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PUPT–1300/ LAVAL-PHY-28/91

arXiv:hep-th/9112063v1 20 Dec 1991PUPT–1300/ LAVAL-PHY-28/91Integrability of the Quantum KdV equationat c = −2.P. Di Francesco1Joseph Henry Laboratories,Princeton University,Princeton, NJ 08544.andP.

Mathieu2 and D. S´en´echal3D´epartement de Physique,Universit´e Laval,Qu´ebec, Canada, G1K 7P4.We present a simple a direct proof of the complete integrability of the quantum KdVequation at c = −2, with an explicit description of all the conservation laws.12/911 Work supported by NSF grant PHY-8512793.2 Work supported by NSERC (Canada) and FCAR (Qu´ebec).3 Work supported by NSERC (Canada)

The seminal work of Zamolodchikov [1] on integrable perturbed minimal models hasbrought to light the relevance of quantum KdV (qKdV) equations in conformal field theory[2][3]. For instance the integrals of motion which are preserved by the φ(1,3) perturbationare known to be exactly equivalent to the qKdV conservation laws [1][3][4].

(By ‘qKdVequation’ we mean the KdV equation formulated canonically via the quantum form ofits second Hamiltonian structure). The full integrability of the qKdV equation has beenwidely expected but proven rigorously only very recently by Feigin and Frenkel [5].

Byusing the relationship between the qKdV equation and the quantum sine-Gordon equationthey have been able to establish the existence of an infinite number of mutually commutingconservation laws. However, their argument is not constructive, and it remains an inter-esting problem to find an explicit form for these conservation laws.

We already know thatone cannot expect a recursive description of the commutation laws `a la L´enard since theqKdV equation is not bi-Hamiltonian[3].A step in this direction was made in [6][7] where it was shown that the conservationlaw of dimension 2k −1 is exactly proportional to the vacuum singular vector of the non-unitary minimal models with (p, p′) = (2, 2k + 1), 4 whose expression is known explicitly[9].Another result is presented here: we display the explicit expression of all conservationlaws at c = −2 and prove their commutativity.The form of these conservation lawshas been guessed some time ago by Sasaki and Yamanaka[2]. For this purpose we use arepresentation of the energy-momentum tensor in terms of two complex fermions of spin1, for which the dynamics is very simple.

It turns out that the conserved densities arebilinear in the fermionic fields. It is then a straightforward exercise to reexpress them interms of the energy-momentum tensor, thus recovering the explicit form of the conserveddensities.Although the relation between singular vectors and qKdV conservation laws is easilygeneralized to extended conformal algebras[6] (N = 1, 2 supersymmetry, WN, etc.) this isnot so for the result described here.

Nevertheless, since the two complex spin 1 fermionsprovide a representation of the full W∞algebra [10] (which is linear and thus not universal)one could reinterpret the conserved quantities in terms of fermionic fields in a WN contextand recover a partial form of the conserved densities of the sl(N) qKdV equation (by this4 This was first observed in [8].1

we mean that some, but not all, of the coefficients of the various terms in the conserveddensities are fixed). This will be illustrated below.At this point we are convinced that it will be very hard to go beyond these results,that is to fully characterize in a constructive way the qKdV conservation laws for arbitraryvalues of c.Let us now turn to the derivation of the main result of this note.

The qKdV equationreads˙T = [T, H],H =Idw (TT)(w)(1)where the dot stands for a time derivative and the parentheses for normal ordering, i.e. (AB)(w) =Idxx −wA(x)B(w).

(2)We will make extensive use of the following rearrangement lemma[11]:(A(BC)) = (B(AC)) + (([A, B])C)(3)where the normal ordered commutator is easily computed from a generic OPE,A(z)B(w) =XrCr(w)(z −w)r(4)to be([A, B])(w) =Xr>0(−1)r+1r!∂rCr(w)(5)At c = −2, T can be represented by the bilinearT = (φψ)(6)where φ and ψ are both fermions of spin 1 with OPEφ(z)ψ(w) =−1(z −w)2,ψ(z)φ(w) =1(z −w)2(7)This is of course nothing but a ghost representation (see the last remark at the end of thisletter). From the above rearrangement lemma, one easily finds that(TT)(z) = 12(φ′′ψ + φψ′′)(z)(8)2

where a prime stands for a derivative w.r.t. the complex coordinate.

The evolution of thefermionic fields can be calculated from˙φ = [φ, H],˙ψ = [ψ, H](9)which yields˙φ = φ′′′,˙ψ = ψ′′′(10)The dynamics of these fermionic fields is thus extremely simple. This system of equationshas an infinite number of integrals of the motion.

However, we are only interested in thosethat can be rewritten in terms of T. A infinite set of such conserved quantities isHk+1 =Idz(φ(k)ψ)(z)(11)where φ(k) = ∂kz φ. For k even this can be reexpressed in terms of T as follows:H2n−1 = 2n−1nIdz(←T n)(z)(12)where the notation (←T n) means a nesting of the normal ordering towards the left:(←T n) = (.

. .

(((TT)T)T) . .

.T)(n factors)(13)This is exactly the form of the first few conservation laws obtained in [2] for the casec = −2. On the other hand, for k odd the conserved integrals (11) cannot be expressed interms of T.We will prove (12) by showing that the exact expression for (←T n) is(←T n) = n2nφ(2n−2)ψ + φψ(2n−2)(14)for n > 1, using a simple recursive argument.

From (8), we see that (14) is satisfied forn = 2. Let us suppose that it is true for n and calculate(←T n+1) = ((←T n)T) = n2nn(φ(2n−2)ψ) + (φψ(2n−2))o(φψ)(15)Let us consider the first term: (φ(2n−2)ψ)(φψ), to which we apply the lemma (3) withA = (φ(2n−2)ψ) and B = φ:(φ(2n−2)ψ)(φψ) = (φ((φ(2n−2)ψ)ψ)) + (([(φ(2n−2)ψ), φ])ψ)(16)3

A further rearrangement of the first term of the r.h.s. yields(φ(2n−2)ψ)(φψ) = (φ(ψ(φ(2n−2)ψ))) + (φ([(φ(2n−2)ψ), ψ])) + (([(φ(2n−2)ψ), φ])ψ)(17)Again, with some rearrangement, the first term on the r.h.s.

of the above equation can bewritten as(φ(ψ(φ(2n−2)ψ))) = −(φ(φ(2n−2)(ψψ))) + (φ(([ψ, φ(2n−2)]+)ψ))(18)(The minus sign comes from the fermion anticommutation, and [A, B]+ = AB + BA).Since (ψψ) = 0 and ([φ(n), ψ(m)]+) = 0, this term vanishes. The second piece in (15) canbe treated similarly, with the result:(φψ(2n−2))(φψ) = −(([(φψ(2n−2)), ψ])φ) −(ψ([(φψ(2n−2)), φ]))(19)The needed normal ordered commutators are([(φ(m)ψ), ψ]) = (−1)mm + 2 ψ(m+2)([(φ(m)ψ), φ]) = 12φ(m+2)([(φψ(m)), φ]) = (−1)mm + 2 φ(m+2)([(φψ(m)), ψ]) = 12ψ(m+2)With these results one readily obtains that(←T n+1) = n + 12n+1 (φ(2n)ψ + φψ(2n))(20)which is exactly the form (14).

This recursive argument proves (14), whose integrationestablishes the equivalence between (11) and (12).It is straightforward to check that all the conservation laws (11) commute with eachother. This provides a direct proof of the complete integrability of the qKdV equation atc = −2.Now some remarks are in order.

First, if we rearrange the conserved densities withthe usual normal ordering nested towards the right, one finds that (except for H5) theconservation laws are in no way simpler for c = −2 than for any other value of c. It isreally the nesting towards the left which makes their expressions in terms of T simple.55 The symmetry breaking between left and right nesting orders stems from the lemma (3)which has no simple equivalent in the other nesting order.4

This somehow looks like an accident with no analog for the extended cases, as far as wecan see. More explicitly, consider (T(TT)):(T(TT)) = 18(φ(4)ψ + φψ(4)) + 12(φ′′ψ′′)(21)Up to a total derivative this is equal to (←T 3).

However, if we introduce another factor ofT on the l.h.s. of (21), quartic terms appear.

In order to recover the conserved density(φ(6)ψ) we have to introduce additional terms, namely (T ′′(TT)) and (T ′′T ′′), reobtainingin this way the usual form of the qKdV conserved densities of degree 8 evaluated at c = −2.Secondly, it was pointed out in [10] that at c = −2, all the generators of the W∞algebra (which is W1+∞with the spin-1 field decoupled) can be expressed as bilineardifferential polynomials in two spin-1 complex fermions. It turns out that by consideringthe first N −1 fields as fundamental and the remaining ones as composites, one recoversthe WN algebra.

This is certainly a remarkable result given that W∞is a linear algebra.It is thus natural to ask whether one could in this way recover all the conservation laws ofthe sl(N) qKdV equation. Unfortunately, as we will see with a few examples, this is notthe case.

Nevertheless, it is interesting to notice that we do get some information in thesense that in a candidate multi-parameter expression for a conserved density, written interms of the WN generators, we can fix a certain number of these parameters simply byrequiring that the conserved quantities be of the form (φ(k)ψ).To be more explicit, let us consider the W3 algebra, generated by T, given in (6), andthe spin-3 field W whose expression in terms of the fermions φ and ψ is [10]W =1√6(φ′ψ −φψ′)(22)The sl(3) analog of the qKdV equation is the quantum Boussinesq (qBsq) equation˙T = [T, H],˙W = [W, H](23)withH =Idz W(z)(24)In terms of the fermionic fields the evolution equations (with a trivial rescaling of the timevariable) translate into˙φ = φ′′,˙ψ = −ψ′′(25)5

Here again the integrals (11) are conserved quantities. Now let us see whether the integralsHk for k ̸= 0 (mod 3) can be uniquely reexpressed in terms of T and W to yield the qBsqconservation laws at c = −2.

The first three qBsq conservation laws:Hdz T,Hdz W andHdz (TW) are readily seen to be of the formH(φ(k)ψ) with k = 0, 1, 3. The next one isIdz(WW) + 13b2(T(TT)) −160(2 −9b2)(T ′T ′)(26)where b2 = 16/(22+5c).6 The term (T(TT)) has already been calculated above (eq.

(21))and is equal to 34(φ(4)ψ), up to total derivatives. On the other hand we have(WW) = 16 ((φ′ψ)(φ′ψ) −(φ′ψ)(φψ′) −(φψ′)(φ′ψ) + (φψ′)(φψ′))(T ′T ′) = ((φ′ψ)(φ′ψ) + (φ′ψ)(φψ′) + (φψ′)(φ′ψ) + (φψ′)(φψ′))The second and third terms of these two expressions will produce an non-vanishing quarticcontribution (φ(φ′(ψψ′))) and these should then be eliminated.This fixes the relativecoefficient of the two terms (WW) and (T ′T ′), i.e.

(WW) + 16(T ′T ′) = 49(φ(4)ψ) + total derivativeIn fact, −(2 −9b2)/60 is indeed 1/6 when c = −2. However it is clear that the relativecoefficient of (T(TT)) and (WW)+ 16(T ′T ′) is not fixed since both expressions are propor-tional to (φ(4)ψ) modulo total derivatives.

An analysis of the next conservation law showsthat some but not all of the coefficients are fixed. Similarly, in the W4 case, the spin-4field (called V ) is proportional to [10]:V = (φ′′ψ −3φ′ψ′ + φψ′′)(27)The first three conservation laws (11) can be written asHT,HW andH[V + γ(TT)], withγ not fixed.Thirdly, concerning the WN case, one could expect that there is a value of c = c(N) atwhich the conserved quantities become very simple when rewritten in terms of a suitablenumber of ghost pairs.

From the point of view of Hamiltonian reduction [12] the WNanalog of c = −2 for N = 2 isc(N) = −N 4 + 2N 3 −N(28)6 The corresponding expression in [3] has a misprint in it.6

The corresponding minimal model is (p, p′) = (1, N). For W3, this requires a representationof T and W in terms of three pairs of ghosts.

However, this does not yield a simplificationof the conservation laws and does not represent a great advance when compared to thegeneric Feigin-Fuchs representation in terms of two bosonic fields, valid for all values of c.Let us now come back to the qKdV case.It is well known [13] that the energy-momentum tensor can be expressed in terms of a pair of anticommuting ghosts b and cwith weights λ and 1 −λ, and OPEb(z)c(w) =1z −w,c(z)b(w) =1z −was follows:T = (1 −λ)b′c + λbc′The corresponding central charge is 1 −12(λ −12)2. Previously we considered the caseλ = 0, with φ = b′ and c = ψ.

Let us now see what happens when λ ̸= 0. Modulo a totalderivative, one finds that(TT) = ( 43λ(1 −λ) + 1)(b′′′c) −2λ(1 −λ)(b(b′(cc′))(29)Hence for λ ̸= 0 (or λ ̸= 1, which is equivalent) (TT) contains quartic terms, The evolutionequation for the b, c fields takes the form (with time rescaled)˙b = ( 43λ(1 −λ) + 1)b′′′ + 2λ(1 −λ)[2(b(b′c′)) + (b(b′′c))](30)and a similar equation for ˙c.

Unless λ = 0 or 1, these are complicated coupled evolutionequations, for which (11) are no longer conserved quantities. This illustrates clearly inwhich sense the central charge c = −2 is special.7

References[1]A.B. Zamolodchikov, JETP Lett.

46 (1987) 160; Int. Jour.

Mod. Phys.

A3 (1988)4235; Adv. Stud.

in Pure Math.(1989).[2]R. Sasaki and I. Yamanaka, Adv.

Stud. in Pure Math.

16 (1988) 271.[3]B.A. Kuperschmidt and P. Mathieu, Phys.

Lett. B227 (1989) 245.[4]T.

Eguchi and S.K. Yang, Phys.

Lett. B224 (1989) 373; T.J. Hollowood and P. Mans-field, Phys.

Lett. B226 (1989) 73.[5]B.

Feigin and E. Frenkel, Free field resolution in affine Toda field theories, RIMS-Harvard preprint (october 1991).[6]P. Di Francesco and P. Mathieu, PUPT-1284/LAVAL-PHY-26/91 (september 1991).[7]T.

Eguchi and S.K. Yang, Phys.

Lett. B335 (1990) 282; A. Kuniba, T. Nakanishi andJ.

Suzuki, Nucl. Phys.

B356 (1991) 750;[8]P.G.O Freund, T.R. Klassen and E. Melzer, Phys.

Lett. B229 (1989) 243.[9]L.

Benoit and Y. Saint-Aubin, Phys. Lett.

B215 (1988) 517; M. Bauer, P. DiFrancesco, C. Itzykson and J.B. Zuber, Nucl. Phys.

B362 (1991) 515.[10]H. Lu, C.N.

Pope, X. Shen and X.J. Wang, Phys.

Lett. B267 (1991) 356.[11]F.A.

Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Nucl. Phys.

B304(1988)348.[12]M. Bershadsky and H. Ooguri, Comm.

Math. Phys.

126 (1989) 49.[13]D. Friedan, E. Martinec and S. Shenker, Nucl.

Phys. B271 (1986) 93.8

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