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Factorization and Topological States

arXiv:hep-th/9108027v2 2 Sep 1991————————————————————————-TIT/HEP–173STUPP–91–120August 10, 2018Factorization and Topological Statesin 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-point amplitudes in two-dimensional c = 1 quantumgravity is understood in terms of short-distance singularities arising from the op-erator product expansion of vertex operators after the Liouville zero mode integra-tion. Apart from the tachyon states, there are infinitely many topological statescontributing to the intermediate states.

It is very important to understand the non-perturbative results of matrix mod-els [1,2] from the viewpoint of the usual continuum approach of the two-dimensionalquantum gravity, i.e. the Liouville theory [3-6].

Recently correlation functions onthe sphere topology have been computed in the Liouville theory [7-12]. These re-sults are consistent with those of matrix models [13-15].

So far only conformalfield theories with central charge c ≤1 have been successfully coupled to quantumgravity.The c = 1 case is the richest and the most interesting. It has been observed thatthis theory can be regarded effectively as a critical string theory in two dimensions,since the Liouville field zero mode provides an additional “time-like” dimensionbesides the obvious single spatial dimension given by the zero mode of the c = 1matter [16].

Since there are no transverse directions, the continuous (field) degreesof freedom are exhausted by the tachyon field. In fact, the partition function forthe torus topology was computed in the Liouville theory, and was found to giveprecisely the same partition function as the tachyon field alone [17,18].

However,there are evidences for the existence of other discrete degrees of freedom in thec = 1 quantum gravity. Firstly, the correlation functions obtained in the matrixmodel exhibit a characteristic singularity structure [14].

In the continuum approachof the Liouville theory, Polyakov has observed that special states with discretemomenta and “energies” can produce such poles, and has called these operatorsco-dimension two operators [19]. More recently, the two-loop partition function hasbeen computed in a matrix model and evidence has been noted for the occurrenceof these topological sates [20].

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 has been most clearlyrevealed through the factorization analysis of scattering amplitudes.

On the otherhand, the factorization and unitarity of the Liouville theory has not yet been wellunderstood.The purpose of this paper is to understand the factorization of c = 1 quan-tum gravity in terms of the short-distance singularities arising from the operator2

product expansion (OPE) of vertex operators. Since we are interested in the shortdistance singularities, we consider correlation functions on a sphere topology only.We find that the singularities of the amplitudes can be understood as short-distancesingularities of two vertex operators and that infinitely many discrete states con-tribute to the intermediate states of the factorized amplitudes, apart from thetachyon states.

These are the co-dimension two operators of Polyakov [19] andpresumably are topological in origin. We have also explicitly constructed some ofthe topological states.Let us consider the c = 1 conformal matter realized by a single bosonic field(string variable) X coupled to the two-dimensional quantum gravity.

After fixingthe conformal gauge gαβ = eαφˆgαβ using the Liouville field φ, the c = 1 quantumgravity can be described by the following action on a sphere [3-6]S = 18πZd2zpˆgˆgαβ∂αX∂βX + ˆgαβ∂αφ∂βφ −Q ˆRφ + 8µ eαφ,(1)where the parameters are given by Q = 2√2, α = −√2 . Since the correct cos-mological term operator [4] in the c = 1 case should be φ eαφ rather than eαφ, therenormalized (correct) cosmological constant µr is given by the following proce-dure: one should replace µ and α by µr/(2ǫ) and (1 −ǫ)α and take the ǫ →0 limit[8,10].

We have also set the “Regge slope parameter” α′ = 2. The gravitationallydressed tachyon vertex operator with momentum p has conformal weight (1, 1):Op =Zd2zpˆg eipX eβ(p)φ,β(p) = −√2 + |p|.

(2)We have chosen the plus sign in front of |p|, following the argument of refs. [4,5].We see that the Liouville zero mode can be regarded as an “imaginary time” andthe exponent β(p) as “energy”.The N-point correlation function of the vertex operators (2) is given by a path3

integral⟨Op1 · · · OpN⟩=ZDXDφVSL(2,C)Op1 · · · OpN e−S= Γ(−s)−αZNYi=1hd2zipˆgi1VSL(2,C)* NYj=1eipjX(zj)+X×*µπZd2wpˆg eα˜φ(w)s NYj=1eβj ˜φ(zj)+˜φ,(3)where VSL(2,C) is the volume of the SL(2, C) group and powers of the string cou-pling constant g−2stfor the sphere topology are omitted. The expectation valuewith ˜φ denotes the path integral over the non-zero mode ˜φ of the Liouville fieldφ = φ0 + ˜φ, after the zero mode φ0 integration.

The defining formula for s can beregarded as “energy-momentum conservation”NXj=1pj + sq + Q = 0,(4)where pj = (pj, −iβj), q = (0, −iα) and Q = (0, −iQ) are two-momenta fortachyons, “cosmological terms”, and the source.For a non-negative integer s,we can evaluate the non-zero mode ˜φ integral by regarding the amplitude as ascattering amplitude of N-tachyons and s “cosmological terms”. After fixing theSL(2, C) gauge invariance (z1 = 0, z2 = 1, z3 = ∞), an integral representation forthe N-tachyon amplitude is given by* NYj=1Opj+= 2πδNXj=1pj1−αΓ(−s) ˜A(p1, · · ·, pN),(5)˜A =µπs ZNYi=4d2zisYj=1d2wjNYi=4|zi|2p1·pi |1 −zi|2p2·piY4≤i

(6)4

In spite of the non-analytic relation (2) between energy β and momentum p, weneed to continue analytically the formula into general complex values of momentain order to explore the singularity structure. Hence we will define the tachyon tohave positive (negative) chirality if (β +√2)/p = 1(−1) irrespective of the actualvalues of momentum [19].

It seems to us that the operators with β < −√2 in eq. (3) are free from the trouble noted in [4,5] since φ0 has already been integratedout.

The physical values of momenta are reached by analytic continuation in s,since s is related to other momenta through energy-momentum conservation (4).For generic physical values of momenta, one finds a finite result for the N-tachyonamplitudes. However, the result is different in different chirality configurations,since the amplitude is non-analytic in momenta.If p1 has negative chirality and the rest p2, · · · , pN positive chirality, the am-plitude is given by [7-11]˜A(p1, · · · , pN) = πN−3[µ∆(−ρ)]sΓ(N + s −2)NYj=2∆(1 −√2pj),(7)where ∆(x) = Γ(x)/Γ(1−x).

The regularization parameter ρ is given by ρ = −α2/2and is eventually set equal to −1 after the analytic continuation (in the centralcharge c). We should replace the combination µ∆(−ρ) by the renormalized cos-mological constant µr, since the correct cosmological term is φ eαφ.

The amplitudeexhibits singularities at pj = (n + 1)/√2, n = 0, 1, 2, · · ·, but has no singularitiesin other combinations of momenta contrary to the dual amplitudes in the criti-cal string theory.These poles for n = 1, 2, · · · will be shown to correspond totopological states as argued by several people [14,19].The amplitudes with one tachyon of positive chirality and the rest negative aregiven by changing the sign of pj. On the other hand, if each chirality has two ormore tachyons, ˜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 consider non-negative integer s in the following. This property has been explicitly demonstrated5

for the four- and five-tachyon amplitudes in the Liouville theory [8,19], and hasbeen 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 notethat our assertion is consistent with the argument for vanishing S-matrix in ref. [14]: they absorbed the ∆(1 ±√2 p) factor in the amplitude to a renormalizationfactor of vertex operators, which becomes infinite if there is only one tachyon ineither one of the chiralities [8-10].

Because of this infinite renormalization, theirrenormalized amplitudes vanish even if there is only one tachyon in either one ofthe chiralities.In order to understand the poles of the amplitudes in terms of short-distancesingularities in the OPE, we shall consider the case of s = non-negative integersby choosing the momentum configuration appropriately. These amplitudes at non-negative integer s represent so-called “bulk” or “resonant” interactions [14,19].Here we shall take the case of s = 0 for the N-tachyon amplitude with only onenegative chirality tachyon (p1), and examine the s = positive integers case at theend.First we shall illustrate the origin of short-distance singularities in the simplestcontext by expanding the integrand of the four tachyon scattering amplitude withs = 0 (we fix z2 = 0, z3 = 1, z4 = ∞and call z1 = z)˜A(p1, · · ·, p4) =Zd2z |z|2p1·p2 |1 −z|2p1·p3≈Z|z|≤ǫd2z |z|−2√2 p2∞Xn=0Γ(1 −√2 p3)n!

Γ(−√2 p3 −n + 1)(−z)n2≈∞Xn=0(−1)n(n! )2πn + 1 −√2 p24Yj=3∆(1 −√2 pj).

(8)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. Furthermore we find6

that successive poles are due to successive terms in the OPE: eip1·X(z1) :: eip2·X(z2) : ∼∞Xn=0 1n!2|z1 −z2|2p1·p2+2n : eip2·X(z2)∂n ¯∂neip1·X(z2) :,(9)with X = (X, φ). The pole at p2 = 1/√2 (n = 0) is due to the tachyon intermediatestate.

The higher level poles (n ≥1) are due to the topological states which wediscuss shortly.We next examine short-distance singularities in amplitudes with five or moretachyons. By the same token, we consider the short-distance singularities due to theOPE (9) of two vertex operators p1 and p2 at z1 and z2.

Because of the kinematicalconstraint, these singularities give poles in p2 at (n + 1)/√2, n = 0, 1, 2, · · ·. Theresidues of these poles are given by kinds of dual amplitudes with N −2 tachyonsp3, · · ·, pN, and an intermediate particle of two-momentum p = p1 + p2 (Fig.

1).Similarly to the four tachyon case, the p2 = 1/√2 (n = 0) pole is due to the tachyonintermediate state with negative chirality. In fact we find that the residue of thepole p2 = 1/√2 is precisely given by the N −1 tachyon amplitude with a single(intermediate state) tachyon p having negative chirality and the rest p3, · · ·, pNhaving positive chirality˜A(p1, p2, p3, · · · , pN) ≈π(N −3)(1 −√2p2)˜A(p, p3, · · ·, pN).

(10)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 tachyonamplitude 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). (11)The residues of these poles in fact correctly reproduce the residues of the poles in7

the full amplitude. It is rather difficult to compute the short-distance singularitiesexplicitly to an arbitrary level except for the four-point amplitude that we havealready worked out in eq.

(8). Therefore we content ourselves with the computationof lower level singularities in explicitly demonstrating that the singularities of theamplitudes all come from the short-distance singularities of p1 and pj.Since the short-distance singularities should come from terms in the OPE, wenext examine the operators responsible for these singularities.For higher levelpoles, it has been pointed out that there are only null states at generic values ofmomenta [19,14].

However, there are exceptional values of momenta where the nullstates degenerate and new primary states emerge as a result. These new primarystates are called co-dimension two operators by Polyakov [19], and special statesor topological states by other people [14,18].

We can construct vertex operatorsfor these topological states in the following way. The Virasoro generators Lm forthe c = 1 quantum gravity are the sum of the generators of the free scalar X andthose of the Liouville field φ.

The condition for the existence of the pole at level nis given byp · (p + Q) = 2(1 −n). (12)We should note that there are two branches of the solution for the condition (12)β = −√2 ±pp2 + 2n.

(13)Although Seiberg has noted trouble with the lower sign due to the zero mode φ0[4,5], we consider both cases here, since we are considering OPE of vertex opera-tors consisting of non-zero mode ˜φ only (φ0 integration gave the s “cosmologicalterms”). We shall call the upper sign solution S- (Seiberg) type and the lower A-(anti-Seiberg) type.

We should construct the field of conformal weight (1, 1) bytaking linear combinations of monomials of derivatives of X multiplied by eip·X.For instance, at level n = 1 we find only one field with weight (1, 1) at generic8

values of momentum, i.e. p ̸= 0V (1) = p · ∂X p · ¯∂X eip·X = −L−1 ¯L−1eip·X.

(14)The above state is clearly null.However, the situation changes at p = 0. For the S-type, the operator vanishesat p = 0.

Therefore we can construct a new operator by a limitV(1,1) = limp→0V (1)p2= ∂X ¯∂X. (15)We easily find that this field is primary and not null.This kind of a peculiaroperator exists only at a discrete momentum and hence it is called co-dimensiontwo.

This is precisely the “graviton”, i.e. the first topological state which gives riseto the pole at p2 = 2/√2 in eq.

(8). 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. (16)At level two, we find two independent fields of weight one for the holomorphic part.The two fields for the holomorphic part areV (2) =L−2 + 32L2−1eip·X,V (3) = L−114i [(8 −p · Q)p −2Q] · ∂X eip·X.

(17)Both fields are null. These two operators are linearly dependent at a special valueof the momentum and we obtain a topological state.

For instance, the (2,1) topo-9

logical 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). (18)We find exactly the same situation for the antiholomorphic 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 the null descendants at level n [15, 18].The momentum p of the initial primary state and the level n are specified by twopositive integers (r, t) and thus the energy β of the topological state is also givenbyp = r −t√2,n = rt,β = −2 ± (r + t)√2.

(19)The upper (lower) sign corresponds to the S-(A-)type solution. We find that theseoperators with (r, t) = (1, 1) and (2, 1) differ from our operators (15) and (18)respectively, only by a certain amount of null operators.The OPE suggests that there may be other short-distance singularities inother combinations of momenta if one considers other combinations of vertexoperators approaching to the same point.

For instance, short-distance singular-ities corresponding to k vertex operators approaching each other, say z1, · · ·, zk,should give poles in p2 + · · · + pk. It is most convenient to fix reduced variablesuj = (zj −z2)/(z1 −z2) (j = 1, · · ·, k) to take the short-distance limit z1 →z2.10

The amplitude exhibits short-distance singularities whose residues are given by aproduct 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

(20)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).If p is theintermediate state momentum flowing into the right side blob, the correspondingmomentum for the dual amplitude of the left side blob can be regarded as −p−Q.This implies that the chirality of the tachyon intermediate state is the same forboth dual amplitudes. In the case of the intermediate topological state, the type(S or A) of the intermediate state for one dual amplitude turns out to be oppositeto the other dual amplitude.

In the present case of pinching together the singlenegative chirality tachyon with the positive chirality tachyons, energy-momentumconservation (4) dictates that the intermediate state tachyon has negative chirality.On the other hand, the tachyon amplitudes are non-vanishing only if a singletachyon has one of the chiralities and the rest have opposite chirality. Thereforethe dual amplitude with the reduced variables vanishes except when it is the three-point function (k = 2).

This is precisely the case we have evaluated already in eq.(10). As for the intermediate topological states, kinematics dictates that it is of S-type (A-type) for the dual amplitude with the original variables (reduced variables).The three-point function with the A-type topological state (k = 2) is nothing but11

the OPE coefficient (9) which we have seen non-vanishing. Four- and more- pointfunctions with the A-type topological state (k ≥3) are more difficult to compute.The topological state of level n consists of a linear cmbination of monomials ofderivatives of X multiplied by a vertex operator eip·X.

Both the number of ∂andthe number of ¯∂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 donot specify the coefficients of the monomials, we obtain an operator containing the(r, t) topological state together with certain amount of null states. We have takensuch an operator as a substitute for the (r, t) topological state of A-type at the leveln = rt, and have explicitly evaluated the dual amplitude with the topological statefor the case of four-point function.

We have found it to vanish. This amplitudearises as the left side blob in Fig.

2 contributing to the pole of level n = rt in thecase of k = N −r −1 = 3. We conjecture in general that the A-type topologicalstate gives vanishing dual amplitude except for the three-point function (k = 2).This property is presumably related to the Seiberg’s finding that only the S-typeis physical.

Only in the three-point dual amplitude (k = 2), we can simply regardthe factor for the blob of particles pinched together (left side blob in Fig. 2) as thecoefficient of the OPE rather than the dual amplitude.Other possibilities are short-distance singularities from the pinching of k tachyonsall with positive chirality, say p2, · · · , pk+1.

We again find that the chirality of theintermediate state tachyon is negative. Hence the dual amplitude with the originalvariables contains two tachyons with negative chirality p1, p besides N −k −1 pos-itive chirality tachyons pk+2, · · ·, pN.

Hence the amplitude vanishes except for thethree-point function (N = k + 2). The non-zero result of the three-point functiongives rise to poles in the momentum pN of the single tachyon with the positivechirality.

We can regard these poles to be the same short-distance singularities asobtained in zN →z1. Kinematics shows that the intermediate topological states inthis case is of A-type (S-type) for the dual amplitude with the original variables (re-duced variables).

According to our conjecture above, the dual amplitude with theoriginal variables again vanishes except for the three-point function (N = k + 2).12

The non-zero result can be interpreted in the same way as the tachyon intermediatestate.These observations explain why there are only sigularities 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 with s =positive integer can be obtained from the s = 0 case as follows: we consider theN + s tachyon scattering amplitude and take a limit of vanishing momenta for stachyons and multiply by (µ/π)s. There is one subtlety: at the vanishing momenta,the chirality is ill-defined [8-10].

We define the vanishing momenta taking limit fromthe positive chirality tachyon. In the limit, we obtain an s-th power of a singularfactor µ∆(0), which should be replaced by the renormalized cosmological constantµr.

In this way we find that the short-distance singularities of the amplitudes withnon-vanishing s can be obtained correctly once the short-distance singularities inthe s = 0 amplitude are correctly obtained.Using the previous argument, wefind that the only non-vanishing short-distance singularities are from the OPE oftwo vertex operators for tachyons. Short-distance singularities from one or more“cosmological term operators” approaching the tachyon vertex operators give avanishing value for the residue.AcknowledgementsOne of the authors (NS) thanks Y. Kitazawa and D. Gross for a discussion onthe Liouville theory.

We would like to thank Patrick Crehan for a careful readingof the manuscript. This work is supported in part by Grant-in-Aid for ScientificResearch from the Ministry of Education, Science and Culture (No.01541237).13

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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.16

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