator of two ofthese elements of g, and we find that we recover the commutation relations (4.29) exactlyfor µ(a) = (2a)!, and soW am ∼= Ea−m,a(−kλ)a+m(a + m)!

(4.32)completes the identification of the vacuum preserving modes of the W-algebra with thealgebra g itself, up to quadratic terms in the fields W. Moreover the modes L1, L0, L−1 areclearly associated to I+, I0, I−. If one examines the commutation relations more carefullyand normalises the algebra correctly according to (2.7), one sees that indeed the couplingsf abcare O(1) in the central charge, thus bearing out our expectations for a classical W-algebra.This clarifies the relationship between the linearised vpa and the generalised Drinfeld-Sokolov reduction.

It follows that there exists at least one classical W-algebra for every19

g and every integral su(1, 1) embedding. If this W-algebra is unique, then the Drinfeld-Sokolov reductions completely saturate the possibilities for bosonic integrally weightedW-algebras.

We return to this point in the conclusion.5Lie algebras and sl(2) embeddingsIn this section, we shall use the theory of finite Lie algebras and their three-dimensionalsubalgebras together with what we have learnt in the preceding sections to look at variousaspects of reductive W-algebras. We briefly comment on a possible relation between au-tomorphisms of the linearised vpa and homomorphisms of the W-algebra.

Then we givea complete list of positive-definite W-algebras without Kac-Moody components by classi-fying all su(2) embeddings of semi-simple Lie algebras with trivial center. We enumeratethe algebras which, in addition, contain no generating spin-2 fields besides the Virasoroalgebra.First, let us consider the case where the su(1, 1) is not a maximal subalgebra of g;that is there exists some algebra h such that su(2) ⊂h ⊂g.

In the limit that c →∞we expect that some subalgebra of generating fields which includes the Virasoro algebracloses. This, however, does not imply that these generating fields generate some W-algebraassociated with su(2) ⊂h, which is a subalgebra of the W-algebra associated with su(2) ⊂g.

As an example we can consider the algebra WA3 associated with the principal su(2)embedding in A3 which is generated by fields of wieght 2, 3, and 4. However we can writesu(2) ⊂B2 ⊂A3 in this case, where the spin 2, 4 fields are associated with the firstembedding.

An inspection of the operator product expansion of the spin 4 field with itselfreveals that it contains a term which is the composite field associated with the square ofthe spin 3 field. The coupling to this term vanishes in the limit that c →∞, but the spin2, 4 algebra does not close on itself for finite central charge, and is distinct from WB2which we would associate with the first embedding.

However, we suspect that a weakerstatement is true. If τ is an automorphism of g for which h is the stable subalgebra, thisgives an automorphism of the vacuum preserving modes in the limit that c →∞.

Sinceτ(L) = L it defines some homomorphism which maps a generating primary field to somelinear combination of primary fields of the same weight. We shall assume that this canbe used to define a homomorphism on the Verma module of the W-algebra associatedwith g. The subalgebra which is stable under this homomorphism will contain only thosegenerating fields associated with h, but will in general require additional generators whichwill be composites in all the generating fields of the algebra.

In the example above, if wechoose the simple roots of A3 to be e1 −e2, e2 −e3, e3 −e4 then the automorphism whichpreserves its B2 subalgebra is given bye1→−e4e2→−e3and the induced homorphism on the WA3 algebra is simply spin-3 →−spin-3. The subal-gebra which is stable under this homomorphism is the smallest closed algebra containingthe spin 2, 4 fields.

If it is the case that the resulting W-algebra does not have a goodclassical limit, which we know to be true in the case that we reduce a super W-algebra20

in this way, then this sort of consideration may provide a powerful tool for constructingdeformable algebras which do not have a good classical limit.As a second topic of interest, we shall now give a complete list of the possibilities forbosonic W-algebras with good classical limit which have no Kac-Moody components. Ifwe want there to be no spin 1 fields in the W-algebra we require that there are no singletsin the decomposition of the adjoint representation of g under su(1, 1).

Alternatively, if wedecompose g w.r.t. the I0 member of the preferred sl(2), we may express this by sayingthat dimg1 = dimg0, or by saying that the centraliser of su(1, 1) in g is zero.

We classifyall the examples of su(1, 1) embeddings where this is so.First we need to borrow some notation from Dynkin [28]. For ease of exposition weshall revert to the real compact form of the Lie algebras, since the form of the algebra is notimportant for the argument.

A regular subalgebra of g is a subalgebra whose root systemis simply a subset of the root system of g. A subalgebra of g is called an R-subalgebra if itis contained in some proper regular subalgebra of g. Otherwise it is called an S-subalgebra.S-subalgebras have the properties that(i) they are integral(ii) dimg1 = dimg0.Thus the condition that an su(2) subalgebra by an S-subalgebra is sufficient for producinga W-algebra without Kac-Moody components, but as it turns out it is not necessary. Inthe other cases there must exist some proper regular subalgebra of g which contains su(2)and moreover its rank must be equal to that of g, since otherwise it is easy to prove thatsome memeber of the Cartan subalgebra will commute with it.

The classification of allsu(2) subalgebras of the exceptional Lie algebras whose centraliser vanishes has been givenin [28]. We reproduce these results in Table 1.

The algebra g is given in the first column.The second column gives the characteristic specifying the su(2) embedding while the thirdsummarises the weights (with degeneracies in parentheses) of the generating fields of thecorresponding W-algebra. The next column gives the minimal subalgebra(s) of g whichcontain the su(2).

A P in the final column indicates that the embedding is principal. Wenow deal with the remaining classical examples.

(i) su(n)The only su(2) S-subalgebra of An is given by the principal embedding. Any other can-didate is a subalgebra of one of the maximal regular subalgebras of su(n) and hence wecan write su(2) ⊂su(p) ⊕su(q) ⊕u(1) ⊂su(n).

Furthermore it is clear that su(2) ⊂su(p) ⊕su(q) so that the u(1) factor commutes with it. Thus no other algebra has zerocentraliser.

(ii) sp(n)Again the only su(2) S-subalgebra of sp(n) is the principal subalgebra.For the othersubalgebras we may writesu(2) ⊂c(p1) ⊕c(p2) ⊂c(n)p1 + p2 = n(5.1)If the copy of the su(2) is not principal in one of the c(pi) we can decompose this further21

till we havesu(2) ⊂c(p1) ⊕..... ⊕c(pi) ⊂c(n)Xjpj = n(5.2)where the copies of su(2) in each c(pi) are principal.We can decompose the adjointrepresentation of c(n) with respect to this direct sum of c(pi) within which the su(2) areprincipal. We find thatadjc(n) =(adjc(p1) ⊗1 ⊗.

. .

⊗1) ⊕... ⊕(1 ⊗. .

. ⊗1 ⊗adjc(pi))Lj,k1 ⊗.

. .

1 ⊗2pj ⊗1 . .

. 1 ⊗2pk ⊗1 .

. .

⊗1 ,where we have denoted the 2p dimensional representation of c(pi) by 2pi. The adjointrepresentation of c(pi) contains no singlets when decomposed with respect to a principalsu(2).

The 2pi representations deomposes into a single irreducible representation of thissu(2) and so the tensor product ...2pi ⊗...2pj.. decomposes with respect to the diagonalsu(2) to give |2pi −2pj|⊕...⊕2pi +2pj. From this we can see that adjc(n) contains a trivialrepresentation of the diagonal su(2) subalgebra if and only if pj = pk for some j ̸= k.(iii) so(n)The argument in this case is a little more involved.

bn again possesses no other su(2) S-subalgebras other than its principal. dn possesses int[(n −2)/2] S-subalgebras, but in factnone of these are maximal (not even the principal) and they correspond to the embeddingssu(2) ⊂so(p) ⊕so(q) ⊂so(2n) where p + q = 2n and p, q are both odd.

For our purposesit will be convenient to think of so(4) as simple. Its principal subalgebra is maximal.Now starting with any su(2) embedding in so(n) it is straightforward to show thatsu(2) ⊂so(p1) ⊕so(p2) ⊕.... ⊕so(pi) ⊂so(n)(5.3)where pi = 3, 4, 5, 7, 9, .... and the copy of su(2) in each simple ideal is principal.ForLj so(pj) to be maximal in so(n) and hence to have trivial centraliser we need thatn −Pj pj > 2.

Again we can decompose the adjoint representation of so(n) with respectto this direct product of principal su(2) subalgebras and we find thatadjso(n)=Mi1 ⊗1 ⊗. .

. ⊗adjso(pi) ⊗1 ⊗.

. .

⊗1Lj,k1 ⊗. .

. 1 ⊗pj ⊗1 .

. .

1 ⊗pk ⊗1 . .

. ⊗1 ,where we have denoted the pj dimensional representation of so(pj) by pj.

Again this willhave a trivial representation with respect to the diagonal su(2) if and only if pj = pk forsome j ̸= k. Notice that if one of the pj = 4, the associated embedding is non-integral.This is the only example of a W-algebra with no Kac-Moody components which is notpositive-definite and can be obtained in this way. The results for su(2) embeddings inclassical algebras are summarised in Table 2.It is also straightforward to extend the above analysis to search for W-algebras with noKac-Moody components and no other spin-2 fields than that associated with the Virasoroalgebra.

If there exists some algebra h such that su(2) ⊂h ⊂g, h is not simple, and22

su(2) is embedded diagonally in more than one of the simple ideals of h then there existsmore than one spin 2 field. This is because the decomposition of the direct product of Ncopies of su(2) with respect to its diagonal subalgebra contains N spin-1 representations.The cases where no such h exist are easy to read offfrom the tables, and are markedwith a check-mark in Table 1.

For the classical algebras, it is clear that only the principalembeddings result in only one spin-2 field.23

GIndexCharacteristicSpinsHspin-2E84000010002(10),3(10),4(10),5(6),6(4)A4 ⊕A40E88810001002(4),3(4),4(5),5(3),6(6)E6 ⊕A207(2),8(3),9B4 ⊕B3E812001001002(3),3,4(5),5(3),6(3)B5 ⊕B207(3),8(3),9,10(2)E816011001002(4),3,4(2),5(3),6(3)E7 ⊕A108(3),9(3),10(2),12E818401001012(3),3,4,5,6(4),7(2)B6 ⊕A108(3),9,10,11,12(2)E823211001012(2),3,4(2),6(3),7E7 ⊕A108(2),9(2),10,11,12(2),14E828010101012,3,4,5,6(2),8(3)B7✓09,10(2),12(2),14,15E840011101012(2),5,6(2),8,9,10(2)E7 ⊕A1012,14(2),15,18E852011010112,4,6,8,9,10,12(2)E8✓114,15,18,20E876011110112,6,8,10,12,14,15,18E8✓120,24E8124011111112,8,12,14,17,20,24,30E8✓P1E7391001002(6),3(4),4(5),5(3),6(3)A5 ⊕A20D6 ⊕A1E7631001012(4),3(2),4(3),5(2),6(4)D6 ⊕A107,8E71111101012(2),3,4(2),5,6(3)D6 ⊕A108(2),9,10E71591010112(2),,4,5,6(2),8(2)F4 ⊕A119,10,12Table 124

GIndexCharacteristicSpinsHspin-2E72311110112,4,6(2),8,9,10,E7✓112,14E73991111112,6,8,10,12,14,18E7✓P1E636101012(3),3(3),4(2),5(2),6(2)A5 ⊕A10E684110112,3,4,5,6(2),8,9C4, G2✓1E6156111112,5,6,8,9,12F4✓P1F41201=>002(6),3(4),4(2)A3 ⊕A1F43601=>012(3),3,4,5,6(2)C3 ⊕A1F46011=>012,3,4,6(2),8B4✓F415611=>112,6,8,12F4✓PG241≡>02(3),3A1 ⊕A1G2281≡>12,6G2✓PTable 1 (cont)GIndexSpinsHAnn(n+1)(n+2)62,3,4,...,n+1AnBnn(n+1)(2n+1)32,4,6,...,2nBnCnn(2n+1)(2n−1)32,4,6,...,2nCnDnn(n−1)(2n−1)32,4,6,...,2n-2,nBnCnPini(2ni+1)(2ni−1)3Pi 2, .., 2ni+Li CniPi>j |ni −nj| + 1, ..., ni + nj + 1Pi ni = n, ni ̸= njso(N)Pini(ni−1)(ni+1)12Pi 2, .., ni−12+ 2 Pi:ni=4 2+Li so(ni)+Pi:ni=4 2Pi>j |ni −nj| + 1, ..., ni + nj + 1Pi ni = n, n −1, ni ̸= njni ∈3, 4, 5, 7, .Table 225

6ConclusionsIn this paper we have found a connection between a general class of W-algebras andfinite Lie algebras. A crucial role in our arguments was played by the vacuum-preserving-algebra(vpa) which is the closed subalgebra of modes which annihilate both right and leftvacuua.

For ‘linear’ W-algebras one finds that the vpa contains a finite subalgebra whichprovides a useful tool for studying the properties of theories invariant under these W-algebras. To extend this idea to more general non-linear quantum W-algebras, it becamenecessary to consider deformable W-algebras which are defined for a range of c values inthe classical limit c →∞, and the subclass of algebras which behave ‘well’ under thislimit.

A natural criterion which arises is that of positive-definitness of the W-algebra,which essentially ensures that all the fields are important to the structure of the algebra.Reductive algebras are algebras which have positive-definite classical limits. As a resultwe were able to assign to each reductive W-algebra a finite Lie (super-)algebra and anembedding of su(1, 1).

The field content of the W-algebra is encoded in this embedding,with each representation of su(1, 1) in the decomposition of the Lie algebra being associatedto one of the Virasoro primary fields; the weight of that field being equal to (1 + thedimension)/2. By considering the structure of the commutation relations of the W-algebra,combined with the Virasoro Ward identities, we were also able to show that this finitealgebra was restricted to be of the form of a direct sum of a semi-simple algebra andan abelian algebra, namely a reductive Lie algebra.

This condition places considerablerestrictions on the possible field contents of W-algebras and on their commutation relations.As an example of the ideas presented, we considered the classical Poisson bracket algebras ofgeneralised Drinfeld-Sokolov type. The analysis here held out our theoretical predictions –to each such W-algebra we were able to assign a finite Lie algebra and an su(1, 1) embeddingin that algebra.

Conversely we used the construction to demonstrate the existence of aclassical W-algebra associated with each such pair.Our work suggests that W-algebras can be divided broadly into three categories: the re-ductive algebras considered in this paper, other deformable algebras, and ‘non-deformable’algebras which are only associative for specific values of c. There are several questionswhich present themselves concerning each category.Firstly, although we have demonstrated the existence of a classical W=algebra asso-ciated with each finite algebra and su(1, 1) embedding, it is not clear as yet whether onecan actually find a quantum W-algebra for each such embedding. The quantisation ofthese models has a lengthy history and is by no means over yet [29–32], although thereseem to be good arguments in favour of their existence.If one can find quantum W-algebras which satisfy the conditions of section 2, namely having a good classical limitwhich is positive-definite, then the question obviously arises, are they unique?

That is,to each such embedding can one uniquely ascribe a quantum W-algebra? We know of nocounterexamples.Our conditions, although they catch many of the W-algebras which have been studiedto date which have proven useful in conformal field theory, still exempt many W-algebras.Indeed we present such an exception with fields of spins 2,4, and 6.

In section 5 we havepresented what we hope will be a useful approach to the study of these algebras, namelyautomorphisms of Lie algebras which preserve a subalgebra. For the case we presented this26

was a Z2 automorphism which preserved the B2 subalgebra of A3. We hope that we canextend this to other cases.

Certainly the idea of dividing out by a finite group action isnot new, but rather the idea that we may be able to ascribe a W-algebra uniquely to eachsuch action, and even reconstruct the larger algebra from the smaller, is. There are goodreasons to believe that the absence of a good classical limit implies strong constraints onthe unitary representations of W-algebras, and we hope to return to this and other topicsin the future.As even more distant projects we can mention the idea that one may be able to showthat each W-algebra which occurs for a specific set of c-values is simply the extension ofa deformable W-algebra by primary fields of integer spin.

Thus, it may be instructive tolook for ‘maximal’ deformable subalgebras of such W-algebras.PB would like to thank the DOE of the U.S.A.for support under grant number DEFG02-90-ER-40560 and the NSF of the U.S.A. under grant PHY900036, and GMTW would liketo thank the SERC of the U.K. for a research assistantship. GMTW would like to thankthe EFI for hospitality at the initiation of this work.

PB and GMTW would like to thankthe Institute of Theoretical Physics at the University of California, Santa Barbara, forhospitality, and where they were in part supported by the National Science Foundationunder Grant No. PHY89-04035.

GMTW would like to thank Trinity College Cambridgefor a Rouse-Ball travelling studentship.We gladly thank H. G. Kausch for many useful conversations during the course ofthis project. GMTW would like to thank C. F. Yastremiz for useful comments on themanuscript.27

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