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Kontsevich - Miwa Transform of the

arXiv:hep-th/9111051v2 28 Nov 1991Kontsevich - Miwa Transform of theVirasoro Constraintsas Null-Vector Decoupling Equations ∗A.M.SemikhatovTheory Division, P.N.Lebedev Physics Institute53 Leninsky prosp., Moscow SU 117924 USSR(November 25, 1991)AbstractWe use the Kontsevich–Miwa transform to relate the Virasoro constraints on integrablehierarchies with the David-Distler-Kawai formalism of gravity-coupled conformal theories.The derivation relies on evaluating the energy-momentum tensor on the hierarchy atspecial values of the spectral parameter. We thus obtain in the Kontsevich parametrizationthe ‘master equations’ which implement the Virasoro constraints and at the same timecoincide with null-vector decoupling equations in an ‘auxiliary’ conformal field theory onthe complex plane of the spectral parameter.

This gives the operators their gravitationalscaling dimensions (for one out of four possibilities to choose signs), with the α+ beingequal to the background charge Q of an abstract bc system underlying the structure ofthe Virasoro constraints. The formalism also generalizes to the N-KdV hierarchies.∗This is a revised version of the author’s virtual paper A Kontsevich - Miwa Transform of theVirasoro Constraints on KP and Generalized KdV Hierarchies (Oct. 1991) which consisted to aconsiderable degree of arithmetical errors and is herewith nullified.1

1IntroductionThe interest in Matrix Models [3, 13, 21] has been stimulated, besides their applicationsto matter+gravity systems [8, 20, 7, 18], also by intriguing relations they have with completelyintegrable equations and the intersection theory on the moduli space of curves [12, 39, 31, 25, 40].On the other hand, as to the foundation of the matrix model approach by itself, a challengingproblem is the direct proof of its equivalence to the conformal field theory formalism for quantumgravity [23, 10, 6]. Assuming that this equivalence exists, as is suggested by a circumstantialevidence, one then has to believe that certain ingredients of conformal field theory satisfyintegrable equations.

These, however, seem to be a long way from the equations which areknown to hold for conformal field theory correlators [11, 24].As we will show, the role of an ‘intermediary’ on the way from non-linear KdV-like equationsto conformal field theory is played by the Virasoro constraints on integrable hierarchies (whichin another guise are recursion relations in topological theories) [17, 8, 22, 29]. The Virasoroconstraints are the heart of the matrix models’ applications to both gravity-coupled theoriesand the intersection theory.

The case studied in most detail is the Virasoro-constrained KdVhierarchy whose relation to the intersection theory on moduli space of Riemann surfaces hasbeen discussed in [39].To reveal this role of the Virasoro constraints, we will adopt the approah which has provedfruitful in establishing the relation between matrix models and the universal moduli space: Thatis, we borrow from the matrix model due to Kontsevich the choice of independent variables.The Kontsevich matrix model provides a combinatorial model of the universal moduli space[25] and, as such, serves as an important step in demonstrating the KdV hierarchy in theintersection theory on the moduli space. Note, however, that the Kontsevich model is not ofthe form of the matrix models considered previously, which raises the question of its equivalenceto one of the “standard” models.

The crucial point in studying this equivalence is, again, theproof of the Virasoro constraints satisfied by the Kontsevich matrix integral [40, 27]. Oncethe constraints are established, one is left with “only” the proof that they specify the modeluniquely.The Kontsevich model by itself, as well as the existing derivations of the Virasoro constraints,appear to be tied up to the KdV hierarchy and thus to the l = 2 minimal models.

There exist,at the same time, matrix models of other minimal conformal theories coupled to gravity, whichcorrespond to higher generalized ‘KdV’ hierarchies [14] 1. Although neither the interpretationof Virasoro-constrained N-KdV hierarchies in terms of moduli spaces, nor the correspondingKontsevich-type matrix integrals are known, it would not be natural if the N = 2 case wereexceptional.

Thus another problem is whether a unified approach exists which allows one torecast Virasoro constraints on N-KdV (and hopefully other) hierarchies into Ward identities ofa would-be Kontsevich-type matrix integral.In this paper we take as a starting point the Virasoro constraints in the usual parametriza-tion and then investigate whether they can be recast into the Kontsevich variables. Thereare obvious similarities between Kontsevich’ parametrization and the Miwa transform used in1 In this paper we will restrict ourselves to the series associated to sl(N) Kac-Moody algebras (and thereforethe A-series minimal models [4], and we will call these the N-KdV hierarchies.

Virasoro constraints on the N-KdV hierarchies admit a unified treatment, which is in turn a specialization of a general construction applicableto hierarchies of the r−matrix type [38].2

the KP hierarchy [32]. We thus attempt to proceed with a general Miwa transformation.

Aswe will see, the Virasoro constraints are not reformulated nicely unless one restricts the Miwatransformation to the Kontsevich one, yet we find it instructive to see at which step the Miwaparametrization fails to work.Pulling back the Virasoro generators to the Kontsevich parametrization seems only possiblefor the combinationPn≥−1 Lnz−n−2 of the Virasoro generators, and only at special valuesof the spectral parameter z. The resulting relations are candidates for the Ward identitiescorresponding to a Kontsevich-type matrix integral.

For the N-KdV hierarchies these relationsare the analogues of the “master equation” of ref.[26]. Further, we show in the KP case thatour version of the master equation (which, though very similar to, is not quite the same as theone from [26]) happens to be nothing but an equation on correlation functions in an ‘auxiliary’conformal field theory, stating the decopling of a certain null vector.

This is in the classicalspirit of [2], yet the applicability to Virasoro-constrained hierarchies seems to be new. Recall inparticular that according to the basic matrix models ideology, the resulting equations that holdon an abstract (spectral-parameter) CP1 are in fact non-perturbative.

It is thus very reassuringto recover in our approach the results of refs. [10, 6]!2Virasoro action on the KP hierarchy2.1.

The KP hierarchy is described in terms of ψDiffoperators [5] as an infinite set ofmutually commuting evolution equations∂K∂tr= −(KDrK−1)−K,r ≥1(2.1)on the coefficients wn(x, t1, t2, t3, . .

.) of a ψDiffoperator (more precisely, a ψDiffsymbol) Kof the form (here and in the sequel, D = ∂/∂x)K = 1 +Xn≥1wnD−n(2.2)Introduce a ‘matrix model potential’ξ(t, z) =Xr≥1trzr(2.3)The wave function and the adjoint wave function are then defined byψ(t, z) = Keξ(t,z),ψ∗(t, z) = K∗−1e−ξ(t,z)(2.4)where K∗is the formal adjoint to K. The wave function ψ is an eigenfunction of the Laxoperator:Qψ(t, z) = zψ(t, z),Q ≡KDK−1(2.5)functions.

The basic property of the wave functions is their relation to the tau function:ψ(t, z) = eξ(t,z)τ(t −[z−1])τ(t),ψ∗(t, z) = e−ξ(t,z)τ(t + [z−1])τ(t)(2.6), (2.7)3

where,t ± [z−1] = (t1 ± z−1, t2 ± 12z−2, t3 ± 13z−3, . .

. )(2.8)2.2.

Now we introduce a Virasoro action on the tau function: The Virasoro generatorsread,Lp>0=12p−1Xk=1∂2∂tp−k∂tk+Xk≥1ktk∂∂tp+k+ (a0 + (J −12)p) ∂∂tpL0=Xk≥1ktk∂∂tk+ 12a20 −124Lp<0=Xk≥1(k −p)tk−p∂∂tk+ 12−p−1Xk=1k(−p −k)tkt−p−k + (a0 + (J −12)p)(−p)t−p(2.9)They satisfy the algebra[Lp, Lq] = (p −q)Lp+q + δp+q,0(−p3)(J2 −J + 16)(2.10)which shows, in particular, the role played by the parameter J. (Shifting L0 as L0 7→L0−12(J2−J + 16) we recover in (2.10) the standard central term −δp+q,0(p3 −p)(J2 −J + 16).) It will bequite useful to introduce an “energy-momentum tensor”T(u) =Xp∈Zu−p−2Lp(2.11)Using this to deform the tau function asτ(t) 7→τ(t) + δτ(t) = τ(t) + T(u)τ(t),(2.12)we translate this action into the space of dressing operators K. The result is [33] that K getsdeformed by means of a left multiplication,δK = −T(u)K,where T(u) is the energy-momentum tensor in another guise, now a pseudodifferential operator2T(u) = (1 −J)∂ψ(t, u)∂u◦D−1 ◦ψ∗(t, u) −Jψ(t, u) ◦D−1 ◦∂ψ∗(t, u)∂u(2.13)Thus, T(u) reproduces the structure of the energy-momentum tensor of a spin−J bc theory[15].

In its own turn, T(u) can be expanded in powers of the variable u, which was introducedin (2.11) and has now acquired the role of a spectral parameter, asT(u) =Xp∈Zu−p−2Lp(2.14)2We have chosen the irrelevant parameter a0 = 12, see [33] where a0 = N + 12.4

This gives the individual Virasoro generators (which are a particular case of the general con-struction applicable to integrable hierarchies of the r−matrix type [38])Lr ≡K(J(r + 1)Dr + PDr+1)K−1)−,P ≡x +Xr≥1rtrDr−1(2.15)2.3. The KP hierarchy can be reduced to generalized N-KdV hierarchies [5] by imposingthe constraintQN ≡L ∈Diff(⇒QNk ∈Diff, k ≥1)(2.16)requiring that the Nth power of the Lax operator be purely differential.

Then, in a standardmanner, the evolutions along the times tNk , k ≥1, drop out and these times may be set tozero. The rest of the tn are conveniently relabelled as ta,i = tNi+a, i ≥0, a = 1, .

. .

, N −1.As to the Virasoro generators, only LNj are compatible with the reduction in the sense thatthey remain symmetries of the reduced hierarchy without imposing any further constraints [35].The value of J can be set to zero [35, 36]. Then, after a rescaling, the generatorsL[N]j= 1N (KXa,i(Ni + a)ta,iDN(i+j)+aK−1)−(2.17)span a Virasoro algebra of their own.Again, we find it very useful to construct the energy-momentum tensor corresponding tothese generators.

Recall that the spectral parameter of the N-KdV hierarchy is ζ = zN. ThenT[N](ζ)(dζ)2≡Xj∈Zζ−j−2L[N]j (dζ)2=NKXb,j(Nj + b)tb,jDNj+b 1z2δ(DN, zN)K−1−(dz)2(2.18)whereδ(u, v) =Xn∈Zuvn(2.19)denotes the formal delta function.

The essential point is that δ(z, D) is a projector onto aneigenspace of D with the eigenvalue z. Then it is obvious thatδ(DN, zN) = 1NN−1Xc=0δ(z(c), D),z(c) = ωcz,ω = exp 2π√−1N!

(2.20)Using this we bring the above energy-momentum tensor to the formT[N](E) =N−1Xc=0ωc∂ψ(t, z(c))∂z◦D−1 ◦ψ∗(t, z(c)) = 1NN−1Xc=0ω2cT(z(c))(2.21)where the following notations have been used: recall that the spectral parameter of an N-KdVhierarchy lies on a complex curve defined in C2 ∋(z, E) by an equation zN = P(E). Then,ψ and ψ∗are defined on this curve, and after the projection onto CP1 yield N wave functionsψ(a)(t, E), distinct away from the branch points.

That is, we have definedψ(a)(t, E) = Keξ(t,z(a)) ≡w(t, z(a))eξ(t,z(a)),ξ(t, z(a)) =Xj,btb,j(z(a))Nj+b(2.22)5

Note a striking similarity between (2.21) and the energy-momentum tensor of conformal theorieson ZN−curves [1].2.4. To conclude this review, we outline the basic steps of how the Virasoro action on theN−KdV tau function, by the generators of the type of (2.9), can be recovered from the “energy-momentum tensor” (2.21).

The usual way to derive objects pertaining to the tau function isthrough the use of the equationresK = −∂log τ,whenceδ∂log τ = −resδK = resT[N](z)K = resT[N](z)(2.23)The residue of T[N](z) is immediately read offfrom (2.21). To the combination of wave functionsthus appearing we apply the formula∂ 1u −zeξ(t,u)−ξ(t,z)τ(t + [z−1] −[u−1])τ(t)!= ψ(t, u)ψ∗(t, z)(2.24)(it follows directly by applying the vertex operator exp Pr≥11r(z−r −u−r) ∂∂tr to the bilinearidentity of the KP hierarchy [5] and then evaluating the integral as a sum over residues).

Itfollows from (2.24) by expanding it at z →u that∂ψ(t, z)∂zψ∗(t, z)=∂(121τ(t)1z∇(t, z)1z∇(t, z)τ(t) +1τ(t)∂ξ(t, z)∂z1z∇(t, z)τ(t)+121τ(t)1z∇(t, z)τ(t) + 12 ∂ξ(t, z)∂z!2(2.25)This expression by itself would lead us back to the KP Virasoro generators (2.9) (withJ = 0). Now the N−KdV generators follow according to the formula (2.21), by substitutingz 7→z(c) and summing over c. The sum over c plays the role of a projector onto the identity ofthe group of Nth roots of unity.

Therefore,N−1Xc=0ω2c∂ψ(t, z(c))∂z(c)ψ∗(t, z(c)) = N∂12N−1Xa=1Xi,j≥0(Nj + a)(N(j + 1) −a)ta,itN−a,jzN(i+j+1)−2+121τ(t)N−1Xa=1Xi,j≥0z−N(i+j+1)−2∂2τ(t)∂ta,i∂tN−a,j+1τ(t)N−1Xa=1Xi,j≥0(Ni + a)ta,izN(i−j)−2 ∂τ(t)∂ta,j(2.26)6

from which the Virasoro generators can be read offasn > 0 :L[N]n=1N12N−1Xa=1n−1Xi=0∂2∂ta,i∂tN−a,n−i−1+ 1NN−1Xa=1Xi≥0(Ni + a)ta,i∂∂ta,i+n,L[N]0=1NN−1Xa=1Xi≥0(Ni + a)ta,i∂∂ta,i,n < 0 :L[N]n=1N12N−1Xa=1−n−1Xi=0(Ni + a)(−N(i + n) −a)ta,itN−a,−i−n−1+1NN−1Xa=1Xi≥−n(Ni + a)ta,i∂∂ta,i+n(2.27)These generators act on the tau functions τ(t) of the N-KdV hierarchy.3Miwa–Kontsevich transformNow we are going to use the same strategy as was used to derive (2.27), but this time inthe Miwa–Kontsevich parametrization of the times of the hierarchy. As in the above, we startwith the simplest case, the KP hierarchy.The Miwa reparametrization of the KP times is accomplished by the substitutiontr = 1rXjnjz−rj(3.1)where {zj} is a set of points on the complex plane and nj are integers.

This parametrizationputs, in a sense, the times and the spectral parameter on equal ground. It may in some casesbe conceptually advantageous to write (3.1) astr = 1rXz∈CP 1nzz−r(3.2)where nz is nonvanishing for only a finite (countable) set of points.

Then the tau functionbecomes a functional τ[n] of a function n on CP1 which must be from the class of functions insome sense close to linear combinations (with integer coefficients) of delta-functions. On theother hand, the way Kontsevich has used a parametrization of this type implied setting all thenj equal to unity.

We will in fact see why a restriction of this kind is necessary, but this willrequire working with the general nj for as long as possible.The Miwa substitution turns out very inconvenient with regard to the use of the standardmachinery of the KP hierarchy (e.g., proceeding along the usual chain (tau function) 7→(wavefunction) 7→(dressing operator), etc. ); instead, it serves to construct a quite different, “discrete”formalism for the KP and related hierarchies [32].

Now, the above expressions (2.15), (2.17) and(2.21) for the Virasoro generators involve all the standard ingredients such as the wave functions,the spectral parameter etc., which complicates re-expressing them in the Miwa parametrization.That is, taking (3.2) seriously, and even viewing it astr = 1rZCP 1 dµ(z)z−rn(z)(3.3)7

one can formally define the wave functions asψ[n](z) =Yj 1 −zzj!nj1τ[n]eδδn(z)τ[n](3.4)Short-‘distance’ expansion as in (2.24) – (2.25) would then require making sense out of thisformula and similarly out of expressions such as∂∂zδδn[z]. Even this, however, would not be quitesatisfactory, as one would still have had to express the result in terms of the derivatives withrespect to zj , which are the parameters of the Kontsevich model: for us, the tau function mustbe a function τ{zj} of points scattered over CP1.There are two circumstances, however, that save the day.

First, we will be interested notin all the Virasoro generators, but rather in those with non-negative (and, in addition, −1)mode numbers Ln≥−1 3. Picking these out amounts to retaining in T(z) only terms with z tonegative powers, i.e., the terms vanishing at z →∞.

This part of T(z) is singled out asT(∞)(v) =Xn≥0v−n−1 12πiIdzznT(z) =12πiIdz1v −zT(z)(3.5)where v is from a neighbourhood of the infinity and the integration contour encompasses thisneighbourhood.Second, a crucial simplification will be achieved by evaluating T(∞)(v) only at the pointsfrom the above set {zj} (one has to take care that they lie in the chosen neighbourhood). Weuse the formulas (2.23), (2.21), (2.13) and (2.24) to find the variation of the tau function τ{zj},which amounts to evaluating, for a fixed index i,12πiIdz1zi −z((1 −J)∂ψ{zj}(z)∂zψ∗{zj}(z) −J ψ{zj}(z)∂ψ∗{zj}(z)∂z)≡∂1τ T (zi)τ(3.6)or, after the use of (2.24),1τ T (zi)τ =12πiIdz1zi −z (1 −J) ∂∂u −J ∂∂z!1u −zeξ(t,u)−ξ(t,z)τ(t + [z−1] −[u−1])τ(t)!regu=z(3.7)where reg implies subtracting the pole−1(u−z)2 , and everything has to be reexpressed throughthe {zj} variables.

This latter task, however, will be achieved not until the final stage of thederivation. Now we perform an expansion using time derivatives acting on the tau functionand thus find:T (zi)=12πiIdz1zi −z(J −12)1zXr≥1z−r−1 ∂∂tr+ 12Xr,sz−r−s−2∂2∂tr∂ts+Xjnj1zj −zXr≥1z−r−1 ∂∂tr+ 12Xjnj + n2j(zj −z)2 + 12Xj,kj̸=knjnk(zj −z)(zk −z)−JXjnj(zj −z)2 + (J −12)Xr≥1z−r−2r ∂∂tr(3.8)3 It is these Virasoro generators that are used to define Virasoro-constrained hierarchies, simply as Ln = 0,n ≥−1.8

Evaluating the residue is the crucial step which allows one to bring (3.8) to a tractable formin terms of the zj . As the integration contour encompasses all the points {zj}, the residues atboth z = zi and z = zj , j ̸= i, contribute to (3.8).

The residue at zi consists of the followingparts: first, the terms with the first-order pole contributeJ −12 −12ni 1ni1zi∂∂zi−12n2i∂2∂z2i+ 1niXj̸=inj1zj −zi∂∂zi−J −12 −12ni Xr≥1rz−r−2i∂∂tr−12Xj̸=inj + n2j −2Jnj(zj −zi)2−12Xj̸=ik̸=ik̸=jnjnk(zj −zi)(zk −zi)(3.9)where we have substitutedXr,sz−r−s−2i∂2∂tr∂ts=1n2i∂2∂z2i+ 1n2i1zi∂∂zi−1niXr≥1z−r−2ir ∂∂tr,Xr≥1z−r−1i∂∂tr=−1ni∂∂zi(3.10)Next, second-order poles occur in the double sum over j, k in (3.8):12πiIdz1zi −zXj̸=injni(zj −z)(zi −z) =Xj̸=ininj(zi −zj)2Now, to get rid of the ∂/∂tr-terms in (3.9) which cannot be expressed through ∂/∂zj, we setthe coefficient in front of these equal to zero:ni =12J −1 ≡1Q(3.11)Then the contribution of the residue at z = zi equalsT (i)(zi) =−12n2i∂2∂z2i+ 1niXj̸=inj1zj −zi∂∂zi−12Xj̸=ik̸=ik̸=jnjnk(zj −zi)(zk −zi) −12Xj̸=inj + n2j −2Jnj −2ninj(zj −zi)2(3.12)Similarly, each of the residues at zj, j ̸= i, contributesT(j)(zi) = −1zj −zi∂∂zj+1zj −ziXk̸=jnjnkzk −zj+ 12nj + n2j −2Jnj(zi −zj)2(3.13)and thus,T (zi)=T (i)(zi) +Xj̸=iT(j)(zi)=−12n2i∂2∂z2i+ 1niXj̸=i1zj −zi nj∂∂zi−ni∂∂zj! (3.14)9

(We have used the identityXj̸=iXk̸=ik̸=j1(zj −zi)njnk(zk −zj) = 12Xj̸=iXk̸=ik̸=jnjnk(zj −zi)(zk −zi) . )Now, the above treatment can be applied equally well to each of the T (zj), and thus (3.11)must hold for all the nj .

Finally,T (zi) = −Q22∂2∂z2i−Xj̸=i1zj −zi ∂∂zj−∂∂zi! (3.15)We thus see that, indeed, in order that the L≥−1−Virasoro generators translate into the{zj} variables, one has to restrict the general Miwa transform (3.1) to a Kontsevich form withall the nj equal to each other 4.From the above derivation of (3.15) we see that zi is nothing but a value taken by the spectralparameter and thus the trick, described in 2.3, with building up invariants with respect to ZNapplies here as well.

That is, to perform the reduction to an N-KdV hierarchy, it suffices tosubstitutezi 7→ωcziand then sum over ZN as in (2.21) 5. We thus arrive atT [N]i≡1NN−1Xc=0ω2cT (z(c)i )=−Q22∂2∂z2i−Xj̸=i1zNj −zNi zjzN−2i∂∂zj−zN−1i∂∂zi!

(3.16)Note that zN ≡ζ can be viewed as a spectral parameter of the N-KdV hierarchy, as the N-KdV Lax operator L (see (2.16)) satisfies Lψ(t, z) = zNψ(t, z). In terms of these variables, theoperator (3.16) becomes, up to an overall factor,−Q2N2ζi∂2∂ζ2i−Q2(N −1)2∂∂ζi+Xj̸=i1ζj −ζi ζj∂∂ζj−ζi∂∂ζi!

(3.17)4Restricting to only integer nj would fix two (equivalent) values J = 0, 1 of conformal spin of the underlyingabstract bc system. For our purposes in Sect.4, however, we will need more general nj and J.5 Clearly, having defined the reduced T −operator as (see (2.21))T [N]i=12πiIdz1zi −z1NN−1Xc=0ω2cT(ωcz),one continues this as=12πi1NN−1Xc=0Idzω−czi −ω−cz ω2cT(z) =12πi1NN−1Xc=0ω2cIdzωczi −zT(z) = 1NN−1Xc=0ω2cT (ωczi).10

When imposing Virasoro constraints on the N−reduced hierarchy, it is these ζi that are candi-dates for eigenvalues of the “source” matrix in a Kontsevich-type matrix integral, at least forQ2 = 1. We consider the reformulation of the Virasoro constraints in more detail in the nextsection.4A la r´echerche de Liouville perduObviously now, if one starts with the Virasoro-constrained KP hierarchy, i.e.,T(∞)(z) = 0,(4.1)one ends up in the Kontsevich parametrization with the KP Virasoro master equation (cf.ref.

[26])T (zi).τ{zj} = 0(4.2)The above derivation of (4.2),(3.15), with the zj (which in the alternative approach are theeigenvalues of the ‘source’ matrix in a matrix integral) viewed as coordinates on the spectralparameter complex plane, suggests an interpretation of the master equation in terms of aconformal field theory living on this complex plane. First, it is natural to assume that (with apossible ‘background’ insertion at infinity)τ{zj} = limn→∞⟨Ψ(z1) .

. .

Ψ(zn)Φ(∞)⟩(4.3)with the pre-limit correlators satisfying,−Q22∂2∂z2i+nXj=1j̸=i1zi −zj ∂∂zj−∂∂zi!⟨Ψ(z1) . .

. Ψ(zn)Φ(∞)⟩= 0(4.4)Further, one can imagine a conformal theory of a U(1) current j(z) and an energy-momentumtensor T(z):j(z) =Xn∈Zjnz−n−1,T(z) =Xn∈ZLnz−n−2(4.5)[jm, jn]=kmδm+n,0[Lm, Ln]=(m −n)Lm+n + d + 112 (m3 −m)δm+n,0[Lm, jn]=−njm+n(4.6)(We have parametrized the central charge as d + 1).

Then, in the standard conformal fieldtheory setting [2], let us look for a null vector at level 2:|Υ⟩=αL2−1 + L−2 + βj−2 + γj2−1 + ǫj−1L−1|Ψ⟩(4.7)where Ψ is a primary field with conformal dimension ∆and U(1) charge q. We will in factneed the specific case γ = 0.

Then (4.7) is a null vector whenα=k2q2 ,β=−qk −12q ,ǫ=−1q ,∆=−q2k −12(4.8)11

with q given by,q2k =d −13 ±q(d −25)(d −1)24(4.9)(so that,∆=1 −d ∓q(d −25)(d −1)24. )(4.10)Factoring the state (4.7) out from the Hilbert space leads in the usual manner to the equationk2q2∂2∂z2 −1qXj1zj −z q ∂∂zj−qj∂∂z!+ 1qXjq∆j −qj∆(zj −z)2⟨Ψ(z)Ψ1(z1) .

. .

Ψn(zn)⟩= 0(4.11)where Ψj are primaries of dimension ∆j and U(1) charge qj. In particular,k2q2∂2∂z2i+Xj̸=i1zi −zj ∂∂zj−∂∂zi!⟨Ψ(z1) .

. .

Ψ(zn)⟩= 0(4.12)This is to be compared with (4.4) (with the insertion at the infinity disregarded)6. We thusarrive at the identificationQ2 = −kq2 =13 −d ±q(d −25)(d −1)6(4.13)and therefore find ourselves in the friendly realm of minimal models [2, 11, 16], tensored with theU(1) current.

Moreover, the theory on the z-plane also incorporates the gravitational dressingof the matter theory. To see this let us first examine closer the constraint |Υ⟩= 0.

Writingthe Hilbert space as (matter) ⊗(current) ≡M ⊗C, |Ψ⟩= |ψ⟩⊗|˜Ψ⟩, we introduce the matterVirasoro generators ln by,Ln = ln + ˜Ln ≡ln + 12kXm∈Z: jn−mjm :(4.14)They then have central charge d. Singling out the j-independent terms, we write|Υ⟩= k2q2l2−1 + l−2!|Ψ⟩+ . .

. (4.15)By virtue of (4.13) the term written out explicitly is by itself a null vector, and can thereforebe set to zero in M. As to the other terms on the RHS of (4.15), we substitute˜L−1|˜Ψ⟩= qkj−1|˜Ψ⟩,˜L−2|˜Ψ⟩=qkj−2 + 12kj2−1|˜Ψ⟩,˜L2−1|˜Ψ⟩= qkj−2 + q2k2j2−1!|˜Ψ⟩6 Note that the energy-momentum tensor T introduced in (4.5) appears to have a priori nothing to do withthe energy-momentum tensor on the hierarchy we have started with.

In terms of the latter tensor, eq. (4.4)comprises the contribution of all the positive-moded Virasoro generators, while out of T (z) only L−1 and L−2enter in the equivalent equation (4.7).12

Then, with the coefficients chosen as in (4.8), all the other terms cancel out, and thus theellipsis in (4.15) vanishes.We are thus left with the null vector k2q2l2−1 + l−2!|ψ⟩(4.16)in the matter Hilbert space M. The dimension of |ψ⟩in the matter sector is found fromL0|Ψ⟩=l0 + 12kj20|Ψ⟩and equalsδ = ∆−12kq2 =5 −d ∓q(1 −d)(25 −d)16(4.17)Viewing this as the ‘bare’ dimension we see that tensoring with the current j is equivalent tothe gravitational dressing: evaluating the gravitational scaling dimension according to [6, 10],ˆδ± = ±√1 −d + 24δ −√1 −d√25 −d −√1 −d(4.18)we findδ+ = 38 ±d −4 −q(1 −d)(25 −d)24(4.19)The sign on the RHS corresponds to that in (4.17) and the previous formulae. In particular,choosing the lower signs throughout, we haveˆδ+ = ∆+ 12(4.20)Therefore, up to the shift by 1/2 (which seems somewhat misterious), the dimension ∆withrespect to the full Virasoro algebra acting in the tensor product space M ⊗C, is equal to thescaling dimension of the gravity-dressed operators.

Thus, the role of the gravitational dressingis effectively played by tensoring with the theory defined by[jm, jn] = kmδm+n,0,jn>0|0⟩= 0,j0|0⟩= q|0⟩with central charge 1 and negative q2/k. The reason for a current to appear at all is thatit serves to represent the hierarchy flows and thus signifies the “hierarchical” origin of thetheory.

This also provides a new insight into the theory of completely integrable evolutions:for Virasoro-constrained hierarchies these amount to the Liouville dynamics in the conformalgauge.To return to the relation with the formalism of [6, 10], recall that the Coulomb-gas realizationof the matter theory requires introducing a scalar field ϕ with the energy-momentum tensorTm = −12∂ϕ∂ϕ + iQm2 ∂2ϕ(4.21)13

Then the matter central charge is equal to 1 −3Q2m, and equating this with d we invert (4.13)asd = 1 −3(Q2 −2)2Q2,(4.22)and thusQ2m = Q −2Q!2(4.23)On the other hand, the parameter Q was introduced initially in the Virasoro constraints (2.9)(where J = Q+12 ) as the background charge of an abstract bc system (cf. eq.(2.13)).

Now, ithas to be tuned asQ2 =Qm ±qQ2m + 822=Qm ± QL22=−QL ∓Qm22(4.24)whereQL =qQ2m + 8(4.25)is the background charge of the ‘Liouville’ scalar field [10, 6] with the energy-momentum tensorTL = −12∂φ∂φ −QL2 ∂2φ(4.26)Equivalently, one sees that (for the respective sugns in (4.13)),Q2 = α2∓(4.27)where eα+φ is the gravitational dressing of the identity operator. This establishes the physicalmeaning of Q (note that Q enters explicitly in the Kontsevich transform through (3.11)).

- Itlooks like the bc system underlying eqs. (2.13) and (2.15) describes (upon imposing the Virasoroconstraints) a ‘mixture’ of the matter and Liouville theories.5Concluding remarks1.

It remains an open problem to represent the N−reduced master equation as a Wardidentity of a matrix integral.2. Our approach was based on a general construction of Virasoro generators on the phasespace of integrable hierarchies [36, 38], and, as the N-KdV hierarchies do not seem so muchformally distinguished in any way, it must apply also to other Virasoro-constrained hierarchies,including the “discrete” ones, e.g.

Toda [19, 28, 36]. This may be especially interesting in viewof the lack of a “discretized” version of the Kontsevich model (which does by itself seem tobe ‘discrete’), while, on the other hand, Virasoro constraints on the Toda hierarchy [19, 29]have been shown [34] to undergo a continuum limit into Virasoro constraints on a KP hierarchyobtained from Toda also as a result of the scaling.

It would be interesting to investigate whatkind of a Kontsevich-type matrix integral the corresponding master equation may be derivedfrom.14

3. Various aspects of the conversion of Virasoro constraints into decoupling equations wouldbe interesting, in particular, from the ‘Liouville’ point of view.

¿ The Kontsevich-type matrixintegral whose Ward identities coincide with our master equation may thus provide a candidatefor a discretized definition of the Liouville theory.It was implicitly understood in Sect.4 that the matter central charge d should be fixed to theminimal-models series; then factoring out the null-vector leads to a bona fide minimal model(and our ψ thus becomes the ‘21’ operator). Now, thinking in terms of the minimal models, howcan the higher null-vectors be arrived at starting from the Virasoro-constrained hierarchies?

Ifthese vectors correspond to higher symmetries of Virasoro-constrained hierarchies, then thewhole Kac table must have a relation to the W∞algebra.Acknowledgements. I am grateful to O.Andreev, A.Zabrodin and A.Mironov for usefulremarks and to A.Subbotin and R.Metsaev for valuable suggestions on the manuscript.15

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