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Operator Product Expansion and Topological

arXiv:hep-th/9111049v1 25 Nov 1991TIT/HEP–179STUPP–91–122February 5, 2018Operator Product Expansion and TopologicalStates in c = 1 Matter Coupled to 2-D GravityNorisuke SakaiDepartment of Physics, Tokyo Institute of TechnologyOh-okayama, Meguro, Tokyo 152, JapanandYoshiaki TaniiPhysics Department, Saitama UniversityUrawa, Saitama 338, JapanAbstractFactorization of the N-tachyon amplitudes in two-dimensional c = 1 quantumgravity is studied by means of the operator product expansion of vertex operatorsafter the Liouville zero mode integration. Short-distance singularities between twotachyons with opposite chiralities account for all singularities in the N-tachyonamplitudes.

Although the factorization is valid, other possible short-distance sin-gularities corresponding to other combinations of vertex operators are absent sincethe residue vanishes.Apart from the tachyon states, there are infinitely manytopological states contributing to the intermediate states. This is a more detailedaccount of our short communication on the factorization.

§1 IntroductionRecent studies in matrix models1),2) have made significant progress in thenonperturbative treatment of two-dimensional quantum gravity and string the-ory. To make further progress, it is important to clarify these nonperturbativeresults of matrix models from the viewpoint of the usual continuum approach ofthe two-dimensional quantum gravity.The most standard way of treating thecontinuum theory of quantum gravity in two-dimensions is the so-called Liouvilletheory.3)−6) Several works have recently appeared to compute the correlation func-tions on the sphere topology using the Liouville theory and the technique of analyticcontinuation.7)−12) These results are consistent with those of matrix models.13)−15) Sofar only conformal field theories with central charge c ≤1 have been successfullycoupled to quantum gravity.In a recent publication,16) we have reported the result of the factorization anal-ysis of the c = 1 quantum gravity.

The purpose of this paper is to give a moredetailed account of the analysis for understanding the factorization of c = 1 quan-tum gravity in terms of the short-distance singularities arising from the operatorproduct expansion (OPE) of vertex operators.The c = 1 case is the richest and the most interesting. It has been observedthat this theory can be regarded effectively as a critical string theory in two di-mensions, since the Liouville field zero mode provides an additional “time-like”dimension besides the obvious single spatial dimension given by the zero mode ofthe c = 1 matter.17) We have a physical scalar particle corresponding to the cen-ter of mass motion of the string.

Though it is massless, it is still referred to as a“tachyon” following the usual terminology borrowed from the critical string theory.Since there are no transverse directions, the continuous (field) degrees of freedomare exhausted by the tachyon field. In fact, the partition function for the torustopology was computed in the Liouville theory, and was found to give precisely thesame partition function as the tachyon field alone.18),19) However, there are indica-tions of the existence of other discrete degrees of freedom in the c = 1 quantum2

gravity. Firstly, the correlation functions obtained in the matrix model exhibit acharacteristic singularity structure.14) In the continuum approach of the Liouvilletheory, Polyakov has observed that special states with discrete momenta and Liou-ville energies can produce such poles, and has called these operators co-dimensiontwo operators.20) Moreover the two-loop partition function has been computed ina matrix model and evidence has been noted for the occurrence of these topo-logical sates.21) Most recently, the symmetry governing such topological states arebeginning to be understood.22),23) It is clearly of vital importance to pin down therole played by these topological states as much as possible.

In the critical stringtheory, the particle content of the theory and unitarity have been most clearlyrevealed through the factorization analysis of scattering amplitudes. On the otherhand, the factorization and unitarity of the Liouville theory have not yet been wellunderstood.Since we are interested in the short distance singularities, we consider corre-lation functions on a sphere topology only.

We find that the singularities of theamplitudes can be understood as short-distance singularities of two vertex oper-ators of tachyons with opposite chiralities. We also find that the other possibleshort-distance singularities corresponding to other combinations of tachyons areabsent since the residue vanishes.

It is found that infinitely many discrete statescontribute to the intermediate states of the factorized amplitudes, apart from thetachyon states. These are the co-dimension two operators of Polyakov20) and pre-sumably are topological in origin.

We have also explicitly constructed some of thetopological states.Recently we have received two papers where other groups have discussed thesubject partly overlapping with ours.24),25) Their results are consistent with ours.In particular, the paper by Di Francesco and Kutasov has shown the decouplingof various topological states in certain kinematical configurations which was onlypartially demonstrated in our analysis.3

§2 Correlation Functions on a SphereIn this paper we consider c = 1 conformal matter realized by a single bosonicfield (string variable) X coupled to two-dimensional quantum gravitySmatter[g, X] =14πα′Zd2ξ√ggαβ∂αX∂βX,Z =ZDgαβDX1Vgaugee−Smatter[g,X],(2.1)where α′ is the “Regge slope parameter” and Vgauge is the volume of the groupof diffeomorphisms. This is the so-called “noncritical string” in one dimension.We use the method of Ref.

3). In this approach the metric is parametrized by adiffeomorphism f and the Liouville field φ representing the freedom of local Weylrescalingf∗gαβ = eφˆgαβ(τ),(2.2)where ˆgαβ(τ) is a reference metric that depends on the moduli parameters τ of theRiemann surface.

The diffeomorphism invariance allows us to choose the conformalgauge. On a sphere there is no moduli and the gauge condition isgαβ = eφˆgαβ.

(2.3)In the following we shall set α′ = 2 for convenience. The matter action in theconformal gauge (2.3) isSmatter[ˆg, X] = 12πZd2z∂X ¯∂X,(2.4)where we have used a holomorphic variable z = ξ1 + iξ2 and an anti-holomorphicvariable ¯z = ξ1 −iξ2.

If we want to consider c < 1 case, we need to introduceadditional terms with a parameter α0 = −p(1 −c)/12Smatter,α0[ˆg, X] = iα04πZd2zpˆg ˆRX + iα02πZdsˆkX,(2.5)where ˆR is the curvature of the two-dimensional surface with the metric ˆgαβ andˆk is the geodesic curvature along the boundary of the surface parametrized by s.4

After converting the Liouville field path integral measure into the translationallyinvariant measure for a usual scalar field,3) we can treat the Liouville field almostas a free field except for the nontrivial dynamics of the zero mode due to thecosmological term with the cosmological constant µ.4),5) The Liouville action isgiven by the following action involving parameters Q, α and µ3)−6)SL[ˆg, φ] = 18πZd2zpˆgˆgαβ∂αφ∂βφ −Q ˆRφ + 8µ eαφ−Q4πZdsˆkφ= 12πZd2z∂φ¯∂φ −Q4ˆRφ + 2µ eαφ−Q4πZdsˆkφ. (2.6)The correlation functions of X and φ are given by⟨X(z, ¯z)X(w, ¯w)⟩= ⟨φ(z, ¯z)φ(w, ¯w)⟩= −log|z −w|2.

(2.7)The energy-momentum tensor is given byT(z) = −12(∂X)2 −12(∂φ)2 −12Q∂2φ. (2.8)If we want to consider c < 1 case with α0 ̸= 0 in Eq.

(2.5), we should add iα0∂2Xto T(z).The parameter Q is fixed by requiring3)−6) that the theory does not depend ona choice of the reference metric ˆgαβ. One finds in the present case of c = 1Q = 2√2.

(2.9)If we want to consider the c < 1 case with α0 ̸= 0 in Eq. (2.5), we should useQ =p(25 −c)/3 = 2q2 + α20 instead.The other parameter α is determinedby requiring that the naive cosmological term operator eαφ is a primary field of5

dimension (1, 1)−12α(α + Q) = 1. (2.10)In the present case of a one-dimensional string (c = 1), one findsα = −√2.

(2.11)If we consider the c < 1 case, we should use α+ = −Q/2 + |α0| instead.We have introduced a naive cosmological term 8µ eαφ.Although the naivecosmological term operator eαφ has conformal weight (1, 1), it has been arguedthat the operator does not correspond to a genuine local operator in the case ofc = 1 because of the Liouville zero mode path integral.4) The correct cosmologicalterm operator in the c = 1 case is given by φ eαφ rather than eαφ. We can obtainthe correlation functions with the correct cosmological term operator µrφ eαφ fromthose with the naive cosmological term operator µ eαφ by the following limitingprocedure: we replace α by (1−ǫ)α and µ by µr/(2ǫ) and take the ǫ →0 limit.8),10)This procedure is guaranteed to give the correct result, since the naive cosmologicalterms decouple from the correlation functions as we will explain later.The gravitationally dressed tachyon vertex operator Op with momentum p isgiven byOp =Zd2zpˆg eipX eβ(p)φ.

(2.12)We see that the Liouville zero mode can be regarded as an “imaginary time” andthe exponent β(p) as “energy”. The gravitational dressing of the tachyon vertexoperator is determined by requiring that the dressed operator has conformal weight(1, 1):12p2 −12β(β + Q) = 1.

(2.13)There are two solutions for the Liouville energy β(p) = −Q2 ± |p|. It has beenargued4),5) that the Liouville zero mode path integral is well-defined only if β >6

−Q/2. Hence we choose in the case of c = 1β(p) = −√2 + |p|.

(2.14)For c < 1, p2/2 in Eq. (2.13) should be replaced by p(p −2α0)/2 and the solutionbecomes β(p) = −Q2 ± |p −α0|.The N-point correlation function of the tachyon vertex operators (2.12) on thesphere without boundary is given by the path integral⟨Op1 · · · OpN⟩=ZDXDφVSL(2,C)e−Smatter[ˆg,X]−SL[ˆg,φ] Op1 · · · OpN=ZNYi=1hd2zipˆgi1VSL(2,C)Deip1X(z1) · · · eipNX(zN)EX×Deβ1φ(z1) · · ·eβNφ(zN)Eφ ,(2.15)where VSL(2,C) is the volume of the SL(2, C) group which is generated by theconformal Killing vectors on the sphereVSL(2,C) =Zd2zad2zbd2zc|za −zb|2|zb −zc|2|zc −za|2.

(2.16)In Eq. (2.15), we have omitted powers of the string coupling constant g−2st for thesphere topology, since we will deal with correlation functions on the sphere only.It is convenient to separate the zero modes in the path integral: the zero modeφ0 of the Liouville field (φ = φ0 + ˜φ) and the zero mode X0 of the matter field(X = X0 + ˜X) and perform the zero mode integrations first.

The integration overthe zero mode φ0 of the Liouville field gives* NYi=1eβiφ+φ=ZDφ e−SL[ˆg,φ] eβ1φ · · · eβNφ=1−αΓ(−s)ZD ˜φ e−SL,0[ˆg,˜φ]µπZd2wpˆg eα˜φs NYi=1eβi ˜φ,(2.17)7

where s is given for the sphere ass = −1α Q +NXi=1βi!. (2.18)The non-zero mode part of the Liouville action SL,0 is given by the free field action.Integration over the matter zero mode X0 gives the momentum conservationNXj=1pj = 0.

(2.19)It is convenient to introduce two-momenta for tachyons pj = (pj, −iβj), for thenaive cosmological terms q = (0, −iα) and for the source Q = (0, −iQ) respectively.The definition of s together with the momentum conservation allow us to writedown “energy-momentum conservation” using two-momentaNXj=1pj + sq + Q = 0. (2.20)Let us note that, in the case of c < 1, the source two-momentum becomes Q =(−2α0, −iQ) and the momentum conservation is modified to PNj=1 pj = 2α0.Hence the two-momentum conservation (2.20) is unchanged.

Therefore we obtainthe N-point correlation function after the zero mode integration* NYj=1Opj+= 2πδNXj=1pjΓ(−s)−αZD ˜XD ˜φVSL(2,C)NYj=1OpjµπZd2wpˆg eα˜φ(w)se−S0= 2πδNXj=1pjΓ(−s)−αZNYi=1hd2zipˆgi1VSL(2,C)×* NYj=1eipj ˜X(zj)+˜X*µπZd2wpˆg eα˜φ(w)s NYj=1eβj ˜φ(zj)+˜φ,(2.21)The expectation values with ˜φ, ˜X denote the path integral over the non-zero modes8

˜φ, ˜X with the free actionS0 = 12πZd2z∂˜φ¯∂˜φ + ∂˜X ¯∂˜X. (2.22)For a non-negative integer s, we can evaluate the non-zero mode ˜φ integralby regarding the amplitude as a scattering amplitude of N-tachyons and s naivecosmological terms.

* NYi=1eβiφ+φ= 1−αΓ(−s)µπs ZsYj=1d2wjY1≤j

It is convenient to factor out the momentum conservation and to definethe normalized amplitude ˜A* NYj=1Opj+= 2πδNXj=1pj1−αΓ(−s) ˜A(p1, · · ·, pN). (2.24)The normalized N-tachyon amplitude is given by˜A =µπs ZNYi=4d2zisYj=1d2wjNYi=4|zi|2p1·pi |1 −zi|2p2·piY4≤i

(2.25)In spite of the non-analytic relation (2.14) between energy β and momentump, we need to continue analytically the formula into general complex values of9

momenta in order to explore the singularity structure. Therefore it is convenient todefine the chirality of tachyons: the tachyon has positive (negative) chirality if thetachyon energy-momentum satisfies (β +√2)/p = 1(−1) irrespective of the actualvalues of momentum.20) It seems to us that the operators with β < −√2 in Eq.

(2.15) are free from the trouble noted in Refs. 4), 5) since the Liouville zero modeφ0 has already been integrated out.

The physical values of momenta are reachedby analytic continuation in s, since s is related to other momenta through (2.18).For generic physical values of momenta (s ̸= 0), one finds a finite result for theN-tachyon amplitudes (bothDQNj=1 OpjEand ˜A(p1, · · · , pN) are finite). However,the result is different in different chirality configurations, since the amplitude isnon-analytic in momenta.Let us consider the kinematical configuration where all tachyons except onehave the same chirality.

If p1 has negative chirality and the rest p2, · · ·, pN positivechirality, momentum conservation readsp1 + p2 + · · · + pN = 0(2.26)and energy conservation dictates that (α = −√2)−p1 + p2 + · · · + pN =√2(N + s −2). (2.27)Thus one obtains the kinematical constraintsp1 = −N + s −2√2,β1 = N + s −4√2.

(2.28)It has been shown that the N-tachyon amplitude is given in this kinematical con-figuration by7)−11)˜A(p1, · · · , pN) = πN−3[µ∆(−ρ)]sΓ(N + s −2)NYj=2∆(1 −√2pj),(2.29)where ∆(x) = Γ(x)/Γ(1 −x). The regularization parameter ρ is given by ρ =−α2/2 and is eventually set equal to −1 after analytic continuation (in the central10

charge c). We see immediately that the insertion of the naive cosmological termoperator always gives vanishing correlation functions, since ∆(−ρ) = Γ(−ρ)/Γ(1+ρ) vanishes at ρ = −1.

This decoupling of the naive cosmological term operatorguarantees the validity of our procedure in computing the correlation functionwith the correct cosmological term operators φ eαφ: we replace α by (1 −ǫ)αand µ by µr/(2ǫ) and take the ǫ →0 limit.8),10) In effect, we should just replacethe combination µ∆(−ρ) by the correct (renormalized) cosmological constant µr.For the case of negative chirality for p1 and positive chirality for the remainingp2, · · ·, pN, the momentum of p1 is fixed because of the kinematical constraints(2.28). As a function of the other momenta p2, · · ·, pN, the N-tachyon amplitudeexhibits singularities atpj = n + 1√2 ,j = 2, · · · N; n = 0, 1, 2, · · ·,(2.30)but has no singularities in other combinations of momenta contrary to the dualamplitudes in the critical string theory.

The first pole will be shown to correspondto tachyon as an intermediate state in the next section. Other higher level poles forn = 1, 2, · · · will be shown to correspond to topological states as argued by severalpeople.14),20)Let us examine other kinematical configurations.

The amplitudes with onetachyon of positive chirality and the rest negative are given by changing the sign ofpj in Eq. (2.29).

On the other hand, if each chirality has two or more tachyons, thenormalized amplitude ˜A is finite for generic momenta but has the factor 1/Γ(−s).Hence ˜A vanishes for more than two tachyons in each chirality, when we considernon-negative integer s in the following. This property has been explicitly demon-strated for the four- and five-tachyon amplitudes in the Liouville theory,8),20) andhas been argued to be a general property using the matrix model.14) Therefore wetake it for granted that the tachyon scattering amplitudes ˜A vanish for non-negativeinteger s, unless there is only one tachyon in either one of the chiralities.Let us note that our assertion is consistent with the argument for vanishingS-matrix by Gross and Klebanov14): they absorb the ∆(1 ±√2 p) factor for the11

individual momenta into a renormalization factor of tachyon vertex operators. Thismomentum dependent renormalization factor is harmless if the momenta are atsome generic values (p ̸= (n + 1)/√2).

However, the two-dimensional kinematicsforces one of the renormalization factors to be infinite, if there is only one tachyon ineither one of the chiralities,8)−10) since the kinematical constraints fix the momentumof the tachyon to be at one of the poles (2.30). In the case of negative chiralityfor p1 and positive chirality for the rest p2, · · ·, pN, the momentum p1 is fixed tobe (2.28).

Because of this infinite renormalization, the renormalized amplitudes ofGross and Klebanov vanish even if there is only one tachyon in either one of thechiralities.Let us suppose that we are interested in the case of vanishingly small valuesof the cosmological constant µ. We note that, even if the cosmological constantµ is infinitesimal, the cosmological term µeαφ can become arbitrarily large forsufficiently large negative values of the Liouville field φ (α < 0 in our convention).Therefore the Liouville field in the path integral is suppressed for large negativevalues.

Since the large positive values of the Liouville field should be cut offasan ultraviolet or short-distance cut-off, the tachyon field space is restricted to alarge but finite volume (proportional to ln µ) .4),5) It is important to remember thatwe cannot neglect the cosmological constant completely even if it is infinitesimallysmall. On the other hand, the contribution to the amplitudes proportional to ln µis given by the tachyon interaction in the bulk and hence is insensitive to thedetails of the cut-offof the Liouville field space.

The nonlinearity of the Liouvilleinteractions remains only in the form of the restricted field space. We are preciselyinterested in this bulk interaction of tachyons.

The finite values of the normalizedscattering amplitudes ˜A correspond to a divergent correlation function at s =positive integers.Since the correlation functions are given by ˜A multiplied by(µ)sΓ(−s), the divergent correlation functions are more properly interpreted asthe logarithmically divergent contribution as we let µ →0. Therefore the finitetachyon scattering amplitudes ˜A at s = non-negative integers represent the tachyoninteraction proportional to the volume of the Liouville field space.14),20),8) Hence they12

are often called the bulk or resonant amplitudes.§3 Higher Level OperatorsSince the short-distance singularities should come from terms in the OPE, wefirst examine the operators responsible for these singularities. We will consider theOPE of vertex operators after the zero mode integration of φ and X in Eq.

(2.21).Therefore the operators we consider in this section consist of only non-zero modeX = (˜φ, ˜X), which have the free action (2.22). The simplest operator is eip·X, whichis the tachyon operator (2.12) with zero modes omitted.

For higher levels, it hasbeen pointed out that there are only null states at generic values of momenta.20),14)However, there are exceptional values of momenta where the null states degenerateand new primary states emerge as a result. These new primary states are calledco-dimension two states by Polyakov,20) and special states, topological states ordiscrete states by other people.14),19) We can construct vertex operators for thesetopological states in the following way.The energy-momentum tensor for X is given byT(z) = −12∂X · ∂X −12iQ · ∂2X= −12∂˜X∂˜X −12∂˜φ ∂˜φ −12Q∂2 ˜φ(3.1)and the anti-holomorphic component given by (3.1) with ∂replaced by ¯∂.

Theysatisfy the Virasoro algebra of the central charge 26. We should construct the fieldof conformal weight (1, 1) with respect to this energy-momentum tensor by takinglinear combinations of monomials of derivatives of X multiplied by eip·X.

Theoperator at level n has n ∂’s and n ¯∂’s for each monomial. The condition for theconformal weight to be (1, 1) at level n is12p · (p + Q) + n = 1.

(3.2)13

We should note that there are two branches of the solution for the condition (3.2)β = −√2 ±pp2 + 2n. (3.3)Seiberg has noted that the vertex operator with β > −√2 gives an ill-definedintegration over the Liouville zero mode.4),5) Hence the lower sign in Eq.

(3.3) isforbidden by this condition. However, we consider both cases here, since we areconsidering vertex operators consisting of non-zero mode ˜φ only.

In fact, the s naivecosmological terms are a result of the Liouville zero mode φ0 integration. We shallcall the upper sign solution S- (Seiberg) type and the lower sign A- (anti-Seiberg)type.At level n = 1 the general form of the vertex operator with one ∂and one ¯∂isV = ζµν∂Xµ ¯∂Xνeip·X.

(3.4)For this operator to be a primary field of the unit conformal weight, the OPE withthe energy-momentum tensor (3.1) must beT(z) V (w) ∼1(z −w)2 V (w) +1z −w ∂V (w),(3.5)which gives the conditions on the polarization tensor(p + Q)µζµν = 0 = ζµν(p + Q)ν(3.6)and the on-shell condition (3.2) with n = 1. A similar condition should also besatisfied for the anti-holomorphic part in order for the operator to be a (1, 1)primary.

Solving these conditions we find only one primary field with weight (1, 1)at generic values of momentum, i.e. p ̸= 0V (1) = p · ∂X p · ¯∂X eip·X = −L−1 ¯L−1eip·X.

(3.7)The above state is clearly null. However, the situation changes at p = 0.

For theS-type, the operator vanishes at p = 0. Therefore we can construct a new operator14

by a limitV(1,1) = limp→0V (1)p2= ∂X ¯∂X. (3.8)We easily find that this field is primary and not null.This kind of a peculiaroperator exists only at a discrete momentum.Since there are two kinematicalconstraints to specify the state, one for the energy and the other for the momentum,the state is called co-dimension two.20) As for the A-type at p = 0, we find that the(1, 1) operator condition does not constrain the polarization tensor multiplying theoperator ∂X ¯∂X eip·X.

Hence we again obtain a new primary fieldV ′(1,1) = ∂X ¯∂Xe−2√2φ. (3.9)At level two, the general form of operators is given by (we present only theholomorphic part)V =ζµ∂2Xµ + ζµν∂Xµ∂Xνeip·X.

(3.10)The OPE (3.5) gives the conditionsζµµ + i(3Q + 2p)µζµ = 0,ζµ −i(p + Q)νζνµ = 0(3.11)and Eq. (3.2) with n = 2.

We find two independent solutions to these conditionsfor the holomorphic partV (2) =L−2 + 32L2−1eip·X,V (3) = L−114i [(8 −p · Q)p −2Q] · ∂X eip·X. (3.12)Both fields are null.At a special value of the momentum these two operatorsare linearly dependent and we obtain a topological state.

For instance, the (2,1)15

topological state of S-type is given byV(2,1) = limp→1√26√2p −1√2(V (2) −V (3))= (13 ∂X∂X −∂φ∂φ −6 i ∂X∂φ −√2 i ∂2X −√2 ∂2φ) e1√2 i (X−iφ). (3.13)We find exactly the same situation for the anti-holomorphic part.

We can con-tinue to explore (1, 1) operators at higher levels similarly. We expect that these(1, 1) operators are null fields for generic values of momenta, and that, at specialvalues of momenta, these null states are not linearly independent, namely theydegenerate.

Then we obtain a new primary state from a limit of an appropriatelinear combination of these null states. We expect to have both S-type and A-typetopological states.There are other procedures to obtain topological states.These states werefound to originate from the gravitational dressing of the primary states in thec = 1 conformal field theory which create null descendants at level n.15),19),26) Themomentum p of the initial primary state and the level n are specified by twopositive integers (r, t)p = r −t√2 ,n = rt.

(3.14)Since the level n corresponds to the n-th derivatives, we find that the energy β ofthe topological state is determined by the (1, 1) conformal weight condition (3.3)and is given byβ = −2 ± (r + t)√2. (3.15)The upper (lower) sign corresponds to the S- (A-) type solution.

For example,(r, t) = (1, 1) and (2, 1) operators of S-type areV(1,1) = ∂X ¯∂X,V(2,1) =∂X∂X + 1√2i∂2X ¯∂X ¯∂X + 1√2i¯∂2Xe1√2(iX+φ). (3.16)We find that these operators coincide with our operators (3.8) and (3.13) respec-16

tively up to only a certain amount of null operators.§4 OPE and FactorizationTo understand the poles of the amplitudes in terms of short-distance singu-larities in the OPE, we shall consider the case of s = non-negative integers bychoosing the momentum configuration appropriately.As we explained before,these amplitudes at non-negative integer s represent the so-called bulk or res-onant interactions.14),20) The only nonvanishing N-tachyon amplitudes ˜A at s =non-negative integers are for the kinematical configuration where one of the chiral-ities have only a single tachyon and the rest opposite chirality. Here we shall takethe case of s = 0 for the N-tachyon amplitude with only one negative chiralitytachyon (p1), and examine the s = positive integers case at the end.First we shall illustrate the origin of short-distance singularities in the simplestcontext of the four tachyon scattering amplitude with s = 0.

As in Eq. (2.25) wefix z2 = 0, z3 = 1, z4 = ∞and set z1 = z to find˜A(p1, · · ·, p4) =Zd2z |z|2p1·p2 |1 −z|2p1·p3.

(4.1)The short-distance singularities corresponding to z1 ∼z2 (z →0) can be exhibitedby expanding the integrand around z = 0˜A(p1, · · · , p4) ≈Z|z|≤ǫd2z |z|2p1·p2∞Xn=0Γ(1 + p1 · p3)n! Γ(p1 · p3 −n + 1)(−z)n2≈∞Xn=0πn + 1 + p1 · p2Γ(1 + p1 · p3)n!

Γ(p1 · p3 −n + 1)2. (4.2)We see that the so-called noncritical string of the c = 1 quantum gravity exhibitsexactly the same type of short-distance singularities as the familiar critical string17

theory. The kinematics at the pole reflects the peculiarities of two-dimensionalphysicsp1 = (−√2, 0),p2 = (n + 1√2, −in −1√2),pj = (pj, −i(−√2 + pj))j = 3, 4.

(4.3)We can express the short-distance singularities in terms of these momenta to find˜A(p1, · · ·, p4) ≈∞Xn=0(−1)n(n! )2πn + 1 −√2 p24Yj=3∆(1 −√2 pj).

(4.4)This shows that all the singularities in p2 in the full amplitude are correctly ac-counted for by these short-distance singularities near z1 ∼z2. Similarly, the singu-larities in p3 and p4 are accounted for by the short-distance singularities for z1 ∼z3and z1 ∼z4 respectively.

Hence we see that all the singularities in the amplitudeare nothing but the short-distance singularities associated with the opposite chi-rality tachyons approaching each other.We can understand these short-distance singularities by means of the OPE oftwo vertex operators: eip1·X(z1) :: eip2·X(z2) : ∼∞Xn=0 1n!2|z1 −z2|2p1·p2+2n Vn(z2). (4.5)The operators on the right hand side are given byVn = : eip2·X∂n ¯∂neip1·X := : (−p1 · ∂nX p1 · ¯∂nX + · · ·) eip·X :,(4.6)where p = p1 + p2.

It can be shown that Vn is a primary field with a conformalweight (1, 1) when p satisfies Eq. (3.2).

Integration of Eq. (4.5) by z1 over theregion |z1 −z2| ≤ǫ gives singularities in the momentum p∞Xn=0 1n!2π12p · (p + Q) + n −1Vn(z2).

(4.7)To discuss the OPE in a general context, we now consider the N-tachyon ampli-tude where p1 has negative chirality and p2, · · ·, pN have positive chirality. From18

the energy-momentum conservation (2.26) and (2.27) for this kinematical configu-ration, the two-momentum of the tachyon 1 takes a fixed valuep1 =−N −2√2, −iN −4√2. (4.8)This kinematical constraint implies for the intermediate state momentum p =p1 + p212p · (p + Q) + n −1 = (N −3)(1 −√2p2) + n.(4.9)At the pole p2 = 1/√2 (n = 0), the intermediate state momentum becomesp =−N −3√2 , −i−√2 + N −3√2.

(4.10)Hence we find that the pole at p2 = 1/√2 (n = 0) in Eq. (4.7) is due to the tachyonintermediate state V0 of negative chirality.

The higher level poles (n ≥1) are dueto the topological states in Vn (n ≥1) that we have discussed in the previoussection. These topological state operators can take various appearances dependingon the amount of null states.

If we letN −3 = t,n = rt,(4.11)then the singular factor in Eq. (4.7) for n ≥1 becomes112p · (p + Q) + n −1 =1t(r + 1 −√2p2).

(4.12)Hence we obtain the polep2 →r + 1√2 . (4.13)In this case the intermediate state momentum becomesp =r −t√2 , −i−√2 + r + t√2=r −t√2 , −i−√2 +sr −t√22+ rt.

(4.14)This two-momentum is precisely the two-momentum (3.14) and (3.15) of the topo-19

logical state at level n = rt of the S-type. Therefore we find that the (r, t) topo-logical state is contained in Vn if the number of tachyons N and the level of theintermediate state n are specified by Eq.

(4.11) and that the operator must be ofS-type. Thus we find that each (r, t) primary topological state appears as an inter-mediate state of the N-tachyon amplitude at level n exactly once.

This togetherwith the above tachyon singularity explains all singularities in Eq. (2.30).Let us discuss the residues of these short-distance singularities.

These polescan be associated with the intermediate tachyon or topological states appearing inthe OPE. As illustrated in Fig.

1 the residues of these poles are given by a productof two parts.One of them, the right side blob in Fig. 1, has N −2 tachyonswith incoming momenta p3, · · · , pN, and an intermediate particle with incomingmomentum p = p1 + p2, and is given by a kind of dual amplitude.

The other parthas two tachyons with incoming momenta p1, p2 and the intermediate particle asshown in the left side blob in Fig. 1.

First we examine the pole at p2 = 1/√2,namely at the lowest level (n = 0). This pole is due to the tachyon intermediatestate with negative chirality.

In fact we find that the residue of the pole p2 = 1/√2is precisely given by a product of the tachyon three point function and the N −1tachyon amplitude with a single (intermediate state) tachyon p having negativechirality and the rest p3, · · ·, pN having positive chirality˜A(p1, p2, p3, · · · , pN) ≈π(N −3)(1 −√2p2)˜A(p, p3, · · ·, pN)= ˜A(p1, p2, −p)2πp · (p + Q) −2˜A(p, p3, · · · , pN). (4.15)This shows that the factorization is valid similarly to critical string theory.

Bysymmetry, we can explain the lowest poles in each individual momentum pj = 1/√2as the tachyon intermediate state in the OPE of p1 and pj.For higher level poles, we explicitly evaluate the residue of the short-distancesingularities up to p = 3/√2 and up to N = 5. For instance, the five tachyon20

amplitude has short-distance singularities at p2 = 2/√2 and 3/√2˜A(p1, · · ·, p5) ≈−π22(2 −√2 p2) +π28(3 −√2 p2)5Yj=3∆(1 −√2 pj). (4.16)The residues of these poles correctly reproduce the residues of the poles in the fullamplitude.

Each term of Eq. (4.16) can also be written as˜A(p1, p2; −p)2πp · (p + Q) + 2(n −1)˜A(p; p3, · · · , pN),(4.17)where n = 1 and 2 for the first and the second terms respectively and −p andp are the momenta of the intermediate particles.

We have explicitly verified thatthese amplitudes appearing in the residue agree with those obtained from the threeand N −1 point amplitudes with one topological state of momentum −p and p.It is rather difficult to compute the short-distance singularities explicitly to anarbitrary level except for the four-point amplitude that we have already workedout in Eq. (4.2).

Therefore we content ourselves with the computation of lower levelsingularities in explicitly demonstrating that the singularities of the amplitudes allcome from the short-distance singularities of p1 and pj.The OPE suggests that there may be other short-distance singularities in othercombinations of momenta if one considers corresponding combinations of vertexoperators approaching to the same point. For instance, short-distance singularitiescorresponding to k vertex operators approaching each other, say z1, · · ·, zk, shouldgive poles in p2 + · · · + pk.

It is most convenient to fix reduced variables uj =(zj −z2)/(z1 −z2) (j = 1, · · ·, k) in taking the short-distance limit z1 →z2.The amplitude exhibits short-distance singularities whose residues are given by a21

product of two dual amplitudes (Fig. 2)˜A(p1, · · · , pN) ≈1VSL(2,C)Z|z1−z2|≤ǫd2z1d2z2kYi=3d2uiNYj=k+1d2zj |z1 −z2|p·p+Q·p−4×Y1≤i

(4.18)The dual amplitude with the original variables zi (i = 2, k + 1, · · ·, N) has N −kpositive chirality tachyons pk+1, · · · , pN and the intermediate state particle p (rightside blob in Fig. 2), whereas the dual amplitude with the reduced variables uj(j = 3, · · ·, k) has the intermediate state particle −p−Q and k tachyons p1, · · · , pkwhose chiralities are positive except p1 (left side blob in Fig.

2).It is important to clarify the kinematical constraints on the intermediate stateswhen the amplitudes are factorized. In the left side blob in Fig.

2, the incomingmomenta of tachyons p1, · · ·, pk are balanced by the incoming momentum −p =−p1 −· · · −pk.However, if we want to interpret the left side blob as a dualamplitude coming from a path integral with our action (2.4) and (2.6), we need toassign the incoming two-momentum of the intermediate particle to be −p−Q, sincethe action dictates that an external source two-momentum Q should be present.Therefore if p is the intermediate state momentum flowing into the right side blobin Fig. 2, the corresponding momentum for the dual amplitude of the left sideblob should be regarded as −p −Q.

Consequently, if the intermediate state is atachyon, the chirality of the tachyon for the left side blob is the same as that ofthe tachyon for the right side blobp = (p, −i(−√2 ± p)),−p −Q = (−p, −i(−√2 ± (−p))). (4.19)Similarly, if the intermediate state is a topological state, the type (S or A) of theintermediate state for the left side blob is the opposite to that of the intermediate22

state for the right side blobp = (p, −i(−√2 ±pp2 + 2n)),−p −Q = (−p, −i(−√2 ∓pp2 + 2n)). (4.20)In the present case, we have only a single tachyon with the negative chiral-ity whose two-momentum is determined by the two-dimensional kinematics (4.8).Since we pinch together the single negative chirality tachyon with the positivechirality tachyons p1, · · · , pk, we find the intermediate particle to have the two-momentump =−N + 2√2+kXj=2pj, −iN −2 −2k√2+kXj=2pj.

(4.21)Since k ≤N −2, it is tachyon ifkXj=2pj = k −1√2(4.22)and its chirality is always negative. In this case, the left side blob has more thantwo tachyons for each chirality.

On the other hand, the tachyon amplitudes arenon-vanishing only if a single tachyon has one of the chiralities and the rest haveopposite chirality. Therefore the dual amplitude with the reduced variables (leftside blob in Fig.

2) vanishes except when it is the three-point function (k = 2).This is precisely the case we have evaluated already in Eq. (4.15).As for the intermediate topological states, kinematics dictates that it is of S-type for the dual amplitude with the original variables corresponding to the rightside blob in Fig.

2p =−N −k −1√2+n√2(N −k −1),−i−√2 + N −k −1√2+n√2(N −k −1)(4.23)for the level n topological state. As we have already explained, the intermediatestate −p −Q for the right side blob is of A-type.

The three-point function with23

the A-type topological state (k = 2) is nothing but the OPE coefficient (4.5)that we have seen non-vanishing. Four- and more- point functions with the A-type topological state (k ≥3) are more difficult to compute.The topologicalstate of level n consists of a linear combination of monomials of derivatives of Xmultiplied by a vertex operator eip·X.Both the number of ∂and the numberof ¯∂should be n for each monomial.

The two momentum p is given by ((r −t)/√2, −i(−2 −r −t)/√2) for the (r, t) topological state of type A. If we do notspecify the coefficients of the monomials, we obtain an operator containing the(r, t) topological state together with a certain amount of null states.We havetaken such an operator as a substitute for the (r, t) topological state of A-typeat the level n = rt, and have explicitly evaluated the dual amplitude with thetopological state for the case of four-point function.

We have found it to vanish.This amplitude arises as the left side blob in Fig. 2 contributing to the pole oflevel n = rt in the case of k = N −r −1 = 3.

We conjecture in general thatthe A-type topological state gives vanishing dual amplitude except for the three-point function (k = 2).Only in the three-point dual amplitude (k = 2), wecan simply regard the factor for the blob of particles pinched together (left sideblob in Fig. 2) as the coefficient of the OPE rather than the dual amplitude.

Weshould emphasize that this decoupling of the topological states of the A-type is verycrucial in explaining the simple singularity structure of the N-tachyon scatteringamplitudes. Let us note that the decoupling of these states has been shown by DiFrancesco and Kutasov in a recent preprint.25) This property is presumably relatedto Seiberg’s finding that only the S-type is physical.

These decoupling propertiesof both tachyon and topological states originate partly from a peculiarity of thetwo-dimensional kinematics (one dimension from the matter X zero mode and theother from the Liouville zero mode), but they also seem to be a manifestation ofthe large symmetry characteristic to the c = 1 matter coupled to quantum gravity.Hence they are worth studying further.Other possibilities are short-distance singularities from the pinching of k tachyonsall with positive chirality, say p2, · · ·, pk+1. Actually the short-distance singular-24

ities due to the pinching of tachyons all with positive chirality can be regardedas the same short-distance singularities as the pinching of the other tachyons in-stead. Namely these limits are equivalent to configurations in which the tachyonsp1 and pk+2, · · ·, pN are pinched rather than p2, · · · , pk+1 by the SL(2, C) in-variance.

Since one of the other tachyons has the opposite chirality, the presentcase is actually the same situation as the previously analyzed case: pinching to-gether the single negative chirality tachyon with the opposite chirality tachyons.Such configurations have already been discussed above and we need not to considerthem.These observations explain why there are only singularities in the individualpj, and none in any combinations of momenta, although the factorization of theN-tachyon amplitudes is valid through the OPE as we have seen.Let us finally discuss the case of s = positive integer. The amplitudes withs = positive integer can be obtained from the s = 0 case as follows: we considerthe N + s tachyon scattering amplitude and take a limit of vanishing momentafor s tachyons and multiply by (µ/π)s. There is one subtlety: at the vanishingmomenta, the chirality is ill-defined.8)−10) We define the vanishing momenta takingthe limit from the positive chirality tachyon.

In the limit, we obtain an s-th powerof a singular factor µ∆(0), which should be replaced by the correct (renormalized)cosmological constant µr. This procedure gives the insertion of the correct cosmo-logical term operator φ eαφ, as we explained earlier.

In this way we find that theshort-distance singularities of the amplitudes with non-vanishing s can be obtainedcorrectly once the short-distance singularities in the s = 0 amplitude are correctlyobtained. Using the previous argument, we find that the only non-vanishing short-distance singularities come from the OPE of two vertex operators for tachyons.Short-distance singularities from one or more naive cosmological term operatorsapproaching the tachyon vertex operators give a vanishing value for the residue.25

AcknowledgementsOne of the authors (NS) thanks Y. Kitazawa and D. Gross for a discussion ofthe Liouville theory. We would like to thank Patrick Crehan for a careful readingof the manuscript.

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94 (Springer-Verlag, 1979).Figure Captions1) The factorization of the N-tachyon amplitude by the OPE of the operators1 and 2. The signs + and −denote the chirality of the tachyons.2) The factorization of the N-tachyon amplitude by the OPE of the operators1, · · ·, k. The signs + and −denote the chirality of the tachyons.28

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