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Ground Rings and Their Modules in

arXiv:hep-th/9111048v1 23 Nov 1991PUPT–1293, RU–91–49Ground Rings and Their Modules in2D Gravity with c ≤1 MatterDavid KutasovJoseph Henry Laboratories,Princeton University, Princeton, NJ 08544, USAEmil Martinec* and Nathan SeibergDepartment of Physics and AstronomyRutgers University, Piscataway, NJ 08855-0849, USAAll solvable two-dimensional quantum gravity models have non-trivial BRST cohomologywith vanishing ghost number. These states form a ring and all the other states in thetheory fall into modules of this ring.

The relations in the ring and in the modules havea physical interpretation. The existence of these rings and modules leads to nontrivialconstraints on the correlation functions and goes a long way toward solving these theoriesin the continuum approach.11/91* On leave of absence from Enrico Fermi Inst.

and Dept. of Physics, University of Chicago,Chicago, IL 60637

Recently, using matrix model techniques, a number of non-critical string models havebeen solved exactly [1] [2] [3] [4] [5]. Some of these models were shown to be equivalentto certain topological field theories [6] and they exhibit unexpected relation to integrablesystems.

Despite some progress in the continuum Liouville description of these theories[7] [8] [9] [10] [11] [12], their surprising integrability is yet to be understood. In the flatspace version of these theories the null vectors in degenerate Virasoro representations leadto Ward identities and to the solution of the theories [13].

This fact has led many peopleto conjecture that the simplicity and solvability of these quantum gravity models shouldbe associated with these null vectors. In this paper we take a step towards a completesolution of the models using the null vectors.The first consequence of the existence of degenerate Virasoro representations in thematter sector of these theories is the appearance of infinitely many new states in the BRSTcohomology [14] [15].

The standard physical fields have the form T = c¯cOeαφ where O is amatter primary field. In the (p, p′) (with p > p′) minimal models there are (p−1)(p′ −1)/2such fieldsTn,n′ = c¯cOn,n′e[1+ pp′ −pn′−p′np′] γ2 φ(1)(γ =q2p′p ) labeled by n = 1, ..., p −1 and n′ = 1, ..., p′ −1 with pn′ −p′n > 0.

In the(non-compact) c = 1 theory there is a continuous set of operatorsTq = c¯ceiqX/√2e(2−|q|) γ2 φ(2)(γ =√2) referred to as tachyons, labeled by the momentum q and infinitely many ‘specialstates’ for integer q labeled by an integer s ≥1Dq,s = c¯ceiqX/√2Pq,s(∂X, ...) ¯Pq,s(¯∂X, ...)e[2−(|q|+2s)] γ2 φ(3)where Pq,s is a polynomial in derivatives of X of dimension |q|s + s2. We will refer to atachyon with integer momentum Tq = Dq,s=0 as a special tachyon.

In the expressions forthe operators (1)-(3) we used the bound on the Liouville exponent of [8].RingsIn the interesting papers [14][15] Lian and Zuckerman have shown that the null vectorsin the matter and Liouville representations lead to more states with other ghost numbers.In the c = 1 system, these have the same X and Liouville momenta as (3) but havevanishing ghost numbers (in our convention the physical states have ghost number one).1

In fact, there are three sets of such states with ghost numbers (1, 0), (0, 1) and (0, 0). Thestates with (1, 0) and (0, 1) lead to conserved currents [16] and the (0, 0) states lead to aring [16].

We will denote these operators Rq,s.In the minimal models each highest weight state has two primitive null vectors, sothe ground ring has twice as many elements as the number of matter primaries.Thevanishing ghost number operators Rn,n′ are labeled by n = 1, ..., p−1 and n′ = 1, ..., p′ −1(without the standard identification of (n, n′) with (p −n, p′ −n′)). They are given bypolynomials in Virasoro generators of Liouville and matter sectors, as well as the modesof the ghost number current, acting on exp[−((n −1) + pp′ (n′ −1)) γ2 φ]On,n′.

Note thatOn,n′ ≡Op−n,p′−n′ but Rn,n′ ̸≡Rp−n,p′−n′ due to the different null vectors used in theirconstruction. The first of them R1,1 is the identity operator.

Explicit construction of someof the other operators was given in [17]R2,1 =|bc −1γ (LL−1 −LM−1)|2e−γφ/2O2,1R1,2 =|bc −γ2 (LL−1 −LM−1)|2e−(p/p′)γφ/2O1,2(4)For example, in pure gravity (p = 3, p′ = 2) R2,1 = |bc −1γ ∂φ|2e−γ2 φ. There are alsooperators with arbitrarily larger negative ghost number.Note that unlike the c = 1system there are no (1, 0) or (0, 1) operators and hence there are no conserved currents.As pointed out by Witten [16] the vanishing ghost number operators are special be-cause they lead to a ring structure.

The ring multiplication is obtained by considering theoperator product expansion of two vanishing ghost number operators Rm and Rm′ in theBRST cohomology. Since the product is BRST invariant, and has vanishing ghost number,it can be written asRm(z)Rm′(w) =Xm′′f m′′m,m′Rm′′(z) + [Q, O](5)for some operator O depending on m, m′, z and w. Here we have used the fact thatRm(z) has dimension zero.

Ignoring the BRST commutator in (5) we find a ring withstructure constants f m′′m,m′. Below we will examine whether these BRST commutators canbe dropped in correlation functions.Treating the Liouville field as free Witten [16] has shown that in the non-compactc = 1 system the ring is generated bya+ = R1,1 = |bc −1γ (∂φ −i∂X)|2e−γ2 (φ−iX)(6)2

and its conjugatea−= R−1,1 = |bc −1γ (∂φ + i∂X)|2e−γ2 (φ+iX)(7)i.e. Rn,s = a(|n|+n)/2+s−1+a(|n|−n)/2+s−1−and it has no relations.

The generators a± havea beautiful interpretation [16] as the phase space coordinates of the free fermions of thematrix model description of this model, and the whole ring is then identified as functionson phase space. Note that the scaling of a± is e−γφ/2; i.e.

they scale like inverse length.This is precisely the expected scaling of λ and its time derivative ˙λ. There is at leastone matrix model operator with these quantum numbers [18]1gstrRdXψ†λ3ψe±iX/√2 butthere may be others.

The power of λ in the operator does not lead to the wrong scalingbehavior because of the factor of the string coupling in the vertex which scales like thesquare of the length.The operators Jq,s and ¯Jq,s related to Rq,s with ghost numbers (1, 0) and (0, 1),are almost conserved.Their divergences ¯∂Jq,s and ∂¯Jq,s are BRST commutators.Ifthese commutators can be ignored, these operators are holomorphic and anti-holomorphiccurrents [16] and lead to a symmetry W′. W′ is the subalgebra of the algebra W of areapreserving diffeomorphisms of the a± plane that preserves the lines a+ = 0 and a−= 0.W transformations closely related to those of this symmetry were first noted in the matrixmodel in [19].

They were modified and identified as symmetries of the matrix model in[18] where their relation to the special states was also explained (see also [20]). In thecontinuum approach this symmetry was related to the special states also in [21].

Theelements of W′ do not have to preserve the lines a± = 0 pointwise but only as a set. Theselines were interpreted in [16] as the Fermi surface of the matrix model and W′ is then thesubalgebra of the matrix model symmetry W which is preserved by the ground state [16].We now return to the minimal models.

The matter content of the ground ring operatorRn,n′ (On,n′), and the CFT fusion rules constrain the multiplication table of the ring.Assuming that the Liouville field is free and examining the Liouville momenta of Rn,n′, itappears that the ring is generated by R1,2 and R2,1Rn,n′ = Rn−12,1 Rn′−11,2(8)(We did not check this expression explicitly in the most general case.) Unlike the c = 1system, n and n′ are bounded, and therefore there must be some relations in the ring.Examining the Liouville momentum we conclude Rp′−11,2= 0 and Rp−12,1 = 0.

It is amusingto note that the relation Rp′−11,2= 0 is the relation in the underlying chiral ring in the LG3

description of the topological field theory at the point p = 1 [6]. It would be interesting tounderstand the role of the other relation in that context.The generator R2,1 scales like the eigenvalue of the matrix model (inverse length) andthe other generator R1,2 scales like the conjugate momentum (length to the power −p/p′).Motivated by the interpretation of the ring at c = 1 and this scaling behavior, we wouldlike to interpret these generators as the eigenvalue and its conjugate momentum in thematrix model.

These are precisely the operators Q and P in Douglas’ [4] derivation of thestring equation. We therefore propose the identification R2,1 = Q and R1,2 = P and thefinite ring as functions on this “phase space.” Note that this “phase space” is not standardbecause Q and P can be raised only to finite powers.

It should be pointed out that thematrix model operators corresponding to R2,1 and R1,2 are generally believed to be givenby the fractional powers Q2p+p′p′+and Qp+2p′p′+of Q and not by Q and Qp/p′+respectively.The apparent discrepancy with the scaling properties is again resolved by recalling theextra factor of the string coupling, which scales as the [ (p+p′)p′]th power of Q. It is curiousthat both the c = 1 and the minimal models have an operator which scales like inverselength.

Such an operator, e−12 γφ, plays a fundamental role in the Backlund transformationin Liouville theory [22] [23] and provides the relation to its SL(2, R) symmetry [23] [24].The discussion above generalizes to the fermionic string. Again, the BRST cohomologycan be analyzed, and when there are degenerate representations there is nontrivial BRSTcohomology at ghost number zero (and all negative ghost numbers at ˆc < 1) [25], exceptthat in this case the generating elements are in the Ramond sector.

For c < 1 the matterhighest weights On,n′ are Ramond for n−n′ odd and Neveu-Schwarz for n−n′ even. HenceR1,2 and R2,1 are Ramond operators.

At c = 1 the special states are easily constructedusing super-SU(2) current algebra as in [16] (where the odd half-integer spin states arein the Ramond sector). The ring multiplication table is identical to the bosonic case, thestandard Z2 symmetry of Ramond-Neveu-Schwarz is identical to the Z2 of even vs. oddpolynomials in λ.

Since the generators are Ramond fields, any representation (see below)contains both Neveu-Schwarz and Ramond states.Note that the construction of the ground ring uses very little of the structure of thetheory, simply that it consists of two sectors: Liouville and matter, and the matter sectorhas degenerate representations. From a null state, general BRST arguments of the typegiven by Witten [16] predict the existence of BRST cohomology at ghost number zero.

Infact one can interpret the program of [23] for c > 1 as a study of this sector of the stringHilbert space. Indeed, the Liouville momenta studied there are precisely at the special4

values given by the Kaˇc formula. Clearly, these operators form a closed operator algebra.It is crucial that the Liouville exponent for these fields is always real even for c > 1.

Itis not clear to us why one is allowed to ignore all the other states in these theories and itremains to be seen what the physical interpretation of these states is for c > 1.ModulesNow, consider the other operators in the BRST cohomology. The operators of fixedghost number form a module (a representation) of the ring.

To see that, consider theoperator product expansion of Rm and an operator in the cohomologyRm(z)Vi(w) =Xi′T i′m,iVi′(z) + [Q, O](9)where the sum over i′ is over the fields in the cohomology with the same ghost number asVi. As in (5), we first ignore the BRST commutator on the right hand side and concludethat the coefficients T i′m,i represent the ring multiplication.

It is sometimes the case thatthis representation is not faithful; i.e. the matrices T i′m,i satisfy more relations than theunderlying ring and represent a quotient of it.We now examine the various modules which are present in these theories.

We startwith the c = 1 system and consider a tachyon state Tq with generic (not integer) momentumq. An easy free field calculation shows that for every fractional part of q and every sign ofq there is a separate module.

For q > 0a+Tq = q2Tq+1 + [Q, O+q+1]a−Tq = 0 + [Q, O−q−1](10)and for q < 0a+Tq = 0 + [Q, O+q+1]a−Tq = q2Tq−1 + [Q, O−q−1](11)None of these modules is faithful. For q > 0 the ring generator a−is represented by zeroand for q < 0 a+ is zero.

This fact has a simple interpretation in the matrix model. Asexplained by Polchinski [26], the tachyons can be thought of as ripples on the Fermi surface.Therefore, they satisfy the equation of the Fermi surface which for vanishing cosmologicalconstant are a+ = 0 for q < 0 and a−= 0 for q > 0.Similarly, the tachyons Tq are not in a faithful representation of the symmetry algebraW′.Since the anti-holomorphic part of Jq,s is the anti-holomorphic parts of Rq,s =5

a(|q|+q)/2+s−1+a(|q|−q)/2+s−1−, only Jq,s=1 act non-trivially and even of these, the negative qJ’s annihilate the positive momentum tachyons and vice versa. In terms of the underlyingphase space the interpretation of this fact is interesting.

The J’s generate the algebra W′of reparametrizations of the filled Fermi sea. It has a subalgebra W′′ of transformationswhich leave the Fermi surface invariant pointwise.

Since the tachyons Tq “live” on theFermi surface, W′′ acts trivially on them and the tachyons represent only the quotientW′/W′′ which is essentially a Virasoro algebra.For integer values of q the relations are different than (10)(11). For q positivea+Tq = q2Tq+1 + [Q, O+q+1]a−Tq = Dq+1,s=1 + [Q, O−q−1]a+Dq,s = A+q,sDq+1,s + [Q, O+q+1,s]a−Dq,s = A−q,sDq−1,s+1 + [Q, O−q−1,s](12)where A±q,s are calculable coefficients.

Similar relations hold for q < 0. Note that the zeromomentum tachyon Tq=0 is annihilated both by a+ and by a−.

However, the cosmologicalconstant operator φTq=0 is in the same module with the special tachyons and the specialstates. We conclude that the special states, the special tachyons and the cosmologicalconstant are all in one module.

Unlike the tachyon module, here the relation a+a−= 0 isnot satisfied. This relation was interpreted on the tachyon module as a consequence of thefact that the tachyons “live” on the Fermi surface.

Similarly we would like to argue thatsince it is not satisfied for the special tachyons and the special states, these are not rippleson the Fermi surface. We conclude that some of the deformations of the potential cannot berepresented as a change in the state of the system.

This observation is consistent with theMinkowski space interpretation of this theory. Rotating X to Minkowskian signature, allthe states in the theory are deformations of the Fermi surface [26].

Indeed, for MinkowskianX momentum and for macroscoipic Liouville states there are no special states in the BRSTcohomology.For c < 1, the ghost number zero states are at values of h such that there are nullvectors in their Verma modules as well, leading to ghost number −1 BRST cohomology viathe same argument that produced the ground ring. This structure repeats at each stage,leading to (p −1)(p′ −1) dimensional cohomology at all negative ghost numbers related tothe tower of inclusions of null modules inside one another in the matter sector [14] (notethat these physical states are not dressed null states).

The ground ring acts within the6

ghost number n Hilbert space; thus it is a representation (module) of the ground ringmodulo BRST commutators. In the fermionic string there is again a tower of inclusions ofnull modules, and hence BRST cohomology at every ghost number.Each BRST module at negative ghost number is a faithful representation of the groundring R. There are as many states in the BRST module as elements of R, which acts ina nondegenerate way.

This is not so for the physical state module at ghost number one,which is half the size due to the identification On,n′ ≡Op−n,p′−n′. The physical statemodule P cannot be a faithful representation and there must be extra relations definingthe action of R on P.These take the formRa1,2Rb2,1 = 0,a + b = [(p′ −1)p/p′] .

(13)There are two interesting submodules of this module. The ring action on the fields T1,n′with n′ = 1, ..., p′ −1 are annihilated up to BRST commutators by R2,1 and satisfyR1,2T1,n′ = T1,n′+1 + [Q, O].

Similarly, Tn,p′−1 are annihilated up to BRST commutatorsby R1,2 and satisfy R2,1Tn,p′−1 = Tn−1,p′−1 + [Q, O] for 1 ≤n < p −pp′ .These areanalogous to the tachyon modules in the c = 1 system (10)(11).Correlation FunctionsIf it is legitimate to drop the BRST commutators in (9) in correlation functions, wederive a set of identities for the amplitudes:< RmVi1...Vin >=Xi′T i′m,i1 < Vi′Vi2...Vin >=Xi′T i′m,i2 < Vi1Vi′...Vin >= ...(14)Note that these are not Ward identities. The latter would involve a sum of n terms ineach of which one of the operators in the correlation function is modified.

Here we have anequality between pairs of correlation functions. As we will see, (14) is not always satisfied.Correspondingly, the BRST commutators in (5) and (9) do not necessarily decouple.

Thestandard proof of their decoupling proceeds by moving the BRST charge from the BRSTcommutator to all the other operators in the correlation function.This has the effectof generating total derivatives on moduli space. The original BRST commutator fails todecouple when these total derivatives do not integrate to zero.

This phenomenon can beequivalently described in terms of contact terms at the boundaries of moduli space. It7

leads to violations of (14); the modules are deformed, with the structure constants T ijkacquiring dependence on the couplings one turns on in the action.Due to the structure of the ring for non-compact c = 1 described above, it is enoughto consider in this caseA(±)(q1, ..., qn) ≡⟨a±(z)Tq1...Tqn⟩(15)where P qi±1 = 0. It is implied in (15) that n−3 of the positions of Tqi are integrated over,and the appropriate T are stripped of the c¯c factors in (2).

The strategy for extractinginformation from (15) is to note that on general grounds A(±)(qi) is independent of z;therefore we can compare its value as a± approaches two different (unintegrated) Tqi. Thiswill give a set of relations between different amplitudes (14).It is convenient to consider first amplitudes, in which the {qi} satisfy a “resonance”condition P(2 −|qi|) = 5.

Such amplitudes are proportional to the volume of space-timeand possess integral representations which have been studied before [27] [10][11]. We willnow show that many of their properties are simple consequences of the action of the ringon the tachyon modules.Consider the general such correlation function ⟨Tq1..TqnTp1..Tpm⟩where qi > 0, pi < 0.It is known [10][11] that for n, m ≥2 these amplitudes vanish.

To derive this fact fromthe ring we evaluateAn,m(qi, pj) = ⟨a+(z)Tq1(0)Tp1(1)Tq2(∞)nYi=3Zd2ziTqi(zi)mYj=2Zd2wiTpj(wj)⟩(16)in two different limits. As z →0 we can replace a+Tq1 by Tq1+1 using (10).

It is importantthat the BRST commutator in (10) does not contribute to (16). Commuting Q to Tqi (Tpj)we find a total derivative in zi (wj).

One can show that it integrates to zero; indeed, it isreadily verified that near all boundaries of moduli space the integrand goes to zero in anappropriate region in momentum space (and is analytically continued to vanish everywhereelse). Hence, as z →0 we findAn,m(qi, pj) = q21⟨Tq1+1nYi=2TqimYj=1Tpj⟩(17)On the other hand, as z →1, we use (11) (again, one can explicitly verify that the boundaryterms due to the BRST commutators vanish) and conclude that An,m(qi, pj) = 0 (forn, m ≥2).

Comparing to (17) we find the desired result.8

The cases n = 1 (any m), and m = 1 (any n) have to be discussed separately:1) m = 1: In this case, momentum conservation and the “resonance” condition enforcep = p1 = −(n −2). As we saw before, for integer p1 (11) should be replaced by (12) i.e.acting with a+ produces one of the special states (3).

Although one can proceed this way,a much more useful relation is obtained by exchanging Tp1 ↔Tq2, such that (16) takes theformAn,1(qi) = ⟨a+(z)Tq1(0)Tq2(1)Tp1(∞)nYi=3Zd2ziTqi(zi)⟩(18)In this case it is easy to see that we can use the naive form of (14) to findq21⟨Tq1+1Tq2nYi=3TqiTp⟩= q22⟨Tq1Tq2+1nYi=3TqiTp⟩(19)Redefining Tq = Γ(1−|q|)Γ(|q|) ˜Tq, we conclude thatF(q1, ...qn) = ⟨˜Tq1... ˜Tqn ˜Tp⟩(20)is periodic in all its arguments (subject to the constraint Pni=1 qi = n −1):F(q1 + 1, q2, ...) = F(q1, q2 + 1, ...) = ...(21)An explicit evaluation [11] yields F(qi) = const, but one can not determine the periodicfunction F from the action of the ring. The algebraic reason for this ambiguity is that thetheory has a number of different modules.

The relation of the ring cannot determine the“reduced matrix elements” of different modules.2) n = 1: In this case we have to be careful with the BRST commutators in (10), (11).As z →0, one can use (10) naively; hence, A1,m = ⟨Tq1+1Qmi=1 Tpi⟩. On the other hand,as z →1, we find a BRST commutator which does not decouple.

The point is that sinceq1 is fixed kinematically (q1 = m −2), an on shell tachyon arises in the channel whereall wi simultaneously approach zero.This can be shown to lead to a finite boundarycontribution of the appropriate total derivative. Hence here a+Tp ̸= 0 (p < 0).

For aquantitative analysis it is more convenient to replace a+ by a−in (16), and imitate theprocedure of the first case.To emphasize the ambiguity of the ring relations (14) in c = 1 by a periodic function(21), it is useful to consider the open c = 1 string theory on the disk [28]. The qualitativeconsiderations used above are valid there as well.

Equation (10) takes the form (for q > 0)a+Tq =qTq+1 + [Q, V +]a−Tq =[Q, V −](22)9

and similarly for q < 0. The analysis of which correlation functions vanish is quite differentin this case; its conclusions are in agreement with [28].

The case ⟨a+Qni=1 TqiTp⟩withqi > 0, p < 0 leads, as above, to⟨nYi=1TqiTp⟩=nYi=11Γ(qi)G(q1, ..., qn)(23)where G is a periodic function of the qi. G is actually a complicated function of momenta[28]: G(qi) = Qn−1l=11sin π(q1+q2+...+ql).

It does not seem to be obtainable from the action ofthe ring.So far we have only discussed “resonant” amplitudes in which µ is in a sense zero.In generic amplitudes (“finite µ”) the situation is more involved. The BRST commutatorterms in (9) - (11) cannot be ignored.

One can still study the deformations of the Fermisurface in the presence of tachyon perturbations.As conjectured in [16], the equationa+a−= 0 should be modified for non-zero µ to a+a−= µ. (Note that this relation isnot an operator relation in the theory.) Unlike [16], from our point of view the relationa+a−= 0 is obtained as a relation in the tachyon module (10)(11).

Following [16], weconjecture that it is also modified to a+a−= µ. Indeed, one can show (using the methodsof [11]) that the three tachyon amplitude with generic momenta qi satisfies< (a+a−−µ) ˜Tq1 ˜Tq2 ˜Tq3 >= 0(24)In the presence of more tachyons the operator (a+a−−µ) does not vanish.

For example< (a+a−−µ) ˜Tq1 ˜Tq2 ˜Tq3 ˜Tq4 >= µ12P|qi|−1(25)and more complicated expressions for higher n point functions. This fact has an obviousinterpretation in the spirit of [26] and [16].

In the presence of more tachyons the Fermisurface is deformed and no longer satisfies a+a−= µ. As explained after equation (14),from the world-sheet point of view, this deformation can be understood as a contact termleading to non-zero correlation functions for the BRST commutators in (9) - (11).To summarize, the main point in this note is the importance of the relations in thering and in its non-faithful modules.

These relations constrain the correlation functions.However, in order to fully utilize the ring and its relations, we have to get better controlof the contact terms at the boundaries of moduli space.In the infinite radius c = 1system the tachyons are small deformations of the Fermi surface in the matrix model, and10

therefore, the relations in the tachyon module have a natural interpretation as determiningthe location of the Fermi surface. The contact terms should therefore be associated withthe deformation of the Fermi surface due to the presence of other tachyons.

We expectthem to appear as multi tachyon states in the right hand side of a±Tq. An important openproblem is to explicitly determine these contact terms.In the minimal models the ring and its relations should be more powerful than in thec = 1 system.

The ambiguity in the correlation functions in the c = 1 theory stems fromthe existence of infinitely many modules and the relations in the ring cannot determine the“reduced matrix elements.” In the minimal models the physical, ghost number one fieldsare all in one module and therefore a similar ambiguity is not present. Unfortunately, forthese theories we do not have an interpretation of the relations in the module analogousto the Fermi surface at c = 1.

An interesting relation R1,2R2,1 = 0 in the submodulesof T1,n′ and Tn,p′−1 is similar to the equation a+a−= 0 in the tachyon module.Anamusing possibility is that this relation will be deformed to R1,2R2,1 = gstr which isreminiscent of the tree level string equation Q0P0 = gstr in terms of the constant (zerothorder in derivatives) terms in the KdV operators P and Q.Since we know that thestring equation is analytic in the matrix model coupling constants, tk, and that these areanalytic in the conformal field theory couplings [12], we expect that the string equationcan be computed perturbatively in these couplings using free field techniques. The non-analyticity of the solution will arise only from the solution of this equation.

We hope thata better understanding of this issue will lead to the entire KdV structure and the Virasoroand W constraints of these theories and will make the connection of Liouville theory tothe matrix model and to topological field theory complete.Note added: After the completion of this work we learned that I. Klebanov andA. Polyakov had obtained some of our c = 1 results using another approach and that P.Bouwknegt, J. McCarthy and K. Pilch had independently found the ground ring in theˆc ≤1 fermionic system.It is a pleasure to thank T. Banks, M. Douglas, B. Lian, G. Moore, A. Polyakov, S.Shenker, C. Vafa, H. Verlinde, E. Witten, A.B.

Zamolodchikov and G. Zuckerman for usefuldiscussions. This work was supported in part by DOE grants DE-FG05-90ER40559, DE-AC02-76ER-03072 and DE-AC02-80ER-10587 and an NSF Presidential Young InvestigatorAward.11

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