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Symmetries and Special States

arXiv:hep-th/9112061v2 8 Feb 1992PUPT-1301December, 1991Symmetries and Special StatesinTwo Dimensional String TheoryUlf H. Danielsson 1Joseph Henry LaboratoriesPrinceton UniversityPrinceton, NJ 08544USAAbstractWe use the W∞symmetry of c = 1 quantum gravity to compute matrix model special statecorrelation functions. The results are compared, and found to agree, with expectations from theLiouville model.1Work supported by a Harold W. Dodds Fellowship

1IntroductionThe last few years have seen tremendous developements in the understanding of two dimen-sional quantum gravity and therefore non critical string theory. The first success came fromfield theory, [1], where the case with matter of central charge c ≤1 coupled to quantumgravity was solved.

Later these theories were also solved by the powerful matrix models,both for c < 1 [2], and for c = 1 [3]. They allow for exact, non perturbative solutions whereone sums over all genus.

This is an important improvement over field theory where highergenus contributions are extremely difficult to calculate. Most results have therefore beenlimited to the sphere or in some cases the torus.These models are important for two reasons.

First they give solutions of two dimensionalquantum gravity and can serve as toy models for higher dimensional theories. Indeed, thematrix model provides directly the exact Wheeler de Witt equation summed over all genus,[4].

Clearly an important object for further study. Second, and perhaps even more inter-esting, they describe non critical string theories.

The most physical example is clearly thecase c = 1. It is now commonplace to identify the Liouville mode as an extra dimension, [5],and thereby obtaining a theory of strings moving in a two dimensional target space.

Naivelyone would expect this theory to be very simple, just a single massless scalar particle, usuallyreferred to as the tachyon. This can be argued by choosing lightcone gauge.

Fortunately,this is not the whole story. Instead there are remnants of the massive excited string modespresent for certain discrete values of the momentum, [4, 6, 7, 8].

They are usually calledspecial or discrete states. Clearly it is very interesting to study these extra states if onewants to learn about truly stringy phenomena.The special states have been the object for several recent studies.

Both using Liouvilletheory and matrix models. An important discovery has been a huge set of symmetries.

Thesesymmetries obey a W∞algebra, which can be thought of as a generalization of a Virasoroalgebra. This has been shown in the Liouville theory with two different methods.

In [9] theW∞was found by explicitly calculating the operator product expansion of the special states.An important tool for doing so is the usual SU(2) symmetry known for a long time. In[10] the symmetry was instead found from the construction of the ground ring.

This gives arepresentation of the current algebra with the currents acting on a set of ghost number zero,spin zero fields. These fields can be shown to be primary, [11], and correspond to specialstates in addition to the standard ones.

The meaning of these and other special states of nonstandard ghost numbers has as yet not been fully clarified. From the matrix model point ofview, the emergence of the symmetry has been more gradual [10, 12, 13, 14].

In [10] it washowever clearly realized that the W∞simply is generated by the matrix eigenvalue and itsconjugated momentum through the Poisson bracketts.1

In this work we will use the W∞to study the special operator correlation functions.Notations and conventions will be much the same as in a previous paper, [6]. We will beable to obtain many of the results in [6] using the simplifications the W∞symmetry provides.In section 2 we make some initial comments on Ward identities and symmetries relatingto the results obtained in [6].

We will also make some comments on possible generalizationsto non harmonic matrix model potentials based on a generalized Wheeler de Witt equation.In section 3 we calculate some matrix model special state correlation functions using the W∞symmetry. We also make some comments on how to identify the counterparts of the Liouvillemodel special states.

Section 4 gives some illustrations of the structure one encounters athigher genus. Finally, section 5 is devoted to a comparison with Liouville theory.

Althoughthe success of the matrix model and its agreement with Liouville theory hardly is in doubt,it is important to make the connection as explicit as possible. In particular, it is so farnot clear how to explicitly extract the space time structure from the matrix model.

Onewould like to be able to study nontrivial space times like the recently discovered black holesolution [15]. We will not be able to adress this question here, but we will be able to makea comparison between our matrix model results and some Liouville theory expectations.

Inthe cases which we will examine we will find perfect agreement.2Ward Identities and SymmetriesLet us now consider the matrix model and its special states and operators. The matrixmodel represents the string theory Riemann surfaces by Feynman diagrams of interactingmatrix variables which triangulate the surfaces.

In the uncompactified case, or at least forlarge enough radius, we can simply integrate out most of the degrees of freedom. The onlyremaining will be the matrix eigenvalues.

They will then behave as non interacting fermionsin the matrix model potential. Obvious candidates for special state correlation functionsare then correlation functions of powers of the matrix model eigenvalues.

Such objects werestudied in [6] and the expected poles for discrete momenta were found. In this section wewill review and extend some of the results of [6] for Ward identities.

These Ward identitiescan be used to recursively obtain the correlation functions.The recent developements revealing the W∞symmetries indicate however that this isnot the whole story. One should also consider correlation functions involving powers of theconjugate momentum.

In this section we will show the existence of this W∞symmetry whichgreatly will simplify the subsequent calculations.Ward identities for correlation functions are in general obtained by changes of variablesin the path integral. Examples of such Ward identities were obtained in [6] from simple2

coordinate changes in the matrix eigenvalue. They can be thought of as generated by com-mutators, or classically i.e.

on the sphere, Poisson bracketts with pλm. They obviously obeya Virasoro algebra which is part of a W∞algebra generated by all monomials pnλm.

Wemay also introduce time dependence and consider generators with certain momenta q, i.e.pnλmeiqt. Let us as an example make a pλkeiqt variation of the one puncture function.

Thetwo puncture function is schematically given by< PP >= ImZ ∞0dTZdλG(λ, λ; T)(1)where the calculation is done at the Fermi surface. We will shift its energy to zero, henceputting the Fermi energy as a constant term in the potential.

G is the path integral givenby, in Euclidean time,G(λ1, λ2) =Z λ2λ1[dpdλ]e−β Rdt(p ˙λ−12p2+U(λ))(2)where U(λ) =Pp tpλp is the potential. The variation of the partition function would haveinvolved a sum over all states in the Fermi sea up to the Fermi level.

By inserting a puncture,i.e. taking a derivative with respect to the Fermi energy, we restrict ourselves to the Fermisurface.

Next we perform the variation of < P >. The measure, as given by (2), is invariantunder the change of variables.

(Clearly we are not supposed to differentiate with respect to twhen changing variables in the measure.) The change in the action give rise to the followingidentity among two point functions:< (Zdt(iqλkp + kλk−1p2 + ∂U∂λ λk)eiqt)T >= 0(3)The puncture is now a tachyon, T, carrying away the momentum.

The piece with a single pis evaluated by integrating over p in the path integral obtaining a ˙λ which then is partiallyintegrated. We then switch to a Hamiltonian formulation, remembering that we should useWeyl ordering.

We finally obtain:−12k(k −1)(k −2) < Ok−3T >q,g−1 +Xp(2k + p)tp < Op+k−1T >q,g(4)+ q21 + k < Ok+1T >q,g= 0where q indicates momentum and g genus. We have introduced the notation Ok for λk.

Thedifferent genus for the first term is due to a p, λ commutator which arises when we want to3

evaluate the p2 against the wave function of the Fermi surface. This gives an ¯h, which is thesame as the genus coupling constant.

Following [4] we may define the loop operator givenbyw(l) = elλ(5)It corresponds to cutting out a hole in the surface with a boundary of length l. The reasonis as follows. If we insert a power n of the original matrix eigenvalue m on the surface thiscreates a little hole, the length of the boundary being proportional to n (the number of legs)and the lattice spacing a.

For fixed n the length clearly shrinks to zero in the double scalinglimit. To get a finite length we must also take n to infinity.

Introducing λ as m expandedaround the top of the potential, we find in the double scaling limitmn ∼(1 + aλ)l/a →elλ(6)We may then Fourier transform to obtain a differential equation in the loop length. We get[Xptp(l2 ∂p∂lp + p2l ∂p−1∂lp−1) + t0l2 −l4 + q2] < w(l)T >= 0(7)where the third term are of order one higher in the string coupling and do not contribute onthe sphere.

t0 is the Fermi energy µ with the appropriate number of β’s absorbed. In the caseof the usual harmonic oscillator potential, where t0 = βµ, the resulting equation is in factthe Wheeler de Witt equation obtained in [4] with a more indirect matrix model method.On the sphere this is one of the most striking verifications of the equivalence of the Liouvilleand matrix models.

At zero momentum the Wheeler de Witt equation is just the Fouriertransform of the Gelfand-Diiki equation for the resolvent for the Schr¨odinger operator. Thiswas the way in which the zero momentum version of (4) was derived in [6].

If we want to becareful, see section 5, we need to rescale λ by √−t2 to get a dimensionless l. This is neededfor the exact correspondence between the above result and the mini superspace canonicalquantization of Liouville theory. Recall that, [7], t2 = −12α′ .

From (7) one might try to drawsome conclusions about the Liouville theory correspondence to the more general potentialsabove. Clearly the last term, which corresponds to the matter piece, does not change while wechange potential.

Instead it is the piece which would be expected to arise from a canonicallyquantized kinetical term for the Liouville mode which gets modified. Hence one is lead tothe conclusion that these more general models (however with p independent potentials) maycorrespond not to modifications of the matter theory but rather to different theories for theLiouville part.

This is also consistent with the point of view for which this paper will argue,that the special states must be represented using both λ’s and p’s, not just the λ’s.4

In principle all correlation functions may be calculated with the help of Ward identitiesderived in this way. However, it is more convenient to make use of the large set of symmetriesin the theory.

As shown by Witten in [10] we may change basis to (p −λ)n(p + λ)m and,for certain time dependence, obtain transformations which leave the action invariant. Thesetransformations are generated by, in Minkowsky time,W r,s = (p + λ)r(p −λ)se(r−s)t(8)Again we get the W∞algebra{W r1,s1, W r2,s2} = (r1s2 −r2s1)W r1+r2−1,s1+s2−1(9)generated, classically, by the Poisson bracketts.

For a general momentum q in (8) we findwhen acting on the Minkowsky action S =R (p ˙λ −12(p2 −λ2)){W r,s, S} = (r −s −q)W r,s(10)Hence a symmetry for appropriate discrete values of imaginary momentum. This is equivalentto saying that W = W(p, λ, t) is a solution ofdWdt = ∂W∂t + {H, W} = 0(11)Expressed in terms of the initial conditions, p0 and λ0, we have W = W(p0, λ0), i.e.

anytime independent function of the initial conditions.The generators (8) are then simplyobtained through evolution in time. Hence the transformations can be understood as timeindependent canonical transformations of the initial conditions.A more convenient way of labeling the generators is through their SU(2) quantum num-bers J = (r + s)/2 and m = (r −s)/2.

With the definitionWJ,m = (p + λ)J+m(p −λ)J−me2mt(12)one gets{WJ1,m1, WJ2,m2} = 2(m1J2 −m2J1)WJ1+J2−1,m1+m2(13)In Euclidean time, which is what we will be using, one should take p →ip and t →it in(12). There is also an extra i in the structure constant of (13) and the eigenvalue of (10).5

3Matrix Model Correlation FunctionsIn this section we will calculate correlation functions involving the operators W as definedabove. The W∞symmetries will help us organize the Ward identities.

As an example, letus start with the two point function. Using< PP >= 1πIm∞Xn=01En + t0(14)and simple perturbation theory we get:< W1W2P >= 1πImXtimeordXn,k< n | W1 | k >< k | W2 | n >En + t01ip1 + Ek −En(15)Since the W’s are of the form (8), continued to Eucledian time, they are simply raising orlowering operators in the inverted harmonic oscillator.

This means that only a few of thematrix elements are nonzero. Since W raises by 2m = r −s we get< W1W2P >= 1πImXn< n | [W1, W2] | n >En + t01i(p1 −2m1)(16)We have reduced the two point function to a one point function using the commutationrelations.

If we restrict ourselves to the sphere, use the algebra given by (9) (with an extrai in the structure constant for Eucledian time) and directly calculate the one point functionwe get< W1W2 >= 2(m1J2 −m2J1)2m1 −p11πµJ1+J2J1 + J2| log µ | 2J1+J2−1(17)or equivalently< W1W2 >=m12m1 −p11πµJ1+J2 | log µ | 2J1+J2(18)The simplest way to obtain the one point function on the sphere is to use the classicalFermi liquid picture introduced by Polchinski [17]. We simply need to do the phase spaceintegral:< W >=ZdpdλW(p, λ)(19)This was also discussed in [16].

To make everything well defined we need however to introducean extra puncture, i.e. take a derivative with respect to the cosmological constant.

Doingthat the integral over the whole Fermi sea becomes just an integral over the Fermi surface:< WP >=IW(p, λ)(20)6

The integral is to be performed along a hyperbola p2−λ22= µ. Although the answer is reallyinfinite, we know that the piece nonanalytic in µ is1π | log µ | for < PP >, i.e.whenwe just integrate 1.The other zero momentum operators simply involve integrations of( p2−λ22)n = µn, again constants along the Fermi surface hyperbola, so we get< W n,nP >= 1πµn | log µ | 2n(21)and from this (17) follows.

Our conventions are such that α′ = 1. Another approach, whichis convenient when calculating correlation functions of nonzero momentum, is to continueto the upside down oscillator where the Fermi surface is a circle.

In that case, however, weneed to remember to put the Liouville volume | log µ | in by hand. Parenthetically we maynote how a general correlation function may be obtained in this way.

For instance the twopoint function is obtained by perturbing the hamiltonian and hence the Fermi surface byone of the operators. If we integrate the other operator against the change in Fermi surfacewe get the correlation.Let us now consider the more complicated case of a three point function.

Again pertur-bation theory gives us1πImXtimeordXn,m,k< n | W1 | m >< m | W2 | k >< k | W3 | n >En + t01ip1 + Em −En1ip3 + En −Ek(22)Let us make the sum over time orderings more explicit. We find1πImXn{< n | (W1W2W3 + W3W2W1) | n >En + t01i(p1 −2m1)i(p3 −2m3) + perm}(23)This may after some straightforward manipulations be rewritten as< W1W2W3 >=< [W1, [W2, W3]] >(p1 −2m1)(p3 + 2m1 + 2m2) +< [W2, [W1, W3]] >(p2 −2m2)(p3 + 2m1 + 2m2)(24)For the sphere we now use the algebra (13) and the explicitly calculated one point functionto get< W1W2W3 >= (2m1 −p1)m2m3J1 + (2m2 −p2)m1m3J2 + (2m3 −p3)m1m2J3(2m1 −p1)(2m2 −p2)(2m3 −p3)×1πµPNn Jn−1 | log µ | 2PNn=1 Jn(25)7

The general higher point function can be obtained recursively from the three point byuse of (9) and (10). To get the N point function with an additional operator WN we varythe N −1 point function with WN knowing that the total variation is zero.

The variationconsists of two terms. One from varying the action as given by (10) and a sum of terms fromvarying the other operators as given by (13).

Each of the terms in this sum is obtained byshifting pi →pi +pN, mi →mi +mN and Ji →Ji +JN+1 −1. We also need to multiply withthe Clebsch-Gordan coefficient 2(JimN −JNmi).

It is an easy exercise to check that (25) isobtained by applying this procedure to (17) or (18). There is one subtlety in the variationalprocedure which should be noted.

The insertion of an operator W in the pathintegral does notinvolve only the operator itself, but also a delta function for its position in eigenvalue space.In general the delta function will also contribute to the variation. Luckily its contributionwill be zero by invariance properties for the W∞generators.Using this method one can write down several different recursion relations.

One simpleexample is:< TJ,JWJ1,m1NYi=2TJi,Ji >= 2J(J1 −m1)2J −p< WJ+J1−1,J+m1NYi=2TJi,Ji >(26)We will come back to this relation later, when we compare with the Liouville model results.Rather than considering these general expressions, let us look at a couple of importantexamples where the form of the general N point function is particularly simple.The first example is the N point function of special tachyons. It is given by=2 QNn=1 JnQNn=2(2Jn −pn)1πdN−3dµN−3µPNn=1 Jn−1 | log µ | 2PNn=1 Jn(27)The quantum numbers have been chosen as mn = Jn for n > 1 and m1 = −J1.

This is justthe pole part of the general tachyon correlation function as computed both in the matrixmodel [18] and in the Liouville theory [18, 19], up to a factorized normalization factor. Theproof is by varying the three point.

We can not just vary the three point tachyon correlationfunction, since some of the J’s are really m’s in disguise and J and m vary differently. Insteadwe start with the general three point and make an arbitrary number of tachyon variations.

Asimplification is that we at each step only have to vary the single negative chirality tachyon.It is only from there were we will get a nonzero Clebsch-Gordan coefficient. Following theprescription above, performing N −3 variations we get8

= [(2m1 −p1 +NXi=4(2mi −pi))m2m3(J1 +NXi=4Ji −N + 3)+(2m2 −p2)(m1 +NXi=4mi)m3J2 + (2m3 −p3)(m1 +NXi=4mi)m2J3]×QNi=4(Ji(m1 + PNj=i+1 mj) −(J1 + PNj=i+1 Jj −N + i)mi)(2m1 −p1 + PNi=4(2mi −pi)) QNn=2(2mn −pn)1πµPNn=1 Jn−N+2 | log µ | 2PNn=1 Jn(28)The product in the denominator is the product of all the Clebsch-Gordan coefficients of thevariations. Note that each get shifted by the successive variations.

By the use of momentumconservation and evaluating the m’s as J’s, the formula (27) is proved. This derivation showshow the combinatorical factor from the µ derivatives is a consequence of the W∞symmetry.If we want to consider the zero momentum operators, we have to be careful.

The Clebsch-Gordan coefficients are zero in this case but these zeroes cancel precisely the momentum polesand leave a finite result. Also, we need to consider both signs of the m quantum numberwhen we take the m →0 limit.

This gives a necessary extra factor of two. We get=NXn=1Jn1πdN−3dµN−3µPNn=1 Jn−1 | log µ | 2PNn=1 Jn(29)We will use induction for the proof.

We find=Xk2(JN+1mk −JkmN+1)PN+1n=1 Jn −12mN+1 −pN+1×1πdN−3dµN−3µPN+1n=1 Jn−2 | log µ | 2PN+1n=1 Jn−1 + (mN+1 →−mN+1)(30)If we then put pN+1 = 0 and use that the sum of all m’s must be zero the result follows.This can also be checked by an explicit phase space calculation.Given these expressions we may check the correlation functions calculated in [6] andindependently in [4]. These were correlation functions of pure powers of the matrix eigenvaluewithout any momentum powers.

In terms of the W’s they are given byOn =nXk=0 nk! (−1)kW n,k−n 12n(31)9

From this it follows that the two point function is given by< OnOm >q=12n+mnXk=0mXl=0 nk! ml!

(−1)k+l < W n−k,kqW m−l,l−q>(32)Using (18) and some simple algebra we find1212 (n+m)1πµ12 (n+m) | log µ |nXk=0 nk! mn+m2−k!4( n2 −k)24( n2 −k)2 −q2(33)In precise agreement with [6] recalling our convention α′ = 1.We can now understandwhy the O operators gave correlation functions with sets of poles and were, dependingon momentum, capable of exciting several special states [6].They were, in fact, linearcombinations of all special operators of a given gravitational dimension i.e.

spin J. Theabove construction with the generators (8) of the W∞disentangles the correlation functions.This means that the matrix model operators to be identified with the Liouville model specialstates are those defined in (12).Finally let us consider the meaning of the momentum poles. As emphasized in [23] weshould not treat the poles in (25) and the | log µ | asymmetrically since the source of the| log µ | is also a momentum pole.

In fact, all the poles should be thought of as cut of by| log µ |. A general N point function (without zero momentum operators) would then have| log µ |N.

This proliferation of logaritms was also noted in [19].4Higher GenusWe have so far basically just treated the sphere, which means, in the matrix model, that wehave been working at the classical level. The W∞has been generated by Poisson bracketts.Nothing can however stop us from considering the full quantum theory, i.e.

all genus. It isjust a matter of algebra to compute for instance the two or three point functions using (16)and (24) respectively.More interestingly, the algebra changes at the quantum level.

There is a deformation with¯h, the genus coupling, as parameter when we use commutators instead of Poisson bracketts.If we define our W’s using Weyl ordering, which is natural from the pathintegral point ofview, the algebra may be conveniently represented using the Moyal bracketts [20]{W1, W2}M = 2¯h sin ¯h2( ∂∂p1∂∂λ2−∂∂p2∂∂λ1)W1W2(34)10

which is a deformation of the usual Poisson brackett. From this we might conclude that alsothe Liouville theory operator product expansions should receive higher genus corrections.Presumably from handles getting caught inside the contour integrals defining the operatorproduct expansions.Another way to exhibit the quantum deformation is through a generalized loop operator[16], instead of (5) we introducew(k, l) = ekp+lλ(35)where p and λ are the conjugate variables.

It is a very old result, [22], that these operatorsobey the algebra[w(k1, l1), w(k2, l2)] = 1¯h sin ¯h(k1l2 −l1k1)w(k1 + k2, l1 + l2)(36)with ¯h →0 giving back a W∞. Interestingly it can be shown [21] that (36) is a representationof SU(N) with ¯h = 1/N.

This is reminiscent of the original unitary symmetry of the matrixmodel.Let us give an explicit example of a two point function to all genus. We choose thecorrelator between spin J = 3/2, m = 1/2 and J = 3/2, m = −1/2.

To do that we need tocalculate < W2,0P >. This is easy.

We have< W2,0P >= 1πIm∞Xn=0(2En)2 + 1En + t0(37)The extra term +1 comes from Weyl ordering. If we keep only terms nonanalytic in t0 thisreduces to< W2,0P >= (4t20 + 1) < PP >(38)To evaluate our two point we use the Moyal brackett to calculate[W3/2,1/2, W3/2,−1/2] = 6W2,0 −4¯h2W0,0(39)(16) and (38) then finally give< W3/2,1/2W3/2,−1/2P >= (48t20 + 4) < PP > 1/21 −p(40)The same procedure may be used to calculate arbitrary correlation functions.11

5Comparison with Liouville TheoryWe would like to understand the W∞structure from the Liouville theory point of view. Asshown in [10] and by more direct methods in [9], we indeed have the same algebraic structurepresent.

Therefore one would expect the comparison of the Liouville and the matrix modelto be straightforward. As we will see, the situation is more subtle.

Let us first consider avery special case which also give us the opportunity to clarify some important points.There are some very simple examples of correlation functions easily computable justusing Liouville notions and no matrix model techniques. These are correlation functionsinvolving the dilaton.

We will in fact be able to obtain some results to all genus simply fromdimensional arguments. Consider the Liouville partition function (or space time free energy)E(∆) = limR→∞1RZDXDφe−R(−t2∂X ¯∂X+∂φ¯∂φ+QRφ+∆eαφ)(41)where t2 = −12α′ , Q = 2√2, α = −√2 and R is the radius of the target space for the matterfield X.

∆is the world sheet cosmological constant, dimensionless from the point of view ofspace time. The only dimensionful quantities are R and α′.

In the noncompact case we haveinfact only α′ at our disposal. From dimensional grounds and KPZ scaling we must haveE(∆)g ∼(−t2)1/2∆2(1−g)(42)at genus g. E(∆) is the generator of connected amplitudes (in space time).

Let us do aLegendre transform to obtain a generating functional for 1PI amplitudes with respect to thepuncture, i.e. the zero momentum tachyon.

This means taking away any pinches. We haveE(∆) = ∆µ −Γ(µ)(43)withµ = ∂E∂∆= (−t2)1/2(∆+ ...)(44)µ has the dimensions of energy.

HenceΓ(µ) ∼(−t2)12(2g−1)µ2−2g(45)In the 1PI generator Γ we should of course regard µ as independent of t2. Since t2 derivativesshould generate dilaton insertions we find the following 1PI amplitude relation< O2...O2 >g=1(2t2)nnYp=1(2g + 1 −2p) <>g(46)12

which is identical to what was obtained in [6] using the matrix model recursion relations(generalizations of the zero momentum Wheeler de Witt equation).As noted in [6] thedilaton one point function involves a factor 2g −1 rather than the expected 2g −2. Fromabove it is clear that this discrepancy is simply due to including the overall (−t2)1/2 in (42).There is a further subtlety in how the dilaton is defined.

As we have seen a pure λ2 is notwhat we would expect to identify with the dilaton. Instead we should have p2 −λ2 if wekeep the algebraic structure in mind.

These are in general different in correlation functions.We we will return to this shortly.It is important to realize the difference between connected and 1PI amplitudes. In thematrix model 1PI amplitudes are the natural objects, in Liouville theory it is more commonto treat the connected ones.

Often the distinction is not very clearly made. Indeed if weconsider generic nonzero momentum the difference is very easy to deal with.

It amountsto a renormalization of the cosmological constant. We can simply replace ∆by µ, [23].

Atzero momentum we must be much more careful. In this case we may have internal puncturepropagators, i.e.

pinches. Some examples of this were obtained in [6].

This will turn out tobe important later on.Already at this point we may find traces of the W∞structure. In fact, the seeminglyinnocent representation of the puncture and the dilaton as µ and t2 derivatives respectivelyis a reflection of the W∞.

Let us give a formal argument for this. First the puncture.

Writethe SU(2) quantum numbers of the puncture as J and m which both will be taken to zero.Choose one of the operators in (25) to be a puncture. We get< W1W2P >= (J1 + J2)m12m1 −p11πµJ1+J2−1 | log µ | 2J1+J2(47)which by comparing with the two point function (17) shows how the puncture is representedas a µ derivative.

The case of the dilaton is equally simple. Proceeding as above we find< W1W2D >= [(J1 + J2)m12m1 −p1+2m21(2m1 −p1)2] 1πµJ1+J2 | log µ | 2J1+J2+1(48)If we introduce explicit t2’s in the two point function we can write it as< W1W2 >=m12m1 −p1/(−2t2)1/21π(µ(−2t2)1/2)J1+J2 | log µ | 2J1+J2(49)Taking a t1/22derivative we indeed reproduce (48).

We must now return to the issue of howprecisely the dilaton is defined. Recall the original matrix model action:βZdt[p ˙λ −12p2 −t2λ2](50)13

with β dimensionless, t2 having the dimension of energy squared and p2 and λ2 the dimensionsof energy and one over energy respectively. To obtain (49) as a generating functional fordilaton insertions with the above definition of the dilaton we should rescale λ and p to makethem dimensionless.

We findβZdt[p ˙λ −1√2(−t2)1/2(p2 −λ2)](51)Hence the matrix model dilaton should be represented by (−t2)1/2 derivatives. This is therescaling eluded to in section 2 in the context of the Wheeler de Witt equation.To obtain the general special operator correlation function in the Liouville theory onewould like to use the group theoretic information provided by the W∞or, for given spin J, theSU(2) symmetry.

The states in the Liouville theory are given by combinations W(z) ¯W(¯z)of the Liouville theory version of the special states W(z). Given this it is tempting to believethat we have a representation of a W∞× W∞symmetry (left times right).

This is howeverin general not correct. In the uncompactified case the left and the right moving states mustbe the same.

The symmetry group is broken down to just the diagonal subgroup. This isachieved in two steps.

First the gravitational dressing must be the same for left and right,otherwise we would be unable to screen using the cosmological constant which treats left andright in the same way. This means that we always must have the same spin J for left andright.

We get a reduction to the diagonal of the piece transverse to SU(2) × SU(2). Thisis true even for the compactified case.

If we in addition are considering the uncompactifiedcase, the left and right moving momenta must be the same and hence the m quantumnumbers. Consequently we just have a representation of the diagonal W∞.

This is in preciseagreement with the matrix model, where we indeed only see one W∞. There is howeveran apparent paradox here.If we would use the free field contractions in computing thecorrelation functions the results would seem to disagree since from this point of view leftand right are still independent.We will return to this important point further on, anddiscover that there in fact seems to be no contradiction.The symmetry may then be used to determine all correlation functions given the specialtachyon correlation functions which may be computed using other means.

The reason isthat all J and m dependence of any correlation function is given by some combination ofClebsch-Gordan coefficients. For given J’s we need the Clebsch-Gordan coefficients of SU(2),the 3j symbols, to get the m dependence.

In fact we have already seen the agreement forthe tachyon correlation functions and if we believe that the group theoretic structure isthe same in the matrix model and in the Liouville theory, we know that the expressionsobtained in the matrix model must agree with Liouville theory. To be more explicit let ushowever look at an example, the three point function, to see how the invariance properties14

determine the correlation functions. The three point function is obtained by consideringcoupling (J1, m1), (J2, m2) and (J3, m3) (with m1 + m2 + m3 = 0) to (J1 + J2 + J3 −2, 0).

Acomplication is that there are in general several different channels to sum over. This is truealready for the three point function.

The reason is that we really should think of the threepoint function as a four point function. The fourth leg carries the excess Liouville momentum,i.e.

J quantum number, into the vacuum. This is a consequence of the non conservation ofLiouville momentum.

Let us use the tachyon three point function for normalization. It isgiven by< T1T2T3 >=J1J2J3(2m2 −p2)(2m3 −p3)1πµJ1+J2+J3−1 | log µ | 2J1+J2+J3+1(52)where we have kept the normalization choice of (27).

Tachyons 2 and 3 are of positive chiralitywhile tachyon 1 has negative chirality. There are two possible channels corresponding toeither p2 = m2 or p3 = m3, i.e.

1 and 2 coming together or 1 and 3 coming together. Thegroup theoretic factor in each case is simply proportional to a product of 3j symbols.

Onefor each vertex. For the 1-2 channel: J1J2Jm1m2m3!

JJ3J′−m3m30! (53)where J = J1 + J2 −1 and J′ = J1 + J2 + J3 −2.

Just retaining the m dependence andadding the two channels we find(2m2 −p2)(J1m2 −J2m1)m3 + (2m3 −p3)(J1m3 −J3m1)m2 =−(2m1 −p1)J1m2m3 −(2m2 −p2)J2m1m3 −(2m3 −p3)J3m1m2(54)which agrees with (25) after using (52) to fix the normalization and J dependence. Thisshould come as no surprise since the calculations are almost identical.Another convenient way to obtain more general correlation functions is through factor-ization.

This is really already implicit in our previous calculations. In fact, if we look at(28) we see the complete factorization of the tachyon correlation function into a productof three point functions, each given by a 3j symbol, times a single zero momentum onepoint function.

This last piece represents the extra leg in any correlation function whichabsorbs excess Liouville momenta. One may note that these three point functions in factinvolve states of the wrong dressing.

This was also pointed out in [24]. Strictly speakingthe expression in (28) is just for one channel, the one where 1 fuses with 2 then with 3 etc.All channels give however identical contributions and can not be distinguished.

Clearly thetachyon correlation function is consistent with the single W∞factorization result.15

As has been remarked, this seems to be in contradiction with what to expect from thenaiive free field calculations in Liouville theory. From such a calculation you would expect toget a different result, all Clebsch-Gordan coefficients squared, one from the left and one fromthe right.

We will however show that the results in the end turn out to be consistent. Let usbegin by considering the tachyon correlation function as computed in [19].

As we have seenthe result is in complete agreement with the matrix model results. On the other hand wehave seen how the matrix model organizes its correlation functions using a single W∞.

Letus consider the Liouville calculation more carefully. The result of [19] is obtained througharguments of analyticity and symmetry.

In particular the by now well known factorizedproduct of gamma functions is found [18] with a certain unknown coefficient independentof the particular momenta. This coefficient is then determined by sending all the momenta,except three, to zero.

This reduces the expression to a three point function with N −3extra punctures. Since the three point function is possible to evaluate directly, the generalresult follows.

The extra N −3 punctures is simply represented as µ derivatives. This isthe Liouville derivation of the expression (27).

The important point is that the use of a µderivative for inserting a puncture is a consequence of having just one W∞! This means thatthe calculation in [19] automatically incorporates this feature.For the more general case with nontachyonic special states, we return to the recursionrelation (26).

Let us redefine the fields according toWJ,m =2J2m −p˜WJ,m(55)Then the recursion relation takes the form< ˜TJ,J ˜WJ1,m1NYi=2˜TJi,Ji >= (J1 −m1)(J + J1 −1)J1< ˜WJ+J1−1,J+m1NYi=2˜TJi,Ji >(56)In [25] these very same recursion relations were obtained in the case J = m = 1/2 usingLiouville methods. The coefficient in front of the right hand side were shown to be of theform (2J1 −1)C2, where C stands for the appropriate Clebsch-Gordan coefficient.

The firstfactor comes from comparing with the purely tachyonic case where the answer is obtainedfrom a simple Veneziano like integral. To see the agreement one uses the Clebsch-Gordancoefficients of the special operator algebra as obtained in [9].CJ3,m3J1,m1,J2,m2 =A(J3, m3)A(J1, m1)A(J2, m2)(J1m2 −J2m1)(57)whereA(J, m) = −12[(2J)!

(J + m)! (J −m)!

]1/2(58)16

At J = m = 1/2 one finds C2 = J1−m12J1which then leads to (56). This is an important checkon the equivalence between the Liouville and matrix model approaches.An everywherepresent difficulty in these comparisons is, however, the fact that we are really sitting righton the momentum poles.

Clearly one needs to carefully regularize all expressions.Let us give some further illustrations in the case of puncture and dilaton insertions.We begin with the puncture. Starting with a general correlation function and inserting apuncture does not change the Veneziano like integral which has to be calculated.

When weinsert a puncture we also must remove one of the screening insertions. The only thing whichchanges is the zero mode part of the calculation.

We recall the resultZdφemφ−∆e−φ ∼Γ(−m)∆m(59)If we start with Γ(−m)µm we end up with Γ(−m + 1)µm−1 = −mΓ(−m)µm−1 when weremove a screening insertion. Comparing with (47) this shows the origin of the W∞relatedfactor in the tachyon correlation function.It is a consequence of the changing numberof puncture screening operators needed.For the tachyons the issue of connected or 1PIamplitudes is trivial for our case with just one tachyon of differing chirality.

There can’t beany internal punctures just from kinematics. This is no longer the case when we turn to thedilaton.

The crucial point is that the dilaton can be represented as a t2 derivative. Usuallythis is precisely equivalent to using the ordinary free field contractions giving Veneziano likecorrelation functions.

Inserting a dilaton in some tachyon correlation function means takingderivatives with respect to t2 (i.e. 1/α′).

For dimensional reasons all tachyon momenta areaccompanied by an t2. Without explicit t2’s one could write k ∂∂k for the dilaton.

For a dilatoninsertion in a nonzero momentum correlation function the dominating pole contributioncomes from letting the t2 derivative act directly on the poles. All other terms are clearlyless singular.

This gives the second term in (48). At zero momentum we must also considerthe dependence from the t2’s which go together with the µ’s.

The latter is a consequenceof dealing with 1PI rather than connected correlation functions. One could in fact obtainthe result by considering the explicit combinations of 1PI amplitudes into connected ones.To be more precise, if we want to obtain the 1PI amplitude from the connected amplitude,we must amputate external puncture legs but also subtract of the diagrams with internalpuncture propagators.

In particular we need to subtract a diagram where a puncture goesoffand converts into a dilaton. This diagram therefore involves a puncture insertion andgives a contribution corresponding to a µ derivative.

This is then simulated by an explicitt2 accompaning the µ’s to assure the proper subtraction. This corresponds to the first termin (48), which only becomes relevant compared to the second term at zero momentum.Otherwise we will get one power less of | log µ |’s.We need not restrict ourselves to a17

dilaton among special tachyons, the same reasoning works for a dilaton inserted in a generalcorrelation function. From these two examples we can conclude that the 1PI nature of thecorrelation functions is very important in the case of zero momentum.6ConclusionsWe have investigated the structure of special operator correlation functions in c = 1 quantumgravity.

Due to the presence of a W∞symmetry the calculations become very simple. Wehave also investigated the connection between the Liouville and the matrix model, indicatingthe agreement for the correlation functions.An important point is the existence in the uncompactified Liouville as well as matrixmodel formulation of c=1 of just one W∞.

From the Liouville point of view this is somewhatobscure since the operator product expansion and its Clebsch-Gordan coefficients seem togive a structure corresponding to two W∞’s. Fortunately the final outcome of the explicitcalculations are identical.

Further work is however needed to establish the full equivalence.A part of the problem is that many of the calculations are so ill defined. The reason isthat we are sitting right on the discrete momentum poles.

Especially in the Liouville theorythis is a big technical problem. Often we must rely on guesswork concerning ill definedanalytical continuations.

It is very doubtful if many of the results would have been obtainedcorrectly without knowing the answers in advance, given by the much more powerful matrixmodel.Important issues for future research are to investigate multicritical points of matrix mod-els with generalized potentials. It is natural to consider even non quadratic dependence onthe momentum.

Such perturbations could arise from adding the special states as we haveseen. Another important point is to identify the ’wrongly’ dressed special states in the ma-trix model.

Such states have negative gravitational dimensions and are important in thecontext of the two dimensional black hole.AcknowledgementsI would like to thank David Gross and Igor Klebanov for numerous discussions.18

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