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PUPT–1284/ LAVAL-PHY-26/91

arXiv:hep-th/9109042v1 24 Sep 1991PUPT–1284/ LAVAL-PHY-26/91Singular Vectors and Conservation Lawsof Quantum KdV type equationsP. Di Francesco1Joseph Henry Laboratories,Princeton University,Princeton, NJ 08544.andP.

Mathieu2D´epartement de Physique,Universit´e Laval,Qu´ebec, Canada, G1K 7P4.We give a direct proof of the relation between vacuum singular vectors and conservationlaws for the quantum KdV equation or equivalently for Φ(1,3)-perturbed conformal fieldtheories. For each degree at which a classical conservation law exists, we find a quantumconserved quantity for a specific value of the central charge.

Various generalizations (N =1, 2 supersymmetric, Boussinesq) of this result are presented.09/911 Work supported by NSF grant PHY-8512793.2 Work supported by NSERC (Canada) and FCAR (Qu´ebec).

1. IntroductionThe equivalence between the Poisson Bracket associated with the second Hamiltonianstructure of the Korteveg-de Vries (KdV) equation, and the classical form of the Virasoroalgebra has the following implication: there exists an infinite number of integrals in involu-tion, whose densities are differential polynomials of the classical energy-momentum tensor[1].

The latter corresponds to the KdV field, and these integrals are the KdV conservationlaws. It is natural to conjecture that this result remains valid in the quantum case [2].This is equivalent to conjecture the complete integrability of the quantum KdV (qKdV)equation formulated canonically via the Virasoro algebra [3].

In the context of Confor-mal Field Theories, these integrals are trivially conserved due to ¯z independence of theirdensities. However, for minimal models, it was conjectured [4] that they remain conservedwhen the theory is driven off-criticality by the (thermal) Φ(1,3) perturbation [3] [5].The first nine qKdV (or equivalently Φ(1,3) perturbed minimal models) conservationlaws have been constructed and checked to commute [6].

As was already clear from thefirst few, they have the same structure as their classical counterparts. In particular thereexists a unique conservation law for each odd degree (=dimension or spin).

The essentialdifference, apart from normal ordering, is that the various coefficients are now polynomialsin the central charge c, with degree fixed by the dimension of the conservation law.In the study of the Φ(1,3) perturbed (p, p′) = (2, 2k + 1) non unitary minimal models,it has been observed that the conservation laws with dimension a multiple of 2k −1vanish: their density was found to be exactly proportional to the singular vector of level2k in the vacuum representation (modulo a total derivative) [7]. This connection betweenvacuum singular vectors and qKdV conservation laws is truly remarkable.

From the explicitexpression for these vectors [8], we get an explicit expression for the qKdV conservationlaws for each odd degree, albeit for a different (and unique) value of c in each case. Thatthe density of this conservation law is somewhat trivial (being a singular vector) doesnot blemish the very non-trivial fact that this conservation law commutes with the qKdVHamiltonian (before the singular vector is modded out!).

Therefore a proof of the aboveobservation boils down to a constructive proof of the existence of a conservation law ofdimension 2k −1 for a value of c outside the classical range (c very large), namely forc = 1 −3(2k −1)2/(2k + 1). This results in further support for the complete integrabilityof the qKdV equation.1

Such a proof has been given in [9], using the Feigin-Fuchs representation.Then,commutativity with the qKdV Hamiltonian is equivalent to commutativity with the Sine-Gordon HamiltonianHsin α−φ (φ a free field, φ(z)φ(w) ∼−log(z −w), α−= −(2p/p′)12and α+ = (2p′/p)12 ). On the other hand, the vertex operators e±iα+φ have respectivedimension 1 and (2p′/p) −1.The latter is integer only if p = 2.In that case, theanticommutator ofHe−iα+φ withHeiα+φ produces, by construction, a qKdV conservationlaw of dimension p′ −2 = 2k −1, which, also by construction, has density proportional tothe vacuum singular vector of dimension 2k.Another argument has been given in [10].

It explains the vanishing of some integralsin the (2, p′) minimal model as a result of an enhancement of symmetry. This is based on alevel-rank type duality for coset models, which translates into the equivalence of particularWn models for different values of n. Denoting a (in general non-unitary) minimal Wnmodel, characterized by two integers (p, p′), by W (p,p′)n,this equivalence reads: W (n,p′)n≡W (p′−n,p′)p′−n.

(Notice that in both cases, p takes the smallest allowed value, which is thedual Coxeter number of the corresponding SL(p) algebra). For the particular case n = 2(W2=Virasoro), this shows that any (2, p′) minimal model is equivalent to some Wp′−2model.

The integrable hierarchy associated with the latter is the SL(p′ −2) generalizedKdV equation [11], for which classical conservation laws exist at each degree not a multipleof p′ −2. It is natural to expect that no additional conservation law arises in the quantumcase.

Therefore the W (2,2k+1)2conservation laws of dimension a multiple of 2k −1 shouldvanish. This being a purely quantum effect, the conservation laws can vanish only if theirdensity is a vacuum singular vector.In this letter we present a third and more direct proof.It does not rely on theFeigin-Fuchs representation (and on the qKdV-Sine-Gordon relationship).

Also it doesnot require any a priori knowledge of symmetry enhancement. In the latter respect it canthus be applied to models for which no duality relation is known.

Our essential point isto show that the relation between vacuum singular vectors and qKdV conservation laws isencoded in a very simple way in the structure of the first few submodules of the vacuumVerma module. We then give the N = 1, 2 supersymmetric extensions of this result.

Inthe N = 1 case, the related conservation laws are those of the superconformal minimalmodels (p = 2, p′ = 4k), perturbed by the superfield bΦ(1,3). These are just the conservationlaws of the quantum version of the supersymmetric KdV equation [12].

For N = 2, thevacuum singular vectors yield conservation laws of the minimal models perturbed by thelowest dimension generator of the chiral ring. These models are naturally associated to2

a N = 2 supersymmetric KdV equation [13] [14]. Finally, from the KNS duality [10], itis also clear that a similar situation is to be expected for all W (n,p′)nmodels, for n ≥3.We show explicitly in the W3 case how the structure of the first vacuum submodulesdirectly implies a relation between singular vectors and conservation laws of the quantumBoussinesq equation.2.

Virasoro minimal modelsThe qKdV equation is given by [3]:∂tT = [T, H](2.1)whereH =I(TT)dζ(2.2)and the parenthesis indicates normal ordering:(AB)(w) =Idζζ −wA(ζ)B(w). (2.3)In terms of modes, one has [15]:(AB)(z) =∞Xn=−∞(z −w)−n−hA−hB(AB)n(w)(2.4)with(AB)n =Xm≤−hAAmBn−m +Xm>hABn−mAm(2.5)hA and hB are the conformal dimensions of A and B respectively.

The qKdV conservationlaws are integrals Hn−1 =Hhndζ such that [Hn−1, H] = 0.Hence a necessary andsufficient condition for hn to be a conserved quantity is that:I0dwIwdz(TT)(z)hn(w) = 0(2.6)Expanding (TT)(z) in modes, one sees that this is equivalent to [6]:Idw(TT)−3hn(w) = 0(2.7)3

In other words, (TT)−3hn must be a total derivative, i.e of the form L−1(...). The explicitform of (TT)−3 in terms of Virasoro modes (T(z) = P z−n−2Ln) is:(TT)−3 = 2Xm≥0L−m−2Lm−1(2.8)We now proceed to prove the following result: for the minimal models (p = 2, p′ =2k + 1), the vacuum singular vector at level 2k, denoted by v2k, is a qKdV conserveddensity.

This amounts to prove that(TT)−3v2k ≃0(2.9)where the symbol ≃means equality modulo total derivative. Since the action of the positiveVirasoro modes vanish on a singular vector, this is equivalent to(2L−2L−1 + 2L−3L0)v2k ≃2(2k −1)L−3v2k ≃0(2.10)The key point of our argument is to notice that v2k is itself degenerate at level 3.

Thissingular vector v2k+3 reads:v2k+3 = (L−3 −1k + 1L−1L−2 +12(k + 1)2 L3−1)v2k(2.11)Moreover v2k+3 also belongs to the Verma module built on L−1I, the level 1 descendentof the vacuum. Therefore, after quotienting the vacuum tower by its first descendent (i.e.setting L−1I = 0), we find that v2k+3 = 0, which immediately results in:L−3v2k ≃0,(2.12)and completes the proof of the above statement.It is very easy to convince oneself that this scenario can only apply to minimal modelswith p = 2.

Indeed the first singular vectors in the vacuum module of a generic (p, p′)model occur at levels 1, p′ −1 and p′ + 2p −2. To see this, recall that the singular vectorsof the vacuum module arise at levels [16]:(2npp′ + p ± p′)2 −(p −p′)24pp′,n ∈Z(2.13)To make contact with qKdV conservation laws, a gap of dimension at most 3 is requiredbetween vp′−1 and vp′+2p−2.

This is possible only for p = 2 (one has p ≥2 in the Virasorocase). Notice that we need only concentrate on these first singular vectors, as vp′−1 is theonly singular vector of the vacuum module which is not a descendent of L−1I.4

3. N=1 superconformal minimal modelsIntroducing the super energy-momentum tensor:bT(Z) = 12G(z) + θT(z)(3.1)where Z stands for the doublet (z, θ), one can write the supersymmetric qKdV equationas [12](dZ = dzdθ and D = θ∂z + ∂θ):∂t bT = [ bT, bH](3.2a)bH =IdZ( bTD bT)(3.2b)Normal ordering is defined by:( bA bB)(Z2) =IdZ2θ12Z12bA(Z1) bB(Z2)(3.3)with θ12 = θ1 −θ2 and Z12 = z1 −z2 −θ1θ2.

Modes of the superfields are introduced inthe natural way as:bA(Z1) =Xr∈Z+h ˆAZ−r−h ˆA12bAr(Z2) + θ12Xn∈Z+h ˆA+ 12Z−n−h ˆA−1212bAn(Z2)(3.4)Supersymmetric qKdV conservation laws bHn−12 =HdZbhn are then characterized by theconditionIdZ2IdZ1( bTD bT)(Z1)bhn(Z2) = 0(3.5)By expanding bTD bT in modes and using the super Cauchy theorem:IdZ1θa12Z−n−112f(Z1) = 1n! (∂nD1−af)(Z2),(a = 0, 1)(3.6)one can rewrite (3.5) as:( bTD bT)−3bhn = G−12 (...) ≃0(3.7)Now for the superconformal minimal models (p = 2, p′ = 4k), the first three singularvectors of the vacuum tower arise at level 12 (bv 12 = G−12 I), 2k −12 and 2k + 1.

The levelsare given by (2.13) divided by 2 [17]. Let us now prove that bv2k−12 is a conserved densityfor (3.2a).

In that case, (3.7) reduces to(4k −2)L−3bv2k−12 ≃0(3.8)5

On the other hand the explicit expression for the singular vector bv2k+1 is:bv2k+1 = (G−32 −12k L−1G−12 )bv2k−12(3.9)Now we observe that bv2k+1 is a descendent of both bv2k−12 and G−12 I. Therefore it vanishesin the quotient of the vacuum module by its first singular vector.

Applying G−32 to ther.h.s. of (3.9), we obtain finally:k + 1kL−3bv2k−12 ≃0(3.10)which is the desired result.Here again, one can ask whether p = 2 is the only case for which some conservationlaws are given by singular vectors.

In fact the level difference between the second and thirdsingular vectors for generic (p, p′) models is p −12. Thus with p = 3 we still get a leveldifference smaller than 3.

The relevant vectors are bvp′−12 and bvp′+2, withbvp′+2 = (G−52 + aG−32 L−1 + bG−12 L2−1 + cG−12 L−2)bvp′−12for some values of a, b and c. Hence G−52 bvp′−12 ≃0 but this does not imply (3.8). Thereforep = 2 is indeed the only possibility.4.

N=2 superconformal minimal modelsThere are three integrable perturbations of the N = 2 superconformal minimal modelswithc = 3(1 −2k + 2)(4.1)corresponding to perturbations by the chiral fields ˜Φl of dimension l/2(k +2), with l = 1, 2and k [13][14]. Each of these cases can be associated to a N = 2 supersymmetric qKdVequation [14].

For the l = 1 perturbed theory, there are conservation laws for all values ofthe degree not a multiple of k+1, in which case the conserved density is proportionnal to avacuum singular vector [13]. The Hamiltonian of the corresponding N = 2 supersymmetricqKdV equation is [14]:˜H =IdZ[( ˜T( ˜T ˜T)) + c −316 ( ˜T[D+, D−] ˜T)](4.2)6

Here we have introduced the N = 2 super energy-momentum tensor˜T(Z) = J(z) + 12θ−G+(z) −12θ+G−(z) + θ+θ−T(z)(4.3)and Z now stands for (z, θ+, θ−), dZ = dzdθ−dθ+, the superderivatives being D± =∂θ∓+ θ±∂z. Normal ordering is now defined by:( ˜A ˜B)(Z2) =IdZ1θ+12θ−12Z12˜A(Z1) ˜B(Z2)(4.4)with Z12 = z1−z2−θ+1 θ−2 +θ−1 θ+2 .

Finally our normalizations are fixed by the commutator:{G+r , G−s } = 2Lr+s + 2(r −s)Jr+s + c3(r2 −14)δr+s,0. (4.5)Proceeding as before we find that a conserved density ˜hn of the N = 2 super qKdVequation ∂t ˜T = [ ˜T, ˜H] must satisfy( ˜T( ˜T ˜T)) + c −316 ( ˜T[D+, D−] ˜T)θ+θ−−3˜hn ≃0(4.6)Here the symbol ≃means equality modulo terms of the form G±−12 (...), and the superscriptθ+θ−reminds that this mode refers to the θ+θ−component of the superfield inside thebrackets.We now argue that (4.6) is always satisfied by hk+1 = ˜vk+1, the third vacuum singularvector (the two first ones are ˜v±12 = G±−12 I).

In that case, (4.6) can be rewritten (after afair amount of algebra):−32k + 1k + 2[L−3 −4(k + 2)J−1J−2]˜vk+1 ≃0(4.7)On the other hand, ˜vk+1 is itself degenerate at level 2 [18], and the singular vector reads:˜vk+3 =−2(k + 1)(k + 2)J−2 + 2(k + 2)L−1J−1 −k + 22G−−12 G+−32+ k + 22G+−12 G−−32 + G−−12 G+−12 L−1 −L2−1˜vk+1(4.8)Moreover, this vector is also a descendent of G±−12 I, therefore vanishes in the quotient ofthe vacuum module by its first two singular vectors. Acting then on (4.8) by J−1 yields,modulo total superderivatives:k + 24[L−3 −4(k + 2)J−1J−2]˜vk+1 ≃0(4.9)This is equivalent to (4.7) and completes our proof.7

5. W3 minimal modelsThe quantum version of the Boussinesq equation is naturally defined in terms of theW3 algebra by [3]:∂tT = [T, H]∂tW = [W, H](5.1)whereH =IdzW(5.2)As before T denotes the energy-momentum tensor and W is a spin 3 conserved current.The conservation laws Hn−1 =Hdzhn are characterized by the condition:W−2hn ≃0(5.3)This condition is satisfied by the vacuum singular vector vp′−2 of the W (p=3,p′=3k±1)3min-imal models .

This vector is itself doubly degenerate at level 2, and one of the singulardescendents reads:vp′ = [W−2 −2p′ −1L−1W−1]vp′−2(5.4)Like all singular vectors in the vacuum module except vp′−2, vp′ is a descendent of acombination of L−1I and W−1I [19]. As W−1I = 12[W0, L−1]I = 12W0L−1I, the vector vp′vanishes in the quotient of the vacuum module by L−1I.

This means that W−2vp′−2 ≃0,and thatHvp′−2 is a conservation law for the quantum Boussinesq equation (5.1).6. ConclusionWe have presented a new and simple proof of the relation between a vacuum singularvector and some conservation laws for qKdV type equations or equivalently for a specialclass of off-critical minimal models.

For the quantum systems considered, this implies theexistence, for a specific value of the central charge, of an integral which commutes withthe defining Hamiltonian, at every degree for which a classical conservation law exists.Going back to the general problem of finding the conservation laws for these quantumequations, recall that these are expected to be polynomials of c with degree fixed by thedimension of the law. We just provided one particular value of each of these polynomials,for a value of c function of the dimension of the conservation law.

This suggests that theactual conservation laws around these values of c (and therefore for all values of c, dueto polynomiality) might be some simple deformation of the corresponding singular vectorexpressions. Although quite appealing, this program admittedly lacks a systematic line ofattack.8

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